[SC] PPA Amicus Brief for SC Appeal (07/29/09)

August 4th, 2009


SC Appeal Amicus Brief (07/29/09)

STATE OF SOUTH CAROLINA COUNTY OF CHARLESTON Town of Mount Pleasant, vs. Robert L. Chimento, Scott Richards, Michael Williamson, Jeremy Brestel, and John Taylor Willis, Defendants-Appellants. IN THE CIRCUIT COURT FOR THE NINTH CIRCUIT TICKET Nos. 98045DB, 98050DB, 98040 DB, 98035DB 98043DB BRIEF OF AMICUS CURIAE THE POKER PLAYERS ALLIANCE IN SUPPORT OF DEFENDANTS Thomas C. Goldstein Christopher M. Egleson Jonathan H. Eisenman Akin Gump Strauss Hauer & Feld LLP 1333 New Hampshire Ave., NW Washington, D.C. 20036-1564 (202) 887-4000 Kenneth L. Adams Adams Holcomb LLP 1875 Eye Street NW Washington, D.C. 20006 (202) 580-8822 TABLE OF CONTENTS TABLE OF AUTHORITIES…………………………………………………………………………………………….. ii STATEMENT OF INTEREST……………………………………………………………………………………………1 ARGUMENT…………………………………………………………………………………………………………………..1 I. II. South Carolina Law Does Not Prohibit Gaming Unless Chance Predominates Over Skill ……………………………………………………………………………………………………………..2 Poker Matches Are Contests of Skill ………………………………………………………………………..7 A. B. Making Correct Decisions In Poker Requires A Diverse Array Of Sophisticated Skills That Games Of Chance Do Not. ………………………………………9 Skilled Players Beat Simple Players In Simulated And Real Poker Play. ………….13 CONCLUSION………………………………………………………………………………………………………………19 i TABLE OF AUTHORITIES Cases Page(s) In re Allen, 377 P.2d 280 (Cal. 1962)……………………………………………………………………………………………3, 8 City of Myrtle Beach v/ Juel P. Corp., 344 S.C. 43, 543 S.E.2d 538 (2001)………………………………………………………………………………..4 D’Orio v. Startup Candy Co., 266 P. 1037 (Utah 1928) ……………………………………………………………………………………………….3 Darlington Theatres v. Coker, 190 S.C. 282, 2 S.E.2d 782 (1939)………………………………………………………………………………….5 Harris v. Missouri Gaming Comm’n, 869 S.W.2d 58 (Mo. 1994)…………………………………………………………………………………………….3 Indoor Recreation Enters., Inc. v. Douglas, 235 N.W.2d 398 (Neb. 1975) …………………………………………………………………………………………3 Johnson v. Collins Entm’t Co., 333 S.C. 96, 508 S.E.2d 575 (1998)………………………………………………………………………………..5 Kraus v. City of Cleveland, 19 N.E.2d 159 (Ohio 1939) ……………………………………………………………………………………………5 Las Vegas Hacienda, Inc. v. Gibson, 359 P.2d 85 (Nev. 1961) ……………………………………………………………………………………………….3 Midwestern Enters., Inc. v. Stenehjem, 625 N.W.2d 234 (N.D. 2001) ……………………………………………………………………………………………5 Monte Carlo Parties, Ltd. v. Webb, 322 S.E.2d 246 (Ga. 1984) …………………………………………………………………………………………….5 Morrow v. State, 511 P.2d 127 (Alaska 1973) …………………………………………………………………………………………..3 Nuckolls v. Great Atl. & Pac. Tea Co., 192 S.C. 156, 5 S.E.2d 862 (1939)…………………………………………………………………………………4 Pennsylvania v. Dent, No. 2008-733, slip op. (Pa. Ct. Com. Pl. Jan. 14, 2009) …………………………………………………….1 Pennsylvania v. Irwin, 636 A.2d 1106 (Pa. 1993) …………………………………………………………………………………………….5 Pennsylvania v. Two Elec. Poker Game Machs., 465 A.2d 973 (Pa. 1983) ………………………………………………………………………………………………..7 PGA Tour, Inc. v. Martin, 532 U.S. 661 (2001) ……………………………………………………………………………………………………. 8 ii Rorrer v. P.J. Club, Inc., 347 S.C. 560, 556 S.E.2d 726 (Ct. App. 2001) ………………………………………………………………..2 State v. Blackmon, 304 S.C. 270, 403 S.E.2d 660 (1991) ………………………………………………………………………………….6 State v. Cutler, 274 S.C. 376, 264 S.E.2d 420 1980) …………………………………………………………………………………..6 State v. Lane, 82 S.C. 144, 63 S.E. 612 (1909)…………………………………………………………………………………….2 State v. Stroupe, 76 S.E.2d 313 (N.C. 1953) ……………………………………………………………………………………………3 United States v. Santos, 128 S. Ct. 2020 (U.S. 2008)……………………………………………………………………………………………….6 Video Gaming Consultants, Inc. v. Dep’t of Revenue, 342 S.C. 34, 535 S.E.2d 642 (2000)………………………………………………………………………………..2 Statutes S.C. Code § 16-19-40…………………………………………………………………………………………………..1, 19 Opinions S.C. Att’y Gen. Op., 2004 WL 235411 (Jan. 22, 2004) ……………………………………………………..6, 7 S.C. Att’y Gen. Op., 2001 WL 957740 (Aug. 2, 2001)………………………………………………………….6 S.C. Att’y Gen. Op., 1995 WL 805729 (Sept. 5, 1995)………………………………………………………….6 1978 S.C. Op. Att’y Gen. 226, 1978 S.C. Op. Att’y Gen. No. 78-201 (1978) …………………………..6 Books David Apostolico, Machiavellian Poker Strategy: How to Play Like a Prince and Rule the Poker Table (2005)……………………………………………13 Doyle Brunson, Doyle Brunson’s Super System: A Course in Power Poker (2002)…………………13 Gus Hansen, Every Hand Revealed (2008)…………………………………………………………………………13 Dan Harrington, Harrington on Hold ’Em: Expert Strategy for No Limit Tournaments (2005) ………………………………………………………….13 Eric Lindgren, World Poker Tour: Making the Final Table (2005) ……………………………………….13 Daniel Negreanu, Power Hold’em Strategy (2008)……………………………………………………………..13 Blair Rodman & Lee Nelson, Kill Phil: The Fast Track to Success in No-Limit Hold ’Em Poker Tournaments (2005)………………………………………………………………13 iii Jonathan Rowson, CHESS FOR ZEBRAS: THINKING DIFFERENTLY ABOUT BLACK AND WHITE (2005) …………………………………………………………………………………………………………………………8 David Sklansky, The Theory of Poker (1994) …………………………………………………………………….13 David Sklansky, Tournament Poker for Advanced Players (2002)………………………………………..13 Articles Noga Alon, Poker, Chance & Skill ……………………………………………………………………………………15 Peter Borm & Ben van der Genugten, On a Measure of Skill for Games with Chance Elements (1996). ………………………………………………………………………………………….15 Anthony Cabot & Robert Hannum, Poker, Public Policy, Law, Mathematics, and the Future of an American Tradition, 22 T.M. Cooley L. Rev. 443 (2005) ………………….18 Benedict Carey, At the Bridge Table, Clues to a Lucid Old Age, N.Y. Times (May 22, 2009) ……4 Rachel Croson, Peter Fishman & Devin G. Pope, Poker Superstars: Skill or Luck? CHANCE (Vol. 21, No. 4, 2008) ……………………………………………………………………………….16, 18 Laure Elie & Romauld Elie, Chance and Strategy in Poker (Sept. 2007) (unpublished manuscript) ………………………………………………………………………….15 Shawn Patrick Green, Online Poker: Interview With Annette ‘Annette_15’ Obrestad, CardPlayer.com (Aug. 12, 2007) …………………………………………………………………………………………17 Paco Hope & Sean McCulloch, Statistical Analysis of Texas Hold’Em (March 4, 2009) ……..9, 16 Trevor Hughes, Definition Clears Man of Gambling Charges, Coloradoan (Jan. 30, 2009) ……………………………………………………………………………………………2 Patrick Larkey et al., Skill in Games, 43 MANAGEMENT SCIENCE 596 (May 1997)……9, 13, 15, 17 Howard Lederer, Why Poker Is a Game of Skill (May 6, 2008) (unpublished manuscript) ……………………………………………………………………….11 Michael A. Tselnik, Check, Raise, or Fold: Poker and the Unlawful Internet Gambling Enforcement Act, 35 Hofstra L. Rev. 1617 (Spring 2007) …………………………………18 Abraham J. Wyner, Chance and Skill in Poker (Apr. 17, 2008) (unpublished manuscript) ……………………………………………………………………..15 Websites World Poker Tour Stats, Website ……………………………………………………………………………………….8 iv STATEMENT OF INTEREST Amicus curiae the Poker Players Alliance is a nonprofit organization whose members are poker players and enthusiasts from around the United States. The Alliance works to protect the legal rights of poker players, and has been involved in these proceedings since trial. The group’s membership has a direct interest in the outcome of this case, because it will determine whether they are permitted to play poker in South Carolina. ARGUMENT Defendants-appellants were charged with playing cards in a “house used as a place of gaming” in violation of S.C. Code § 16-19-40. As the state and Appellants agree, “gaming” in the statute means “gambling.” As Appellants also explain in their brief, and as we explain below, under South Carolina law poker is “gambling” if the outcome is determined predominantly by chance rather than skill. For the reasons we explain, this appeal presents a narrow legal question: to resolve the case in Appellants’ favor this Court need only rule that a home game of poker in particular does not violate the statute. In this brief, amicus explains why the question under the statute is whether skill predominates over chance in poker, why skill does in fact predominate over chance in poker, and why Appellants’ particular conduct here therefore does not violate the statute. A Pennsylvania court recently addressed the same question in almost identical circumstances. See Pennsylvania v. Dent, No. 2008-733, slip op. at 14-15 (Pa. Ct. Com. Pl. Jan. 14, 2009) (attached for the Court’s convenience as Ex. A). In Dent, the court concluded that the defendants were not engaged in unlawful gambling activity because the game they were playing, Texas Hold ’Em, is a game in which skill predominates over chance. Notably, although Pennsylvania positive law does not define “unlawful gambling,” the Dent Court proceeded by using the common law American rule, also known as the “predominance test” or the “dominant factor 1 test,” to determine that Texas Hold ‘Em is not unlawful gambling. Likewise, a Colorado jury recently acquitted a man who had hosted regular informal poker tournaments at a local bar after the man defended his actions by demonstrating that skill predominates over chance in poker. See Trevor Hughes, Definition Clears Man of Gambling Charges, Coloradoan (Jan. 30, 2009) (attached as Ex. B).1 This court should follow the course set in those cases and hold that the relevant question is whether the game at issue is a game of chance, and that since—as the trial court found—poker is not a game of chance, playing poker in a private home does not violate Section 16-19-40. I. South Carolina Law Does Not Prohibit Gaming Unless Chance Predominates Over Skill Appellants were charged with violating Section 16-19-40 by playing poker in a private home. To show that playing poker in such a setting violated the statute, the Town had to prove that poker is a game in which the outcome is determined predominantly by chance rather than skill. That is true because Section 16-19-40 prohibits playing “any game with cards or dice” in “any house used as a place of gaming.” The Town could only show that the house in question was being “used as a place of gaming” if there was “gaming” taking place at the house. As an initial matter, “gaming” in Section 16-19-40 means “gambling,” as both parties here agree. The courts have consistently treated the terms as equivalents, as in State v. Lane, 82 S.C. 144, 144, 63 S.E. 612, 613 (1909), and Video Gaming Consultants, Inc. v. Dep’t of Revenue, 342 S.C. 34, 44, 535 S.E.2d 642, 648 (2000), in which the Supreme Court used them interchangeably. See also Rorrer v. P.J. Club, Inc., 347 S.C. 560, 566, 556 S.E.2d 726, 729 (Ct. App. 1 http://www.coloradoan.com/article/20090130/NEWS01/901300328, last accessed Feb. 9, 2009. 2 2001) (purpose of statute allowing “right to recover excessive gambling losses” is to “punish excessive gaming.”). This appeal therefore turns on whether poker is gambling. The common test for whether an activity is “gambling” is the so-called “dominant factor” test, which has been adopted by the high courts of Nebraska, California, Nevada, North Carolina, Utah, Missouri, and Alaska, among other courts. See, e.g., Indoor Recreation Enters., Inc. v. Douglas, 235 N.W.2d 398, 400 (Neb. 1975); In re Allen, 377 P.2d 280 (Cal. 1962); Las Vegas Hacienda, Inc. v. Gibson, 359 P.2d 85, 87 (Nev. 1961); State v. Stroupe, 76 S.E.2d 313, 316-17 (N.C. 1953); D’Orio v. Startup Candy Co., 266 P. 1037, 1038 (Utah 1928); Harris v. Missouri Gaming Comm’n, 869 S.W.2d 58, 62 (Mo. 1994); Morrow v. State, 511 P.2d 127, 129 (Alaska 1973); see also Dent, slip op. at 14-15. Under that test, an activity is not gambling if skill predominates over chance in determining the outcome of the activity. The court below recognized that the courts of a number of states follow the dominant factor test (Op. at 3), and recognized that “if [it] knew that this State follows that test in this factual circumstance the decision would be simple” because “Texas Hold-em is a game of skill” (id.). The court, however, declined to apply the dominant factor test, but not on the basis of any affirmative legal conclusion about what test to apply in determining what constitutes “gambling.” The court erred in declining to decide whether the dominant factor test applies under the laws of this State. But the effect of what the court ruled, instead—that South Carolina’s policy is to suppress gambling by suppressing all card and dice games (Op. at 4)—is that Section 16-19-40 can be read to suppress a game of Monopoly or bridge. It is true that the plain language of the Section could be read to prohibit “any game with cards or dice,” but as the Supreme Court observed, “[h]owever plain the ordinary meaning of the words used in a statute may be, the courts will reject that meaning when to accept it would lead to a result so plainly absurd that it could not pos- 3 sibly have been intended by the Legislature . . . .” Broadhurst v. City of Myrtle Beach Election Comm’n, 342 S.C. 373, 380, 537 S.E.2d 543, 546 (2000). The statute itself provides a qualifier to “any game with cards or dice,” e.g., that the game must also be played in a “house used as a place of gaming.” But without a functional definition of gaming, there is no reason why Section 16-19-40 would not apply to a game like Monopoly, an absurd result that a proper interpretation of the law must avoid. Otherwise, what would prevent a SWAT team from hauling a group of retirees away from a penny-ante game of bridge, also a game of skill,2 as one hauled Appellants from their five-dollar-a-hand poker tables? That absurdity can be avoided by reading the “unlawful games” statute in conjunction with the dominant factor test. However, instead of applying the dominant factor test, the court noted that it lacked a “clear guideline from the Legislature or from the majority of this Supreme Court” and so would not “set itself to definitively conclude that this State will or does follow the ‘Dominant Test’ Theory.” Op. at 4. Because the trial court did not believe it had sufficient guidance to apply the dominant factor test, this Court is obliged to answer the legal question that the court below, for lack of “clear guidance,” left unaddressed. The trial court feared trespassing in the Legislature’s domain by using a common law test to make sense of Section 16-19-40, but in doing so, the court neglected the presumption that the Legislature acts against the background of the common law, and—unless explicitly indicated—in accordance with the common law. E.g., Nuckolls v. Great Atl. & Pac. Tea Co., 192 S.C. 156, 161, 5 S.E.2d 862, 864 (1939); see, e.g., City of Myrtle Beach v. Juel P. Corp., 344 S.C. 43, 48, 543 S.E.2d 538, 540 (2001) (applying the common law timeframe for “abandonment” when an ordinance using the term failed to otherwise 2 See, e.g., Benedict Carey, At the Bridge Table, Clues to a Lucid Old Age, N.Y. Times (May 22, 2009), available at http://www.nytimes.com/2009/05/22/health/research/22brain.html (discussing the mental challenge 4 define it). The recent opinions of at least two Justices in Johnson v. Collins Entertainment Co., 333 S.C. 96, 508 S.E.2d 575 (1998), offer strong evidence that the dominant-factor test governs under South Carolina law. In Johnson, the question was whether video gaming machines constituted a “lottery” under the Constitution. The majority decided that the machines were not a lottery, establishing a principle that the term “lottery” in particular is to be narrowly construed. Id. at 102. The majority opinion, therefore, did not reach the more general question of what constitutes gambling. Justice Burnett in dissent, however, read the term “lottery” more broadly, finding support in the Court’s earlier decision Darlington Theatres v. Coker, 190 S.C. 282, 2 S.E.2d 782 (1939). Under Darlington Theatres, Justice Burnett understood the term lottery to describe any game involving “(1) [t]he giving of a prize, (2) by a method involving chance, (3) for a consideration paid by the contestant or participant,” Johnson, 333 S.C. at 109, 598 S.E.2d at 582, thus reading the term “lottery” to be expansive enough to cover gambling in general.3 From there, Justice Burnett (unlike the majority) had to determine what degree of chance is required to satisfy the second prong of the test. Justice Burnett reasoned that the American rule should apply (as it does in a majority of jurisdictions—by court ruling), and concluded that “where the dominant factor in a inherent in playing bridge, the skill necessary to play, and the positive mental effect of being “engrossed in some mental activities like cards.”). Chance, consideration, and prize are widely recognized as the three elements of gambling. See, e.g., Midwestern Enters., Inc. v. Stenehjem, 625 N.W.2d 234, 237 (N.D. 2001) (“The three elements of gambling are generally recognized as consideration, prize, and chance.”); Monte Carlo Parties, Ltd. v. Webb, 322 S.E.2d 246, 248 (Ga. 1984) (“The crime of gambling, in Georgia, consists of three elements: consideration, chance, and prize.”); Pennsylvania v. Irwin, 636 A.2d 1106, 1107 (Pa. 1993) (“The three elements of gambling are (1) consideration; (2) a result determined by chance rather than skill; and (3) reward.”); Kraus v. City of Cleveland, 19 N.E.2d 159, 161 (Ohio 1939) (“there is involved in the game three elements of gambling, namely, chance, price and a prize.”). 3 5 participant’s success or failure in a particular scheme is beyond his control, the scheme is a lottery.” Id. at 113, 584.4 Justice Burnett’s logic is convincing, and the wealth of authority that he collects persuasively demonstrates that the consensus view in America law in cases of this sort is that the dominant-factor test states the proper question. See id. at 114, 584 n.10 (collecting cases). Justice Burnett’s view is further bolstered by the fact that he was joined as to the proper “legal standard” by Justice Toal, see id. at 120, 588, and by the fact that there is no indication that any other Justice would disagree. The Attorney General has furthermore “consistently stated that the test of whether a particular game is a game of chance or skill is governed by the so-called ‘predominance’ test.” S.C. Att’y Gen. Op. dated Jan. 22, 2004 (citing S.C. Att’y Gen. Ops. dated Aug. 2, 2001; Sept. 5, 1995; Dec. 5, 1978).5 In the absence of any indication to the contrary, the necessary conclusion from the Justices’ analysis and that of the Attorney General is that under the law of this State, as under the majority of others’, the question of whether an activity constitutes gambling turns on whether “the dominant factor in a participant’s success or failure in a particular scheme is beyond his control.” Id. at 113, 584 (Burnett, J., dissenting). At a minimum, the municipal court should have applied the rule of lenity in construing the statute. As the Supreme Court has consistently held, “when a statute is penal in nature, it must be construed strictly against the State and in favor of the defendant.” State v. Blackmon, 304 S.C. 270, 273, 403 S.E.2d 660, 662 (1991); see also State v. Cutler, 274 S.C. 376, 378, 264 S.E.2d 420, 421 (1980) (same); see generally United States v. Santos, 128 S. Ct. 2020, 2025 4 As Justice Burnett noted, under the alternative common law rule, the British rule, a game requiring any amount of skill at all would be outside the ambit of “gambling.” See id. at 112-13, 583-85. In the opinion cited, the Attorney General was correct in its assessment of the law, i.e., that the dominant factor test applies. He erred, however, in his determination that “card games, such as poker are generally games of 5 6 (2008) (“[t]he rule of lenity requires ambiguous criminal laws to be interpreted in favor of the defendants subjected to them”) (collecting cases). The municipal court was of the view that it lacked “clear guidance” on the meaning of the statute at issue here. If the court itself lacked the “guidance” necessary to interpret the statute, Appellants certainly lacked that guidance as well, and the court should have construed the statute to permit Appellants’ behavior. It should, in short, have held that playing poker in a house renders that house a gambling house only if poker is, in fact, gambling under the dominant-factor test. II. Poker Matches Are Contests of Skill Under the dominant-factor test, poker is not gambling. As the magistrate judge held here, “the evidence and studies are overwhelming” that “Texas Hold-em is a game of skill.” Op. at 3. Indeed, at trial, the Town did not dispute that poker is a game of skill, and the trial court’s holding that poker is in fact a game of skill is not at issue on this appeal. An understanding of poker and how it differs from games of chance nevertheless constitutes important background in this case. For the Court’s convenience, therefore, amicus presents below an account of the skill involved in playing poker, drawing upon the trial transcript, academic studies, and amicus’s own experience with the game. As is true for similar games like golf, billiards, and bridge, when good poker players play against bad players, the good players consistently and routinely prevail. Players who enter golf and bridge tournaments pay a fee to enter, and earn a cash reward if they win, but these games are contests of skill because their outcome is determined principally by skill. See Two Elec. Poker Game Machs., 465 A.2d at 977 (“[i]t cannot be disputed that football, baseball and golf require substantial skill, training and finesse” even though “the result of each game turns in part chance.” S.C. Att’y Gen. Op. dated Jan. 22, 2004 (2004 WL 235411). For the reasons explained in the text, and as 7 upon luck or chance”); In re Allen, 377 P.2d 280, 281 (Cal. 1962) (bridge requires skill and is not a “game of chance”). So too with poker. To be sure, there is some accumulation of luck over the course of a poker match that will affect how individual players perform. That is also true, for example, of golf, where “changes in the weather may produce harder greens and more head winds for the tournament leader than for his closest pursuers” or a “lucky bounce may save a shot or two.” PGA Tour, Inc. v. Martin, 532 U.S. 661, 687 (2001). But, as in golf, skill is nonetheless dominant in poker play.6 The fact that every hand of poker involves multiple decision points (at each of the multiple rounds of betting), multiple decisions at each decision point (bet, call, raise, or fold), and innumerable factors that call for skill to evaluate each of those decisions (for example, the player’s own cards, the odds of his hand improving, his sense of the strength of the other player’s hand, his sense of the other players’ perception of him), establishes that poker is a contest of skill. Two general methods of determining the role of chance in an activity have developed in state courts to analyze the issue of whether a game is one of skill or chance. The first method is to evaluate the game’s structure and rules. If the structure and rules allow sufficient room for a player’s exercise of skill to overcome the chance element in the game, the game is one of skill and the gambling laws do not apply. See, e.g., In re Allen, 377 P.2d at 281-82 (holding the card game of bridge to be one predominantly of skill). A second approach, which the scientific comthe court below found (Op. at 3), Texas Hold ’Em is game of skill, not of chance. To appreciate the role that chance plays in almost every game, it is important to keep in mind just how few games exist in which luck plays no role whatsoever. Chess is the prototypical example of a game of pure skill, because both players have perfect information regarding the other’s pieces and all that matters is how skillfully a player deploys them. The important point for present purposes is that games of pure skill are exceedingly rare; at least some degree of luck plays a substantial role in almost every game we play. In fact, between two equally matched chess players, the coin flip to determine who plays black or white may have an effect on the outcome. See e.g. Jonathan Rowson, CHESS FOR ZEBRAS: THINKING DIFFERENTLY ABOUT BLACK AND WHITE at 193 (Gambit Publications 2005) (“the conventional wisdom is that White begins the game with a small advantage and, holding all other factors constant, scores approximately 56% to Black’s 44%.”) 6 8 munity favors, is an empirical approach that examines the actual play of the game. Using the well-accepted premise that in a game predominated by skill the more skillful players will consistently perform better (see, e.g., Patrick Larkey et al., Skill in Games, 43 MANAGEMENT SCIENCE 596 [May 1997] [attached as Ex. C.]), this approach looks for specific instances over repeated trials to see if in fact the “more skillful players tend to score better than less skillful players.” Id. at 596. Each method independently—and certainly both methods when taken together—confirms that the game of poker is a game of skill. A. Making Correct Decisions In Poker Requires A Diverse Array Of Sophisticated Skills That Games Of Chance Do Not. The essence of poker is correct decision-making. Each time it is a player’s turn to act, he must choose among several decisions, typically whether to bet, raise, or fold. During the course of a single session, a player will have to make hundreds of those decisions. In order to make the optimal decision the player must take into account a variety of factors. The importance of decision-making in poker cannot be understated: in a recent statistical analysis of millions of actual poker hands, the players’ decisions alone rather than the cards dealt accounted for the result in 76% of all the hands played. See Paco Hope & Sean McCulloch, Statistical Analysis of Texas Hold ’Em at 5 (March 4, 2009).7 In other words, in those 76% of hands, all but one player folded, making the remaining player the hand’s winner, and the actual cards were never revealed. Moreover, according to this report, in roughly 50% of hands that do play to a showdown,8 a player who would have won had he stayed in will have folded, meaning that in only 12% of 7 http://www.cigital.com/resources/gaming/poker/100M-Hand-AnalysisReport.pdf, last accessed July 23, 2009. A “showdown” is when all of the cards have been dealt and the players still in the hand expose their hold cards and the best hand wins the pot. It is only at the showdown where the winner is determined by the fall of the cards rather than by which players have folded in response to the moves of other players. 8 9 hands—that is, half of the 24% that play to showdown—does the player who was dealt the “luckiest” hand win. With player decisions deciding close to 90% of all poker hands, the players who consistently make good decisions will win. Those who do not will generally lose. In all, as expert witness and champion professional poker player Michael R. Sexton testified at trial, “[t]he object of poker is to make correct decisions.” Tr. at 149. The “luck of the cards” has little to do with one’s decision-making skills. To make the right decisions consistently, poker players must employ a range of skills. And by skill, we do not mean simply a sophisticated knowledge of odds. Knowledge of the odds is simply a prerequisite to competent poker play. To be skilled at poker, players must develop an ability to directly influence the way an individual hand turns out—who collects the pot at the end, and how much is in the pot. “Successful players must possess intellectual and psychological skills. They must know the rules and the mathematical odds. They must know how to read their opponents’ ‘tells’ and styles. They must know when to hold and fold and raise. They must know how to manage their money.” Dent, slip op. at 14-15; see generally id. (concluding that these skills determine the outcome in poker and that it therefore is not gambling under Pennsylvania law). See also generally Sexton Testimony, Tr. at 154 (describing skills involved in deducing what other players are holding); Sexton Testimony, Tr. at 164 (describing how a player exploits how other players perceive him). Of course it is true that individual moves in poker are called “bets.” But that vocabulary is misleading. The “bet” is not a wager on a chance event. Unlike poker “bets,” true wagers do not alter the outcome of the event. A bet on the Super Bowl does not change the score; bets at a blackjack table are made before the cards are dealt; bets on roulette wheels are placed before the ball is dropped. Bets at a poker table are different. What is called a “bet” in poker is really a 10 “move” like a move in any other game: it is a strategic maneuver designed to provoke a desired reaction from an opponent. The importance of these moves is heightened because, in typical complex poker games, a player must contend with a large number of decision-making stages and a variety of possible courses of action at each stage. In each hand of Texas Hold ’Em, a player has four principal decision-making opportunities: the first after he receives his down cards, and the next three as the common cards are turned over in three stages. At each stage the player has available to him many courses of action. The focus of each decision is how worthwhile it is to risk additional chips relative to the chance of winning all the chips in the pot in that hand. These decisionmaking stages reduce the element of chance in the game, since logical decision-making at each of these stages allows the player to control whether, and how much, he wins or loses. To make optimal moves at each of these stages, players must be mathematicians, observers of human nature, and capable deceivers. Poker players use their “bets” principally to communicate with, manipulate, and intimidate their opponents.9 Skeptics sometimes say that no amount of skill can turn a deuce into an ace. It is true that skill cannot change the cards. But skill allows the player with the deuce to make his opponent believe he has an ace, causing his opponent to fold a hand that would have won the pot. So skill also means that a good player will lose less with a deuce and win more with an ace than a bad one. See Sexton Testimony, Tr. at 175. Indeed, as noted, more than 75% of all hands are won when one player bets and all remaining players fold in response. See Hope et al. at 5; see also Howard Lederer, Why Poker Is a Game of Skill (May 6, 2008) (unpublished manuscript, attached as Ex. D). 9 As noted, poker is sometimes thought to be gambling simply because the vernacular of poker resembles that of gambling—players make “bets” as they play. But this Court should avoid that mistake, and should look be- 11 Even in that subset of hands that do go to a showdown, the players typically are not betting on the outcome of a chance event. For example, when a poker player bets as a bluff, he is not hoping that his cards will prove to be better than his opponents’ cards. Instead, the player hopes to win the pot by convincing his opponent to fold the best hand. As noted above, in roughly 50% of hands that do play to a showdown, a player who would have won had he stayed in will have folded, meaning that in 88% of hands the player who eventually won the hand did so by “convincing” his competitor to fold. That fact attests to the skill required of the winning player in bluffing his competitor into folding. See Paco, Statistical Analysis at 5. Of course, a player trying to chase another player out may get called and lose. But what he was betting on was not what cards his opponents held—the essence of gambling. He was betting to influence what his opponents would do—the essence of strategy. Furthermore, the outcome of a hand of poker is not only who wins and who loses, but how much each player wins or loses. A player’s assessment of his own cards and what cards the other players are holding will affect whether and how much the player bets. So even in the 12% of hands that reach a showdown and in which the best hand dealt wins the pot, the players’ skill will determine how much is won and how much is lost. Thus, in every single hand played, the skill of the players determines the outcome of the game. The importance of skill in poker is further demonstrated by the fact that a novice poker player can improve his talents and raise the level of his game through study and accumulating game experience. See Sexton Testimony, Tr. At 150. After only a short time, a player can acquire basic game skills, such as learning when to fold and how to make the basic calculations. yond the labels to the way the game is played. A “bet” on a poker hand is not a wager, because poker hands are not usually resolved by a chance event. 12 The more a person continues to practice and learn, the more his skills will improve, something that is also true for chess, golf, and bridge players.10 Together, the specific skills required to play poker, the demonstrated fact that poker hands are won by maneuvering rather than in a showdown the vast majority of the time, and the fact that in every hand the players’ skill determines the amounts won and lost by each player, show that skill is required to be a winning poker player. B. Skilled Players Beat Simple Players In Simulated And Real Poker Play. Several recent studies have definitively demonstrated that a player must be skilled in order to win at poker. Indeed, every single study to examine this issue has reached the very same conclusion: poker turns on skill. Until quite recently, any rigorous analysis of whether skill or chance predominated in poker could involve only an assessment of the rules of play themselves, because no research had assembled a statistical assessment of the role of skill in poker. The subject has now received academic attention, and the studies uniformly confirm that skill determines the outcome in poker games. This reflects an evolving understanding, and popularization, of the sophistication of the game of poker. In one recent game-theoretical study, for example, the author used a computer simulation to prove that a combination of the skills discussed above is required in order to win consistently at poker. See Larkey, supra. For his 2001 paper on “Skill in Games,” Professor Larkey built a computer model of a simplified version of poker. See id. The “general behaviors mandated for A significant literature is available to help the novice player develop. See, e.g., Gus Hansen, Every Hand Revealed (2008); Daniel Negreanu, Power Hold’em Strategy (2008); David Apostolico, Machiavellian Poker Strategy: How to Play Like a Prince and Rule the Poker Table (2005); Dan Harrington, Harrington on Hold ’Em: Expert Strategy for No Limit Tournaments (2005); Eric Lindgren, World Poker Tour: Making the Final Table (2005); Blair Rodman & Lee Nelson, Kill Phil: The Fast Track to Success in No-Limit Hold ’Em Poker Tournaments (2005); Doyle Brunson, Doyle Brunson’s Super System: A Course in Power Poker (2002); David Sklansky, Tournament Poker for Advanced Players (2002); David Sklansky, The Theory of Poker (1994). 10 13 player success” at this simplified game were: (a) observation, (b) memory, (c) computation, (d) knowledge of the random device, (e) misleading opponents about the actual strength of your position, and (f) correct interpretation and forecasts of opponents’ behaviors. Id. at 597. To evaluate the relative importance of these areas of skill, singly and in combination, the authors programmed twelve different robot players who would compete against one another. Each was programmed to use a different combination of strategies. Id. The simplest robot only knew the rules of the game—when to bet and how much it was allowed to bet—but aside from that essentially played randomly and without regard to its hand. A second robot understood the relative values of the hands. It would bet aggressively when it was dealt a good hand, and hold back when it got a bad hand. It ignored its opponents, while three other similar robots made conservative or aggressive assumptions about what the other player’s hands contained. Another robot bluffed aggressively. The more sophisticated robots watched their opponent’s betting patterns and made deductions about what those opponents were likely to be holding. Some of these robots would bluff by playing randomly a small percentage of the time in order to confuse other opponents capable of watching and learning. The authors ran a tournament that pitted each robot player against each other player in 100 one-on-one games. Over the course of the tournament, the random-play robot won only 0.4% of its games. It lost $546,000. The four robots that dominated the contest were the ones capable of sophisticated calculations about their odds of winning. The robot that could only calculate odds came in fourth. The robot that could calculate odds and that also bluffed occasionally came in third. But the two most successful robots of all were the robots that most closely emulated real poker players. A robot that not only calculated odds but also observed fellow players and adjusted its style of play came in second at $400,000. The best robot of all calcu- 14 lated odds, learned about its opponents, and bluffed occasionally in order to throw its competitors off track. Even in the simplified game of poker designed for the study, with simple hands and only two rounds of betting, the best robot was the robot with the essential skills that every poker player learns, practices, and tries to master. It calculated the odds it was playing against, which was essential to its success. But it outperformed the others by deceiving its competitors with strategic bluffs while learning about and adjusting to its competitors’ style of play. It won 89% of the hands it played, and earned $432,000. See Larkey at 601, table 2. A substantial number of other studies—including every study ever to have addressed the issue—reach the same conclusion as Professor Larkey. • Noga Alon, Poker, Chance and Skill. Professor Alon provides a detailed analysis of several simplified models of poker in order to allow a precise mathematical analysis. Though simplified, these models capture many of the main properties of sophisticated poker play. The article concludes that skill is the major component in deciding the results of a long sequence of hands because knowledge of hand probabilities is a learned skill fundamental to determining and implementing an advanced strategy; and an advanced strategy will earn more than a strategy of an unskilled player in the long run. As the common practice is to play many hands, the conclusion is that poker is predominantly a game of skill. (Attached as Ex. E). Laure Elie & Romauld Elie, Chance and Strategy in Poker (Sept. 2007) (unpublished manuscript). The Elie study expands on Professor Alon’s work by testing its hypothesis not on a simplified version of poker, but on games with 2 or 4 players (up from Alon’s two-player model), with or without blind betting, and with constant or variable stakes. Using computer simulation, Elie & Elie confirmed that the quality of a player’s strategy—the skill with which the player plays the game—has an overriding influence over the game’s outcome. (Attached as Ex. F.) Abraham J. Wyner, Chance and Skill in Poker (Apr. 2008) (unpublished manuscript). Reviewing the Alon and Elie & Elie studies, Professor Wyner concludes that both studies accurately described a salient fact about the game of poker: a skilled player who can calculate the odds and bet and bluff on that basis has a substantial advantage over players who lack these skills. (Attached as Ex. G.) Peter Borm & Ben van der Genugten, On a Measure of Skill for Games with Chance Elements (1996). In order for laws restricting games of chance to be sensibly applied, Borm and van der Genugten argue that some threshold level of skill must be established 15 • • • beyond which games cease to be games of chance and become games of skill. They developed a scale by which a game of pure chance ranks “0” and one of pure skill ranks “1,” and then sought to rank a series of games on that scale. For a “0” game, a the odds of a beginner winning are the same as those the most advanced player winning; in a “1” game, the most optimal player can always win. Blackjack, considered a game of chance, is ranked 0.16. Based on their mathematical model, the authors conclude that an extremely simplified “poker” game, with three players playing with only four cards, valued at 10, 20, 30, and 40, has a skill level more than double that of blackjack. (Attached as Ex. H.) • Rachael Croson, Peter Fishman & Devin G. Pope, Poker Superstars: Skills or Luck? 21 Chance, No. 4, 25-28 (2008). The authors compared data from 81 poker tournaments and 48 Professional Golfers’ Association Tournaments in an effort to determine whether the success achieved by the elite poker players—individuals who have finished in the top 18 of at least one high-stakes Texas Hold’em tournament—is due to skill or luck. Analysis of the data led the authors to conclude that poker seems to involve a significant amount of skill because success in a given tournament can be predicted based on past success in tournament play. The authors also found that there are quantifiable skill differentials between elite poker players which are similar to skill differentials between comparably elite golfers. (Attached as Ex. I.) Gerard Cohen, Consultation on Professor Alon’s Poker, Chance and Skill. Professor Cohen confirms the validity of Professor Alon’s conclusions. According to Cohen, players must adapt their strategies to the number of players (by betting less often and with a hand that is stronger as this number increases). Moreover, the skilled player must take into account in his or her strategy the position and the order of players around the table. The importance of using these skills in real poker play, which is even more complex than in Alon’s case studies, leads him to the conclusion that skill is predominant in determining poker outcomes. (Attached as Ex. J.) Zvi Gilula, Expert Opinion. Professor Gilula concludes that winning a poker tournament is depends significantly more on the participants’ strategic capabilities and understanding than on luck. He notes that players must learn to: evaluate, within a predetermined interval of time, the strength of the hand that he holds in each stage of the game; mask his own strategy; evaluate his opponents’ strategies; and translate the insights which arise from using these other abilities into a rational decision making policy. The effect of these abilities is that the probability for an insightful player with strategic skills to win a poker tournament, when playing against a player who does not have these skills, is much higher than 50%. (Attached as Ex. K.) Paco Hope (Cigital Inc.) & Sean McCulloch, Statistical Analysis of Texas Hold’Em (Mar. 4, 2009). Hope and McCulloch examine 103 million hands of a particular poker variant—Texas Hold’ Em—played on PokerStars. For each hand analyzed, they ask whether the hand ended in a showdown, and if so, whether the player with the best two cards won the hand. They conclude that in the majority of cases—75.7% of the time— the game’s outcome is determined with no player seeing more than his or her own cards and some or all of the community cards. In those hands, all players folded to a single 16 • • • remaining player, who took the pot. In the remaining 24.3% of hands that go to a showdown, where cards are revealed to determine a winner, only 50% are won by the player who, had everyone stayed in the game, would have held the winning hand. The remaining hands are won by a player with an inferior hand, because the player with the best hand folded. From this, the authors determine that the winner in a majority of games is determined by something other than randomly drawn cards. The number of identifiable skills required to excel at poker and the simulations and studies just discussed all predict that, in real life, the more skilled players will win. In fact, that is what actual poker play makes clear. The best poker players beat other poker players as often as the best golfers beat other golfers, if not more often. It is true that poker has a “random device” (see Larkey at 597) that introduces short term uncertainty into each hand, but over time the randomness of the cards evens out and all players eventually get the same share of good and bad hands. Their results differ based on how skillfully they play those hands. A striking example of the limited role that the cards play in determining the outcome of poker matches may be found in the recent story of Annette Obrestad, a 19-year-old poker prodigy who beat 179 other players—without looking at her own cards (except one peek on one hand). See Shawn Patrick Green, Online Poker: Interview With Annette ‘Annette_15’ Obrestad, CardPlayer.com (Aug. 12, 2007).11 Obrestad’s feat shows it is the player’s skill rather than the deal of the cards that determines the outcome of poker play.12 The same result is demonstrated by comparing the results of recent golf and poker tournaments. In the 25-year period beginning with 1976 and ending in 2000, 21 different players won the World Series of Poker. One player won three times in that span (Stu Ungar), and three more players won twice (Johnny Moss, Doyle Brunson and Johnny Chan). Three of these repeat 11 http://www.cardplayer.com/poker-news/2536-online-poker-interview-with-annette-39-annette_15-39obrestad, last accessed July 22, 2009. 17 winners won back-to-back wins in consecutive years (Brunson, Ungar and Chan). Fourteen of the twenty-one were “repeat finalists” who finished among the top ten in one or more of the other tournaments. In the same period, there were twenty-two different winners of the PGA Championship, and three multiple winners. Only Tiger Woods won back-to-back titles. Fifteen of the twentytwo champions made it into the top ten in another Championship. These numbers confirm that poker requires as much skill as golf to win consistently. Accord Croson, Fishman & Pope, supra, at 14 (Ex. I at 3-4). Two recent legal analyses reached the same conclusion. See Anthony Cabot & Robert Hannum, Poker, Public Policy, Law, Mathematics, and the Future of an American Tradition, 22 T.M. Cooley L. Rev. 443 (2005) (conducting Texas Hold ’Em simulations to determine that skilled opponents beat unskilled ones); Michael A. Tselnik, Check, Raise, or Fold: Poker and the Unlawful Internet Gambling Enforcement Act, 35 Hofstra L. Rev. 1617, 1664-65 (Spring 2007). As expert witness Professor Hannum testified at trial, “the consensus” view among members of the scientific community “is in agreement with my opinion that skill is the predominant factor in poker.” Tr. at 208. It is precisely because poker requires roughly the same amount of skill as golf that poker tournaments now rival golf tournaments in popularity on television. The only people who watch anyone play roulette on television are casino security guards. People only watch lottery drawings to see if they have won. But poker matches are spectator events because, as in any game that people tune in to watch, it is fun to watch good players get beaten by even better players. Like golf, poker is a game won and lost predominately on the basis of the skills of the players. This example also refutes the conclusion that the “chance” of what a player is dealt as initial hole cards has a substantial affect on outcome; it cannot affect someone who never looks at them. 12 18 Appellants in this case were playing a game of skill. They were not engaged in unlawful gambling. CONCLUSION For the foregoing reasons, the Court should reverse the decision of the Mount Pleasant Municipal Court, hold that S.C. Code § 16-19-40 does not prohibit playing poker in a private home, and dismiss the action against Appellants. Respectfully submitted this 30th day of July, 2009. By: _/s/ Thomas C. Goldstein Thomas C. Goldstein Christopher M. Egleson Jonathan H. Eisenman Akin Gump Strauss Hauer & Feld LLP 1333 New Hampshire Ave., NW Washington, D.C. 20036-1564 (202) 887-4000 Kenneth L. Adams Adams Holcomb LLP 1875 Eye Street NW Washington, D.C. 20006 (202) 580-8822 19 JAN-16-2009 FRI 02:31 PM CAMPANA &LOVECCHIO FAX No. 570 326 3498 p. 002 COMMONWEALTH OF PENNSYLVANIA vS DIANE A. DENT IN THE COURT OF COMMON PLEAS FOR THE 26TH JUDICIAL DISTRICT, COLUMBIA COUNTY BRANCH, PENNSYLVANIA CRIMINAL DIVISION CASE NO; 733 OF 2008 Defendant ********************************** COMMONWEALTH OF PENNSYLVANIA VS IN THE COURT OF COMMON PLEAS FOR THE 26TH JUDICIAL DISTRICT, COLUMBIA COUNTY BRANCH, PENNSYLVANIA CRIMINAL DIVISION WALTER WATKINS Defendant CASE NO: 746 OF 2008 ,’.’ 1(‘”‘ ·,1 ‘ IY\ THOMAS LEIPOLD, ESQUIRE, Attorney for the CornrnonwealthC’, COMMONWEALTH OF PENNSYLVANIA vs WALTER WATKINS Defendant IN THE COURT OF COMMON PLEAS FOR THE 26TH JUDICIAL DISTRICT, COLUMBIA COUNTY BRANCH, PENNSYLVANIA CRIMINAL DIVISION CASE NO: 746 OF 2008 ORDER AND NOW, this 14 t h day of January 2009, defendants’ The cases against Motions for Writ of Habeas Corpus is GRANTED. the defendants are DISMISSED. The property seized from defendant Watkins shall be returned to him forthwith. BY THE COURT: 15 Page 1 3 of 6 DOCUMENTS Copyright 2009 Fort Collins Coloradan All Rights Reserved Fort Collins Coloradoan (Colorado) January 30, 2009 Friday SECTION: LOCAL; Pg. 3A LENGTH: 578 words HEADLINE: Definition clears man of gambling charges BYLINE: TREVOR HUGHES BODY: TrevorHughes @coloradoan.com When Windsor resident Kevin Raley started helping to organize friendly poker tournaments in a Greeley bar, he never thought he’d end up in court facing charges of illegal gambling. But following an undercover investigation by the Colorado Bureau of Investigation, Raley and four other players were arrested in August. They were charged with professional gambling and illegal gambling, and faced jail time if convicted. Last week, however, a Weld County jury agreed with Raley’s argument that poker games between friends are just that – poker games between friends. The jury acquitted Raley on a charge of illegal gambling after prosecutors dropped the professional gambling charge before the trial began. “We never believed we were doing anything wrong whatsoever,” Raley said. “It’s entertainment. Some people go to the movies. Some people play golf. I play poker.” The national Poker Players Alliance helped Raley, a software consultant, mount his defense, paying for an expert witness to testify that poker is a game of skill, not chance. Under Colorado law, illegal gambling “means risking any money, credit, deposit, or other thing of value for gain contingent in whole or in part upon lot, chance, the operation of a gambling device, or the happening or outcome of an event, including a sporting event, over which the person taking a risk has no control, but does not include bona fide contests of skill.” The PPA’s expert, professor Robert Hannum of the University of Denver, testified that poker isn’t dependent primarily on chance but on each player’s skill. Hannum is a professor of statistics and is the author of the book “Practical Casino Math.” Hannum said there are many factors that go into how a player plays a game of poker, and few of them are based on Page 2 Definition clears man of gambling charges Fort Collins Coloradoan (Colorado) January 30, 2009 Friday chance. “There are a lot of facets to the skill, in terms of knowing the math and the odds, reading the people, trying to glean what other players’ hole cards might be. But it’s all expressed in the decision they make in how much money, if any, they are willing to invest,” Hannum said. He noted that a skilled poker player will beat an unskilled one “consistently and probably convincingly,” but that true games of chance require no skill. In a statement, the PPA lauded the jury’s decision and said it hoped the outcome would help law enforcement to focus on what it said is “real unlawful activity.” “… The not-guilty verdict cements the rights of Colorado citizens to enjoy the American pastime of poker and will allow law enforcement to use its scarce resources to investigate real unlawful activity in the state, not poker games,” Colorado state PPA director Gary Reed said. Prosecutors dropped the professional gambling charge against Raley before the case went to trial. The illegal gambling charge carried a maximum penalty of a $100 fine. A spokeswoman for Weld County District Attorney Ken Buck said prosecutors felt the case was an “appropriate” one to present to a jury. CBI spokesman Lance Clem said CBI agents got involved at the request of Greeley police. “We thought we helped put together a good case and still feel that way,” Clem said. Raley said he still doesn’t understand why CBI and prosecutors thought targeting him made sense. “The five of us all assumed that once all the facts were known to the DA, they would drop the charges,” Raley said. “We never assumed we would go clear to court.” The three men and one woman arrested with Raley still face charges. LOAD-DATE: February 6, 2009 Copyright © 2001 All Rights Reserved Copyright © 2001 All Rights Reserved Copyright © 2001 All Rights Reserved Copyright © 2001 All Rights Reserved Copyright © 2001 All Rights Reserved Copyright © 2001 All Rights Reserved Copyright © 2001 All Rights Reserved Copyright © 2001 All Rights Reserved Copyright © 2001 All Rights Reserved Copyright © 2001 All Rights Reserved Copyright © 2001 All Rights Reserved Copyright © 2001 All Rights Reserved Copyright © 2001 All Rights Reserved Copyright © 2001 All Rights Reserved Poker, Chance and Skill Noga Alon ∗ 1 Introduction The question if poker is a game of skill or a game of chance received a considerable amount of attention mainly because of its potential legal implications. See, for example, [3] and its many references. Most of the material dealing with the subject focuses on legal issues, and only briefly touches the question from a purely scientific point of view. In the present article we address the question as a scientific one. To do so, we provide a detailed analysis of several simplified models of poker, which can be viewed as toy models of Texas Hold’em, the most popular variant of poker. The advantage of considering these simplified models is that unlike the real game, they are simple enough to allow a precise mathematical analysis, and yet there is every reason to believe that this analysis captures many of the main properties of the far more complicated real game, and enables us to estimate the advantage of skilled players over less skilled ones. The analysis suggests that skill plays an important role in poker. As explained in the second half of the article, this fact, together with the Central Limit Theorem, imply that skill is the major component in deciding the results of a long sequence of hands. As the common practice is to play many hands, the conclusion is that poker is predominantly a game of skill. The article is organized as follows. In Section 2 we describe the rules of Texas Hold’em which is probably the most popular poker game played in casinos and card-rooms throughout the world, as well as in online poker sites. Section 3 contains the basic probabilistic information regarding the odds of the main possibilities in the game. In Section 4 we give a detailed analysis of several simplified versions of poker. Section 5 contains a discussion of ∗ Schools of Mathematics and Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. Email: [email protected] 1 the relevance of the Law of Large Numbers, or more specifically, the Central Limit Theorem, to the determination of the success of skilled and less skilled players in a sequence of games. This is illustrated by considering the simplified versions introduced in Section 4. A summary and concluding remarks appear in the final section 6. 2 The Game There are many versions of poker, here we focus on Texas Hold’em (often called Hold’em, for short). The game is usually played with at most 10 (and at least 2) players. This is the most popular member of a class of poker games known as community card games, which all bear some similarity to each other. Like most variants of poker, the objective in hold’em is to win pots, where a pot is the sum of the money bet by all players in a hand. A pot is won either at the showdown by forming the best five card poker hand out of the seven cards available, or by betting to cause other players to fold and abandon their claim to the pot. The objective of a player is not to win the maximum number of individual pots, but rather to make mathematically correct decisions in order to maximize the expected net amount won in the long run. Here is a rough brief description of the game: Each player is dealt two cards and this is followed by a round of betting. Then the dealer spreads three cards face up (called the flop) in the middle, and this is followed by a second round of betting. The dealer places a fourth card (called the turn) face up and another round of betting follows. Finally, the dealer places a fifth card (called the river) face up and the last round of betting takes place. Each player who has not folded during the betting rounds gets the best hand of five cards among his own two cards plus the five community cards in the center. A more detailed account follows. See, e.g., [9] for several variants and further details. Hold’em is often played using small and big blind bets. A dealer button is used to represent the player in the dealer position; the dealer button rotates clockwise after each hand, changing the position of the dealer and blinds. The small blind is posted by the player to the left of the dealer and is usually equal to half of the big blind. The big blind, posted by the player to the left of the small blind, is equal to the minimum bet. There are several variations on the betting structure, here we describe limit hold’em. In this version bets and raises during the first two rounds of betting (pre-flop and flop) must 2 be equal to the big blind; this amount is called the small bet. In the next two rounds of betting (turn and river), bets and raises must be equal to twice the big blind; this amount is called the big bet. A play of a hand begins with each player being dealt two cards face down from a standard deck of 52 cards. These cards are the player’s hole or pocket cards, they are the only cards each player will receive individually, and they will only (possibly) be revealed at the showdown, making hold’em a closed poker game. After the pocket cards are dealt, there is a ”pre-flop” betting round, beginning with the player to the left of the big blind (or the player to the left of the dealer, if no blinds are used) and continuing clockwise. A round of betting continues until every player has either folded, put in all of their chips, or matched the amount put in by each other active player. After the pre-flop betting round, assuming there remain at least two players taking part in the hand, the dealer deals a flop; three face-up community cards. The flop is followed by a second betting round. This and all subsequent betting rounds begin with the player to the dealer’s left and continue clockwise. After the flop betting round ends a single community card (called the turn) is dealt, followed by a third betting round. A final single community card (called the river) is then dealt, followed by a fourth betting round and the showdown, if necessary. If a player bets and all other players fold, then the remaining player is awarded the pot and is not required to show his hole cards. If two or more players remain after the final betting round, a showdown occurs. On the showdown, each player plays the best five-card hand he can make from the seven cards comprising his two pocket cards and the five community cards. A player may use both of his own two pocket cards, only one, or none at all, to form his final five-card hand. If the five community cards form the player’s best hand, then the player is said to be playing the board and can only hope to split the pot, since each other active player can also use the same five cards to construct the same hand. If the best hand is shared by more than one player, then the pot is split equally among them. The best hand is determined according to the ranking described below. If the significant part of the hand involves fewer than five cards, (such as two pair or three of a kind), then the additional cards (called kickers) are used to settle ties. Note that only the card’s numerical rank matters; suit values are irrelevant in Hold’em. 3 The ranking of the hands is as follows: • Royal Flush (the top hand): The five highest cards, the 10 through the Ace, all five of the same suit. A royal flush is also an ace-high straight flush. • Straight Flush: Any five cards of the same suit in consecutive numerical order. • Four of a Kind: Four cards of the same denomination. • Full House: Any three cards of the same denomination, plus any pair of a different denomination. Ties are broken first by the three of a kind, then the pair. • Flush: Any five non-consecutive cards of the same suit. • Straight: Any five consecutive cards of mixed suits. Ace can be high or low. • Three of a Kind: Three cards of the same denomination. • Two Pair: Any two cards of the same denomination, plus any other two cards of the same denomination. If both hands have the same high pair, the second pair wins. If both pairs tie, the high (fifth) card wins. • Pair: Any two cards of the same denomination. In a tie, the high card wins. • High Card: If no other hand is achieved, the highest card held wins. Texas hold’em (usually with a no-limit betting structure) is played as the main event in many of the famous tournaments, including the World Series of Poker’s Main Event. Traditionally, a poker tournament is played with chips that represent a player’s stake in the tournament. Standard play allows all entrants to ”buy-in” a fixed amount and all players begin with an equal value of chips. Play proceeds until one player has accumulated all the chips in play. The money pool from the players ”buy-ins” are redistributed to the players in relation to the place they finished in the tournament. Usually only a small percentage of the players receive any money, with the majority receiving nothing. As a result the strategy in poker tournaments can be different from that in a cash game. 4 3 Odds and Probabilities Some familiarity with the odds of the various possible combinations in poker is necessary, though certainly not sufficient, for skilled poker play. The ranking of hands in poker is determined according to their frequencies as 5-card poker hands. These frequencies can be easily computed. There are are given below. The numbers of 5-card poker hands: • Royal Flush: 4 1 52 5 = 2, 598, 960 different poker hands. Among these 4 are Royal Flush and 36 are non-royal Straight Flush. These and the numbers of the other hands =4 9 1 4 1 • Straight (non-royal) Flush: • Four of a Kind: • Full House: • Flush: 13 5 4 1 10 1 13 1 13 1 4 3 4 4 12 1 48 1 4 2 = 36 = 624 = 3, 744 − 40 = 5, 108 4 5 1 • Straight: − 40 = 10, 200 13 1 4 3 4 1 12 2 4 2 1 • Three of a Kind: • Two Pair: • Pair: 13 1 4 2 13 2 12 3 13 5 = 54, 912 4 2 11 2 1 4 3 1 = 123, 552 = 1, 098, 240 • High Card: [ − 10](45 − 4) = 1, 302, 540 123,552 2,598,960 Thus, for example, a fraction of number is 52 7 = 0.047539 of all 5 card hands form a Two Pair. More relevant to Hold’em is the corresponding information for 7-card hands. Their total = 133, 784, 560. The number of hands for each possibility of the best 5 card subset is also not difficult to compute. This is done in [1] , and appears below, together with the probability of each possibility in a random 7-card hand. 5 The numbers and frequencies of 7-card poker hands: Royal Flush Straight (non-royal) Flush Four of a Kind Full House Flush Straight Three of a Kind Two Pair Pair High Card 4, 324 37, 260 224, 848 3, 473, 184 4, 047, 644 6, 180, 020 6, 461, 620 .0000323 .000278 .0017 .026 .030 .046 .048 31, 433, 400 .235 58, 627, 800 .438 23, 294, 460 .174 Hence, when playing Hold’em a player should expect to get Three of a Kind or higher once in about 20 hands, and Four of a Kind once in about 600 hands. During the game, a player should be capable of estimating the probability of improving his hand when the turn or river community cards will be dealt. If, for example, the player holds two diamonds, and the flop contains two other diamonds, then there are 9 additional diamonds in the deck, implying that the probability that the next community card will be a diamond is 9/47, and in case it will not, the probability that the last community card will be a diamond is 9/46. A player should also always be aware of the expected winning amount in a game; in general one should bet when the expected value of the gain (which is the amount in the pot after the bet, times the probability of winning) is greater than the wager. Of course, even if the player knows the precise probability, this should be modified from time to time in order not to reveal the strategy of the player; bluffing is a crucial part of the game as will be clear from the analysis of the simplified versions considered in the next section. 4 Simple Variants There is a significant amount of literature on various toy models of poker, starting with the variants discussed in the classical book of Von Neumann and Morgenstern [8]. See, for 6 example, [7], [5], [6]. In most of these articles, however, the authors try to find the best strategy of the players assuming they play optimally. Our treatment here is different, as the main intention is to assess the significance of skill in the game. We therefore investigate the case in which one player is more skilled than the other(s). Although the models we suggest are vast simplifications of the real game, they do seem to capture many of the properties of real poker. 4.1 The Basic Variant Consider a version of Hold’em in which each player gets two face down pocket cards, the flop, turn and river community cards are spread face up in the middle, and only then there is one round of betting. Suppose, further, that in this round each player is allowed to either fold, or bet 1 chip, and these decisions are made simultaneously by all players. If all players fold then nothing happens, if at least one player bets, then the active player with the highest hand wins the pot. Given the 5 community cards, there are m = 47 2 = 1081 possibilities for the two pocket cards of each player, and ignoring equalities, there is a linear order among them. Therefore, a perfect player that sees the community cards and his hole cards, knows precisely the rank of his hole cards among the 1081 possibilities, and hence can compute, in principle, the precise probability that his hand is the highest among all hands of the participants. It is worth noting that knowing these precise probabilities in all cases is not an easy matter, and is probably beyond the ability of a human being, as this requires to memorize a huge table of ranks representing all possible values of the community cards and the player’s hole cards. Yet, it seems that skilled poker players can estimate well the probability in each case. Ignoring the (rather negligible) effect of the fact that the pairs forming the pocket cards of all players should be disjoint, one can model this situation by a game in which the players are dealt random distinct numbers between m = 1081 (the strongest possibility for the pocket cards given the community cards) and 1 (the weakest possibility). As m is a large number this can be further simplified by considering the case in which each player is dealt his hole number; a uniformly chosen random real number in the unit interval [0, 1], where a higher number is considered better than a lower one. In what follows we refer to this game as the basic game. We start with the simplest case, in which there are two players, A (Alice) and B (Bob). In this case, Alice gets a uniform random number xA ∈ [0, 1], and Bob gets a uniform 7 random number xB ∈ [0, 1], where the choices of xA , xB are independent. Each player knows his/her own number, but not the one of the other player, and they have to choose between folding and betting 1 chip. Suppose that Bob is an unskilled player, who plays randomly. That is, for any value of xB , Bob decides to fold with probability 1/2, and decides to bet with probability 1/2. Alice, who is a skilled player, suspects that this is Bob’s strategy, and chooses her strategy in order to ensure maximum expected gain in the game against Bob. To determine the strategy of Alice, let us consider how she should behave when her pocket number is xA = x. If she decides to bet, then the expected number of chips she wins (including her own chip) is 1 1 1 + x2. 2 2 Indeed, with probability 1/2 Bob will fold, and in this case Alice will win her single chip, giving the first term above. With probability 1/2 Bob will decide to bet, in this case with probability x his number xB lies in [0, x) and is thus smaller than Alice’s number, and if so Alice will win two chips. This gives the second term. Alice should bet if and only if her expected win exceeds her cost, which is the 1 chip she bets. Thus, she should choose to bet 1 1 if and only if 2 1 + 2 x2 ≥ 1, that is, if her hole number x = xA is at least 1/2. If, indeed, Bob and Alice follow the above strategies, then at least one of them folds with probability 1 − 1 2 · 1 2 = 3/4, and thus, with probability 3/4 the expected net gain of Alice is 0. The probability that Alice’s net gain is 1 is 1 2 1 xdx = 1/2 3 , 16 and the probability that Alice’s net gain is −1 is 1 2 1 (1 − x)dx = 1/2 1 . 16 Altogether, in a single hand, the expected value of the random variable X describing Alice’s net gain is E(X) = and its variance is V ar[X] = E(X 2 ) − (E(X))2 = 3 1 1 15 + − ( )2 = . 16 16 8 64 3 1 1 ·1+ · (−1) = , 16 16 8 8 We have thus proved the following, where here and in what follows we refer to the player playing randomly as the unskilled player. Proposition 4.1 In a single hand of the basic game with two players, a skilled one and an unskilled one, the expected value of the net gain of the skilled player is 1/8 and the variance of this net gain is 15/64. Note that, not surprisingly, the skilled player has a significant advantage over the unskilled one. 4.2 The Importance of Being Unpredictable Suppose that Bob and Alice play a sequence of hands of the the basic game described above. Bob is likely to realize that Alice’s strategy is better than his random one, and he is also likely to observe that she is betting if and only if her hole number xA is at least 1/2. He can thus decide to adopt Alice’s winning strategy, and bet if and only if his number xB is at least 1/2. However, when he starts doing so, Alice, who is more skilled, realizes that this is the case. She can thus adjust her strategy and choose the optimal response to the new strategy of Bob. It is not difficult to modify the previous computation to this case. Observe, first, that if xA < 1/2, then Alice should not bet, as with the new strategy of Bob this can never lead to any winning. If Alice hole number is x ≥ 1/2, and she decides to bet, then the expected amount she wins is 1 1 1 · 1 + (x − )2 = 2x − . 2 2 2 Indeed, with probability 1 2 Bob’s number xB will lie in [0, 1/2], he will not bet, and Alice 1 2 1 Bob’s number will lie in [ 2 , x) and 1 2 will get her chip back. Similarly, with probability x − in this case Alice’s win will be 2. Therefore, Alice should bet if and only if 2x − ≥ 1, that is, if x ≥ 3 . In case Bob and Alice play according to these new strategies, then the 4 1 random variable describing Alice’s net gain is 0 with probability 1 − 2 · 1 = 7 , it is +1 with 4 8 probability 1 1 3/4 (x − 2 )dx = 3 32 and it is −1 with probability 1 3/4 (1 − x)dx = 1 32 . This gives the following. Proposition 4.2 In a single play of the basic game with two players A and B, where A bets if and only if xA ≥ 3/4 and B bets if and only if xB ≥ 1/2, the expected value of the net gain of A is 1/16 and the variance of this net gain is 31/256. 9 Note that here the losing player is using exactly the same strategy used by the winning player in the previous subsection. This shows that already in this simplified version of the game, a winning player should adjust her strategy to those of the other players. It also shows the importance of bluffing; once your strategy is revealed, the other players can exploit it. These principles hold (in a far more sophisticated way) in real poker; it is crucial for a winning player to stay unpredictable, and to take into account the strategy of the other players. 4.3 More Players In real poker the number of players is often larger than 2. Consider the basic game in which there are n + 1 players denoted by P0 , P1 , . . . , Pn . As our objective is to measure the significance of skill, assume that the first player, P0 , is skilled, and all other players are unskilled and play randomly. Therefore, the players are dealt n + 1 uniform, independent random numbers in [0, 1], where xi is the hole number of Pi , then each of them decides to fold or bet one chip, where all these decisions are taken simultaneously, and finally the active player with the largest number wins the pot. Let us compute the optimal strategy for P0 , assuming all other players play randomly. If x0 = x and P0 decides to bet, then the expected amount of chips he wins is 1 2n n (k + 1) k=0 n k x . k Indeed, the probability that exactly k players among the n unskilled ones decide to bet is n k 2n . If so, then the probability that all their hole numbers will lie in [0, x) is xk , and in this case P0 will win the pot, whose size will be k + 1. Therefore, P0 should bet if and only if 1 2n Since n n (k + 1) k=0 n k x ≥ 1. k (k + 1) k=0 n k d x = x(1 + x)n k dx = (1 + x)n + nx(1 + x)n−1 , it follows that P0 should bet when x0 = x if and only if (1 + x)n + nx(1 + x)n−1 ≥ 2n . 10 In particular, for n = 1 (two players, one skilled and one unskilled), the skilled player should bet if and only if (1 + x) + x ≥ 2, that is, if and only if x ≥ 1/2, as we have already seen in subsection 4.1. If n = 2 (three players), the skilled player should bet if and only if (1 + x)2 + 2x(1 + x) ≥ 4, that is, if and only if x ≥ √ 13−2 3 = 0.535.., and if n = 9 (10 players, 9 of whom are unskilled), the skilled player should bet if and only if his hole number x satisfies (1 + x)8 (10x + 1) ≥ 512, that is, whenever x exceeds 0.685... Here, too, the mathematical analysis of the simplified model reveals a crucial feature of real poker: a skilled player should adjust his strategy to the number of players. In general, when this number grows, the player should fold more often and bet mostly with stronger hands. 4.4 Blinds and Position In the basic model considered in subsection 4.1, there is no nontrivial optimal strategy in the sense of Game Theory, that is, if both players play optimally, then their best (mixed) strategy is to keep folding and never bet. Indeed, as a uniformly chosen random number in [0, 1] is strictly smaller than 1 with probability 1, one can show that for any nontrivial betting policy of one of the players, there is a strategy that beats it. The reason for this is that this simplified version of the game ignores the cost of playing and, more crucially, contains no forced bets (called blinds, or ante in real poker) which are necessary to create an initial stake for the players to contest. We thus discuss here a slightly more realistic model of the game, containing a forced blind bet. In order to enable a rigorous analysis, this model is still far from the real game, and yet its analysis illustrates nicely the fact that in real poker the strategy has to be adjusted to the position and the order in which players have to act. Consider, thus, a model in which there are two players. The game starts with a blind bet of 1 chip by the first player, then the 5 community cards as well as the two pocket cards of each player are dealt. The second player can now either fold or bet 3 chips, and the first player can also either fold or raise his bet to 3, where both players make their decisions simultaneously. If both players fold nothing happens, if one player folds and the other bets, then the active player wins the pot, and if both players bet, the higher hand wins the pot. The choice of the numbers 1 and 3 here is arbitrary, and the analysis can be carried out for different numbers in a similar manner. By the discussion in subsection 4.1, assuming the players can memorize a substantial table of possibilities, the game is well approximated 11 by a version in which the first player makes a blind bet of 1, then the players get uniform, independent, random pocket numbers in [0, 1], and then the second player either folds or bets 3, and the first either folds or increases his bet to 3. The blind bet alternates between the players, as obviously having to start with it is a disadvantage. We call this version of the game the basic game with a blind bet, and analyze it as in subsection 4.1 for two players, a skilled one (Alice) and an unskilled one playing randomly (Bob). There are two cases to consider, depending on the identity of the player posting the blind bet. Assume, first, that Alice is making the blind bet. If her number is x and she decides to bet, then her expected win is 1 2 · 3 + 1x · 6 = 2 3 2 + 3x. Indeed, with probability 1/2 Bob folds 1 2x and then Alice gets back her 3 chips, and with probability Bob bets and his number is smaller than x, and if so Alice wins 6 chips. Alice should bet if and only if she expects to win at least the cost of increasing her bet. As this cost is 2, she should bet if and only if 3 2 + 3x ≥ 2, that is, if and only if x ≥ 1 . If she uses this strategy, then her net gain is 6 1 1 2 1/6 (1 −3 with probability − x)dx = probability 1/2, and +3 with 25 144 . It is −1 with 1 1 35 probability 2 1/6 xdx = 144 . probability 1 2 · 1 6 = 1 12 , 0 with A similar analysis shows that when Bob is posting the blind bet Alice should bet if and only if her number x = xA satisfies 1 2 · 4 + 1 x · 6 = 2 + 3x ≥ 3, that is, if and only if x ≥ 1/3. 2 1 1 2 1/3 (1−x)dx With this strategy the expected net gain of Alice is −3 with probability it is 0 with probability 1/3, it is +1 with probability 1 1 2 1/3 xdx 1 2 = 1, 9 1 · 2 = 3 , and it is +3 with probability 3 = 2 . We summarize these facts in the following. 9 Proposition 4.3 Suppose a skilled player is playing one basic game with a blind bet against an unskilled player. (i) If the skilled player posts the blind bet, then her expected net gain is 25 1 35 1 · (−3) + · (−1) + ·3= 144 12 144 8 and the variance is 1 35 1 733 25 · (−3)2 + · 12 + · 32 − ( )2 = . 144 12 144 8 192 (ii) If the unskilled player posts the blind bet, then the expected gain of the skilled player is 1 1 2 2 · (−3) + · 1 + · 3 = 9 3 9 3 with variance 1 1 2 2 26 · (−3)2 + · 12 + · 32 − ( )2 = . 9 3 9 3 9 12 Note that the skilled player has to use one strategy when posting the blind bet and another one when the second player is posting the blind bet. Indeed, in real poker the strategy has to take the position into account. 5 The Effect of the Central Limit Theorem The analysis of the simplified models of poker discussed in the previous section shows that skilled players have a rather significant advantage over unskilled ones; this advantage becomes more and more prominent as the number of hands played increases. Intuitively that’s a clear fact, as in the long run the cards dealt to all player are similar on average. A rigorous explanation with precise quantitative estimates can be given using the Central Limit Theorem. By (a special case of) the Central Limit Theorem (see, e.g., [4]), the normalized sum of independent uniformly bounded random variables is converging to a normal distribution. A precise version follows. Theorem 5.1 Let M be a positive real, and let X1 , X2 , . . . be a sequence of independent random variables, where each Xi satisfies |Xi | ≤ M , the expectation of Xi is µi and its 2 variance is σi . Define Zn = Then, for every real z, n→∞ n i=1 Xi − n i=1 µi n 2 i=1 σi . lim P rob[Zn ≤ z] = Φ(z) where z 1 2 Φ(z) = √ e−t /2 dt, 2π −∞ is the cumulative distribution function of a standard Normal Random Variable. (1) Applying this theorem to the basic game between a skilled and an unskilled player in the basic game discussed in subsection 4.1, we get the following. Proposition 5.2 In a sequence of n hands of the basic game between a skilled and an unskilled player, the probability that the skilled player will not lead at the end is approximately Φ(− n/15), where Φ(z) is given in (1). 13 The proof is simple. For each i, 1 ≤ i ≤ n, let Xi denote the net gain of the skilled player in the i-th hand. By Proposition 4.1 the expected value of each Xi is µi = 2 is σi = 15/64. Using the notation of Theorem 5.1, put n i=1 Xi 1 8 and its variance Zn = − n/8 . 15n/64 n i=1 Xi Since the random variables Xi are independent (and bounded), the theorem applies and shows that for large n, the probability that is precisely the probability that Zn ≤ is approximately y − n/8 ). 15n/64 n i=1 Xi is the total net gain of the skilled player, the probability he will not lead at the Φ( n i=1 Xi is at most some real number y, which y − n/8 15n/64 As end is precisely the probability that y = 0 in the last displayed equation. ≤ 0. The desired result follows by substituting The above approximation is very accurate already for modest values of n, and certainly for all n > 50. Taking the values of the function Φ from a table of Normal Distribution we conclude that, for example, for n = 60 this probability is Φ(−2) = 0.0227.. and for n = 240 the probability is Φ(−4) = 0.00003167.., that is, smaller than 1/30, 000. For n = 350 the probability the unskilled player wins is already smaller than one in a million. Note that by the same reasoning one can bound the probability that after n games the skilled player will have a net gain of at most y chips. Thus, for example, the probability that after n = 240 hands the skilled player will have a net gain of at most n/16 = 15 chips is roughly Φ((15 − 30)/ 15 · 240/64) = Φ(−2) = 0.0227.. A similar computation for the case of the simple game with a blind bet can be carried out using Proposition 4.3. Proposition 5.3 Suppose a skilled and an unskilled player are playing 2n hands of the basic game with a blind bet, where each player posts the blind bet n times. Then the probability that the skilled player will not lead at the end is approximately √ 19 n Φ(− √ ). 3863 14 We omit the detailed computation and only give two examples. If n = 90 then the prob√ √ ability that at the end the skilled player will not be ahead is about Φ(−19 90/ 3863) = 0.00187.. For n = 140 this probability drops down to less than 0.00016. The discussion above shows that the skill component in poker (at least in the simplified models considered here), which gives some advantage in a single hand, provides a major advantage in a sequence of games. In fact, when the sequence becomes long, as is usually the case in poker games, a skilled player wins against an unskilled one with overwhelming probability. It is instructive to compare the situation here to other games, without restricting the discussion to card games. Consider, for example, tennis. There is certainly an important skill component in tennis, but there is surely also some influence of chance in the game, arising from the impact of lots of random elements, like the wind, the sun, balls hitting hidden bumps in the court, etc. Indeed, without these, a stronger player would beat a weaker one in every point (while serving, say), and this is certainly not the case. In reality, a top-ten player probably wins about 55% of the points in a match against a player ranked 100, that is, the stronger player has an advantage of about 0.1 in a single point. However, since a match consists of 3, 4 or 5 sets, each set consists of at least 6 (and usually more) games, and each game consists of aleast 4 points, in a typical match there are at least 72 points, and often at least twice that number. The Central Limit Theorem thus kicks in, and implies that even a relatively small advantage in a single point becomes a major factor in deciding the final result of the game. The situation in poker is similar. Indeed, poker is different than tennis as it has an inherent element of chance in it, but the influence of this is not necessarily larger, and in fact appears to be smaller, than the influence of chance elements in tennis. The repeated nature of the game reduces considerably the effect of chance, making poker almost entirely a game of skill. 6 Summary and Concluding Remarks By analyzing simplified versions of poker we have seen that although like in essentially almost any other game there is some influence of chance in poker, the game is predominantly a game of skill. Indeed, the discussion in Section 4 shows that in the simplified one-round version of the game, a good player should first be able to master the probabilities in the game sufficiently well in order to be able to translate his pocket cards and the community 15 cards to an accurate rank of his cards among the available possibilities. He should then be able to use this information to estimate the probability of winning. We have seen that the strategy of a wining player should be adjusted to that of the other players, as a strategy that is winning against some player may well be losing against another. The number of players and the position at the table should also be taken into account, and bluffing is important in order not to reveal one’s strategy. Therefore, a significant amount of skill is required to play well any of the simplified versions of the game discussed in Section 4. The real game, is, of course, far more complicated, and there is every reason to believe that skill plays a dominant role in the real version as well. The Central Limit Theorem discussed in Section 5 implies that the significance of skill increases dramatically as the number of hands played grows. As usually the number of hands played is rather large, this fact implies that the end result in a long sequence of hands is determined with near certainty by the skill of the players. The real game is far more complicated than the simplified versions analyzed here, and playing it well requires a lot of skill. A skilled player should be able to assess the strength of his hand as a function of his hole cards, the community cards, the number of players still in the game, their betting strategy and the position at the table. He should be able to assess the model of play of the other players, estimate the probability of improving his hand once the next community cards are revealed, and should be able to hide his strategy by bluffing and leaving his behavior unpredictable. It is not surprising that there is no software that plays poker as well as a good human player, although, for comparison, there are computer programs that play chess at least as well as the very best human chess players. Indeed, in many ways poker requires more human skill than chess, as an optimal strategy depends so crucially on the behavior of the opponents. The challenges of poker have been investigated in papers in Game Theory like [8], [7], [6], and in Artificial Intelligence (see, e.g., [2]), and there are still many intriguing questions concerning the analysis of optimal strategies for the game. In almost every existing game there is an element of skill and an element of chance. As a matter of fact, the principles of Statistical Physics and Quantum Mechanics imply that some influence of chance appears in essentially every phenomenon in our life, not only in games. Despite the inherent element of chance in poker, our analysis of the simplified models suggests that the result of a soccer match, and probably even that of a tennis match, are 16 influenced by chance more than the results in poker played over a long sequence of hands. The main reason some people may feel otherwise is psychological- one tends to associate randomness with cards or dice more than with weather, wind or bumps in a court, even when the latter have a greater effect on the end result. The fact that a significant number of players excel repeatedly in poker tournaments is a further indication that poker is mainly a game of skill. Practice and study do help to improve in poker, and although luck may well play an essential role in a single hand, we believe that skill is the major component, by far, in deciding the results of a long sequence of hands. As the common practice is to play many hands, this strongly supports the conclusion that skill is far more dominant than luck, and that poker is predominantly a game of skill. References [1] B. Alspach, 7-Card Poker Hands, http://www.math.sfu.ca/ alspach/comp20/. [2] D. Billings, A. Davidson, J. Schaeffer and D. Szafron, The challenge of poker, Artificial Intelligence Journal 134 (2002), 201-240. [3] A. Cabot and R. Hannum, Poker: Public policy, law, mathematics and the future of an American tradition, Cooley Law Review, 2006. [4] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, 1971 [5] A. J. Goldman, and J. J. Stone, A continuous poker game, Duke Math. J. 27 (1960), 41–53. [6] V. V. Mazalov and I. S. Makhankov, On a model of two-card poker, Int. J. Math. Game Theory Algebra 11 (2001), no. 5, 97–105. [7] D. J. Newman, A model for ‘real’ poker, Operations Res. 7 (1959), 557–560. [8] J. von Neumann and O. Morgenstern, Theory of Games and Economic Behaviour, Princeton University Press, 1944. [9] Texas Hold’em, Wikipedia, http://en.wikipedia.org/wiki/Texas hold ’em 17 Chance and strategy in Poker Laure Elie* Romuald Elie† September 2007 Summary The aim of this analysis is to quantify the impact of chance versus strategy in the game of Texas Hold’em Poker. It thereby complements N. Alon’s [1] work on this subject by broadening the game model considered. The results were obtained with a theoretical study carried out by digital simulations of virtual poker games. We concluded that, for a sufficiently high number of consecutive games, it is clear that strategy rather than chance is the overiding factor in the outcome of a Texas Hold’em Poker game. Key words: Texas Hold’hem Poker, chance, strategy, Monte Carlo simulations. Professeur, Université Paris Diderot, Laboratoire de Probabilités et Modèles aléatoires, UMR CNRS 7599, [email protected] † Maître de conférence, ENSAE & Université Paris Dauphine, [email protected] * 1 1 Introduction Poker being a game increasingly practised, a question with important legal consequences has come to the fore over the last few years: Is poker a game where strategy prevails over chance? This study endeavours to give an answer which is mathematically rigourous to the question detailed from a legal point of view in [7]. Of course, a very large number of studies suggest various poker strategies (see [3] for example), but very few deal with the question of chance in games results. They are mainly focused on game theory problematics searching for strategic balance between players, or artificial intelligence allowing a progressive adaptation to opponents’ behaviour. The only study which seems to answer this question satisfactorily is N. Alon’s [1]. Studying a simplified version of a game of poker, his analysis concludes that Poker is essentially a game of strategy. Indeed, thanks to the Central Limit Theorem, a powerful tool in the theory of probability, he shows that the strategy employed is a determining factor in the result of a sufficiently large number of games. In this study, we first validate N. Alon’s findings and we then generalise them. Indeed, N. Alon considers a simplified version of the game of poker, taking into account only the last round of the game where all the community cards are known. He studies mainly games between two players: Alice who has a well-defined strategy and Bob who plays in a random manner. He assumes that Alice has an intimate knowledge of the way Bob plays which gives her a considerable advantage. We put ourselves in a real game of Texas Hold’em poker model, in following the pattern of the various stages of the game: Preflop, Flop, Turn, River. In this more general model, even if we only consider two players, the analytical calculations carried out by N. Alon can not be done so we have used digital simulations. In other words we have performed a digital simulation of virtual poker games between Bob, who plays in a random manner, and Alice who follows a well thought-out strategy, and we have analysed the results. In this model which is very close to reality, we draw conclusions that are very similar to N. Alon’s: for a sufficiently high number of games, the strategy employed is a determining factor in the outcome of a game of Texas Hold’em Poker. This study is organised as follows: first, we present the rules and we describe the course of a Texas Hold’em Poker game. Then, we study N. Alon’s results and we suggest a more general game model, for which we specify interesting strategies. Finally, with the assistance of virtual games digital simulations, we estimate the probability of winning using these strategies. The strategies selected are not necessarily the best ones in the end, but they have the advantage of defining simple and realistic decision criteria for a poker player who is able to assess his own skills. These criteria can also be simply adapted to multi-player games. As we will see, the strategies are sufficient to ensure very high probabilities of winning. 2 2 The game of Texas Hold’em Poker We will attempt in this study to consider a type of poker game as close as possible to that of the Texas Hold’em no limit as it is officially described in [6] and the principles of which are laid out in this section. 2.1 The game rounds Texas Hold’em Poker is played with a standard 52-card deck and each game is punctuated by an alternation of card dealing and rounds of betting. There are 4 of these phases and card dealing takes place in the following order. Preflop: Each player is dealt 2 pocket cards face down. Flop: Three community cards (the flop) are now dealt face up. Turn : A fourth community card (the turn) is revealed River : A fifth community card (the river) is revealed At the end of each phase of card dealing, a betting round starts. Players place their bets one after the other and if a player wishes to stay in the game, he or she must at least match the biggest stake. This round of betting stops when all the players who are still in the game have bet the same number of chips. All the bets make up what is called the pot. Finally two scenarios are possible: either there is only one player left in the game and he or she wins the pot or there are several players left and the game moves to the next stage. If, after the last round of betting following the river, several players are still in the game, they are rewarded according to the value of their hand. Each player can then make use of his or her 2 pocket cards and the 5 community cards to make the best 5-card hand possible out of the 7 available. The player with the best hand then wins the pot and, in the event of two or more hands being worth the same, the players concerned split the pot. The various possible poker hands are described in the following section. A game of poker is made up of a succession of rounds of this type, rounds where players take it in turns to bet first. At the beginning players have the same amount of chips available to them and they fold when they have no chips left. In order to encourage players to bet, compulsory stakes are added at the first round of betting (blind or ante) for certain players on the table. 3 2.2 Hand ranking At the end of a poker game, each player still in the game must reveal his or her cards and the strength of his or her cards is determined by the best 5-card hand he or she can assemble out of the 7 available to him or her. In a 52-card deck, there are 2,598,960 5-card unordered hands possible and all these combinations of cards are separated in ten categories according to their probability of appearing: Royal Flush: Ace, King, Queen, Jack and Ten of the same suit. Straight Flush: Any straight with all 5 cards of the same suit. Four of a kind: 4 cards of the same rank. Full house: 3 cards of the same rank together with any 2 cards of the same rank. Flush: 5 cards of the same suit which are not consecutive. Straight: 5 consecutive cards of different suits. Three of a kind: 3 cards of the same rank. Two-pair: 2 cards of the same rank together with another two cards of the same rank. One-Pair: 2 cards of the same rank. High card: Any hand that does not make up any of the above-mentioned hands. The rarer a 5-card hand is the more it is worth. Within each category, hands are ranked according to how high the cards are. All 5-card hands can therefore be ranked amongst each other, with sometimes the possibility of a draw. Let us now consider combinations of 7 cards where the best 5-card hand out of the 7 possible is kept. There are then 133,784,560 possible combinations of 7 unordered cards. The following table shows [1] and [9], for each hand category, the number of 5 and 7-card possible combinations and their probability to occur. 5-card combination Number Probability in % 4 1.5 10−4 36 1.4 10−3 624 2.4 10−2 3 744 0.15 5 108 0.20 10 200 0.39 54 912 2.11 123 552 4.75 1 098 240 42.3 1 302 540 50.11 7-card combination Number Probability in % 4324 3.2 10−3 37 260 2.8 10−2 224 848 0.17 3 473 184 2.60 4 047 644 3.03 6 180 020 4.62 6 461 620 4.83 31 433 400 23.5 58 627 800 43.8 23 294 460 17.4 Royal Flush Straight Flush Four of a kind Full house Flush Straight Three of a kind Two-pair One-Pair High card 4 3 Analysis of N. Alon’s article 3.1 Structure of the analysis In his analysis [1], N. Alon considers a variant of poker game suggested by Von Neumann and Morgenstern [5]. He considers a game with two players, Alice and Bob which works as follows: 1. Each player is ‘dealt’ a random number drawn in even interval [0, 1]. This number represents the value of their cards and is represented with an xA for Alice and xB for Bob. 2. Both players then decide simultaneously to bet 1 chip or to fold. 3. If one of the two players has folded, the game stops and there is no exchange of money. If both players have bet, the player whose cards have the highest value wins the amount wagered by the other, i.e. 1 chip. This simple game variant captures the essence of the last round of a poker game well, when all the community cards are revealed. Indeed, there are then C247 = 1081 possible combinations of two cards for each player. Disgarding the draws and superpositions between the various combinations, these combinations can be ordered. Each player is therefore dealt cards with a value ranging from 1 to 1081. This range, divided by 1081, resembles the values of the card combinations xA and xB considered in the variant. At the end of the game, if both players have bet, the player with the highest range wins the pot. The benefit of considering such a simplified game is that it allows us to perform analytical calculations in an explicit manner and to calculate the players expected winnings. N. Alon mainly considers a two-player game where Bob plays in a random manner and Alice plays in a more strategic way by adapting to Bob’s game. Bob’s random behaviour can be represented with the following pattern: he bets 1 chip with a probability of p:= 1/2 and folds therefore with a probability of 1 − p = 1/2. Alice knows Bob’s strategy and seeks to adapt her strategy to his behaviour. 3.2 The Results N. Alon then demonstrates that the optimal strategy for Alice consists in betting if and only if the value of her cards is above 1/2. He then shows that her winning odds are 1/8 at each game and that the associated variance is 15/64. In other words, Alice’s average winning odds at each game are 1/8 and the manner in which they fluctuate is characterised by a variance of 15/64. Thanks to the Central Limit Theorem, he can then estimate the probability of Alice losing following a sufficiently high number n of consecutive games. The results are very convincing; after for example 350 games, he observes that the number of times Alice looses is less than 1 in a million. He thus naturally comes to the conclusion that for a sufficiently high number of games, strategy is a determining factor in the outcome of a game. 5 He also broadens these results by adding compulsory bets (blinds) in each game and he briefly analyses the instance of a game with more than 2 players. 3.3 The main limitations The results presented in N. Alon’s article are absolutely valid and pertinent but it is nevertheless regrettable that the analysis should be restricted to a simplified version of poker game which only takes the last round of the game into account. It is true that the analytical calculations proposed would be difficult to apply to the complete version of the game. We will therefore broaden N. Alon’s findings to a real poker game model by replacing the analytical calculations with digital simulations. In the same way, the fact that Alice knows Bob’s strategy, in other words the probability p with which he bets one chip, is a problem. Indeed, it is rather surprising that she would be able to identify her opponent strategy that easily. Here, we will aim to construct reasonable strategies which seem adapted to Bob’s various random ways of playing. 6 4 The model considered We will therefore use a similar approach to that of N. Alon, but with a game very close to the real rules of Texas Hold’em Poker. In the first instance, in order to better understand the required strategies, we will limit ourselves to games with 2 players. 4.1 The game We will consider that two players, Alice and Bob are sat around a table of poker. 4.1.1 Stages of the game The game progresses in 4 rounds: Preflop: Alice and Bob are dealt 2 cards each. They simultaneously decide to bet 0 or 1 chip. If both players bet the same amount, the game continues. Flop: the first three community cards are dealt face up. They simultaneously decide to bet 0 or 1 chip. If both players bet the same amount, the game continues. Turn: the fourth community card is revealed. They simultaneously decide to bet 0 or 1 chip. If both players bet the same amount, the game continues. River : the fifth community card is revealed. They simultaneously decide to bet 0 or 1 chip. If both players bet the same amount, they compare hands. The player with the best hand wins the pot or in the event of a draw, the two players share it. Note that the main differences between a real Texas Hold’em Poker game and this version are as follows: – There is no compulsory bet or blind – The amounts wagered at each round are fixed and equal to 1 – At each round the players decide simultaneously whether they want to bet – It’s a 2-player game We will include in section 4.3 some variants of the game described here and they will take the following differences into account: blind, variable bets, several players. 4.1.2 The players Just like in N. Alon’s article, we consider that 2 people play against each other: – Bob who has a random strategy with a probability p of betting and (1 -p) to fold at each round of the game. – Alice who aims to adapt her strategy to Bob’s. She suspects that Bob’s strategy is random but she does not know the probability p which governs his decisions. She makes her decisions by estimating what his pocket cards might be. 4.2 Alice’s optimal strategy Because Alice does not know the probability p characterising Bob’s behaviour, she will devise a strategy not dependent on p. She observes however that at the last round where the pot was not nil, the situation most favourable to Bob is if he never folds, i.e. p=1. To define her strategy, she therefore considers the case where Bob never folds. It is this strategy which she will use later even when p is different from 1. 7 4.2.1 The optimal strategy during the River Alice’s optimal strategy during the river is easy to determine. Let us assume that the pot is worth P, and that Alice has cards which mean that the probability that she will win is X. Knowing that Bob does not fold and discarding the possibilities of a draw, if Alice bets, her chance of winning is X(P + 2) − 1. Her chance of winning being nil if she folds, we can deduce that Alice must bet at the river if and only if X≥ 1 P+2 The interpretation of this boundary is clear and can be read in the following manner: as she wagers one chip in the hope of winning P +2, it is in her interest to play if and only if her probability of winning is greater than 1/(P + 2). Note that the more there is in the pot, the least important it is for Alice to have good cards in order to bet. To estimate her probability of winning X, all Alice has to do is count the number of hands she could beat amongst the C245 = 990 other possible hands. This calculation, easily performed by a computer is of course impossible for a human brain. However for experienced players, it is not difficult to estimate X relatively precisely. In order to adapt to the reality of a player who estimates his or her probability X of winning with possibly one error, we will present in section 5.2.4 the results of digital games where we have artificially added a random measuring error on the estimation of X. 4.2.2 The optimal strategy during the Turn Let us assume that Alice uses in the last round the strategy previously described. Let us then work out what her strategy should be at the previous round. Note that P is the amount in the pot and X is the random variable equal to Alice’s winning odds at the last round of the game. The variable X is random in the sense that it is not yet known; it can indeed have 46 different values depending on the last community card. If Alice decides to bet in this round, her odds of winning are: E [(X(P + 4) − 1)1X≥1/(P+4) − 1] . 8 Accordingly, as she would win nothing by folding, it is in Alice’s interest to bet if and only if 1 + P[X ≥ 1/(P + 4)] P+4 E [X1X≥1/(P+4)] ≥ This time, Alice’s strategy which when analysed appears more complex is in fact also very intuitive. The principle is the following: There is no point in Alice betting at this round if she does not bet at the following one. Therefore she hopes to win P + 4 by betting 1 at this round and 1 at the following. She is testing whether her odds of winning by betting at the last round are greater than the total she has wagered divided by her winnings. Calling R the event where Alice bets during the river, Alice must bet during the Turn if and only if E [X1R] ≥ 1 + P[R] P+4 4.2.3 Optimal strategy during the Flop The same type of rationale can be applied to the flop if a strategy has been defined thereafter. Calling T, the event where Alice bets during the turn and T ∩ R, the event where Alice bets both during the turn and during the river, Alice’s odds of winning pot P are the following: E [(X(P + 6) − 1)1T∩R − 1T − 1] . One can see that Alice will have to bet if and only if her winning probability is greater than the ratio between the potential amount she has wagered and the total pot. 1 + P[T∩R] + P[R] Alice must bet during the Flop if and only if E [X1T∩R] ≥ P+6 4.2.4 Optimal strategy during the Preflop Obviously, the same rationale always apply when none of the community cards are yet available. Calling F the event where: “Alice bets during the Flop” we know that Alice must bet during the Preflop if and only if E [X1F∩T∩R] ≥ 1 + P[F∩T∩R] + P[T∩R] + P[R] 8 4.2.5 Optimal strategy when the stakes vary So far we have assumed that the amount wagered at each round equalled 1 chip. Of course in reality, stakes vary. Taking into account the variability of these stakes renders the previous analysis much more complex. Indeed in this instance the criterion used whereby the odds of winning are maximised does not seem very appropriate. However the results previously obtained can be easily extended to the scenario where the stakes at each round are different but determine ,or vary according to, the pot. Empirically, in a poker game, the closer the river is the more the stakes go up. Let us call mP, mF, mT and mR the stakes for each player respectively during the preflop, during the flop, during the turn and during the river. We deduce that the optimal criteria which Alice must consider at each round of the game for a pot of P are the following: 9 River: X≥ mR P + 2 mR mT + mRP[R] P + 2 * (mT + mR) mF + mTP[T∩R] + mRP[R] P + 2 * (mF + mT+ mR) mP + mFP[F∩T∩R] + mTP[T∩R] + mRP[R] P + 2 * (mP + mF + mT+ mR) Turn: E[X1R] ≥ Flop: E [X1T∩R] ≥ Preflop: E [X1F∩T∩R] ≥ 4.3 The strategies put into action In order to make this analysis more realistic and more reader-friendly, we will simplify the criteria taken into account in Alice’s decision making process. We will therefore not use the optimal strategy which Alice has available to her. Here we will consider only strategies that are merely based on Alice’s hand odds of winning versus another hand. In other words, in each round, Alice knows E [X] where X represents her probability of winning at the last round of the game. Of course, E [X] is recalculated at each round of the game by taking into account the new community cards. For an experienced player, it is easy to know the potential of one’s hand and to see how it evolves as the flop, the turn and the river unfold. 4.3.1 The reference strategy We will consider strategies where, at each round of the game,Alice bets if and only if E [X] > x(P) where P is the value of the pot and x a decision function. We still have to specify the functions xP, xF, xT et xR corresponding to each round of the game: preflop, flop, turn and river. To simplify the presentation, we will assume that the stakes at each round are equal to one chip. First of all, during the river, X is no longer random as all the cards are known and the criteria of optimal strategy can be applied: X≥ 1 P+2 namely xR(P) := 1 P+2 Let us now consider the other rounds of the game. The previous optimal criteria can not be applied but they give a good idea of the threshold function xP, xF et xT that is reasonable to use. Let us assume that at each round of the game, Alice decides to bet thinking that she will not fold at any of the following rounds. Then her decision criteria during the river, the flop and the preflop become respectively E[X] ≥ 2 , P+4 E[X] ≥ 3 P+6 and E[X] ≥ 4 P+8 10 The criteria therefore have the desired make up and will be the ones we adopt. Note also that the pot is inevitably empty when the players are at the preflop so xP := xP (0) = 1/2. Given that the pot has a value of P, we will in fact use Alice’s following decision criteria: 1 Preflop: E[X] ≥ xP := ; 2 Flop: Turn: River: E[X] ≥ xF(P) := E[X] ≥ xT(P) := X ≥ xR(P) := 3 ; P+6 2 ; P+4 1 . P+6 We will now deal with the distinct variants of the game allowing us to better take into account the Texas Hold’em Poker game specifics. We provide strategies when the stakes are varied, when a blind is added or when there are more than 2 players. 4.3.2 Strategy with increasing stakes In a game of Poker, stakes empirically appear to increase with each round of the game. We have already presented the optimal strategy with varied stakes in section 4.2.5. Let us call mP, mF, mT and mR the stakes at preflop, flop, turn and at the river respectively. These stakes are known in advance as being dependent on the pot. By adapting the rationale described in the previous section, for a pot that is worth a given value of P, the following criteria are easily obtained: 1 Preflop: E[X] ≥ xP := ; 2 Flop: E[X] ≥ xF(P) := mF + mT + mR ; P + 2 * (mF + mT+ mR) mT + mR ; P + 2 * (mT+ mR) mR . P + 2 * mR Turn: E[X] ≥ xT(P) := River: X ≥ xR(P) := Alice’s strategy is rather cautious. Intending to play until the last round, Alice chooses a relatively high level of cards to bet. 4.3.3 Strategy with a blind In the game considered so far, players decide simultaneously whether they want to bet or not and they do not have a forced bet (blind). Let us consider a game where each player is forced to wager 1 chip at preflop every other game. This way, folding has an additional cost and the 2 11 player’s roles are asymmetrical. So we isolate 2 cases depending on whether it is Bob or Alice who pays the blind. 1. Bob pays the blind: then Alice’s strategy which is based on the fact that Bob will in any case bet, will not change. 2. Alice pays the blind: in this case Alice is forced to bet at the preflop. She then applies her strategy simply from the flop. To conclude, Alice’s strategy remains unchanged, apart from the fact that every other time, she has no choice but to bet at the preflop. 4.3.4 Strategy for a game with n players We consider here a game where Alice has n opponents who have the same strategy as Bob. These n players bet with a probability of p and fold with a probability of 1-p. Because she does not know p, Alice seeks a strategy based on the assumption that the other players always bet. All the same we assume that she will adapt her strategy to the number of players still in the game at each round. Let us say that we are at the river with a pot of a value of P and a number n of players still in the game. Alice knows X, the probability of her hand beating another hand given the five community cards. Accordingly, discarding cards superposition, her probability of beating all n other players equates to Xn. As she wagers 1 chip in the hope of winning P + n + 1, her criterion of choice is obtained by the operation: Xn ≥ 1/(P + n + 1). The same type of rationale can be applied to the various rounds of the game. The probability X of winning at the last round can simply be replaced by Xn and in the event of winning, the winnings can be adapted to the number of players n still in the game. Calling the different stakes at each round mP, mF, mT and mR, one obtains for a pot of a value of P and n players still in the game the following criteria: 1 n Preflop: E[X ] ≥ xP := ; 2 Flop: E[Xn] ≥ xF(P) := mF + mT + mR ; P + (n + 1) * (mF + mT+ mR) mT + mR ; P + (n + 1) * (mT+ mR) mR . P + (n + 1) * mR Turn: E[Xn] ≥ xT(P) := River: Xn ≥ xR(P) := 12 5 Digital tests In his analysis, N. Alon calculates in a theoretical manner Alice’s winning odds and the variance Y at each game where she uses her strategy. With these odds and variance, he deduces from the Central Limit Theorem the odds of Alice loosing after any random sufficiently high number n of games. In our more general game model, we cannot exactly calculate the odds and variance of this random variable Y. We have therefore chosen to perform digital simulations of virtual games in order to estimate them. These estimation techniques called the ‘Monte Carlo’ methods are represented in section 5.1.2 In order to implement Alice’s strategy in a computer environment, we had to, at each round of the game, calculate her probability of winning E [X]. In order to calculate it at the river, at the turn and at the flop, we have simulated all the possible combinations of cards being revealed so that this probability can be calculated precisely. On the other hand, in order to calculate it during the preflop, when Alice only knows her two pocket cards, we have used tables of already calculated probabilities. These tables depend of course on the number of players and have been taken from [2] and [8]. In order to make Alice’s strategy more realistic, we also present digital results where errors on the calculation of E [X] have been artificially introduced. In this section, we first tackle the mathematical theorical justifications underlying our approach, then we present the digital results obtained. 5.1 Theoretical justification 5.1.1 The Central Limit Theorem We lay out the manner in which, from Alice’s winning odds and variance over a game, we can work out her odds of losing after a given number n of games. The result expressed had already been noticed and used by N. Alon [1]. We start by presenting (a version of) the central limit theorem. Theorem 5.1 Be M a real positive number and Y1, Y2, … a suit of random variables independent of such laws in such a way that each Yi satisfies |Yi| ≤ M. Calling µ et σ2 the odds and the variance of each Xi, then In addition, we have where Φ is the distribution function of normal law defined by : is the sum of Let us assume that Yi represents Alice winning at the ith game. Then, Alice’s winnings over the first n games. Accordingly, Alice will be losing after n games if and only if . As the random variables Yi are independent and of the same law; we can apply the previous theorem and deduce the following result: 13 Proposition 5.1 Be µ and σ2 Alice’s winning odds and variance at each game. The odds of Alice . losing after a sufficiently high number n of games is in the order of In his analysis, N. Alon manages to calculate Alice’s winning odds and variance at each game perfectly. In our more general game model, we cannot do this so we will obtain a digital approximation thanks to the Monte Carlo method. 5.1.2 The Monte Carlo methods Given a specific strategy for Alice, we are seeking to estimate the odds µ and the variance σ2 of her winning Y at each game. The Monte Carlo methods are based on the Central Limit Theorem stated above. Let us consider a pool of n games where Alice’s winnings Yi are available to us. Then, according to (5.1) in the theorm 5.1, we can estimate µ with Note that a standard estimator of Y ‘s variance is given by The idea is to use and instead of µ andσ2. We can demonstrate that the results stated in , see [4] for example. theorem 5.1 stay true when the variance σ2 is replaced by its estimation From this we conclude the following result: Proposition 5.2 Let us consider a pool of N poker games where, for each i≤ N, Yi represents Alice’s winnings at the ith game. Then, the possibility of Alice losing after a sufficiently high number , with n of games is in the order of 5.2 Digital results Here we present the digital results obtained in simulations of a pool of 30,000 poker games played between Alice and one or several random players. The strategies employed are those detailed in section 4.3. They were elaborated in the context of the opponent betting at all the rounds but we will test them in the context of the opponent betting randomly with a probability of p. Alice’s winning odds and variance are unknown and are therefore estimated with the Monte Carlo methods previously described. There is therefore a measuring error on the magnitude of these numbers which is absolutely controllable. The values provided below are not absolutely exact but the important thing is that their order of magnitude are completely valid. In each variant of the game, the conclusion remains the same: for a sufficiently high number of games, strategy is a deciding factor in the outcome of a game. 5.2.1 The reference game Let us consider first of all the reference game for which Alice’s strategy has been presented in section 4.3.1, with a stake of 1 chip at each round. The following table provides Alice’s estimated winning odds and variance at each game for various values of p. With proposition 5.2, we also calculate the odds of Alice losing after 50, 100 and 500 games. 14 P Odds Variance % of chance for 50 games Alice to not be 100 games leading the game 500 games after… 1 0.32 6.7 18.7 10.5 0.25 0.9 0.35 5.2 15.7 7.7 0.07 0.8 0.36 3.9 12.2 5.0 0.02 0.7 0.36 2.8 8.7 2.7 9.10-4 0.6 0.33 2.0 5.3 2.10-2 2.10-5 0.5 0.30 1.4 2.5 2.10-3 3.10-8 It’s very clear that Alice’s strategy gives her a considerable advantage. Figure 1, representing, for various values of p, the odds of Alice having lost money after n games, is also very telling. In the most unfavourable case where p = 1, the number of times when Alice is in a losing position is 1 in 10 after 100 games, less than 3 in a thousand after 500 games, and less than 4 in 100,000 after 1,000 games. Reference game % chance of having lost money Number of games FIG. 1 – % chance for Alice to be in deficit vs the number of games In a game of poker, each player has available to him or her the same initial number of chips and gets ‘knocked-out’ when he or she has no chips left. So in order to get knocked-out, a significant number of games have to be lost without winning too many of them. In this reference game there are exchanges of at most 4 chips at each round . After grading Alice’s winnings distribution for a game, we have also estimated the odds of Alice losing given a specific initial number of chips. In the worst case scenario where p = 1, the results shown in the following table are very convincing. The higher the initial number of chips is, the more games Alice has to loose in order to be knocked-out. Therefore Alice gets knocked-out less often. We can observe for example that the number of times Alice looses is only 6 out of 1,000 with a fairly reasonable initial number of 50 chips. Initial number of chips Odds of Alice being knocked-out 5.2.2 Increase of stakes Let us now consider the extended scenario where the stakes increase at each round of the game. Alice’s strategy in this case is presented in 4.3.2. We are still making the assumption that 10 24% 25 7% 50 0.6% 100 0.005% 15 Bob bets with probability p and we present the digital results for cases where the stake is 1 at preflop, 2 at the flop, 4 at the turn and 8 at the river. The digital results are presented in the following table and in Figure 2. P Odds Variance 50 games % of chance for 100 games Alice to not be 500 games leading the game after… 1 1.5 61 8.9 2.9 10-3 0.9 1.3 50 9.9 3.4 2.10-3 0.8 1.1 41 10.8 4.0 4.10-3 0.7 1 33 11.2 4.2 6.10-3 0.6 0.8 25 11.8 4.7 9.10-3 0.5 0.7 19 12.0 4.9 10-2 In this version of the game, Alice has again a considerable advantage over her opponent. However, her winning variance is high because as many as 30 chips can be wagered in this game. Note that the winning odds decrease with p. Indeed Alice has a rather cautious strategy and the more often Bob folds, the less lucrative the games are for her, insomuch as the important stakes are at the end of a game. Even in the most unfavourable case analysed here, Alice’s odds of losing are less than in the previous reference game. In the case where p = 1, the number of times when Alice is in a losing position is 3 in 100 after 100 games, 1 in 100,000 after 500 games, and less than 1 in a million after 650 games. Game with increasing stake % chance of having lost money Number of games FIG. 2 – % chance for Alice to be in deficit vs number of games 5.2.3 Blind added An important component in the rules of Texas Hold’em Poker is the use of forced stakes, a.k.a. ‘the blind’. They force players to wager and put emphasis on their position around the table. Considering games with a forced stake alternating between the two players of 1 chip at the preflop, we have carried out a digital test of the strategy presented in section 4.3.3. In the following table, we show Alice’s estimated winning odds and variance. As previously, the results are complemented by the odds of Alice not leading after n games, see Figure 3. P Odds 1 0.28 0.9 0.32 0.8 0.36 0.7 0.36 0.6 0.39 0.5 0.38 16 Variance % of chance for 50 games Alice to not be 100 games leading the game 500 games after… 8.4 24.9 16.9 1.6 6.7 18.7 10.4 0.24 5.2 13.2 5.7 2.10-2 4.0 8.5 2.6 7.10-4 2.9 5.4 1.1 2.10-5 2.1 3.2 0.4 2.10-7 Unsurprisingly, Alice’s performance is not as good in this game model as she sometimes has to wait for the second round of the game before she can fold despite the fact that she may have a bad hand. Like in the previous two games, strategy is nevertheless still the overiding factor in the outcome of a game. Indeed, in the most unfavourable case where p = 1, the number of times that Alice is in a losing position is 17 in 100 after 100 games, less than 2 in 100 after 500 games and less than 1 in a thousand after 1,000 games. In order to compare this game with the reference game, we have also graded Alice’s winnings distribution for a game and estimated the odds of Alice losing given an initial number of chips. In the worst case scenario where p = 1, the results are shown in the following table. We can observe for example that the number of times Alice looses in this instance is 3 in 100 for an initial number of 50 chips. Initial number of chips Odds of Alice being knocked-out 10 32% 25 15% 50 3% 100 0.01% Game with blinds % chance of having lost money Number of games FIG. 3 – % chance for Alice to be in deficit vs number of games Game where Alice does not have the precision of a computer 17 % chance of having lost money Number of games FIG. 4 – % chance for Alice to be in deficit vs number of games 5.2.4 Alice does not have the precision of a computer In all the strategies employed until now, we always assume that Alice knows how to perfectly evaluate her hand. An experienced player certainly always has a precise idea of his or her ‘hand in the pocket’ potential, however it does not seem reasonable to expect a player to be able to evaluate his or her hand in such a precise manner. In order to make this analysis more realistic, we have artificially added a random error on the evaluation that Alice makes of her hand. We have mathematically added the odds of winning E [X] calculated by Alice with an independent centered normal law with a variance of 10−2. In other words Alice now estimates her odds of winning and her approximation has a precision of 0.2, 95 times out of 100. For example, if Alice’s odds of winning are 0.6, she will estimate them between 0.4 and 0.8 and will adapt her strategy to her estimation. The results obtained in this way are laid out in the following table and in Figure 4. P Odds Variance 50 games % of chance for 100 games Alice to not be 500 games leading the game after… 1 0.23 5.5 24.4 16.3 1.4 0.9 0.27 4.7 19.3 11.0 0.3 0.8 0.30 3.9 13.8 6.2 2.10-2 0.7 0.33 3.2 9.5 3.2 3.10-3 0.6 0.35 2.6 6.5 1.7 9.10-5 0.5 0.36 2.1 3.5 0.5 5.10-7 We arrive to the same conclusions as previously: even if Alice wins less often because she badly estimates her hand potential, her strategy remains dominant over that of her opponent. This way, in the most unfavourable case where p = 1, the number of times when Alice is in a losing position is 16 in 100 after 100 games, less than 2 in 100 after 500 games, and less than 1 in 1,000 after 1,000 games. In this game model, we have also estimated Alice’s odds of losing given an initial number of chips. In the worst case scenario where p = 1, the results are shown in the following table. In this instance, the number of times that Alice looses is 1 in 100 for an initial number of 50 chips. Initial number of chips Odds of Alice being knocked-out 5.2.5 A 4-player game 18 10 28% 25 10% 50 1.3% 100 0.02% We now consider a 4-player game, where Alice plays against 3 players who have a random strategy characterised by p. Alice uses the strategy laid out in section 4.3.4. We have performed a digital estimation of Alice’s winning odds and variance. In a game of 4 players, proposition 5.2 allows the calculation of Alice’s odds of losing money after n games. The results obtained for various values of p are laid out in the following table and in Figure 5. P Odds Variance % of chance for 50 games Alice to not be 100 games leading the game 500 games after… 1 0.60 20 17.2 9.1 0.1 0.9 0.88 17 6.4 1.5 7.10-5 0.8 0.89 12 3.5 0.5 5.10-7 0.7 0.86 8 1.8 0.2 2.10-9 0.6 0.5 0.82 0.69 6 4 0.8 0.5 3.10-2 10-2 -13 7.10 2.10-14 Once again, the conclusion is similar: for a sufficiently high number of games played, the players’ results are very clearly correlated with their respective strategies. Thus, in the most unfavourable case where p = 1, the number of times that Alice loses money is around 9 in 100 after 100 games, 1 in a thousand after 500 games, and 1 in 100,000 after 1,000 games. 4-player game % chance of having lost money Number of games FIG. 5 – % chance for Alice to be in deficit Vs Number of games 19 Conclusions Here we have analysed the influence of chance on the outcome of poker games between several players, one player having a dominant strategy over the others. In order to determine this dominant strategy, we have assumed that the other players were playing randomly and we have carried out a theoretical analysis over the expected winnings of a strategical player. Stemming from this analysis, we opted for a strategy which was not optimal but which was easy to understand. In order to quantify the performances of a strategic player versus his opponents, we have performed a computer simulation of a pool of virtual poker games. This has enabled us to evaluate the winning odds and variance of a strategic player and to work out his or her chances of winning. We have considered game cases with 2 or 4 players, with or without blind, with constant or variable stakes. We also studied the case where the strategic player estimates his or her hand potential with little precision. In all the game variants, the conclusion remains the same: for a sufficiently high number of consecutive games of Texas Hold’em Poker, the quality of the strategy employed has an overiding influence over the outcome of the game. Our conclusions are therefore similar to N. Alon’s [1] but we have also dealt with broader game models. Furthermore these conclusions are completely consistent with the empirical observation that it is usually the same professional players who reach the final phases of Poker tournaments. 20 References [1] Alon N. (2007). Poker, Chance and Skill. Preprint. [2] Montmirel F. (2007). Poker Cadillac, Fantaisium. [3] Cutler W. (1975). An optimal strategy for Pot-limit Poker, The American Mathematical Monthly. [4] Fishman G.S. (1995). Monte Carlo concepts, algorithms and applications. Springer series in Operation Research. [5] Von Neumann J. & O.Morgenstern (1944). Theory of Games and Economic Behaviour, Princeton University Press. [6] Arrêté du 14 mai 2007 relatif à la règlementation des jeux dans les casinos (Decree of the 14th of May 2007 regarding the regulation of games in Casinos), (2007) Official Journal of the French Republic. [7] Kelly J. M., Dhar Z. & T. Verbiest (2007). Poker and the Law : Is it a Game of Skill or Chance and Legally Does it Matter ? Gaming Law Review 11(3), 190-202. doi :10.1089/glr.2007.11309. [8] Poker, rules and strategies. “http ://www.poker-regle/strategie.com/” rubrique “probabilites”. [9] Poker on Wikipedia. “http ://fr.wikipedia.org/wiki/Poker”. 21 Chance and Skill in Poker Professor Abraham J. Wyner April 17th, 2008 I. Biography of the Author Professor Abraham (Adi) Wyner is Associate Professor of Statistics at the Wharton School of Business. He came to Wharton in 1999, from the University of California at Berkeley, where he was an Assistant Professor and a NSF Post-Doctoral Fellow. Dr. Wyner did his B.S. in Mathematics at Yale University where he won the Stanley Prize for excellence in Mathematics, heading west to complete his doctorate in Statistics on the West Coast, at Stanford University. Professor Wyner’s principle focus at Wharton has been research in Applied Probability, Information Theory and Statistical Learning. He has published more than 20 articles in leading journals in many different fields, including Statistics, Probability, Information Theory, Computer Science and Bio-Informatics. He has received many grants from the NSF, NIH and private industry. Professor Wyner has participated in numerous consulting projects in various businesses. He was one the earliest consultants for TiVo, Inc, where he helped to develop personalization software. Dr. Wyner created some of the first on-line data summarization tools, while acting as CTO for Surfnotes, Inc. More recently, he has developed statistical analyses for banks and marketing research firms and has served as consultant to several law firms in Philadelphia, New York and Washington, D.C. In addition, he has served as statistical faculty advisor for the University Pennsylvania Law School. His interest in sports statistics has led to an ongoing collaboration with ESPN.com and “ESPN: the Magazine” where Dr. Wyner is the PI on the ESPN funded MLB player evaluation research project. He has served as faculty advisor to the Wharton Quant Club, numerous MBA cohorts and the Wharton Gaming club. For several years he taught an undergraduate course in Gaming that was so popular that over 1000 students competed for only 12 slots. II. Is Poker predominantly a game of Skill? In this consultation, I will address the question of whether poker (and more specifically Texas Hold’em Poker) is a game whose outcome is dependent more on skill than on chance, by evaluating two scientific articles where the issue has been analyzed in detail. One is an article by Professor Noga Alon, of Tel Aviv University (which is attached to this opinion as Annex A) and a second is essentially a follow up to Alon’s article, written by Laure Elie and Romuald Elie of the University of Paris (which is attached to this opinion as Annex B). They have applied mathematical techniques to provide scientific evidence to the fact that poker is a game wherein winning is more dependent on skill than on chance. III. Poker, Chance and Skill, by Professor Noga Alon: Noga Alon considers the game of “Texas Hold’Em” for which he provides a detailed and accurate description. Then he calculates probabilities for each type of hand and explains how knowledge of these probabilities is necessary in order to wager in a way that will maximize the expected winnings. This is his first intimation that a skilled player, who is able to calculate probabilities and use those calculations, will have an advantage over a player who cannot. One course of action, rejected by Alon, is to attempt to mathematically quantify the level of skill in a game. Instead, Alon constructs a simplified game of Texas Hold ‘Em poker which he uses a model. The basic argument is that of a fortiori: if it is possible to demonstrate that Skill predominates in simplified Texas Hold ‘Em, than all the more so it will dominate for the actual game. Alon constructs several different single-betting-round games based on a “basic game” constructed to depend on the ranking property of poker. The games are as follows: 1. A two player game involving a beginner “Bob” who plays randomly against an advanced “Alice,” who plays optimally. 2. An extension of the previous involving an “Improved” Bob against “Adapted” Alice, who adapts her advanced strategy to counter Bob. 3. A multiple player extension with Advanced Alice against multiple beginners in a ring game. Alon shows that it is possible in these simple games to calculate exactly the strategy that Alice should play in order to maximize her expected winnings per round. Alon finds such a strategy for all three versions and then he calculates Alice’s expected winnings per round and the variance of her winnings. He then applies the Central Limit Theorem for repeated independent events to calculate (for version 1) the approximate chance that Alice does not have more money than Bob after n rounds of play. As an added twist, he calculates the same probability with a blind bet instead. For the simplest version, Alon shows that Alice’s skill will dominate Bob’s luck based approach. In fact, we have that: • • • Although 7 out of 8 games end in a draw, if there is a winner, then Advanced Alice is 3 times more likely to win than Beginner Bob After 15 rounds of play, the chance that Alice is ahead is about 84%. After 150 rounds, the chance that Alice is ahead is about 99.9%. To summarize, Alon accurately argues that: • • • Knowledge of hand probabilities is a learned skill fundamental to determining and implementing an advanced strategy An advanced strategy will earn more than a strategy of an unskilled player with high probability in the short run An advanced strategy will earn more than a strategy of an unskilled player in the long run, with certainty. So it is abundantly clear that in a simple game which pits and expert against a novice1, the skilled player will dominate quickly. Skill is the deciding and dominant factor. IV. Limitations and Extensions: Chance and Strategy in Poker. By Laure Elie and Romuald Elie. The Alon analysis is of course limited to a basic simplified one round game of pseudopoker. To conclude that poker itself is predominantly skill, one has to accept that the intricacies of actual poker will necessarily favor the skill factor, from which it follows, a fortiori, that real poker is predominantly skill. The argument is a heuristic, but it is compelling. A second limitation is Alon’s choice of players where Bob, who plays with basically no skill at all, challenges advanced player Alice. A more convincing argument would show that skill dominates the outcome of a game involving a highly skilled opponent against a player of modest abilities. This challenge is met by the analysis in the article “Chance and Stategy in Poker” of Laure Elie and Romuald Elie, of the University of Paris. They build upon Alon’s analysis extending the basic game to multiple round play, with pre-Flop, Flop, Turn, and River rounds, which follows the format of Texas Hold’Em itself. They also consider challengers who employ a range of strategies. Much of their article is devoted to developing the multiple-round game and calculating the optimal strategy for Alice. Since the game is too complex to calculate the expectation and variance of the each player’s winnings, the instead simulate millions of rounds using the computer. This method, appropriately called “Monte Carlo” in the statistics literature, is an extremely effective way to approximate (to and desired level of accuracy) difficult to calculate probabilities, averages and variances. The analysis presented in this article examines poker games involving blinds, increasing stakes and tournaments (i.e. “knock-out” games). In each, the optimal (or nearly optimal) player Alice is challenged by a range of opponents indexed by their probability p of calling/betting in a given round. The main conclusions are as follows: 1 On the other hand, when two equally skilled players challenge each other the outcome is determined predominantly due to chance. This is true for all games, including athletic competitions. This is why a poker match involving the world’s best players seems to be often decided by chance. • In the basic game, after only 50 rounds of play Alice has at worst less than a 20% chance of being behind even the most skilled challenger (p=1). After 500 rounds, this chance is less about ¼ of 1%. In a game with increasing stakes, Alice has at worst a 12% chance of behind after 50 rounds even against her most skilled challenger (p= ½ ). After 100 rounds the chance is less than 5%. In tournament style play, Alice has less than a 1/100 of 1% chance of being knocked out when each player starts with 100 chips. That chance increases to at most 32% when the starting stakes are only 10 chips. In a 4 player game, Alice has less than 20% chance of being behind after only 50 games even against 3 modestly skilled opponents (p=1). After 500 plays that chance is less than 1/10 of 1%. • • • The conclusion is very obvious. A skilled player will trounce lesser skilled opponents not only in the long-run, but also, with high probability, after what amounts to a session of only a couple of hours. Furthermore, in tournament play, where the number of rounds is not fixed, the skilled player has a decisive advantage even with modest initial stakes. Skill is the dominant and decisive factor. V. Summary and Conclusion Poker while simple enough to learn and play with only a short lesson, is extremely intricate and complex. A skilled player who is able to calculate correctly the probabilities of different hand configurations and is able to use that knowledge to bet and bluff appropriately has a substantial advantage over players without these skills. The two papers evaluated here ably demonstrate using mathematical analysis and computer simulation exactly how decisive that advantage is. The player who just “hopes to get the cards” will get them from time to time, but even after a single evening of play against a top player, he will be decisively beaten. Skill dominates chance in poker. Gérard Cohen’s consultation on the scientific validity of Professor N. ALON’s consultation: “Poker, Chance and Skill” I. INTRODUCING GERARD COHEN   Gérard Cohen was born in PARIS in 1951 on the 25th of August. His full Curriculum Vitae is featured in Appendix 1. He holds a post-graduate diploma from the National School of Telecommunication (ENST), is a state qualified doctor in Science (Mathematics) and a lecturer at the ENST. Apart from the ENST, he taught in many institutions, in particular at the University of Paris, in the 6th arrondissement to post-graduate pre-doctoral students, as well as in mixed post-graduate lectures, X, in Paris in the 6th arrondissement and in the ENST. He heads up the “Mathematics for IT and Networks” team comprising of 10 lecturers-researchers and 6 doctorate students. The subjects he teaches are mainly the following: – Theory of information and probabilities – Encoding, Complexity and Cryptography He’s a member of various scientific societies, including the French Mathematical Society, the American Mathematical Society, the Who’s who in the World and he’s a Senior member of the IEEE Gérard Cohen is the president and founder of the IEEE chapter on Information Theory. Expert in encoding, information theory, complexity and cryptography he is the author or co-author of 3 books, of over 100 publications in international journals and he has directed around fifteen theses. A complete list of his work can be found in appendix 1. The large number of industrial contracts and of consultations (SNCF, Gemplus, Canal plus, Sagem…) as well as the amount of public financing (CNRS, bi-national projects) he has been entrusted with and his participation in the European Projects of Excellence (see Appendix 1) are all testimonials of his expertise in ‘error detection   and correction’ and cryptography. II. THE QUESTION:   The question posed is the following: what is the scientific validity of the conclusions drawn in Professor N. Alon’s consultation on the subject of “Poker, Chance and Skills”? This consultation, presented in its entirety in Appendix 2, concludes in particular the following: “By analyzing simplified versions of poker we have seen that although like essentially almost any other game there is some influence of chance in poker, the game is predominantly a game of skill.” III. The expertʹs view   If Poker was a game of chance only, a beginner or a computer would play just as well as a champion (i.e. there would not be any champions…). There is therefore undeniably a part of know-how in the game. The challenge here is to show that Poker is primarily a game of skill. However there is no mathematical theory or even quantification of the notion of skill. Professor Alon therefore has to call upon a subtle blend of theorems and arguments to support his demonstration. In particular, in order to be able to formalise the problem, N. Alon makes use of simplified yet fairly realistic versions so that ‘continuity’ arguments play a role and the real problem ‘inherits’ the properties so obtained. The game of poker being of nil sum (what is won by some is lost by others, except for the various fees), the relevant criterion adopted is that of the hope of positive gain. Note however that the aim of the study differs from those of the usual Game Theory   2 where the best strategy is sought in the context of optimal game. Here we want to demonstrate the impact of a player’s superior skill on his expected winnings. After a brief description of the main version of the game, a few calculations of hand probability are carried out. Already at this stage, the influence of the player’s ingenuity is shown: probability calculations and hand ranking, although basic, cannot be done in real time by a human being; the skill here consists in estimating one’s position with a blend of estimation and intuition (as a Chess champion would). At this stage, superior calculation methods such as those done by a computer could still substitute this skill. Then begins the demonstration itself, based on simple variations, of the importance of know-how, as opposed to not-only chance but also simple memory or the ability to make calculations. Initially we estimate the average advantages in rounds involving two opponents of the most skilled player ‘A’ over the other player ‘B’ (propositions 4.1 and 4.2). In this scenario, B is “one strategy behind” of A and A (Alice) is aware of it. In the first case study, B (Bob) plays in a random manner and Alice (knowing it) maximises her winnings prospects: proposition 4.1 determines this maximum value as well as the distribution in relation to this average. In the second case study, Bob has adapted and plays in the same way as Alice did in the previous variant; there again, Alice’s optimal strategy is calculated in proposition 4.2 where the winnings average and dispersion are obtained. However, in accordance with intuition, A’s average winnings reduce as both players progress, based on the realistic assumption that Bob is always one strategy behind from Alice: in both the first two specific examples, the standard average winnings are initially worth 1/8 (instance 4.1), then they decrease to 1/16 (instance 4.2); in order to prevent the winnings from becoming nil, A has to bluff. Later the study broadens to several opponents. The adopted model, legitimate since the purpose is to demonstrate the influence of skill, assumes that A is the only expert and that the other opponents play randomly. The analysis carried out in 4.4 shows that A’s strategy must adapt to the number of players (by betting less often and with a hand that is stronger as this number increases) and it quantifies numerically this adaptation: the probability of A betting thus decreases by 1/2 with 2 players (A and B), by 0.465 with 3 players and by 0.315 for 10 players. Then a demonstration is made (in section 4.4) that the skilled player must take into account in his or her strategy the position and the order of players around the table. This is still the analysis of a simplified version but in which the notion of compulsory betting is integrated. In the numerical variant chosen by the author for illustration purposes, which is easily adaptable, instance 4.3 shows the hand minimum value to bet as well as the expected winnings and their distribution.   3 In addition, making use of the Central Limit Theorem (Law of Large Numbers), Nogal Alon provides an analysis much more subtle than the simple calculation of expected winnings: providing a sufficient number of games is played, a skilled player’s winnings will follow the trend of normal law of which moments can be calculated (average, but also variance, etc…). This means that expected winnings can be precisely measured thanks to the Normal law distribution function. The numerical convergence based on the number of played rounds is quick; it is thus possible to estimate precisely, not only A’s expected winnings but also B’s probability of winning, the probability of A’s maximum winnings being a certain amount in n rounds etc… Finally a few observations convincingly conclude that the real game, much more complex, requires even more know-how. My conclusion is that I confirm the validity of Professor Alon’s demonstration: Skill has a predominant part to play in Poker. Prof. Gérard Cohen IV. APPENDICES  APPENDIX 1: CURRICULUM VITAE AND LIST OF GÉRARD COHENʹS  PUBLICATIONS   APPENDIX 2: N. ALONʹS CONSULTATION: ʺPOKER, CHANCE AND SKILLʺ    4 Expert Opinion Expert’s name: Prof. Zvi Gilula I the undersigned, Prof. Zvi Gilula, have been asked to express my professional expert opinion with regard to the question that has been set forth below. The details of my education and my professional experience are as follows: Details of my education and course of my academic employment: 1978: Ph.D. in Statistics, Hebrew University of Jerusalem. 1978-1980: Post-doctoral study, University of Chicago. 1981-1986: Faculty member with the rank of Lecturer in Statistics, Hebrew University, and Visiting Professor with the rank of Assistant Professor and Visiting Associate Professor, University of Chicago. 1986-1990: Tenured faculty member with the rank of Senior Lecturer in Statistics, Hebrew University, and Visiting Professor with the rank of Full Professor, University of Chicago. 1995-today: Full Professor of Statistics, Hebrew University, and Adjunct Full Professor, University of Chicago. 2005-today: Head of the Department of Statistics, Hebrew University. Special international recognition of professional achievements: 1989: Elected fellow of the American Statistical Association. 1989: Elected fellow of the Royal Statistical Society. Details of my experience: 1986-2007: Associate editor, Journal of the American Statistical Association – one of the four leading periodicals in the world of statistics. 1982-today: Permanent or temporary consultant to a large number of companies and projects, including medical studies in cardiology, gastroenterology, neurology, dermatology and ophthalmology, Hadassah Hospital (Jerusalem), Shaare Zedek Hospital (Jerusalem), Billings Hospital (Chicago). Consultant to drug companies developing new drugs: Hoffman La Roche, Solvay, Pfizer, Agis, Teva. Statistical consultant to various companies in the areas of marketing statistics, consumer loyalty to brands, client satisfaction: TNS-Teleseker (Israel), Forteligent (USA), Navistar (USA), Midas Mufflers (USA). Consultant to the Israel Association of Advertising Companies and Israel Association of Advertisers. Consultant to the Israel Rating Committee. Consultant to the Israel Second Authority for Television and Radio. Cardinal expert opinions in legal proceedings – Samuels v. Israel Ministry of Health, Jerusalem Municipality and Rafa Ltd.; Navistar v. Ford. I have been asked to express my expert opinion on the following subject: Is winning in a multiple-player tournament of the Texas Hold’em card game (hereinafter: “Texas Hold’em”) more dependent on chance than on understanding or skill? Even before presenting the assumptions which underlie my expert opinion and my complete professional analysis, I shall begin by stating that I would like to clarify that my definitive reply is that a Texas Hold’em tournament is a game wherein winning is significantly more dependent on the participants’ strategic capabilities and understanding than on luck. The probability that an insightful player with strategic skills will win a Texas Hold’em tournament, in comparison with a player who does not have these skills, is much higher than 50%. Following is my expert opinion: The assumptions which I used in my expert opinion: 1. The objective of a Texas Hold’em tournament is not to win a single hand, but to accumulate the maximum number of chips and to win the entire tournament. This is because, at the end of the tournament, the prizes are only given to the players ranked in the first places in the entire tournament. In the course of such a tournament, each player is required to play a large number of hands. A Texas Hold’em tournament is played in the following manner: 2.1 Each participant in the tournament pays an identical amount as an entry fee (the “buy-in”), and against that amount, receives an identical quantity of chips. The participant cannot purchase additional chips during the 2. 2 tournament, nor may he resign during the tournament and cash in the chips he still holds. Accordingly, a participant in the tournament does not have any additional monetary expenditure over and above the initial entry fee. 2.2 The amount of the prizes for the tournament winners is determined in advance, as a function of the player’s final ranking in the tournament, relative to all the other participants. In addition, all of the participants in the game play only against each other, and there is no involvement of an external entity (the “house”) in the game. The participants in the tournament are divided into a number of tables; at each table, the participants play against each other. A player who has lost all the chips he received at the beginning of the tournament drops out of the tournament and leaves the table. In this way, the game goes on until each table is occupied by a single winner, who holds the chips of all those who dropped out. This winner goes to another table and plays against winners from other tables. This method continues until one player is left at the last table, who has accumulated all of the chips from the entire tournament. This person is the winner. A single hand of Texas Hold’em, of which hundreds (or more) are played in the course of a tournament, is conducted as follows: 2.5.1 Up to 10 players sit around each gaming table. In each hand, the cards are dealt by another person (known as the “dealer”), so that each player, at the start of each hand, may be located in a different place relative to the dealer. 2.5.2 In every hand, the player seated at the dealer’s left is required to put a predetermined number of chips into the pot (known as the “small blind”). The player seated at that player’s left is also required to put in a number of chips, which is twice as high as the number put in by the first player, seated to his right (known as the “big blind”). 2.3 2.4 2.5 3 2.5.3 In the first deal, each player is dealt two cards, face down; the identity of those cards is known only to the player who receives them (known as “hole cards”). 2.5.4 At the end of the first deal, each of the players sitting around the table has to decide in turn (according to his position relative to the dealer of that hand) which of the following actions he wishes to take: 2.5.4.1 To bet a number of chips equal to the number bet by the player before him (“call”). To announce that he is leaving the hand without risking his chips (“fold”). To bet a number of chips higher than the number bet by the player before him, with no predetermined limit (“raise”), thereby requiring the remaining players who wish to continue playing the hand in question to bet an equal, and possibly a higher, number of chips. It should be clarified that, at each stage of the hand, any player who wishes to continue playing the hand in question must bet a number of chips which is at least equal to that bet by the player who preceded him in the betting round. This procedure, consisting of a round of decision making, within which the players are required to choose one of the alternative courses of action, shall be hereafter referred to as the “first decisions round”. 2.5.5 At the end of the first betting round, the dealer turns up three cards in the center of the table, which are common to all of the players (the “community cards”) and are exposed to everyone playing (these three cards are known as the “flop”). 2.5.6 After the flop is opened, another round of decision making (hereinafter: the “second decisions round”) begins, in which the 4 2.5.4.2 2.5.4.3 players remaining in the game are again required to choose one of the same possible alternatives: to decide whether they wish to withdraw and throw in their hand, in view of the bet placed by the player seated immediately before them (to fold), or to continue in the hand, thereby risking additional chips (and if so, how many). 2.5.7 At the end of the second decisions round, the dealer exposes an additional community card (the “turn”; this is the fourth community card), after which another round of decision making takes place (hereinafter: the “third decisions round”), in which the players remaining in the hand are again required to choose one of the possible alternatives: to withdraw (to fold), or to continue, thereby risking additional chips (and if so, how many). 2.5.8 At the end of the third decisions round, the dealer exposes an additional community card (the “river”; this is the fifth community card), after which another round of decision making takes place (hereinafter: the “fourth decisions round”), in which the players remaining in the hand are required to decide whether they wish to fold and throw in their hand (if another player has made a bet in any amount), or to compete with another player or players, thereby risking additional chips, with a view to winning the hand. 2.5.9 The fourth decisions round can end in one of the following two ways: (a) After one of the players remaining in the game chooses to bet (that is, to risk additional chips), the remaining players in the game choose to throw in their cards (to fold) – in this case, the only player remaining in the game is the winner of the hand (that is, wins all of the chips in the pot), and is not required to expose his hole cards. Two or more of the players choose to remain in the game and not to fold – in this case, the players’ hole cards are exposed, and the player with the best possible combination (whereby each player is required to put together the best (b) 5 possible combination of five cards from among the seven cards available to him: his two hole cards and the five community cards), according to the ranking of hands customary in the game of Texas Hold’em, wins the hand (that is, wins all of the chips in the pot). Details of the scientific analysis: 3. The nature of chance in Texas Hold’em The game of Texas Hold’em includes a single component of “chance”, which is the fact that a player has no control of the cards dealt out of a deck of 52 cards, in each stage of the hand, and the identity of the cards is determined randomly. 4. The components of understanding and ability in Texas Hold’em 4.1 In order to ascertain the components of understanding and ability in Texas Hold’em, we must first establish the objective of the participants in the tournament, and the central strategy (the “optimal strategy”) which must be used in order to win a Texas Hold’em tournament: 4.1.1 In order to win a Texas Hold’em tournament, a player is not required to win the highest possible number of winning hands, but rather, to cumulatively win the highest number of chips. 4.1.2 Accordingly, in order to win a Texas Hold’em tournament, the game strategy to be adopted is one which will maximize the average number of chips which the player wins in the long term (the “maximizing the expected number of won chips”, as this is known in scientific language). 4.1.3 We shall now examine these strategies and consider how the players can make use of their understanding and ability in order to implement them. We shall show that these strategies do indeed have a decisive effect on the outcome of the game (winning one of the prizes). This effect is significantly greater than that of the randomness involved in dealing the cards. 6 Direct probability calculations1 indicate a total of 133,784,560 possibilities for putting together a five card series out of a seven card series sampled from a parent deck of 52 cards. In order to understand the significance of this vast number of possibilities, I shall state that, if we assume that a player plays all day every day, without resting (and if we assume that about six minutes of real playing, including betting time, are necessary in order to create a series of seven cards), this player will need more than 1527 years of playing in order to ensure that he encounters a combination equal to that created in his first game! The distribution of these possibilities, according to the ranked five card combinations in poker, is as follows: Combination Royal Flush Straight (non-royal) Flush Four of a Kind Full House Flush Straight Three of a Kind Two Pair Pair High Card Incidence 4,324 37,260 224,848 3,473,184 4,047,644 6,180,020 6,461,620 31,433,400 58,627,800 23,294,460 Probability (relative incidence) .0000323 .000278 .0017 .026 .030 .046 .048 .235 .438 .174 This table indicates that almost 85% of the possible combinations referred to the three categories with the lowest value in poker. Any combination better than the three combinations with the highest incidence appears to be significantly rare, relative to the more common combinations (on the average, the more common combinations occur 5.67 times more often than the rare combinations). This fact indicates that, in the absolute majority of cases, all of the participating players must cope with the fact that they hold cards of See e.g. an article by Prof. Alspach, Department of Mathematics, University of California in San Francisco (Internet link: http://www.math.sfu.ca/alspach/comp20). 1 7 moderate “strength” or less, and only in rare cases can a player win a hand simply because he was “lucky” enough to get an especially strong initial combination of cards. 4.2 Before addressing the manner in which an intelligent, experienced player can implement an optimal game strategy, it is important to realize that, in every individual hand, throughout the entire tournament, each player is faced with four “watersheds” (these are the four rounds of decision making), at which he is required to make insightful decisions with regard to his course as the hand progresses. At the heart of this insightful decision is the question of the advisability of risking additional chips, as against the chance of winning all of the chips in the pot for the hand. What, then, are the skills which the Texas Hold’em player is required to express throughout the tournament? In order for me to explain, I shall list below the abilities which underlie those skills: 4.4.1 The ability to evaluate, within a predetermined interval of time, the strength of the hand that he holds in each stage of the game. 4.4.2 The ability to make it difficult for his opponents to discover the game strategy which is using in various situations in the tournament. 4.4.3 The ability to evaluate his opponents’ game strategies from the standpoint of probability. 4.4.4 The ability to translate the insights which arise from that set forth above to a rational risk taking and decision making policy. 4.3 4.4 4.5 In light of the range of skills which a player is required to implement in the course of the tournament, it is obvious that a player who relies on luck alone, whereas his opponents are expressing these skills, has a significantly lower probability of winning the tournament. 8 4.6 Once we have ascertained the various types of skills, we shall now proceed to demonstrate how they are expressed in the course of an individual hand and throughout the entire tournament. Rational and insightful use of bluffing: 4.7.1 In his article “Poker, Chance and Skill” (which is attached to this expert opinion as Appendix A), Prof. Noga Alon of Tel Aviv University, who has recently been declared as the 2008 recipient of the Israel Prize in Mathematics, analyzes the importance of rational use of bluffing. According to Prof. Alon’s analysis (on pp. 9 and 10), an insightful player who seeks to increase his winnings must try to prevent his opponents from deciphering his ordinary game strategy and must always remain unpredictable. As Prof. Alon sums up: “It is crucial for a winning player to stay unpredictable, and to take into account the strategy of the other players.” 4.7 4.8 Rational, insightful decision making in each round of decision making: 4.8.1 It is obvious that, in order to obtain the outcome of the game, the player needs to know how to calculate probabilities. At the most basic level, the player is required to calculate the rarity of the various poker combinations. If we add the fact that a player is required to decide on his course of action (to bet or to leave the game) within a predetermined interval of time, the insightful player requires a calculation speed which is beyond the ability to calculate. 4.8.2 However, in order to succeed in the game of Texas Hold’em, the insightful player also has to evaluate the chance that the next community cards to be exposed will significantly improve his situation relative to that of his opponents (these cards are called “outs”). According to his evaluation of that chance and according to the size of the pot, the player must decide how many chips he 9 wishes to risk against the chance of improving his situation and continuing to compete for the pot. 4.8.3 In order to make myself clear, I shall use three different examples, in all of which the target player (the player whom we are examining), after the two face down hole cards and the first three community cards (the flop) have been dealt, holds two pair: (a) Example 1 – in this example, the target player was dealt two hole cards which constitute a pair (two eights), and the flop consists of an additional pair and a different third card (two sixes and a four). This being the case, the target player now has two pair (eights and sixes). The target player is now required to evaluate his chances of improving his cards and bettering his situation. This improvement is possible if the next community card to be exposed (the turn) is a six or an eight, at which point the player will have a strong hand known as a full house (a pair and three of a kind). Because there are two sixes and two eights remaining in the deck (for the purpose of calculation, the cards remaining in the deck and those held by the other players are considered to be the same), he has four possible cards which will improve his hand (that is, four outs). It is easy to see that turning up a six creates a basis for improving the hands of the entire table, whereas turning up an eight creates a significant improvement for the target player. The insightful target player must figure out (within a predetermined interval of time) the degree of risk which arises for him and for the other players, and must evaluate the size of the bet which he will offer. This is a complex operation, which can apparently be optimally implemented by relatively few people. On the other hand, players who are “street smart” (that is, experienced in playing Texas 10 Hold’em) and have a good strategic memory will find it relatively easy to make an approximate calculation at a satisfactory level of approximation. (b) Example 2 – in this example, the target player was dealt two different hole cards (a seven and a ten), and the flop consists of exactly the same cards he holds, plus a different third card (a seven, a ten and an ace). This being the case, the target player now has two pair (sevens and tens). The target player will have the possibility of improving his present situation in one of the following possibilities with regard to the next community card (turn): (1) If the turn shows a seven or a 10, the player will have a strong hand known as a full house (a pair and three of a kind). If the turn shows an ace, the player will have a stronger hand than he had before, because two pair of tens and aces is stronger than two pair of sevens and tens. (2) Because there are two sevens, two tens and three aces remaining in the deck (insofar as they are not held by other players), he has seven possible cards which will improve his hand (that is, seven outs). (c) Example 3 – in this example, the target player was dealt two different hole cards (a four and a six) and the flop consists of two nines and a six. This being the case, the target player now has two pair (sixes and nines). The target player is now required to evaluate his chances of improving his cards and bettering his situation. This improvement is possible if the next community card to be exposed (the turn) is a six or a nine, at which point the player will have a strong hand known as a full house. 11 Because there are two sixes and two nines remaining in the deck, he has four possible cards which will improve his hand (that is, four outs). It is easy to see that turning up a nine creates a basis for improving the hands of the entire table, whereas turning up a six creates a significant improvement for the target player. The insightful target player must figure out (within a predetermined interval of time) the degree of risk which arises for him and for the other players, and must evaluate the size of the bet which he will offer. 4.8.4 A player who is not gifted with the target player’s skills and cannot calculate these possibilities (dividing the number of outs by 47) within a predetermined interval of time will lose more chips than the target player in the long term. 4.8.5 It is important to emphasize that the player can make the calculations described above with regard to any combination of cards opened, after any stage, and that the situations set forth above constitute examples only. 4.8.6 Furthermore, the use of the abilities described above is necessary and possible in all stages of the hand. As set forth above, the player does not control the two hole cards; they are determined by chance. However, once the player has chosen to participate in the hand and thereafter, considerations of improving the series of cards held by the target player come into play in each of the remaining stages of the hand. 4.8.7 In addition to all of that which has been set forth above, the target player is required to decipher his opponents’ game strategy (if they have one) and to conceal his own. This need does not decrease even after the “river” stage, when the game goes into the final betting round. Admittedly, it is no longer possible to improve the series of cards held by the players after this stage; nonetheless, the 12 considerations with respect to the advisability of risk remain in force. 4.8.9 That which has been set forth above shows that the importance of the two concealed hole cards is not decisive, because the playing of the hand actually begins after they have been dealt, when each player is required to make decisions and to implement strategies in accordance with his ability to evaluate his own cards and his expectation of winning at each stage of the game. 4.8.10 Intelligent readers of this document are likely to wonder – and rightly so – how often the situation described above occurs. If this situation is rare and does not represent a frequently occurring situation in Texas Hold’em, then, although it is still correct that skills are required in order to implement a good game strategy in this situation, the rarity of the situation indicates that, most of the time, the target player does not have to have such skills in order to win. In fact, a simple and immediate probability calculation, on the basis of the table given above, shows that the probability of having two pair or better in a random deal of seven cards out of 52 is 0.388. In other words, the situation described above occurs in almost 40% of games, so that the incidence of such a situation is definitely high. If we begin with an opening situation of one pair, for example, the incidence of cases in which probability calculations provide an advantage rises to more than 50%. 4.9 Rational decision making in the light of the player’s location at the gaming table relative to the dealer (“position”): 4.9.1 In accordance with that which has been set forth above, the game of Texas Hold’em is characterized, among other things, by the fact that the players’ positions, relative to that of the dealer, change in every hand, and accordingly, their turn in the internal betting order of players changes as well. Furthermore, an additional 13 characteristic is that two players in each hand must make one “blind bet”.2 4.9.2 The insightful target player must integrate these two unique characteristics into the decision making process, while exploiting, as optimally as possible, his ability to make decisions after all of the other players have made theirs. 4.9.3 Prof. Noga Alon, in his article, analyzed the significance of a player’s location relative to the blind bettors. Prof. Alon’s analyses (on pp. 11 and 12 of his article) show that an insightful player who is not a blind bettor has a significant advantage, with regard to the chance of winning chips, over a non-insightful player who is a blind bettor (an “advantageous situation”). The average number of chips that is won by an insightful player in an advantageous situation is more than five times higher than that of an insightful player who is a blind bettor. 4.9.4 In addition, the player at the “end of the table” (the last player to bet) has a clear advantage of having observed his predecessors’ bets. This advantage is reflected in that player’s ability (if he is insightful) to evaluate the strength of his cards relative to all of the other players, and thereby to adopt an optimal strategy, with conditions of information which no other player has. 4.10 Up to this point, my expert opinion has dealt with the existence of optimal game strategies, the necessity of use thereof and the frequency of need therefor in the course of an individual hand and throughout a Texas Hold’em tournament. The existence of optimal game strategies indicates that, in order to win a Texas Hold’em tournament, during which the player participates in 4.11 A blind bet means that a player must risk a predetermined number of chips before the pocket cards are dealt. According to the description of the game as shown above, in each hand (in clockwise order), there are two players who must make a blind bet (the small blind and the big blind). The purpose of the blind bet is to create a minimum threshold of risk, which – for the remaining players – constitutes a precondition for participating in the game. Naturally, a blind bet presents a disadvantage for the blind bettors in every hand. Blind betting also helps increase the speed of the game and prevents it from going on indefinitely. 2 14 hundreds of hands or more, it is necessary to have a variety of skills which express strategic ability or understanding. 4.12 It should be noted that these skills do not require an especially high level of education, and that they can also be implemented by ordinary people. These skills can be acquired by study and experience. A player who makes use of these skills and who applies the strategies that have been described above will benefit from a definitive advantage over a player who does not have these skills or does not apply the strategies described above. The basic analysis performed by Prof. Alon (on pp. 12 and 13 of his article) provides some insight into the great significance of the component of skill in the game of Texas Hold’em, relative to the component of chance: 4.14.1 According to Prof. Alon’s calculations, in a series of 60 games played by two players, one of whom is insightful (in other words, has the skills set forth in Section 4.4 above) and the other is not (in other words, does not implement game strategies), the probability that the insightful player will lose the series is only about 2%. In a series of 240 games, this probability decreases to three-thousandths of one percent (less than 1:30,000). In a series of 350 games, this probability falls below one in a million!! 4.14.2 In typical tournaments, the player is exposed to many tens, if not hundreds, of games. In fact, if the target player is insightful and his competitors are not, it may be expected that the insightful player will win the overwhelming majority of such tournaments, relative to players who are not possessed of strategic ability and understanding skills. This indicates that the component of skill – and not the component of chance – is that which definitively settles the outcome of the game (that is, the outcome of the tournament). 4.13 4.14 15 5. Empirical evidence in support of the conclusion that Texas Hold’em is a game of skill: 5.1 Empirical evidence which supports the conclusion that a Texas Hold’em tournament is a game in which the outcome (that is, the identity of the winners) is determined by the component of skill, and not by the component of chance, may be found in a research study performed by professors from the Department of Economics of the University of California, Berkeley: Prof. Pope and Prof. Fishman (the latter currently teaches at the University of Pennsylvania, in Philadelphia, PA). In their study, which is attached as Appendix B to my expert opinion, professors examined 81 Texas Hold’em tournaments, in which players’ achievements were documented in terms of their position in ranking of winners at the end of each tournament (hereinafter: “ranking”). the the the the 5.2 5.3 The research question examined in the professors’ article was whether a Texas Hold’em player’s past history in tournaments can predict is reading in future tournaments. In their article (on pp. 6, 7 and 10), the researchers found that, in fact, a Texas Hold’em player’s achievements in the past provide the ability to predict his future achievements. Thus, for example, it was found that a long time player will be able to improve his achievements in the future and attain a higher future ranking than in the past. In addition, the higher a player’s past ranking, the more that player’s ranking will improve in the future. The fact that a positive correlation was found between a Texas Hold’em player’s past and future achievements clearly indicates the considerable importance of the component of skill. Repeatable achievements can only be explained by the existence of non-random factors which bear a causal relationship to those achievements. These non-random factors must be dependent on skill, according to the explanation set forth in Section 4 above. 5.4 5.5 16 5.6 Accordingly, it is not remarkable that there are no lottery or bingo championships, because winning these games depends on random factors (“luck” or “chance”, in the case before us). On the other hand, chess, checkers, bridge, backgammon and Texas Hold’em championships (tournaments) are routinely held, because the outcomes of these games are decided by the player’s skills, and not by chance. Summary and conclusions: In this expert opinion, I have analyzed quantitative and behavioral aspects in the game of Texas Hold’em. This examination shows, beyond all doubt, that there are game strategies which give their users a significant advantage over players who are not aware of those strategies and/or do not use them. The need for the strategies in question is critical in determining the advantage of those who use them; in game situations where these strategies are required, they are common and frequently used. These strategies require the ability to analyze the state of the gaming table (rapid probability calculations), the ability to decipher the opponents’ strategies, the ability to conceal the target player’s strategy from his opponents, and the insightful ability to translate all of these skills into considerations to be weighed when taking a monetary risk. It is accordingly obvious that winning a Texas Hold’em tournament is predominantly based on strategic skills, rather than on chance. The effect of these skills is that the probability for an insightful player with strategic skills to win a Texas Hold’em tournament, when playing against a player who does not have these skills, is much higher than 50%. It should be noted that this conclusion also applies with regard to a series of Texas Hold’em hands which are not played within the formal framework of a tournament, but rather, as an independent series of hands (cash games). This is because the skills which the player is required to express in his playing are basically identical to those used in the course of a Texas Hold’em tournament. Accordingly, and in accordance with the analyses and facts that have been set forth in my expert opinion, I arrive at the definitive conclusion: 17 A Texas Hold’em tournament is a game in which winning depends significantly more on the participants’ skills (that is, their understanding and their strategic ability) than on chance. 18

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