Playing Head-To-Head Texas hold’em
Texas hold’em all in strategy and related problems
Andrew Beal is a billionaire who made his fortune in banking and related ventures. While he is not a mathematician he has a conjecture named after him, the Beal Conjecture.
Today I want to talk not about his conjecture, but his adventures playing Texas hold’em against some of the best players in the world. I believe there is an interesting property of this poker game that should be studied more carefully.
Okay, I guess I should at least state his conjecture. Beal’s conjecture is a generalization of Fermat’s Last Theorem. If
where , , , , and are positive integers and the exponents , and are all greater than , then , and must have a non-trivial common factor. The classic Fermat’s Last Theorem is the special case when .
I believe it is still open, and Beal is offering a $100,000 prize for its resolution. I say, “I believe,” since there are many claims and counter-claims on this conjecture. So before you start to work hard to get the prize please check and be sure the problem is still open.
This problem is definitely one an amateur could solve—I can imagine two approaches. You might find an elementary argument that proves if there is a solution where have no non-trivial common factor, then . This would yield a proof that there are no solutions—in a sense Andrew Wiles would have done all the hard work. Another possible approach is to find a counter-example. I suspect a naive brute force search is unlikely to succeed, but a small insight and enough computer time could find a counter-example.
An example of this is Leonhard Euler in 1732 showed that
This proved that not all Fermat numbers were primes. Euler did not do this by brute search: he proved a simple lemma about the possible divisors of such Fermat Numbers. In particular, he proved every factor of is of the form . This greatly helped his search, since all the calculations were done by hand.
Let’s move from number theory to probability theory and poker.
One of the most popular poker games these days is Texas hold’em. It is played in dorm rooms, on-line, and there are many big money tournaments on ESPN. I believe the interest in the game is driven by its simple rules, by the amount of money available to winners of the major tournaments, and by the feeling that somehow: the game is not that “hard” to play well.
I will not give the complete rules of Texas hold’em here, since there are so many good descriptions on-line. See here for one. Here is a quote explaining the game at a high level:
Texas hold’em is a variation of the standard card game of poker. The game consists of two cards being dealt face down to each player and then five community cards being placed by the dealer—a series of three (“the flop”) then two additional single cards (“the turn” and “the river”), with players having the option to check, bet or fold after each deal, i.e. betting may occur prior to the flop, “on the flop,” “on the turn,” and “on the river.”
I have played poker over the years for tiny stakes, really tiny. I started a standing game at Yale, then one at Berkeley, and finally one at Princeton. We played “dealer’s choice,” and allowed lots of strange and crazy games: from hi-low games to games with names like anaconda, 727, and others with names I no longer even remember.
Let me recall one fun hand that happened at Yale. Stan Eisenstat was sitting to my right and he was the only player in a large pot against another player—I will call X. The game was a crazy form of stud with individual face-down and face-up cards, and with the following property: we each passed cards at the beginning to our neighbors. In this case I had received three cards from Stan. He was showing a very strong hand—it looked like he could have four kings. Player X was showing a possible straight flush:
While I was out of this game, I knew that Stan knew that X could not have the straight flush. I knew this since the key card the was one that Stan had passed to me at the beginning of the game. The card was in the discard pile—it could not be in X’s hand. Theorem. Stan finally called a large raise of X and they both showed their hands. Of course X had a flush, but not a straight flush. And Stan did have his four kings.
Stan was happy as he raked in the pot, and said something about how he thought X was bluffing. I looked at Stan and said, “you passed me the key card, don’t you remember?” Stan said no, he had forgotten. Oh well. That’s why we are not professional card players.
Beal Plays The Pros
Beal likes to play Texas hold’em and wanted to see how well he could do against the world’s best players. Eventually, he worked out a plan: the best players in Las Vegas formed a “team” and Beal played one of the team members in head-to-head hold’em. After a while another team member would take over, but Beal played on alone. These matches happened over a period of time; often Beal would fly into Vegas, play for a day or two and then go back to Dallas.
There were two reasons for the team setup. Beal wanted to play very high stakes—recall he is a billionaire—and he felt that the large stakes would favor him. The pros played serious poker, but never for the type of stakes Beal wanted. So Beal hoped the large stakes might take them out of their comfort-zone. For another, Beal wanted to play head-to-head. He was afraid if he played several of them at once, they might be able to gang up on him. They could even cheat him—it would only take a very small amount of information moving from one player to another to make the game unfair.
Beal turned out to be a very strong player, but the pros were some of the best in the world. He won some money, lost some money, and again won some money. But finally the team prevailed and won a ton of money, and he gave up poker—although not forever. There is a fun book on Beal and the pros written by Michael Craig—it is called “The Professor, the Banker, and the Suicide King: Inside the Richest Poker Game of All Time.”
Alice Plays Bob
Since we do not have real money to play the pros we will consider what happens when Alice and Bob play Texas hold’em. Imagine that Bob is invited to play a head-to-head match of Texas hold’em with Alice. She is a professional poker player, and worse she is a world champion at hold’em. Does Bob have any chance at all in winning?
Let’s be precise and state the rules they will be using for betting. Each hand the ante for each player will be one chip, and they will each start with chips. The betting is no-limit: the player’s bets are limited only by the amount of money they have in front of them. The match is over when one player gets all the chips. Note, I have stipulated a simple ante rather than the “blind bets” actually used in hold’em. This simplifies the discussion without greatly changing the results.
The surprise, I think, is that Bob can follow a strategy which will make sure that Alice will win less much than of the time—even though Bob is a weak player. At the world championship level to have a reasonable chance of winning is cool—I think. There is nothing like this for other games: in chess Bob would be killed, in Backgammon he would very likely lose, the same holds for Bridge or Scrabble, or almost any reasonable game. But not for Texas hold’em. Ken Regan pointed out that a book on poker by James McManus has a chapter on this very issue.
Theorem: For a match of head-to-head Texas hold’em there is a simple strategy that wins with probability against any player, where is independent of .
Here is a sketch of the proof. Bob plays the following simple strategy. Every hand he bets all his chips—in poker terms he “goes all in.” He may for our analysis even tell her that is his strategy—and doesn’t mind always betting first even though that is a major disadvantage in non-extreme poker betting strategies.
Clearly each hand Alice has a simple choice: she can either call Bob’s bet or fold her hand. Consider the first hand. Alice is dealt some hand . She can calculate the probability will win and decide whether or not to go all in. If she does, then the match is over—either Bob or Alice wins. Actually, there always is a small chance of a tie in Texas hold’em, so there is a chance they will essentially start the match over.
Alice could also look at the hand and decide to fold. In this case Bob now has chips and Alice . Then, they go on to the next hand and Alice has the same decision to make.
The key insight is that no matter what hand Alice finally decides to call Bob with there is a positive probability that Bob will win.
An interesting open problem is, what is the exact probability that Bob wins with best play by Alice? The task of computing is compounded by the fact that once Alice folds and Bob has more chips than Alice, an all-in bet that Bob loses will no longer end the match. Bob need only bet as much as Alice’s stack to put Alice all-in, and he will still have the chips by which he was leading. Thus even if Alice wins the first played hand, Bob could still get back in the match and win with a series of doubling wins with his remaining chips. This is more of a complication than enumerating the possible hands and their win probabilities (even taking ties into account), for which published computer analyses are available.
Another point is that, Bob is not revealing any information by his strategy. His play is the same all the time. He might as well choose to do this blind, without looking at his cards, and the probability would still be the same. So whatever Alice’s mind reading skills maybe, they are futile with Bob’s strategy.
Even more, Bob is probably best if he avoids looking at his cards. Since he is a weak player he may have a “tell” that gives Alice information, so he best not even knowing his actual cards.
Alice’s Problem Generalized and Simplified
Here is a simplified problem that is related to Alice’s dilemma. Imagine there are possible hands she can be dealt. The hands have the probability of occurring
Each hand also has a probability of winning . Note, the values do not sum to one. Also we assume that each : that is there is no hand that is always a winner. Finally we stipulate now that a single win by Alice wins the match, even if Bob has chips left over.
The problem Alice faces is this. She gets hands with the given probability distribution and must decide when to bet. Given she will only see hands at most what is her optimal strategy? Note, this generalizes Texas hold’em, but it ignores the issue of chip stack sizes.
Find the exact value of from the theorem. Also solve the problem outlined in the last section. What is the best strategy for Alice? It reminds me of other on-line decision problems such as the Online Secretary Problem, but seems to be different.