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 pokerlike
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. 
b_{n} 
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 nolimit game, B can be thought of as the
effective stack size of the hand. B is the sum off all the
b_{n}. 
In a nolimit game with one round of betting, b ≤ B. In a potlimit 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:
 How often should Player 1 bet with a losing card (bluff)?
 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 gametheoretic 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 multibarrel 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 potsized
bet on each of the two betting rounds. That is b_{1} = a, and
b_{2} = 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 reallife 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 openended 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 Nolimit Hold'em, to the topic of bet sizing in nolimit
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 realworld poker. Some of them have
been discussed in this article, but the paper is littered with numerous
almost throwaway 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 realworld 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.
