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Commentary on The Endgame in Poker


Introduction

It's well known among those who follow tournament poker that Chris "Jesus" Ferguson won the main event at the World Series of Poker in 2000. What's not quite as well known is that in the previous year he was awarded his PhD in computer science from UCLA. Of course, it can't be a huge surprise that "Jesus" would obtain an advanced degree from UCLA, as his father, Thomas, teaches in the Mathematics department of the same school. Both have a professional interest in the branch of mathematics known as game theory, so it's not too surprising that they might collaborate on a paper titled The Endgame in Poker. This paper is available from the UCLA Math Department web site, or can be found in the collection Optimal Play, edited by Ethier and Eadington.

Beginning

The paper begins with some historical background and recommended reading concerning the problem the authors are addressing. For anyone interested in doing a literature search for game theoretical approaches as they have been applied to poker, this list is a great place to start.

The first question that needs to be addressed is what exactly is the "classical endgame" referred to in the paper? It's a poker-like situation that is also referred to as the "classical betting situation" in the literature. Personally, I believe this second label is a better description of the problem, but most of the literature and the Ferguson paper use the former description, so this article will use that terminology.

The classical endgame poker situation is as follows: Two players ante. Player 1 receives a card that is either a "winning card" or a "losing card". Both players know the probability that Player 1 receives a winning card, but only Player 1 knows what kind of card he is dealt on each hand. Player 1 may check or bet. If Player 1 checks, the game is over. If Player 1 bets, Player 2 may call or fold. Player 1 wins if he holds a winning card or Player 2 folds to a bet.

At this point it is convenient to provide a listing of the symbols the Fergusons use in this paper.

Symbol Meaning
P Probability that Player 1 receives a winning card.
a Amount each player antes.
bn The size of the bet Player 1 is allowed to make on the nth betting round. If there is only one betting round, then the subscript is omitted.
B The maximum allowable bet in situations where that's not fixed. In a no-limit game, B can be thought of as the effective stack size of the hand. B is the sum off all the bn.

In a no-limit game with one round of betting, b ≤ B. In a pot-limit game with one round of betting, b ≤ 2a.

While this game isn't likely to become a fixture at anyone's local card room, there are occasions in real poker that approximate this situation. The authors describe some of these situations as well as explaining where this model falls short. One analous situation that one can keep in mind as we go forward is to imagine Player 1 has a very strong draw, like a nut flush or straight draw, and Player 2 has a decent, but vulnerable, made hand, such as top pair, top kicker.

Optimal Play

The Fergusons examine the basic endgame with one or more rounds of betting. In each case, it should be pretty obvious that it will be correct for Player 1 to bet whenever he has a winning card. The open questions for each betting round are:

  1. How often should Player 1 bet with a losing card (bluff)?
  2. How often should Player 2 call when Player 1 bets?

In each case, if P is large enough, then Player 1's optimal strategy is to always bet and Player 2's optimal strategy is to always fold. If this is the case, then on each hand Player 1 will win Player 2's ante. If P is smaller, then it will be correct for Player 2 to call some of the time. This situation is what the authors call the "all strategies active" case.

The authors seek to derive a game-theoretic optimal strategy for each player. The methods they use are familiar to those who have studied game theory. We can find a solution by solving the minimax equation or by using the Indifference Principle. For those who aren't familiar, finding the minimax solution is a method for determining an optimal mixed strategy for a game that can be expressed as a matrix, as in Section 1 of the paper. Using the Indifference Principle is described in the same section of the paper, and this method should be familiar to those who have read Chen and Ankenman's book, The Mathematics of Poker. In this paper, the authors use the Indifference Principle to derive optimal strategies for different games types, starting with a game in which there is one round of betting.

In section 1.1 they discuss the optimum bet size. In the original Basic Endgame problem, the size of the bet, b, was fixed at some number. What if b is not fixed? In that case, the authors prove that Player 1 should always bet the maximum allowed, and whether Player 1 has a winning card or not, he should always bet the same amount.

In section two, the authors generalize this game to two rounds of betting. Note that hand strengths don't change from round to round, or are "static" in the terminology used by Chen and Ankenman. Now on each betting round, Player 1 may bet or pass. If Player 1 passes, the game is over at that point, and since Player 1 would always continue betting if he had a winning card, Player 2 would win the pot at that point. If Player 1 bets, then Player 2 may call or fold. The Fergusons derive an optimal betting frequency for Player 1 that depends on the size of the antes, the bet sizes, and the probability that Player 1 is dealt a winning card. Similarly, they come up with an optimal bet size for each of the two betting rounds.

In the next section, the authors generalize these results to game with an arbitrary number of betting rounds. The math gets rather hairy at this point, but the basic results are extensions of the previous results. Some of these can be expressed in qualitative terms that can be understood by those without an understanding of advanced mathematics.

  • The more betting rounds we add, the more the value of the game favors Player 1, even as the stakes remain the same.
  • As the number of betting rounds increases, it becomes correct for Player 1 to bluff more frequently on the first round.
  • There are times where Player 1 should bet on the first round and then pass a subsequent round. Sometimes Player 1 should bet on the first round without a winning card and then bet again on one or more subsequent rounds. In this simplified poker game, it is part of the optimal strategy to both make multi-barrel bluffs and to bluff on one round and then give up on the next.
  • Sometimes Player 2 should call on one betting round and pass on the subsequent betting round.
  • Similarly, as long as P is not so great that Player 2 should never call, the fraction of the time that she should call decreases on each subsequent betting round.
  • The optimal bet size on each betting round is the same fraction of the pot. What fraction that is depends on the stakes of the game, described by the variable B. For example, if B = 3a and there are 2 rounds of betting, then optimal strategy would call for a half pot-sized bet on each of the two betting rounds. That is b1 = a, and b2 = 2a. This is the same result arrived at in Chapter 19 of The Mathematics of Poker by Chen and Ankenman.

The next section covers the basic endgame with a continuum of betting rounds. It also assumes that Player 1 has the option of betting an infinitesimal bet size. There's no real-life poker analogy to this, and thinking about it is likely to make one's brain hurt. It is a cute mathematical construct, though.

Up until this point in the paper, Player 1 always knows when he has received a winning card, but Player 2 does not. In Section 5, the authors modify the game so that even though Player 2 doesn't know what card Player 1 has, they do have some information in addition to the probability P about what card they might have. The analogous real world situation is that if Player 1 is on an open-ended straight draw in hold 'em, Player 2 might have one of the cards Player 1 needs as a blocker, and that affects the odds that Player 1 can make his hand. This analysis gets really complicated, but this extra factor can lead to an adjustment to the optimal bet size from the game where Player 2 does not have this information.

This result is of special interest to me, since Russ Fox and I devoted a considerable number of pages in our book, Winning Strategies for No-limit Hold'em, to the topic of bet sizing in no-limit hold 'em. None of the many factors we cite for modifying one's bet size are directly applicable to this game, but altering bet size based on how much information each player has about each other's hand strength is a factor we didn't explicitly consider. I plan to spend some time thinking about this issue, though.

Finally, the last section of the paper allows for imprecise knowledge for Player 1. An analogy to real poker is that Player 1 knows how high his hand is, but the possibility exists that Player 2 might have a better hand. Again, the math is very complicated, and in this particular instance the authors don't go all that far in analyzing test cases.

Conclusions

Throughout the paper, but especially in the last couple of situations, the authors present some interesting mathematical insights but gloss over some possible analogies to real-world poker. Some of them have been discussed in this article, but the paper is littered with numerous almost throw-away comments that upon further analysis might lead to fruitful poker strategies.

Of course, it is very dangerous to try to draw direct, mathematical conclusions about real-world poker based on these mathematical constructs. However, even the most basic poker game predicts that bluffing will be an important part of all poker. Similarly, we may gain some insight into poker strategies by examining these "toy" poker games, although expecting precise mathematical solutions to be transferable is perilous at best.

In any case, some day sufficient mathematical methods and computing power may become available, allowing us to find optimal solutions to "real" poker games. When this happens, we shouldn't be surprised if many of the revelations we learn from "toy" poker games don't turn out to be part of the solutions to more complex versions of poker.