Poker and Philosophy

by Bronson, Eric

  It should be noted that, in a quantum-mechanical universe, there is an important sense in which anything can happen at any time. For example, as Michael Strevens points out, there is a very slim chance that a dropped egg will spontaneously reconstitute itself into an unbroken egg (Strevens, p. 243). Similarly, there is a very slim chance that the 9 will transform itself into the A just as the dealer is turning it over. But the probability of this happening is so extremely tiny that we can safely ignore it for all practical purposes.

  Unfortunately, as noted, there is a problem with objective chances if we want probability to guide us in poker. Namely, as a player, you do not know what the objective chances are. In order for you to find out whether the objective probability is 100 percent or zero, you would have to cheat. For example, you might bribe the dealer to perform a “top card peek” and communicate the result to you.11 But assuming that you are not going to cheat, objective chances are not a very useful guide. (You can’t use information that you don’t have.)

  WPT Cam Probabilities

  Of course, since no one knows what the river card will be, no poker expert would ever suggest that objective chances should guide one’s life at the poker table. However, there’s a somewhat similar proposal that is much more reasonable. Namely, you should calculate the mathematical expectation of the various plays using the objective probability that a hand will end up being the winning hand given all of the cards that have been dealt so far. In fact, Mike Sexton, Vince Van Patten, and the viewing public typically use what I will call the WPT Cam probability to evaluate the decisions of the players.12

  During the final table of a WPT event, numbers are displayed on the television screen next to each hand. This number is the probability that the associated hand will end up being the winning hand (if it goes to a showdown). The WPT Cam probability takes into account the community cards and the down cards of all of the players (including the down cards of the players who have folded their hands). This probability seems to be what is relevant to whether a player can expect to make money in the long run.

  Unfortunately, the WPT Cam probability has essentially the same problem as objective chances. Unlike the viewing public, the players at the table do not get to see what the WPT Cam sees. In fact, as a player, you will typically have no idea what cards the other players have folded before the flop. (You can be pretty sure that no one has folded Aces or Kings. But you cannot even be sure how many Aces and Kings have been folded. Someone could easily have folded a King or an Ace with a lousy kicker.) Thus, WPT Cam probability is also not a very useful guide.

  Sklansky Probabilities

  Sklansky makes another proposal for how probability should guide one’s life at the poker table with his Fundamental Theorem of Poker (FTP). According to this theorem, you should try to play the way you would play if you could see your opponent’s cards (Sklansky, pp. 17–18). In other words, you should calculate the mathematical expectation of the various plays using what I will call the Sklansky probability. This is the objective probability that your hand will end up being the winning hand given your opponent’s cards (as well as your cards and the community cards).

  It should be noted that there is actually a second clause to the FTP. Namely, you should also try to get your opponent to fail to play the way she would play if she could see your cards. So, for example, you should use your poker face to keep your opponent from getting any information about your hand. And you might even want to engage in some deception. But let’s just focus here on your decision to call or to fold.

  In my hold’em example above, what should you do if you wanted to follow Sklansky’s advice? Suppose that your opponent actually has a pair of Q’s. In that case, what is the Sklansky probability of your ending up with the winning hand? Well, you have six outs. In particular, the other three A’s in the deck and the other three K’s would give you a hand that beats Q’s. The probability that you will end up with the winning hand is 13.6 percent. (This is your six outs divided by the forty-four cards that have not been seen by you or your opponent.13) Thus, you should call the bet rather than fold.14

  The probability that you will end up with the winning hand can change over time.15 For example, before the flop, the probability that Big Slick will beat Q’s is 43.4 percent; after three blanks on the flop, the probability is 23.9 percent; after another blank on the turn, the probability is 13.6 percent; after another blank on the river, the probability is zero percent. (Here the probability keeps going down, but the probability could jump up and down depending on what community cards are dealt.) It is clearly the probability at the time of your decision that should be used calculate the mathematical expectation of the various plays.

  Unfortunately, there is a problem with Sklansky’s advice. In order to play as if you could see your opponent’s cards, you would have to know what your opponent’s cards are. But you probably do not know that your opponent has Q’s.

  There are many great stories of great poker players putting their opponents on specific hands.16 For example, such players have folded K-K before the flop because they could tell that their opponent had A-A. And then there was the time that three-time World Series of Poker Champion, Stu “The Kid” Unger, correctly called an all-in bet with only 10-high because he knew that his opponent had “4-5 or 5-6” (essentially the only hands that Unger could beat with his 10-high). Nevertheless, not even great players can always read their opponents with this degree of specificity. As a result, you will rarely be able to follow Sklansky’s advice. Thus, Sklansky probability is also not a very useful guide.

  Relative Probabilities

  You rarely know exactly what cards your opponent has. But you’ll typically know something (and often quite a lot) about her cards. Consider a somewhat more realistic scenario (that is, more realistic than knowing that your opponent has Q’s). Your opponent has been playing consistently tight. In particular, she only plays high pairs (10’s, J’s, Q’s, K’s, or A’s). So, you can be pretty sure that she has a high pair now. (In fact, for the sake of simplicity, let’s suppose that you are completely sure that she has a high pair.) But she always plays all high pairs the same way. So, you do not know which high pair she has now. (In fact, let’s suppose that you think that each of these high pairs is equally likely.)

  In this scenario, what is the probability that you should assign to ending up with the winning hand? Well, you have six outs if she has 10’s, J’s, or Q’s. Thus, as noted above, the Sklansky probability would be 13.6 percent. But you only have three outs (the other three A’s in the deck) if she has K’s. Thus, the Sklansky probability would be 6.8 percent. And you have no outs at all if she has A’s. In other words, you would be drawing dead and the Sklansky probability would be zero percent. It seems clear that you should average these probabilities based on how likely the five possibilities are.17 In other words, given that you do not know exactly what cards she has, the probability that you should assign to ending up with the winning hand is only 9.5 percent.18 Thus, contrary to Sklansky’s advice, you should fold rather than call the bet.19

  In this scenario, 9.5 percent is what Beebee and Papineau call the relative probability that you will end up with the winning hand. Since it takes into account all the information that you actually have available to you, the relative probability should be used to calculate the mathematical expectation of the various plays. Also, since it only takes into account information that you actually have available to you, relative probability (unlike the three previous proposals) can be a useful guide to poker. By contrast, if you tried to use the Sklansky probability to calculate the mathematical expectation of the various plays, it would just be a lucky guess if you picked the right probability.

  There are a couple of things to be noted here. First, in this scenario, the relative probability is different from the Sklansky probability. But if you did have enough information to put your opponent on a specific hand, then the relative probability would be the same as the Sklansky probability. Thus, you sho
uld follow Sklansky’s advice if you do know the Sklansky probability. Second, the fact that you can determine the relative probability does not mean that you will automatically do so. In other words, the subjective probability that you assign to your hand winning will not necessarily match the relative probability. For example, you might be an optimist who is always sure (despite evidence to the contrary) that her opponent has nothing. Alternatively, you might be a pessimist who is always sure that her opponent has the nuts.

  The Value of Information

  So, you should use the relative probability rather than the Sklansky probability to calculate the mathematical expectation of the various plays. But there is, nevertheless, a sense in which Sklansky is exactly right. That is, you would be better off if you could determine exactly what cards your opponent has. In fact, according to Phil Hellmuth, “If you know what the other players have when you’re playing no-limit hold’em, then you cannot lose” (Hellmuth Jr. 2004, p. 117). Thus, FTP serves as a useful ideal to shoot for. And philosophy can explain why this is the case.

  Additional information is always valuable when you’re making decisions. It can never hurt you. And it can help you if there’s a possibility that the information would lead you to make a different play. The philosopher of science Jack Good was the first person to formally prove this.20

  Here’s how this result plays out in the more realistic scenario above. Folding is the play with the highest mathematical expectation. And if you found out that your opponent has A’s or K’s, you should still fold. In that case, you are no better off, but neither are you worse off. However, if you found out that she has Q’s, J’s, or 10’s, you should call the bet. In that case, you’re better off.21

  Because information is always valuable, you should gather as much information as possible about your opponent’s hand. For example, how did she bet before the flop and on the flop? Does she have any tells? Perhaps she starts breathing faster or her hands shake if she is dealt K’s or A’s. Even if this information does not determine precisely which cards your opponent has, it is still valuable.

  Since additional information is always valuable, you should certainly gather it if it is free. But there is a limit to how much you should pay for it. For example, in this scenario, you would not want to pay more than $2,727 to get your opponent to show you her cards.22 Such deals would certainly be frowned upon in most poker games anyway.

  But it’s only from the perspective of the player that additional information is always valuable.23 Let me give you an example of how things might look from the perspective of a more highly informed observer of the game. Thanks to the WPT Cam, Mike Sexton knows that you have Big Slick and that your opponent has Q’s. But he also knows that the other players have folded three of your outs. (Three of these players were dealt K’s with lousy kickers.) Finally, since he is great at reading poker players, he knows that you have put your opponent on a high pair, but that you are still not sure which high pair. Using all of this information, he can predict that you are going to fold rather than call the bet given your current uncertainty. (As noted, the relative probability is 9.5 percent, which is less than the threshold of 11.1 percent.) He can also predict that you would call the bet rather than fold if you were to learn that your opponent has Q’s. (As noted above, the Sklansky probability is 13.6 percent, which is greater than 11.1 percent.) However, from Sexton’s perspective, calling is the wrong thing to do. (Since only three of the thirty-six cards that have not been seen by the WPT Cam will give you the winning hand, the WPT Cam probability is 8.3 percent, which is less than 11.1 percent.) Thus, Sexton thinks that it would be bad for you to put your opponent on Q’s.

  This complication aside, FTP is a useful ideal to shoot for. But if we are looking for ideals to shoot for, why should we pick FTP? For example, instead of just trying to play the way you would play if you could see your opponent’s cards, why not try to play the way you would play if you could see the next card in the deck as well? This additional information would certainly be valuable. In fact, knowing what is going to come on the river would tell you for sure whether you would end up with the winning hand. This, however, is clearly an unrealistic ideal to shoot for. You would have to cheat to get this information.

  But there are other possibilities. For instance, you could try to play the way you would play if you could see your opponent’s cards and the cards that have been folded by the other players. This additional information would also be valuable. And it would not necessarily require cheating to get this information. After all, if it’s possible to figure out what cards another player is playing, it may be possible to figure out what cards another player is not playing. For example, perhaps that look of disappointment on the face of the player across the table means that she had to fold an Ace or a King because of a lousy kicker. Without access to the WPT Cam, it might be extremely difficult to determine precisely what cards have been folded. But even incomplete information is valuable.

  If you follow the advice given here, you can expect to make money at the poker table. However, it is important to emphasize that there is no guarantee that you will make money. Just ask the Cincinnati Kid!24

  ________

  1 See David Sklansky, The Theory of Poker (Henderson: Two Plus Two, 1987), pp. 9–16.

  2 See A. Alvarez, Poker: Bets, Bluffs, and Bad Beats (San Francisco: Chronicle Books, 2001), pp. 86–87. For other bad beat stories, see Phil Hellmuth, Jr., Bad Beats and Lucky Draws (New York: HarperCollins, 2004).

  3 There is no guarantee that you will make money in the long run. Life is short and losing streaks can be long. But you can expect to make money in the long run.

  4 See, for example, Phil Hellmuth, “When to Fold Aces Preflop.” Card Player Volume 18, No. 5 (2005), p. 20.

  5 See Kevin Conley, “The Players.” New Yorker Volume 81, No. 20 (2005), pp. 52–58.

  6 See Allyn Jaffrey Shulman, “Gustav Hansen: A Style of His Own.” Card Player Volume 18, No. 3 (2005), pp. 50–54.

  7 Let p be the probability of your ending up with the winning hand. (p × $160,000) – ((1 – p) × $20,000) is greater than zero if and only if p is greater than 11.1 percent.

  8 Joseph Butler, The Analogy of Religion (London: J.M. Dent and Sons, 1906), p. xxv.

  9 See, for example, Rudolf Carnap, “Probability as a Guide in Life,” Journal of Philosophy 44 (1947), pp. 141–48; Helen Beebee and David Papineau, “Probability as a Guide to Life,” Journal of Philosophy 94 (1997), pp. 217–243; Michael Strevens, “Objective Probability as a Guide to the World,” Philosophical Studies 95 (1999), pp. 243–275; and Henry E. Kyburg Jr., “Probability as a Guide in Life,” Monist 84 (2001), pp. 135–152.

  10 David Lewis, “A Subjectivist’s Guide to Objective Chance,” in Richard Jeffrey, ed., Studies in Inductive Logic and Probability, Volume 2, (Berkeley: University of California Press, 1980), pp. 263–293.

  11 Darwin Ortiz, Gambling Scams (New York: Dodd Mead, 1984), p. 12.

  12 At the final table of WPT events, small cameras (the WPT Cam) placed around the table can see each player’s hole cards. Unlike objective chances, the WPT Cam probability is not based on complete information about the world. But the WPT Cam probability is still an objective probability, based on the incomplete information about the world that the WPT Cam can see.

  13 If you have no idea what cards your opponent has, then the probability that you should assign to improving on the river is 6 divided by the 46 cards in the deck that you cannot see. For a clear explanation, look at Mike Matros, The Making of a Poker Player (New York: Kensington, 2005), pp. 23–24. But the FTP requires you to calculate probabilities as if you could see your opponents’ cards. Given that your opponent has Q’s, the probability you should assign to winning the hand is 6 divided by 44.

  14 To be precise, the mathematical expectation of calling the bet when your opponent has Q’s is $4,545 = (13.6% × $160,000) – (86.4% × $20,000). By contrast, the mathematical expectation of folding is only $0.

 
; 15 For more information on this topic, see Lewis 1980, pp. 271ff.

  16 See, for example, Hellmuth, Jr. 2004, pp. 117–132, on “Reading Other Players’ Mail.”

  17 For a poker expert suggesting this sort of calculation, see Matros, pp. 49–50.

  18 9.5% = (3/5 × 13.6%) + (1/5 × 6.8%) + (1/5 × 0.0%).

  19 To be precise, the mathematical expectation of folding when you are in this scenario is $0. By contrast, the mathematical expectation of calling the bet is minus $2,818. In other words, you can expect to lose $2,818 on average if you call the bet.

  20 I.J. Good, “On the Principle of Total Evidence,” British Journal for the Philosophy of Science 17 (1967), pp. 319–322.

  21 To be precise, the mathematical expectation of calling the bet if your opponent has Q’s, J’s, or 10’s is $4,545. See my earlier note 12.

  22 $2,727 = (40% × $0) + (60% × $4,545). If each of the high pairs is equally likely, then 40 percent of the time this information will make you no better off, but 60 percent of the time it will increase your mathematical expectation by $4,545.

  23 See I.J. Good, “A Little Learning Can Be Dangerous,” British Journal for the Philosophy of Science 25 (1974), pp. 340–42.

  24 His Queens full of Tens famously and improbably lost to a Jack-high straight flush. See Richard Jessup, The Cincinnati Kid (Boston: Little, Brown, 1963). Of course, it is debatable whether the Kid really played this hand very well. See for instance Alvarez 2001, p. 92. (I would like to thank Eric Bronson, Andrew Cohen, Terry Horgan, Peter Lewis, Kay Mathiesen, and Bill Taylor for very helpful suggestions in preparing this chapter.)