Full opinion text
MEMORANDUM, ORDER, & JUDGMENT JACK B. WEINSTEIN, Senior District Judge. Table of Contents I. Introduction..............................................................168 II. Facts........................ 170 A. Procedural History....................................................170 B. Evidence on Poker....................................................171 1. Poker in the United States .........................................171 2. Game Play Generally..............................................172 3. Expert Testimony.................................................173 a. Defense Expert...............................................173 b. Government Expert............................................185 c. Defense Expert’s Supplemental Report...........................189 4. Other Evidence...................................................198 5. Conclusions of Other Courts and the States...........................194 6. Compared to Video or “Joker” Poker................................197 C. Evidence at Trial.....................................................198 III. Federal Rule of Criminal Procedure.........................................198 IV. Rules of Statutory Construction ............................................199 A. Generally............................................................199 B. Rule of Lenity........................................................199 V.Federal Gambling Laws ...................................................200 A. Illegal Gambling Business Act..........................................200 1. Statutory Language...............................................200 2. Dictionary Definitions..............................................202 3. Common Law.....................................................202 4. Legislative History................................................203 a. Purpose of the Statute.........................................203 b. Definition of Gambling Generally................................205 c. Discussion of Particular Games..................................207 5. Commission on the Review of the National Policy Towards Gambling.....210 6. Subsequent Mafia Involvement in Poker Games.......................211 B. Other Gambling Statutes...............................................212 1. Contemporary with the IGBA.......................................212 2. Pre-IGBA........................................................213 a. Transporting Gambling Materials................................213 b. Gambling Ships ...............................................215 c. Wire Act.....................................................215 d. Travel Act....................................................215 3. Post-IGBA.......................................................216 a. Indian Gambling Regulatory Act ................................216 b. National Gambling Impact Study Commission Act.................217 c. Unlawful Internet Gambling Enforcement Act of 2006 ..............219 VI.Proof Needed That Business Engaged in “Gambling” Under the IGBA..........219 A. Limited Case Law Interpreting 18 U.S.C. § 1955(b)(2).....................219 B. Statutory Text and Legislative History are Ambiguous ....................221 1. Text.............................................................221 2. Legislative History................................................223 3. Other Federal Statutes.............................................224 C. Rule of Lenity Weighs in Favor of the Defendant.........................224 VIL Poker is Not Gambling Under IGBA........................................224 A. No Controlling Federal Cases..........................................225 B. Only “Games of Chance” Are Gambling Under IGBA......................226 1. Statute is Ambiguous..............................................227 a. Text.........................................................227 b. Dictionary and Common Law Definitions.........................227 e. Legislative History............................................227 d. Other Federal Statutes.........................................228 2. Gambling Not Limited to House-Banked Games ......................229 3. Gambling is Limited to Games Predominated By Chance...............229 C. Poker is Predominated By Skill Rather than Chance ......................231 D. Poker is Not Gambling Under IGBA....................................234 VIII. Conclusion 235 Table of Figures Fig. 1: Winning through time (April 2010 through March 2011) for the top and bottom ten players in terms of total dollar amounts won or lost at $5/$10 stakes.................................................................177 Fig. 2: Win rate comparison: Queen Jack suited (e.g.QA JA)........................180 Fig. 3: Win rate comparison: King Nine offsuit (e.g.KA 9A ).........................181 Fig. 4: Average win rate for players of different predicted skill, for $5/$10 stakes players in the prediction group ...........................................183 Fig. 5: Percentage of the time a higher skilled player (top 50% of skill) would predominate over a lower skilled player (bottom 50% of skill) after a given number of hands at $0.50/$1.00 stakes.....................................184 Fig. 6: Percentage of the time a higher skilled player (top 50% of skill) would predominate over a lower skilled player (bottom 50% of skill) after a given number of hands at $l/$2 stakes..........................................187 Fig. 7: Simulated Cumulative Winnings of Top 10 Winners and Losers (1,000 players, 100,000 trials each)..............................................188 Fig. 8: Average win rates for players of different predicted skill, for $5/$10 stakes, adding rake back in player results.........................................190 Fig. 9: Contribution of skill to poker...............................................193 I. Introduction Defendant Lawrence Dicristina is charged with operating an illegal gambling business involving poker games in violation of the Illegal Gambling Business Act (IGBA), 18 U.S.C. § 1955, and conspiring to do so. See Second Superseding Indictment, Doc. Entry 25, Dec. 9, 2011. The type of poker alleged and proved, to have been played in defendant’s establishment was “Texas Hold’em,” a game described in Part 11(B)(1), infra. When reference is made to “poker” in this memorandum, this is the variant of pokerreferred to. Mr. Dicristina moved to dismiss the indictment on the grounds that a poker room does not fall under the definition of an illegal gambling business proscribed by the federal statute because poker is predominately a game of skill rather than chance. Def.’s Mot. to Dismiss the Indictment, Doc. Entry 69, June 29, 2012. He also contended that whether poker is a game of chance or skill is a mixed question of law and fact to be determined by the jury. Id. Following pretrial oral argument and expert testimony, the court ruled that whether poker constituted gambling under the applicable federal criminal statute would be decided as a matter of law. See Tr. of Daubert Hr’g 89:1-5, July 6, 2012 (“Def. Expert Daubert Hr’g Tr.”). Decision on the motion to dismiss was reserved. The case proceeded to trial, the jury being instructed that poker constituted gambling under the IGBA. Defendant was convicted on both counts. He then renewed his motion for a judgment of acquittal. See Def.’s Mem. of L. in Supp. of his Mot. for a Judgment of Acquittal Under Rule 29 of the Federal Rules of Criminal Procedure, Doc. Entry 92, July 19, 2012. Although the defendant initially raised the issue of whether poker as played in this case is gambling under New York law, see Def. Expert Daubert Hr’g Tr. 85:5-86:1 — the violation of which is an element of the federal offense, see 18 U.S.C. § 1955(b)(l)(i) — he has not renewed this aspect of his motion. See Def.’s Mot. to Dismiss the Indictment, Doc. Entry 69, June 29, 2012. The argument is waived. In any event, it has no merit. New York courts have long considered that poker contains a sufficient element of chance to constitute gambling under that state’s laws. See N.Y. Penal Law § 225.00(2) (“ ‘Gambling.’ A person engages in gambling when he stakes or risks something of value upon the outcome of a contest of chance or a future contingent event not under his control or influence, upon an agreement or understanding that he will receive something of value in the event of a certain outcome.”); N.Y. Penal Law § 225.00(1) (“ ‘Contest of chance’ means any contest, game, gaming scheme or gaming device in which the outcome depends in a material degree upon an element of chance, notwithstanding that skill of the contestants may also be a factor therein.”); In re Plato’s Cave Corp. v. State Liquor Auth., 115 A.D.2d 426, 496 N.Y.S.2d 436, 438 (1st Dep’t 1985), aff'd 68 N.Y.2d 791, 506 N.Y.S.2d 856, 498 N.E.2d 420 (1986) (holding that a Joker Poker video game fell under § 225.00’s definition of gambling in partial reliance on its similarity to poker; “[ajlthough there is a degree of skill and concentration involved in playing poker, ‘the outcome depends in a material degree upon an element of chance,’ i.e., the draw of the cards”); see also Dalton v. Pataki 11 A.D.3d 62, 780 N.Y.S.2d 47, 64 n. 5 (3d Dep’t 2004) (noting that “the term ‘game of chance’ or ‘contest of chance’ ... has been interpreted to include such games as ‘stud’ poker”), aff'd 68 N.Y.2d 791, 506 N.Y.S.2d 856, 498 N.E.2d 420 (1986); People v. Turner, 165 Misc.2d 222, 629 N.Y.S.2d 661, 662 (N.Y.Crim.Ct.1995) (“Games of chance range from those that require no skill, such as a lottery ..., to those such as poker or blackjack which require considerable skill in calculating the probability of drawing particular cards. Nonetheless, the latter are as much games of chance as the former, since the outcome depends to a material degree upon the random distribution of cards.... The skill of the player may increase the odds in the player’s favor, but cannot determine the outcome regardless of the degree of skill employed.”); People v. Dubinsky, 31 N.Y.S.2d 234, 237 (Bronx Cnty. Ct.Spec.Sess.1941) (“There is no doubt that playing ‘Zstud’ poker for money is a game of chance and constitutes gambling.”); id. at 238 (“[T]he courts of many states including our own seem to be unanimous in their holding that where a host receives some consideration or some payment for permitting a card game to be played or other gaming to take place in his premises, that constitutes gambling.”); cf. Katz’s Delicatessen, Inc. v. O’Connell, 302 N.Y. 286, 97 N.E.2d 906, 907 (1951) (holding that “a social game of poker played in a basement room” of a liquor store violated a law which provided that “[n]o person licensed to sell alcoholic beverages shall suffer or permit any gambling on the licensed premises, or suffer or permit such premises to become disorderly”). This series of New York State decisions do not decide the issue now posed: whether a business involving illegal poker games violates the federal IGBA. The defendant’s argument in favor of dismissal is two-fold. First, he claims that even if poker is “gambling” under New York law, not all violations of state gambling laws constitute “gambling” prosecutable under the IGBA. The federal statute defines gambling as “including] but ... not limited to pool-selling, bookmaking, maintaining slot machines, roulette wheels or dice tables, and conducting lotteries, policy, bolita or numbers games, or selling chances therein.” 18 U.S.C. § 1955(b)(2). He submits that this definition limits which state gaming laws trigger prosecution under the IGBA: a business must involve games sufficiently similar to the nine games enumerated in the federal definition in order to be prosecuted as a “gambling business” under the IGBA. Second, he contends that the poker room he operated does not constitute a “gambling business” under this narrow definition — that is to say, that a game run by a “gambling business” must be both house-banked and predominated by chance in order to be sufficiently similar to those enumerated, and that his poker enterprise was neither. The government counters that § 1955(b)(2) does not, by its plain language, restrict what kinds of games constitute gambling under the IGBA. Urged is that the goal of the federal statute was to curb the influence of organized crime, which derived substantial revenues from illegal gambling, and that the federal law was passed to bolster states’ efforts on this front. It follows, so the argument goes, that any gambling activity that is illegal under state law is “gambling” under the IGBA. Gov’t Response in Opp. to Def.’s Mot. to Dismiss the Indictment and Mot. to Preclude Expert Testimony, Doc. Entry 76, July 5, 2012. Both the defendant’s and the government’s interpretations of the statute are plausible. It is unclear from the text and legislative history of the IGBA whether every state gambling offense would permit a federal conviction. See Part VI, infra. It is equally uncertain whether, in enacting the statute, Congress foresaw that poker businesses would be prosecutable under it. See Part VII, infra. In light of these ambiguities, the rule of lenity requires that the defendant’s interpretation be adopted, and his conviction be dismissed. His acts did not constitute a federal crime. Declaring that the federal gambling statute does not cover poker games of the type operated by the defendant does not prevent federal law enforcement agencies from achieving the goals of the statute, as put forward by the government, or encroach on any states’ ability to proscribe these card games. In poker games controlled by the Mob, a violation of state gambling laws would permit federal prosecution under the Racketeer Influenced and Corrupt Organizations Act (RICO). See 18 U.S.C. §§ 1961-1968. Even if the gambling business is not Mob — controlled—as it was not in this case — New York State gambling laws would permit prosecution in state court. See, e.g. N.Y. Penal Law § 225.05. For the reasons stated below, the defendant’s conviction is vacated, and the indictment dismissed. II. Facts A. Procedural History On December 9, 2011, defendant Lawrence Dicristina was charged in a second superseding indictment with operating an illegal gambling business involving poker games in violation of the IGBA (Count Two) and conspiring to do so (Count One). See Second Superseding Indictment, Doc. Entry 25, Dec. 9, 2011. He initially pled guilty to Count Two, the substantive IGBA offense. See Tr. of Guilty Plea, Doc. Entry 40, Dec. 12, 2011. At his sentencing hearing, he moved to withdraw his plea. Doc. Entry 50, May 1, 2012. The plea was withdrawn, and a trial date was set. Id. A week before trial, the defendant moved under Federal Rule of Criminal Procedure 29(b) to dismiss the indictment, contending that poker did not constitute gambling as defined by the IGBA. See Def.’s Mot. to Dismiss the Indictment, Doc. Entry 69, June 29, 2012. An amicus brief in support of his argument was filed by the Poker Players Alliance (PPA). See PPA Mot. to File an Amicus Br. in Supp. of Def.’s Mot. to Dismiss, Doc. Entry 74, July 1, 2012. The defendant also sought to introduce expert testimony at trial to show that poker was a game of skill rather than chance and thus outside the purview of the statute. See Doc. Entry 73, July 3, 2012. Following a Daubert hearing, defendant’s witness was permitted to present his opinion as an expert, but his testimony was excluded from the trial as irrelevant. See Def. Expert Daubert Hr’g Tr. 84:1-2. It was ruled that whether poker constituted gambling was a matter of law that would be decided by the court, rather than by the jury. Id. 89:1-5. Decision on defendant’s motion to dismiss was reserved. Id. He renewed his motion at the close of the government’s case, Tr. of Trial 264:2-3, July 10, 2012, and at the close of evidence, id. 325:18-23, with decision again reserved. After it was instructed that running a poker establishment was illegal, the jury convicted the defendant of both Counts One and Two. See Doc. Entry 91, July 12, 2012. Additional briefing on the motion to dismiss was submitted following the verdict. The government submitted evidence at a post-trial Daubert hearing held on August 10, 2012. See generally Tr. of Daubert Hr’g, Aug. 13, 2012 (“Gov’t Expert Daubert Hr’g Tr.”). Experts for the government and the defendant testified independently and in a joint discussion with the court, submitting reports supplemented by extensive briefs and letters. See id.; Def. Letter Attaching Expert Report App. (Report of Dr. Randall Heeb), Doc. Entry 77, July 6, 2012 (“Def. Expert Report”); Report of Dr. David DeRosa, Doc. Entry 103, Aug. 10, 2012 (“Gov’t Expert Report”); Def.’s Letter Addressing Issues Raised at Aug. 10 Daubert Hearing App. (Supplemental Report of Dr. Randal D. Heeb), Doc. Entry 104, Aug. 13, 2012 (“Def. Expert Supp. Report”); Gov’t Reply to Def.’s Letter Addressing Issues Raised at Aug. 10 Daubert Hearing, Doc. Entry 107, Aug. 17, 2012 (“Gov’t Reply Letter”); Def.’s Letter Reply to Gov’t Aug. 17, 2012 Letter, Doc. Entry 108, Aug. 20, 2012. B. Evidence on Poker Central to the issue of whether poker falls within the ambit of the statute is: 1) whether Congress, at the time of its enactment of the IGBA, understood “gambling” to include poker; and 2) whether poker is, like the other games enumerated in that statute, a game predominated by chance. 1. Poker in the United States Poker has a long history in the United States. See, e.g., “The National Game,” N.Y. Times, Feb. 12, 1875 (describing the spread of literature on poker). The game first appeared in roughly its modern form in the early nineteenth century in New Orleans. James McManus, Cowboys Full: The Story of Poker 51 (2009). It has been embraced by many of our political leaders and other public figures. For example, “Justice Douglas was a regular at President Franklin Roosevelt’s poker parties; Chief Justice Vinson played poker with President Truman.” Cheney v. U.S. Dist. Court for Dist. of Columbia, 541 U.S. 913, 917, 124 S.Ct. 1391, 158 L.Ed.2d 225 (2004) (citing J. Simon, Independent Journey: The Life of William O. Douglas 220-221 (1980); D. McCullough, Truman 511 (1992)). Driven in part by the Internet, which allows for online poker playing, and cable television, which frequently broadcasts poker tournaments, the game has surged in popularity in recent years. Anthony Holden, Bigger Deal: A Year Inside the Poker Boom 1, 10 (2007). In 2006, 8,773 players entered the “main event” in the World Series of Poker, the most prestigious poker tournament in the United States, and more than 44,500 players participated in the tournament at large. Id. at 1. 2. Game Play Generally While there are several major variants of poker, the defendant’s business involved “No-Limit Texas Hold’em,” a game of increasing popularity. See Def. Expert Dauberb Hr’g Tr. 14:24-15:4; Part 11(C), infra. No-Limit Texas Hold’em is currently the most popular variant of poker. See Def. Expert Daubert Hr’g Tr. 14:12-22. Texas Hold’em poker has been described as follows: The game is usually played with at most 10 (and at least 2) players. 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. 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. 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. PPA Mot. for Leave to File an Amicus Br. Ex. B (Noga Alon, Poker, Chance and Skill 2-3 (unpublished manuscript)), Doc. Entry 74, July 3, 2012. The government does not dispute that skill plays some role in poker, particularly in “no-limit” poker games. As one writer has described the matter: The limits, or absence of limits, on how much a player may bet and raise will dramatically affect the game dynamics, including players’ decisions and strategies and the relative balance of luck versus skill in the game.... In no-limit games, a player may bet or raise any amount he has in front of him (table stakes limit betting in a hand to the chips and money on the table).... While the initial distribution of cards and replacement cards are random, the decision on which cards to discard, the methods and steps in betting, the analysis of playing habits of other players, and the management of a player’s chips from hand to hand are all skills.... Anthony Cabot & Robert Hannum, Poker: Public Policy, Law, Mathematics, and the Future of an American Tradition, 22 T.M. Cooley L. Rev. 443, 450-54 (2005). 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. PPA Mot. for Leave to File an Amicus Br. Ex. B (Noga Alon, Poker, Chance and Skill 16 (unpublished manuscript)), Doc. Entry 74, July 3, 2012. Unlike other games commonly considered gambling, such as roulette, craps, or blackjack, the house does not participate in game play during poker games. Cabot & Hannum, supra, at 452. Instead, players compete against each other, and money won or lost is merely transferred from one player to another. The casino provides a dealer, who does not play, and the casino makes money by taking a percentage of each pot, charging an hourly fee, or collecting a flat amount for every hand. The first of these is the most common; a “rake” (percentage extracted) of 5% to 10% is typical. Cabot & Hannum, supra, at 452-53. 3. Expert Testimony a. Defense Expert At a July 6, 2012 pretrial hearing, Dr. Randal D. Heeb, a respected economist, statistician, and player in national poker tournaments, testified as an expert on game theory. See Def. Expert Daubert Hr’g Tr. 12:7-8. He highlighted the number of skilled strategic choices, such as how much to wager, made by poker players in the course of playing a single hand: [Tjhere’s a very large number of decisions that need to be made by a poker player playing any given hand.... The simplest and most obvious choice that a player has to make is simply whether to play a particular hand____ [For example, a hand with a King and a Nine in different suits] is a hand which is particularly easy for unskillful players to play badly, and for skillful players to play well. And the very first decision that a player has to make is simply whether or not to play and how much to bet, and less skillful players seeing a King and a Nine, which seem like relatively high cards, and when the King comes, they will have a pair of Kings, which seems like a relatively strong hand, and yet unskillful players tempted to play that hand are very likely to bet too much and to lose to players with even better hands. Because even though it seems like a good hand, when you happen to get another King, you make a strong hand, it seems like you are likely to win. In fact, you may win relatively many hands. You are not going to win very much money with that hand, and when you lose, you are likely to lose a lot of money. So unskillful players tend to play that hand poorly, and make bad decisions, not only the decisions you play, but also how much to bet, .how to respond to other players, when other players raise them, for example. So a more skillful player would recognize that the only way to win with that hand would be, for example to get a pair [of] Kings, and if nobody else has anything that is a very good hand, you are likely to win almost nothing with that pair of Kings. On the other hand, if another player, instead of starting with a King Nine, had a King Ace, and now when all of the common cards are dealt subsequently in the game, a King appears, both players would have a pair of Kings, but the player with the Ace King would have a much stronger hand and therefore win much more often and much more. The less skillful player would have a tendency to bet too much with the King Nine early, and would also tend to call too often later. Tr. of Hr’g 26:22-28:12, July 6, 2012. Bluffing, raising, and folding require honed skills to maximize the value of the cards dealt by Lady Luck: [Wagering is] also used to try, for example, to force an opponent to fold their hand, and there are two relevant aspects of that strategic play. One would be a bluff. If I think you have the best hand and I want to bet an amount that is going to induce you to give up your hand, you know, that’s one element of the wager. And what’s important about that wager is not that I am betting on the outcome of some outside event, but rather the amount that I choose to bet is carefully strategically chosen in order to influence your behavior, and I choose an amount based on what I think [you] will do, given the amount that I bet, and it can go two different ways. Sort of the novice way to think about this is the more that I bet, the more likely you are to fold, because it’s harder for you to put up more money to call my bet. With more expert players, they would anticipate that effect. And, so, there’s a bluff that has a name among poker players called a “post oak bluff’ where you bet a small amount to make your opponent think you are trying to make them call. So they think about that with their model of how you play, and they decide to fold because of the small amount that I bet. And so the amount that you bet becomes incredibly important. There’s a second way that the amount that you bet, whether or not to bet becomes important. Even if I’m pretty sure I got the best hand, I may want you to fold because, for example, if you have a flush draw and I have the best current hand, I may not want you to get your chance to make your flush. So I will bet an amount designed to make it uneconomic for you to make the call, and if you are a good player, you will recognize that and you will fold. If you are a bad player, you will make an unskillful play that actually wins me money over time. So by making that correct amount of the bet, I’ve influenced the outcome, both immediately in the single hand and over the long haul, the amount that I win over time playing against either skillful or unskillful players. So ... whether or not to bet, whether or not to raise, which is going to bet zero, let the other person bet and raise after the fact, these are all strategic elements which are the essence of poker, and in that sense there’s nothing analogous to that in a game of chance like betting on a football game or betting on the roll of the dice. Id. 50:8-51:24. Position at the table and the habits of fellow players must be taken into account to play successfully. [AJnother very simple thing to notice that unskillful players don’t tend to notice ... is that the position at the table matters. So the person [who] has to act first is at a disadvantage, because they don’t know what the other players are going to do. So the person that acts last, and because you act in turn around the table in a clockwise fashion, the person that acts last has a big advantage. They got to see what other players did before they made their choices. So a skillful player recognizes that advantage and ... changes their strategy, plays more hands, and plays them more aggressively in later positions, but at the same time folds their hands and plays more conservatively, if other players have acted with strength in front of them. So if a player has a bet in an early position, that is a relatively strong play. And so a player that gets to play later, would take that into account if they are a skillful player, and would only play a very strong hand. Yet another way that a skillful player can use relatively simple information is to recognize how skillful their opponents are and change their strategy based on who they are playing against. So if a skillful player bets in an ... early position — and I know that that is a skillful player because I have observed how they played the rest of their hands — I know that they know that that early position is a dangerous position for them to be in. So if a skillful player bets in that position, I ... think they ... have a stronger hand, I react accordingly. Unskillful players don’t even notice this. So, again, that’s a relatively easy play that can be learned in one day of training and a couple of days of practice, which would dramatically improve a player’s results. Id. 37:3-38:9; see also id. 49:7-16 (“In poker, you don’t know the cards that [the other players] have, but they know what cards they have. They don’t know the cards that you have. So you have a model of your opponent and how they react to the situation that they see. They have a model of you in their mind, and how well the players play, make their decisions to use this information, which is generated by the chance mechanism, the way in which they then use the information that they do have to make their strategic choices is what makes poker such an ... interesting game.”). When poker is played live, as it was in the instant case, rather than on a computer, additional skills come into play, such as “the ability to read their opponent, to detect from their opponent’s betting behavior, from their tone of voice, from their reactions, whether or not they have a good hand.” Id. 42:1-4. Dr. Heeb opined that poker differs from other forms of gambling, such as sports betting, because the player .can rely on sophisticated skills to change the outcome of the game. • Id. 49:17-20. [T]he player in a poker game is making all of the decisions, making all the plays, which include whether or not to wager on a particular hand and how much. And, in fact, the act of wagering itself is the essence of the decision. So in one sense in a gamble over any other mechanism, whether it was a bet on a baseball game or a bet on the roll of the dice, the wager itself is completely independent of the event being wagered on. Whereas, in poker, the wager is not in the same sense a wager on the outcome. It is the strategic choice that you are- making. You are trying to influence the outcome of the game, either by the amount that you are wagering, trying to build up and win more money. Id. 49:21-50:8. According to Dr. Heeb, “many people make a living playing poker and win consistently over time” whereas “it is impossible to make a living and to win consistently playing casino games such as roulette” where chance predominates. Def. Expert Report at 11. This fact alone was an independent foundation for his opinion that skill predominates over chance in poker. Id. As shown in Figure 1 below, prepared by Dr. Heeb, the ten most proficient players earn dramatically more money than the ten least proficient players over the course of a year. The most skillful professionals earn the same celestial salaries as professional ball players. The expert for the government, Dr. DeRosa, demonstrated — and Dr. Heeb conceded — that a figure similar to Figure 1 could be obtained by chance tosses of a coin. See Part II(A)(B)(3)(b), infra; Gov’t Reply Letter at 3. But, while Dr. Heeb showed that the same poker players would consistently come out on top in the play of a new set of multiple hands of poker, this consistency could not be demonstrated in a new set of coin tossers by the same tossers. See Part 11(B)(3)(c), infra. According to Dr. Heeb, Figure 1 shows that the ten best players “win consistently” and that “these players’ cumulative amounts won are nearly always increasing,” even though they may have “a few losing days or weeks.” Id. By contrast, the ten worst players “are consistently losing throughout the year. Id. He posits that “[t]he fact that the winning players tend to win consistently and the losing players tend to lose consistently demonstrates that there is a skill differential between these groups.” Id. Dr. Heeb acknowledged that poker also involves an element of chance. Def. Expert Daubert Hr’g Tr. 58:12-13. “On any given hand ... the probabilities are certainly finite.” Id. 62:14-15. The following hypothetical posed by the government to Dr. Heeb on cross-examination is illustrative: Q. ... The lower skilled player has seven deuce offhand, which is ... statistically the wors[t] starting hand in poker; is that correct? A. Yeah, probably. Q. And the higher skilled player has two aces, which is the best starting hand ... in hold’em poker; is that right? A. That’s right. That’s right. Q. And when the hand starts, the high skilled player with the two aces is about an 87 percent to 12 percent favorite.... A. That sounds right. Q. So ... the lower skill player perhaps demonstrating his lack of skills goes all in for $500 with his seven deuce off suit and the higher skill player calls that. Just as a matter of percentage, the lower skilled player has about a 12 percent chance of actually winning that hand, correct? A. At the point that they both made the bet and called the be[t], yes. Q. So about one out of ten times, give or take, in that scenario, the lower skilled player will actually win? A. That’s right, one out of eight actually. Q. ... And at that point the lower skilled player can take his money and go home, right? A. There’s nothing to stop a player from quitting, I guess. Q. And the fact that the higher skilled player is higher skilled, doesn’t get him his money back, right? A. That’s right Id. 70:4-71:10; see also 57:18-24 (“For example, a bad beat [a subjective term for a hand in which a player with what appear to be strong cards nevertheless loses], ... might be a hand in which after the play of the hand, the betting has all been made, I believe that I have or I may even know exactly that I have an 85 percent chance of winning and a 15 percent chance of losing, and a bad beat would be a hand on which the 15 percent chance occurs.”). He acknowledged that poker falls in between chess, which he characterized as an almost pure game of skill, and roulette, which he characterized as a pure game of chance. Id. 44:8-45:9. According to Dr. Heeb: [T]he question then is: How do you know if skill predominates over chance in poker? And the right way to analyze that question is to ask: Over how long does it take for skill to essentially show itself and predominate over the element of chance? And the answer is that it’s sufficiently few number of hands, that a player could reach that number of hands in a few playing sessions. And, again, depending on how skillful that player is, an extremely skillful player, that player’s skill would manifest itself in that player’s results relatively quickly. Id. 45:9-18. Based on his research, Dr. Heeb concluded that skill predominated over chance in determining the outcome of a poker game. He summarized the results of his study of 415 million hands of No Limit Texas Hold’em that were played on-line at the PokerStars website from April 2010 to March 2011. Id. 13:3-9. To verify the reliability of the data he received from PokerStars, he obtained publicly available data from HandHQ, a company that tracks hands played on PokerStars and other online poker sites. Def. Expert Report at 10. Using this outside source, he confirmed that the data received from PokerStars was an accurate records of hands played. Id. Although his information came from internet poker, rather than face-to-face games, Dr. Heeb concluded that the data set he chose was appropriate: The game is a game of skill in exactly the same way, whether it’s played live or played over the internet----So my conclusions ... carry over exactly to when the exact same game is played, whether it is played in person, played with cards, ... or played electronically over the internet. The only difference between playing live and playing in person is that the live game brings in some additional elements of skill which are not available to the internet player. Def. Expert Daubert Hr’g Tr. 41:13-24. Using this sizable data set, Dr. Heeb conducted two different analyses to evaluate the relative effect of skill and luck on players’ success rate in poker. First, he looked at whether a player’s average win rate on all other hands was predictive of their success in a particular kind of hand — for example, a King and a Nine in different suits (the King Nine hand). Id. 22:14-23:3. He divided players into two groups: those whose success rate was above the median, and those whose success rate was below the median. Id. 24:13-22. Players whose success rate was above the median were more successful with the King Nine hand than players whose success rate was below the median. Id. 24:23-25:3. As summarized in Figures 2 and 3, more highly skilled players won more — or lost less — than lower-skilled players when dealt the same hand. Dr. Heeb concluded that a player’s skill had a statistically significant effect on the amount of money won or lost in a particular hand in poker. Id. 25:15-19; 26:11-18; see also id. 28:21-29:3 (The fact that “players who tended to play poorly with all the rest of their cards, also played poorly and lost, or [won] less, with any particular hand, would be sufficient onto itself to demonstrate that the actions of the skillful players are skill, not luck, because if they were lucky, of course the fact that they were lucky on one set[] of cards, would have no influence on how they did with other cards.”). While Dr. Heeb acknowledged that “in a single hand, what cards are dealt is going to be an even more important factor, ... regardless of which hand a player is dealt,” he explained that “when players are dealt the same hand, the more skillful player plays it much better and achieves a better result.” Def. Expert Daubert Hr’g Tr. 39:6-15. But see Gov’t Reply Letter at 6 (contending that “what hand a player is dealt is the predominant determinant of how much he will profit or lose from that hand ”). Second, he randomly divided the same data set into two groups. Id. 30:21-24. He used regression analysis on the first group to create “a skill index that related how skillfully the player played to what their actual win rates were.” Id. 31:19-21. His skill index included “240 statistics about their play. [It wo]uld be equivalent — if I could draw another analogy, and I hope that this isn’t stretching it too far, but a baseball player that decides how often they swing at the first pitch or whether they are waiting for a curve ball or a strike.” Gov’t Expert Daubert Hr’g Tr. 103:4-8. He then applied the skill index to the second analysis group to see whether players who were predicted to be of high skill actually won significantly more money than players predicted to be of low skill. Def. Expert Daubert Hr’g Tr. 33:1-5. He found that “[t]he lowest skill players according to the predicted skill index in fact achieve much worse results. Average players still don’t do very well. Very good players are winning players.” Id. 33:7-10. As illustrated in Figure 4, a player’s win rate increases as his skill level increases. Win rate is calculated as a fraction of the big blind won per hand, on average. Dr. Heeb explained Figure 4 as follows: The line slopes upward to the right, indicating that players with higher predicted skill on average have higher win rates. If poker were a game of chance, there would be no relationship between predicted skill and results. The slope of the line reflects the much higher expected win rate of skillful players compared to less skillful players. This relationship characterizes games of skill. Def. Expert Report at 36. Dr. Heeb then divided the analysis group into players with skill below the median and players with skill above the median. Def. Expert Daubert Hr’g Tr. 33:19-23. As shown in Figure 5 below, he found that “it takes about 900 hands of poker for the high skill group to predominate over the low skill group with 90 percent confidence” and about “1400 hands for 95 percent confidence.” 13. C4 CO CO Dr. Heeb testified that this number of hands is “quite reasonably played in a relatively short amount of time by players that are playing poker seriously” and that “over a longer period of time, skill predominates even more.” Id. 34:19-23; see also Gov’t Expert Daubert Hr’g Tr. 92:22-93:8 (“90 percent of the time the top half [in a $l/$2 game, see Fig. 6, infra] would be ahead after 880 hands, which is about 30 hours of play.... [A]t the World Series [of Poker], a typical one day of play is 10 hours, excluding breaks, so it actually takes about 13 hours to play 10 hours of play. That’s a standard day. A standard tournament takes three days. So in the amount of time it takes players to play one poker tournament,. 90 percent of the more skillful half of the players would be ahead of the less skillful half of the players.”). Based on these analyses, this expert concluded that skill predominates over chance in poker. Id. 18:4-5. He also noted that the “topic ... has been addressed by a number of other researchers, none of whom had the data I have available with the whole cards available to me. All of the results that I have seen, both in published and unpublished papers, are all consistent with the result that skill predominates over chance.” Id. 40:20-25; see also Part 11(B)(4), infra. b. Government Expert The government’s expert, Dr. David DeRosa, who is also a well-qualified econometrician, testified at a post-verdict Daubert hearing on August 10, 2012. See generally Gov’t Expert Daubert Hr’g Tr. Dr. Heeb was present at the hearing to answer questions regarding his report and participate in a technical discussion with Dr. DeRosa and the court. See generally id. Unlike Dr. Heeb, Dr. DeRosa neither has any personal experience with poker, see id. 6:7-14, nor has he independently analyzed the game. Moreover, he noted that he had “not been provided with any of the data or statistical analyses summarized in Dr. Heeb’s report.” Gov’t Expert Report at 2. His testimony was thus limited to a critique of whether “Dr. Heeb’s stated results support his overarching conclusion, namely that skill predominates over chance in poker.” Id. He did not offer any opinions as to the validity of Dr. Heeb’s calculations or his general methodology. Id. Dr. DeRosa acknowledged that “the actions of [poker] players ha[ve] some impact on the outcome [of the game] and poker is not a game of pure chance.” Id. at 10. Nevertheless, he had several criticisms of Dr. Heeb’s work. First, he argued that Dr. Heeb’s analysis erroneously relied on “relative skill rather than likelihood that a player will earn a profit.” Id. at 4. According to Dr. DeRosa: Poker players play poker to win money. A player will consider himself a winner if he earns a profit. The fact that he may lose less money than another player is likely to be of cold comfort. I believe that the proper metric for determining success at a session of poker is whether or not a player profits from playing the game. Id. at 6; see also Gov’t Reply Letter at 1-2 (emphasizing this point). Yet Dr. Heeb’s own data shows that most players lose money at poker. Dr. DeRosa pointed to Figure 4, supra, which he said shows that: [P]layers in the 51st to 75th skill percentile lost approximately .15 to .45 [of the big blind] per hand. In the $5/$10 game Dr. Heeb analyzes [in Fig. 4, supra], this equates to a loss of approximately $1.50-$4.50 per hand. Even top players in the 90th skill percentile appear to have, on average, suffered losses from their poker playing. Only between the 90th and 95th skill percentile does it appear that “skillful” players begin to experience a positive win rate (i.e. have a positive expected return). Gov’t Expert Report at 6-7; see also Gov’t Reply Letter at 3. Dr. DeRosa concluded from the fact that so many players lose money that “if a player were to make a profit at any given session in a game where he faced a negative expected rate of return, such profit would have to be primarily as the result of luck.” Gov’t Expert Report at 7. Because so few players win money, he contended, skill cannot predominate over luck in poker. See Gov’t Expert Daubert Hr’g Tr. 9:7-17 (“I go in with a certain amount of money, and I leave with more money. And if I don’t do that, I am a loser. I’m a loser. So a lot of this ranking stuff is irrelevant because skill should be winning money. And as I read the report and as I read Dr. Heeb’s testimony, probably 95 percent of the people who play this online poker lose money so I don’t understand where the skill is. How could it be skillful playing if you’re losing money? And I don’t consider it skill if you lose less money than the unfortunate fellow who lost more money.”); id. 24:25-25:5 (“But the other striking thing that I found was the idea that 95 percent of these people lose money. So for 95 percent — according to the report that I read. That win rate is a negative number so 95 percent of them lose money. So, Your Honor, isn’t then poker a game of chance, not skill, just on that alone?”). Second, Dr. DeRosa disagreed with Dr. Heeb’s use of cumulative data — i.e., his examination of results over a large number of hands — to evaluate whether skilled players outperform unskilled players. Dr. DeRosa instead stated that the relevant frame of reference for determining whether skill or chance predominates is a single hand. Gov’t Expert Daubert Hr’g Tr. 27:12-18 (“[Tjhere’s no requirement that you have to play forever. It’s not like a game of golf where you have to play all 18 holes. It’s not like a game of tennis where you have to play three sets. You can drop out any time you want. So the fact that if you play one hand chance is the material decider, I would say that says it right there.”). [N]o matter how long a player sits at a table, the probability of receiving any given hand is the same on the next hand before the cards are dealt as it was for every other hand in the game. This is reminiscent of a famous paradigm from basic probability theory. If red has come up 20 times in a row in roulette, it does not mean that “black is due.” The probability of getting red or black (or green) with each spin of the wheel is independent of prior history. From this point of view, Dr. Heeb’s analysis of long-term results is of limited value.... Gov’t Expert Report at 10; see also Gov’t Reply Letter at 2-3. Moreover, he argued, without independent factual support, that the number of hands required for skill to predominate over chance in poker to a reasonable degree of statistical confidence exceeded the number played by an average player. For example, as shown in Figure 6, in order for a skilled player to beat an unskilled player with 95% confidence in a game with $l/$2 stakes, the player would have to play 1,399 hands. “Assuming an average of 30 hands per hour, a player would have to play more than 46.6 hours with no breaks” to reach that number of hands. Gov’t Expert Report at 9. Third, Dr. DeRosa argued that the presence of persistent winners and losers, as illustrated by Dr. Heeb’s chart (shown as Figure 1, supra) does not prove that skill predominates over chance in poker, because “persistent winners and losers (selected after the fact) result normally from random chance variation.” Id. at 11. If one were to chart the performance of players involved in a game whose outcome was determined by purely by chance, one might expect to see similar results. Some players are seen — after the fact — to have consistently won. This does not prove or disprove that their winning was a result of skill. Id. at 12. To demonstrate this point, Dr. DeRosa ran a simulation that produced results similar to having 1,000 players each flip a coin 100,000 times. Id. As with a coin flip, each player had “an equal chance of either winning or losing an equal amount of money on any given trial.” Id. As shown below in Figure 7, his results appear strikingly similar to those of Dr. Heeb in Figure 1. According to Dr. DeRosa, “[t]his experiment shows that in a game of pure chance, in the long run, the top winners exhibit the behavior of persistently winning while the top losers appear to be consistently losing.” Gov’t Expert Report at 13; see also Gov’t Reply Letter at 3-4; Figure 1 and accompanying text, supra. Finally, Dr. DeRosa questioned the validity of Dr. Heeb’s data set, as well as his methodology. Since the data came from an online poker website, Dr. DeRosa was concerned that it might not represent “real hands played by real players” and might include false data points representing instances in which live players — who may have believed that they were playing against another, similarly-situated person — were playing against either a computer or a confederate of the gaming company. See id. at 17; Gov’t Expert Daubert Hr’g 32:5-10; 73:22-74:20; see also Gov’t Reply Letter at 10 (“It should be noted that Dr. Heeb admitted that he never performed any test to determine whether players were colluding or had more information than they should have had (such as knowing what the community cards would be before they were dealt). In a game based on imperfect information, knowledge of even a single card, whether held by an opponent or included in the community cards[,] would be massively valuable.”). He also questioned Dr. Heeb’s decision to categorize “[pjlayers with too few hands played to calculate statistics for all the various hand and position combinations” as “the lowest skill level” in his analysis of whether skilled players played particular hands better than unskilled players. Def. Expert Report at 36 n.7. According to Dr. DeRosa: These players could be ‘skilled’ players who only played a few ... hands and then quit because they were losing.... The assumption ... could cause a self-serving bias in Dr. Heeb’s analysis. A player only plays a few hands because he loses. He is then assumed to be unskilled and therefore the expected loss of unskilled players increases. Gov’t Expert Report at 16. Dr. DeRosa concluded that Dr. Heeb has not proven that skill predominates over chance in poker. Gov’t Expert Report at 3. He did not discuss Dr. Heeb’s analysis of the relative performance of skilled versus unskilled players in playing the same hand. See Fig. 2-3, supra, and accompanying text. c. Defense Expert’s Supplemental Report Following the August 10, 2012 Daubert hearing, Dr. Heeb submitted a supplemental expert report responding to Dr. DeRosa’s critiques and providing additional support for his conclusion that skill predominates over chance in poker. See generally Def. Expert Supp. Report. First, Dr. Heeb argued that his definition of skill, and his evaluation of relative skill levels, was appropriate. He conceded that only “10 percent to 20 percent of the players in any given game are good enough to win consistently.... And that’s represented by the top 6 to 8 percent of players on” Figure 4, supra. Gov’t Expert Daubert Hr’g Tr. 86:5-9. He contested the view, however, that poker players can only demonstrate skill by winning money. Instead, as Dr. Heeb put it, “the question of whether skill predominates requires only an examination of the relative importance of skill.” Def. Expert Supp. Report at 1. Moreover, he added, “[a] big part of the reason that more poker players have losing results than have winning results on a cumulative basis is that in most circumstances, including the data I studied, player pay a fee to the operator of a poker game in the form of a ‘rake,’ which is a small percentage deducted from each pot.” Id. at 2. To demonstrate this, Dr. Heeb adjusted Figure 4, supra, to add the rake payments back to the players’ winnings. In his initial analysis, which included the payment of the rake, as shown in Figure 4, only 28% of players in the $5/$10 game have a positive profit over the course of a year. When the rake is added back, 37% of players have a positive profit, as shown in Figure 8. As before, win rate is calculated as fraction of the big blind won per hand, on average. Dr. Heeb further explained that “[t]he reason that less than half the players earn a profit is that the best players (with or without the rake) win more than their prorata share of the profits, so there are fewer winners than losers.” Id. But see Gov’t Reply Letter at 2 (“[A]nalyzing the data without the rake is misleading because poker is a game where winning or losing is determined by monetary performance in an absolute sense. Unlike games such as chess or golf, wagering money is an integral part of a poker game. Whether a player is successful depends on how much money he has won or lost, but by how well he has faired against the other players.”)/ Dr. Heeb also defended his assignment of players who play very few hands to the low skill group, explaining that, in his experience as a poker player, “players who play only a few hands and quit” — even if they win money before leaving the game, as some do — “are almost always low skill players.” Id. at 7. He noted that, because “the outcome of players’ hands are effectively weighted by the number of hands that they played, ... a player who played only a few hands would get very little weight in the simulation.” Id. Second, Dr. Heeb explained his reliance on cumulative results, and particularly his dependence on results involving large numbers of hands played: In order to assess whether poker is a game in which skill predominates, it is essential to consider the cumulative outcome after a sufficient number of hands. Consider baseball or gold. Baseball is a game of skill on every pitch, even those pitches on which something “lucky” or “unlucky” happens. Nonetheless, it would not be possible to determine with statistical confidence that baseball is a game of skill from a single pitch, or even a single game, even though the score is known and can change with each pitch. Similarly, each stroke in gold can change the score. Poker is a game of skill on every hand that is played, but proving this statistically requires a sufficient number of hands.... ... Typical poker players, even in social games, play multiple sessions at regular intervals, such as playing once or twice weekly. Even after only an eight-hour session of poker (about 240 hands), skillful play can already be distinguished from less skillful play.... [When the 50% most skilled players are compared to the 50% least skilled players] 76% of higher skill players are already ahead after just 240 hands. If I focus on the top 10% of players, compared with the bottom 30% of players, 88% are ahead after just 240 hands. More than 90% are ahead after just 300 hands. This number of hands can easily be played in a single session of poker, even by casual players in a social game.... A serious poker player, even an amateur, can easily play thousands of hands a month in live play. Id. at 3^4. See also Gov’t Expert Daubert Hr’g Tr. 78:22-79:22 (“Dr. Heeb: ... [I]f I look over a large number of plays, a large number of a statistical sample, what I learn from the sample after enough observations is arbitrarily close to the absolute truth. And it’s just a question of how many you have to look at to know what the true underlying distribution or any underlying statistic would be. So depending on how closely you want to know the truth, you may have to look at more and more — a larger and larger sample, more and more observations. But the underlying truth hasn’t ever changed. So we’re trying to detect whether or not this is a game of sMll with a degree of precision.... After enough information, we can determine that the probability of players getting different results, the skillful players’ better results, the less skillful players’ less successful results, the probability of that happening by anything other than a difference in skill between the players becomes ... smaller and smaller the more observations on the data, the more hands that we play. It’s still a game of skill even on one hand. But to say that we have shown that with statistical certainty requires more and more information the more precisely we want to measure it.”); see id. 77:20-78:6 (“Dr. Heeb: The way that I would think about it is that the question is whether or not the game is a game of skill. How many hands do you need to look at in order to discern that with statistical certainty? A more familiar example might be baseball. One pitch in baseball is a skillful event, both skillful by the pitcher and skillful by the hitter. If you wanted to demonstrate that skill, you would look at a lot of pitches, games, for the season. And based on a statistical analysis of a lot of events, you would be able to determine whether or not that was a skillful act. But it’s a skillful act in the execution of just one swing or one p