Poker and Philosophy

by Bronson, Eric

  Thought4: If I bet, he might think3 I’m just betting because I think2 he’ll think1 I must not have A-K.

  We can already see the Davidsonian thesis at work. In order to have good reason to attribute to T.J. the third-order thought you do, you have to assume that you and he agree on many things. Some of the beliefs you must assume you and he share are:

  that A-K is a hand people like to raise with;

  that one strategy players sometimes employ when the flop comes rags after they’ve raised before the flop with A-K is to bet, hoping their opponent will fold;

  that twice before, you didn’t bet with A-K in this situation;

  that T.J. remembers that twice before you didn’t bet with A-K;

  that you remember that twice before you didn’t bet with A-K.

  And so on. If there were any one of these beliefs that you didn’t have or didn’t think T.J. had, you wouldn’t be justified in forming the fourth-order thought.

  The idea that successful thought attribution depends on shared belief explains the surprising fact that an experienced player will sometimes fare quite poorly against a novice. The pro cannot assume, for instance, that the novice knows that some players are more apt to bet with A-K when the flop comes rags than others—or that it’s significant that the flop has come with all low cards. It can thus be difficult for pros to deduce what a novice would think if the pro were to act in such-and-such way. And if the pro did assume his opponent shared some of his own sophisticated thoughts about A-K (which he might well do, before he realizes his opponent is a novice), he might undermine his own cleverness. (In “Odds or Evens,” the kiss of death is being two steps (not one) ahead of your opponent.2) But perhaps I’m just trying to convince myself that I’d have a chance against a pro. After all, the pros would still be able to capitalize on all my physical tells. And unfortunately, I tend to giggle when I have a good hand. (Or do I?)

  Back to your mental sparring with T.J., which might not stop at the fourth order. You might well take it one step further. After thinking4 If I bet, he might think3 I’m just betting because I think2 he’ll think1 I must not have A-K, you might then think6:

  Maybe he’d think5 I’d think4 this, in which case I should bet.

  That’s a thought of the sixth order. Now what about your thoughts—that is, my patient reader’s—in reading this essay? The thoughts involved in your comprehending the last few sentences were of the seventh order. Those sentences were about a thought of the sixth order, so your comprehending them was of the seventh. I could have titled this chapter “Thinking about Thinking about Thinking about Thinking about Thinking about Thinking about Thinking (about Poker)”. (And if you understand why, then I could have added another “Thinking”.)

  You might wonder whether the pros really do think these long, intricate thoughts at the tables. Spelled out explicitly, the sixth-order thought is:

  Maybe he’d think5 I’d think4 that if I bet, he might think3 I’m just betting because I think2 he’ll think1 I must not have A-K, in which case I should bet.

  Do the pros really say all this to themselves at the table? Well, that depends on whether saying all that to themselves is required for their having that thought. Precisely what is involved when someone thinks a thought? Sentences in the mind? Words? Mental images? What is thinking, for that matter? Or, as Oxford philosopher Gilbert Ryle (1900–1976) is said to have articulated the question, “What is it that the man in Rodin’s sculpture is doing?” Thinking about Big Slick of course, but Ryle wasn’t asking what the man in the sculpture is thinking about, but rather what his thinking—or any thinking, really—is, or involves. According to some players, feeling and the “subconscious” play a significant role in their thoughts at the table. Crowd favorite Doyle “Texas Dolly” Brunson writes about a particular situation: “. . . even though I might not consciously do it . . . I recall that this same play came up (or something close to it) and this is what he did or what somebody else did. So I get a feeling that he’s bluffing or that I can make a play here and get the pot. But, actually my subconscious mind is reasoning it all out.”3 But these matters bring us quickly to the heart of some central debates in contemporary philosophy of mind and psychology.

  Thinking about Daniel Negreanu

  By emphasizing the number or height of the levels of thought employed in expert poker games, I don’t mean to nourish the perhaps specious assumption that those who excel at poker think at extremely high levels—as if T.J. Cloutier is so successful because he can go to the sixteenth, while most other pros go only to the fourteenth or fifteenth. My sense, actually, is that the levels are not so high. Not that the thoughts are not complex or impressive, but that more crucial to expert play is making sets of thought attributions, each of which is typically of the fourth order or below, and then inferring what those attributions together imply.

  Consider a hand that professional poker player Dan Harrington deems the most ingenious in recent history.4 It was down to the last two players at the 2004 Plaza tournament, Daniel Negreanu and Freddy Deeb. Negreanu, with $337,700 and in the little blind, has A-7 and calls the $1,600 big blind. Deeb, with $342,300 and in the big blind, has A-K and raises $7,000. Negreanu calls. The flop comes K-6-2. Deeb bets $16,000, and Negreanu calls. The turn is the 4. Deeb checks, Negreanu bets $30,000, and Deeb calls. The river is the 4. Deeb bets $65,000, Negreanu raises $100,000, and Deeb folds.

  Negreanu had the weaker hand but still managed to win the pot against an opponent who held A-K and flopped top pair. How did he do this? You may think Negreanu simply got lucky, that he bluffed on the river as anyone can, and his opponent happened to fold. But this is far from true. In order for a bluff to be successful against an experienced player who holds a decent hand, many things need to be in place. Two crucial ones are:

  (1) that your opponent believes that, given what you’ve done on each round of betting so far, the only hands you could reasonably have would all beat him.

  (2) that your opponent is capable of folding when he believes this.

  The great players are those who can recognize the rare situations in which they know (2) is true and can make (1) true. As Harrington sees it, Negreanu believed that if he raised $100,000 on the river, Deeb would review what Negreanu had done on each round of betting and would conclude that the only cards he could reasonably have, consistent with how Negreanu acted on each round, would make Negreanu either a flush or full house.5 But why raise only $100,000? $100,000 is a small raise given the amount of money already in the pot. The answer is that Negreanu knew that Deeb is particularly leery of “lure” bets (relatively small bets made to lure a call). He also believed Deeb knew that he (Negreanu) is capable of making such a bet when he has a good hand. So Negreanu figured that if he made a small raise, Deeb might think he was trying to lure a call, which would also be consistent with his having a flush or full house. Talk about more moves than a mongoose!

  The reason I’ve traversed this hand so thoroughly is to show that Negreanu’s thinking appears not to involve any thoughts higher than the fourth order. His central thoughts were probably:

  Thought4: Deeb knows3 I’m capable of thinking2 that a smaller bet is more likely to make my opponent think1 it’s worth it for him to call.

  Thought2: Deeb’s particularly likely—more so than other players—to think1 that a small bet is a lure bet and to fold his hand in response.

  Thought1: Deeb can’t beat a flush.

  Thought2: If I raise, Deeb will go1 through everything I’ve done so far in the hand and conclude1 that my most likely hands are a flush and a full house.6

  From this set of thoughts Negreanu reasons that if he bets $100,000, Deeb will fold.

  We can thus distinguish between two skills the expert player must have: the ability to make individual thought attributions, and the ability to deduce from all the attributions he’s made what play would be the most effective. The second is no less important than the first. Yet another skill is the ability to anticipate when
those rare sets that allow for ingenious plays like Negreanu’s might materialize later in the hand. Negreanu had little reason to call on the flop unless he anticipated the possibility of successfully bluffing later on. Indeed, he knew which cards would afford him precisely that opportunity.

  What sets of thought attributions allow for such clever plays? It might be fruitful to lay out a variety of these plays and scrutinize the relations among the sets involved. For instance, Negreanu remarked about a play Erick Lindgren once made against him, “I didn’t think he could be so stupid. But it wasn’t stupid. It was like a step above. He knows that I know that he wouldn’t do something so stupid, so by doing something so quote-unquote stupid it actually became a great play.”7 How does the set of attributions Lindgren made and capitalized on compare to the one Negreanu employed against Deeb? Do they share a similar structure? If valuable sets fall into different groups, would a taxonomy of them help players identify and anticipate them more easily at the table? Are there promising sets that remain undiscovered? This may be fertile territory for the serious poker player.

  My sharp reader may protest: How do you know Negreanu’s thoughts about Deeb didn’t go higher than the fourth order? For instance, maybe Negreanu thought6 about the possibility that Deeb would reason5 that Negreanu had these thoughts4 but concluded6 that Deeb wouldn’t reason5 this way. Or maybe Negreanu thought8 that Deeb would think7 Negreanu would figure6 Deeb would reason5 this way and so would fold to a small bet. Perhaps Negreanu did think all this—it’s consistent with all the data. Only the pros can tell us whether their thoughts go this high. Harrington never intimates that they do. But then again maybe that’s all just to keep us from knowing what these guys really think. You never know in poker—which is why it’s all so interesting to an epistemologist.8

  It is true that no matter what level your present thought about your opponent is at, you can always wonder: “But then again, he might think I think this.” My point is that taking it one step higher than your opponent in this way may not be the central element of poker, as it perhaps is in other games, like Odds or Evens, or even Rock, Paper, Scissors. (Incidentally, it’s becoming potentially lucrative to excel at Rock, Paper, Scissors too. In the spring of 2005, the president of a Japanese electronics company organized a game of Rock, Paper, Scissors between Christie’s and Sotheby’s, in order to settle which auction house would sell his company’s twenty-million-dollar art collection.9 Christie’s won. Christie’s’ people had researched the thought attribution that is typically done in Rock, Paper, Scissors—they talked to eleven-year-olds (really)—whereas Sotheby’s had assumed the game was simply “a game of chance.”)

  Still, if your opponent anticipates what you’re thinking, you’re in trouble. Here are two things to keep in mind when wrestling with how and when to stop those “But maybe he thinks I think this” iterations. First, if you’re playing someone who can out-think you, one tactic is to choose “randomly.” You might use the second-hand on your wristwatch. In “Odds or Evens,” you could throw two fingers when the hand is past the Six, and one when it’s not. (My sister caught on to this one sooner than I would have liked.) Choosing randomly disarms your opponent’s skill entirely. Likewise, if you wish to bet your A-K only twenty-five percent of the time, bet it only when the second-hand’s past the Nine. Second, the more complex your own reasoning, the less confident your opponent can be that your reasoning is that complex, even if the possibility that it is crosses his mind, and so the less prudent it would be for him to risk his chips on his suspicion that it is. Even if it had occurred to Deeb that Negreanu may be making the ingenious move he made, it would have been difficult for Deeb to risk the rest of his chips on this thought. The possibility that Negreanu was thinking everything that he was had to be, from Deeb’s vantage point, more remote than the possibility that Negreanu had a flush or full house. Yet another skill of the great poker player, then, is the ability to detect situations in which, even if your opponent does intuit what you are up to, he cannot prudently act on that intuition.10 Notice, finally, the depth of agreement that must again exist for a play like Negreanu’s to be successful: making such a move requires putting the other person in a position where you both agree on what he has to do.

  How Does Negreanu Do It???

  All of this leads at last to what is the most impressive and, to me, bewildering element of good poker. How do you know what your opponent is thinking? You may think I’ve already addressed this epistemological question, but I haven’t really. I discussed the nature of some of the thoughts you might attribute to your opponent when you have A-K and the flop comes rags, but I didn’t say which of them you should attribute. I explained that expert plays typically involve drawing inferences from a set of thought attributions of the fourth order and below, but I didn’t indicate how and when one is justified in making the individual thought attributions in the first place. Or to put it a different way, how on Earth did Negreanu know all that about Deeb?!! Even if Davidson is right that you need to begin by assuming a great deal of agreement, how much and about what?

  In everyday life, we learn a lot about what people think from what they say. Even in the case of the rabbits, we had the natives’ words to go on. But in poker, no one’s going to tell you anything. And unless you’ve got a friend who already knows your opponent and can tell you how he plays—that he’s loose-aggressive, say, or a calling station, a flounder, or a rock (or that your entire table is a rock garden, for that matter)—the only way to determine how your opponent thinks is to infer it from what he does.

  But inferring how your opponent thinks from what he does is extremely difficult. And the reason goes back to those rabbits. Remember I said that what you say on a particular occasion is a vector of what you believe and what you mean? Similarly, what you do is a vector of what you believe and what you want. You might wait outside the restaurant because you believe your friend is going to be outside and you want to see him, or you might do so because you believe he’s going to be inside and you don’t want to see him. One and the same action could be the product of countless sets of beliefs and wants.

  Suppose the guy two seats to your left raises before the flop. You don’t know anything about him, except that he’s wearing mirrored sunglasses, a green visor, and a t-shirt that says “Lucky Larry,” so you pay close attention to see what you can learn. Before the flop, Larry gets called by one player, and then bets when the flop comes 2-6-9 (just as you were considering doing earlier against T.J. with your A-K). His opponent folds, and Larry wins the pot. What can you conclude about how Larry plays? Not much. His two actions (his raise and subsequent bet) could have been the product of many different sets of beliefs and wants. Larry might have raised with A-K, believed he was weak on the flop and wanted his opponent out of the hand. Or maybe he raised with the beautissimous pocket A-A and wanted a call. Or he had a middle pair (you know, like Snowmen or the Speed Limit), thought it was weak and wanted a fold. He might even have been trying to steal the pot before the flop with nothing but Woolworths, gotten lucky when a 5 and 10 came on the flop and then wanted a call. These are just some of the many combinations of belief and want that could have led Larry to act as he did. And that’s all assuming he’s a decent player, that he’s rational, and so on.

  Now, it is true that in this last example you did not get to see Larry’s cards, and sometimes you do. And then it’s easier. But not much. For instance, suppose Larry did get called on the flop. He and his caller then both checked the turn and river, and Larry showed A-K. Can you conclude now that he’s the kind of player that typically bets A-K when the flop comes rags? Not really. After all, Larry could be betting here to make it appear that he’s that kind of player. Or perhaps Larry has a policy of betting only once in twelve times with A-K and a ragged flop, and this was that once (his second-hand had passed the Eleven). Or maybe the reason he bet is that he had a read on what his caller was thinking. Sometimes you need to think about what your opponent thinks about wha
t another opponent thinks.11

  Of course, the more you see Larry show Big Slick in this kind of situation, the more confident you can be that he’s the sort of player that typically bets A-K when the flop comes rags. You must steer clear here, though, of that vexatious “Fallacy of Unrepresentative Sample”: whenever Larry decides not to bet in such a situation, he will often subsequently fold, and you will not take note of his A-K.12 But more importantly, this precise situation just doesn’t happen often enough. Consider all the things that must be in place: Larry has to be dealt A-K, he has to raise before the flop, one other player (and only one) has to call him, the flop has to come rags, Larry has to bet on the flop, his opponent has to call, and finally, Larry has to show his cards at the end of the hand. Entire tournaments could go by without that happening even once to Larry.

  So how, in just a couple of hours, does anyone ever learn how someone else thinks at the poker table? The answer, I presume, has a lot to do with patterns that players have observed in past experience—patterns like “People who play fewer hands are less likely to bet their A-K when the flop comes rags.”13 How many hands Larry plays is something you can observe in a brief period of time. Equipped with a battery of such heuristics (and no doubt they’re often more sophisticated), you can begin almost immediately to sketch schemas of particular players on the basis of them. Until new information about a player presents itself, at which point you’ll modify your schema in light of it, you attribute and act on the basis of the schema you have. This is another, quite different way in which shared thoughts are involved and assumed in thought attribution at the poker table. If such patterns of thought and play did not exist among poker players, this strategy would not be available.