A federal Court has just ruled that Poker is a game of skill rather than luck opening the door to a huge change in the treatment of Internet gambling in the US
The case UNITED STATES OF AMERICA Vs. LAWRENCE DICRISTINA was decided by the United States District Court for the Eastern District of New York
The case seemed to turn on the testimony of Randal D. Heeb, “a respected economist, statistician, and player in national poker tournaments”, who testified as an expert on game theory.
The opinion which is over 100 pages go into tremendous detail on all the federal laws on gambling and concludes that:
“Neither the text of the IGBA nor its legislative history demonstrate that Congress designed the statute to cover all state gambling offenses. Nor does the definition of”gambling” include games, such as poker, which are predominated by skill.”
If you have interest in this topic you should check out the whole opinion (pdf)
I reprinted the research of the expert witness Randal D. Heeb, as its pretty interesting for anyone who has played poker.
“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:
[T]here’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.
Bluffing, raising, and folding require honed skills to maximize the value of the cards
[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.
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Position at the table and the habits of fellow players must be taken into account to play successfully.
[A]nother 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
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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,
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this consistency could not be demonstrated in a new set of coin tossers by the same tossers. See Part II(B)(3)(c), infra.
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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
– I -~
~ AJ”‘V- f’ -~
I ;.OJ ,_/ ~I
$150,000 I .'” r ,”~.~
– – Winning player 1 – – Winning player 2
- - Winningplayer 6 – Winningplilyer 7
- - Losingpiayer 1 – l osing player 2 – L osi ngpl ayer 6 – Losing player 7
– Winning player 3 – Winningplayer 8 – l osingp/ayer 3
– l osingp!ayer 8
– – Winningplayer4 – – Winning player 5
– Winningplayer9 – Winningplayer 10 – Losingplilyer 4 – l osingp!ayer 5
– Losingplilyer 9 – l osingp/ayer 10
Note: For illustrative purposes, in this figure all the hands played by each player are distributed evenly from left to right across the graph. Thus, for example, the midpoint 01 tile graph marks the midpoint of each player’s playing history. Some players played more hands than others, and they may have played them al different times . However, all of the players depicted in the graph played a high volume of hands.
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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
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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.
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
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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.
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Fig. 2: Win rate comparison: Queen Jack suited (e.g. Q