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Approximating Game-Theoretic Optimal Strategies for Full-scale Poker


Department of Computing Science, University of Alberta
Edmonton, Alberta, T6G 2E8, Canada
Email: darse,burch,davidson,holte,jonathan,terence,duane 

Abstract Due to the computational limitations involved, only simpli-

fied poker variations have been solved in the past (e.g. [Kuhn,
The computation of the first complete approxima- 1950; Sakaguchi and Sakai, 1992 ]). While these are of the-
tions of game-theoretic optimal strategies for full- oretical interest, the same methods are not feasible for real
scale poker is addressed. Several abstraction tech- games, which are too large by many orders of magnitude
niques are combined to represent the game of 2-
player Texas Holdem, having size 
, using
([Koller and Pfeffer, 1997 ]).
[Shi and Littman, 2001 ] investigated abstraction tech-
closely related models each having size . niques to reduce the large search space and complexity of the
Despite the reduction in size by a factor of 100 problem, using a simplified variant of poker. [Takusagawa,
billion, the resulting models retain the key prop- 2000] created near-optimal strategies for the play of three
erties and structure of the real game. Linear pro- specific Holdem flops and betting sequences. [Selby, 1999]
gramming solutions to the abstracted game are used computed an optimal solution for the abbreviated game of
to create substantially improved poker-playing pro- preflop Holdem.
grams, able to defeat strong human players and be Using new abstraction techniques, we have produced vi-
competitive against world-class opponents. able pseudo-optimal strategies for the game of 2-player
Texas Holdem. The resulting poker-playing programs have
1 Introduction demonstrated a tremendous improvement in performance.
Mathematical game theory was introduced by John von Neu- Whereas the previous best poker programs were easily beaten
mann in the 1940s, and has since become one of the founda- by any competent human player, the new programs are capa-
tions of modern economics [von Neumann and Morgenstern, ble of defeating very strong players, and can hold their own
1944]. Von Neumann used the game of poker as a basic against world-class opposition.
model for 2-player zero-sum adversarial games, and proved Although some domain-specific knowledge is an asset in
the first fundamental result, the famous minimax theorem. A creating accurate reduced-scale models, analogous methods
few years later, John Nash added results for -player non- can be developed for many other imperfect information do-
mains and generalized game trees. We describe a general
cooperative games, for which he later won the Nobel Prize
[Nash, 1950 ]. Many decision problems can be modeled using method of problem reformulation that permits the indepen-
game theory, and it has been employed in a wide variety of dent solution of sub-trees by estimating the conditional prob-
domains in recent years. abilities needed as input for each computation.
Of particular interest is the existence of optimal solutions, This paper makes the following contributions:
or Nash equilibria. An optimal solution provides a random- 1. Abstraction techniques that can reduce an  
ized mixed strategy, basically a recipe of how to play in each poker search space to a manageable  
, without
possible situation. Using this strategy ensures that an agent losing the most important properties of the game.
will obtain at least the game-theoretic value of the game, re-
gardless of the opponents strategy. Unfortunately, finding 2. A poker-playing program that is a major improvement
exact optimal solutions is limited to relatively small problem over previous efforts, and is capable of competing with
sizes, and is not practical for most real domains. world-class opposition.
This paper explores the use of highly abstracted mathemat-
ical models which capture the most essential properties of the
2 Game Theory
real domain, such that an exact solution to the smaller prob- Game theory encompasses all forms of competition between
lem provides a useful approximation of an optimal strategy two or more agents. Unlike chess or checkers, poker is a
for the real domain. The application domain used is the game game of imperfect information and chance outcomes. It can
of poker, specifically Texas Holdem, the most popular form be represented with an imperfect information game tree hav-
of casino poker and the poker variant used to determine the ing chance nodes and decision nodes, which are grouped into
world champion at the annual World Series of Poker. information sets.
Since the nodes in this tree are not independent, divide-
and-conquer methods for computing sub-trees (such as the
alpha-beta algorithm) are not applicable. For a more detailed
description of imperfect information game tree structure, see
[Koller and Megiddo, 1992 ].
A strategy is a set of rules for choosing an action at ev-
ery decision node of the tree. In general, this will be a ran-
domized mixed strategy, which is a probability distribution
over the various alternatives. A player must use the same pol-
icy across all nodes in the same information set, since from
that players perspective they are indistinguishable from each
other (differing only in the hidden information component).
The conventional method for solving such a problem is to
convert the descriptive representation, or extensive form, into
a system of linear equations, which is then solved by a lin-
ear programming (LP) system such as the Simplex algorithm.
The optimal solutions are computed simultaneously for all
players, ensuring the best worst-case outcome for each player. Figure 1: Branching factors for Holdem and abstractions.
Traditionally, the conversion to normal form was accom-
panied by an exponential blow-up in the size of the prob-
lem, meaning that only very small problem instances could 4 Abstractions
be solved in practice. [Koller et al., 1994] described an alter- Texas Holdem has an easily identifiable structure, alternat-
nate LP representation, called sequence form, which exploits ing between chance nodes and betting rounds in four distinct
the common property of perfect recall (wherein all players stages. A high-level view of the imperfect information game
know the preceding history of the game), to obtain a system tree is shown in Figure 1.
of equations and unknowns that is only linear in the size of Holdem can be reformulated to produce similar but much
the game tree. This exponential reduction in representation smaller games. The objective is to reduce the scale of the
has re-opened the possibility of using game-theoretic analy- problem without severely altering the fundamental structure
sis for many domains. However, since the game tree itself of the game, or the resulting optimal strategies. There are
can be very large, the LP solution method is still limited to many ways of doing this, varying in the overall reduction and
moderate problem sizes (normally less than a billion nodes). in the accuracy of the resulting approximation.
Some of the most accurate abstractions include suit equiv-
alence isomorphisms (offering a reduction of at most a factor
3 Texas Holdem of  ), rank equivalence (only under certain conditions),
A game (or hand) of Texas Holdem consists of four stages, and rank near-equivalence. The optimal solutions to these ab-
each followed by a round of betting: stracted problems will either be exactly the same or will have
Preflop: Each player is dealt two private cards face down a small bounded error, which we refer to as near-optimal so-
(the hole cards). lutions. Unfortunately, the abstractions which produce an ex-
Flop: Three community cards (shared by all players) are act or near-exact reformulation do not produce the very large
dealt face up. reductions required to make full-scale poker tractable.
Turn: A single community card is dealt face up. A common method for controlling the game size is deck
River: A final community card is dealt face up. reduction. Using less than the standard 52-card deck greatly
reduces the branching factor at chance nodes. Other methods
After the betting, all active players reveal their hole cards
include reducing the number of cards in a players hand (e.g.
for the showdown. The player with the best five-card poker
from a 2-card hand to a 1-card hand), and reducing the num-
hand formed from their two private cards and the five public
ber of board cards (e.g. a 1-card flop), as was done by [Shi
cards wins all the money wagered (ties are possible).
and Littman, 2001] for the game of Rhode Island Holdem.
The game starts off with two forced bets (the blinds) put [Koller and Pfeffer, 1997 ] used such parameters to generate a
into the pot. When it is a players turn to act, they must ei- wide variety of tractable games to solve with their Gala sys-
ther bet/raise (increase their investment in the pot), check/call
(match what the opponent has bet or raised), or fold (quit and
We have used a number of small and intermediate sized
surrender all money contributed to the pot).
games, ranging from eight cards (two suits, four ranks) to 24
The best-known non-commercial Texas Holdem program cards (three suits, eight ranks) for the purpose of studying
is Poki. It has been playing online since 1997 and has earned abstraction methods, comparing the results with known exact
an impressive winning record, albeit against generally weak or near-optimal solutions. However, these smaller games are
opposition [Billings et al., 2002 ]. The systems abilities not suitable for use as an approximation for Texas Holdem,
are based on enumeration and simulation techniques, expert as the underlying structures of the games are different. To
knowledge, and opponent modeling. The programs weak- produce good playing strategies for full-scale poker, we look
nesses are easily exploited by strong players, especially in for abstractions of the real game which do not alter that basic
the 2-player game.
structure. The effect of truncation can be lessened through the use
The abstraction techniques used in practice are powerful of expected value leaf nodes. Instead of ending the game
in terms of reducing the problem size, and subsume those abruptly and awarding the pot to the strongest hand at that
previously mentioned. However, since they are also much moment, we compute an average conclusion over all possible
cruder, we call their solutions pseudo-optimal, to emphasize chance outcomes. For a 3-round model ending on the turn,
that there is no guarantee that the resulting approximations we roll-out all 44 possible river cards, assuming no further
will be accurate, or even reasonable. Some will be low-risk betting (or alternately, assuming one bet per player for the
propositions, while others will require empirical testing to de- last round). Each player is awarded a fraction of the pot, cor-
termine if they have merit. responding to their probability of winning the hand. In a 2-
round preflop model, we roll-out all 990 2-card combinations
4.1 Betting round reduction of the turn and river.
The standard rules of limit Holdem allow for a maximum of The most extreme form of truncation results in a 1-round
four bets per player per round. 1 Thus in 2-player limit poker model, with no foresight of future betting rounds. Since each
there are 19 possible betting sequences, of which two do not future round provides a refinement to the approximation, this
occur in practice.2 Of the remaining 17 sequences, 8 end in a will not reflect a correct strategy for the real game. In par-
fold (leading to a terminal node in the game tree), and 9 end ticular, betting plans that extend over more than one round,

  
      
in a call (carrying forward to the next chance node). Using
 ,  ,  ,  , 
and capital letters for the second player, the tree of possible
such as deferring the raise of a very strong hand, are lost
entirely. Nevertheless, even these simplistic models can be
useful when combined with expected value leaf nodes.
betting sequences for each round is: Alex Selby computed an optimal solution for the game of
kK kBf kBc kBrF kBrC kBrRf kBrRc kBrRrF kBrRrC preflop Holdem, which consists of only the first betting round
bF bC bRf bRc bRrF bRrC bRrRf bRrRc followed by an EV roll-out of the five board cards to deter-
We call this local collection of decision nodes a betting mine the winner [Selby, 1999 ]. Although there are some se-
tree, and represent it diagramatically with a triangle. rious limitations in the strategy based on this 1-round model,
With betting round reduction, each player is allowed a we have incorporated the Selby preflop system into one of our
maximum of three bets per round, thereby eliminating the last programs, PsOpti1, as described later in this section.
two sequences in each line. The effective branching factor of In contrast to truncating rounds, we can bypass certain
the betting tree is reduced from nine to seven. This does not early stages of the game. We frequently use postflop mod-
appear to have a substantial effect on play, or on the expected els, which ignore the preflop betting round, and use a single
value (EV) for each player. This observation has been verified fixed flop of three cards (see Figure 1).
experimentally. In contrast, we computed the corresponding It is natural to consider the idea of independent betting
postflop models with a maximum of two bets per player per rounds, where each phase of the game is treated in isolation.
round, and found radical changes to the optimal strategies, Unfortunately, the betting history from previous rounds will
strongly suggesting that that level of abstraction is not safe. almost always contain contextual information that is critical
to making appropriate decisions. The probability distribution
4.2 Elimination of betting rounds over the hands for each player is strongly dependent on the
Large reductions in the size of a poker game tree can be ob- path that led to that decision point, so it cannot be ignored
tained by elimination of betting rounds. There are several without risking a considerable loss of information. However,
ways to do this, and they generally have a significant impact the naive independence assumption can be viable in certain
on the nature of the game. First, the game may be truncated, circumstances, and we do implicitly use it in the design of
by eliminating the last round or rounds. In Holdem, ignor- PsOpti1 to bridge the gap between the 1-round preflop model
ing the last board card and the final betting round produces a and the 3-round postflop model.
3-round model of the actual 4-round game. The solution to Another possible abstraction we explored was merging two
the 3-round model loses some of the subtlety involved in the or more rounds into a single round, such as creating a com-
true optimal strategy, but the degradation applies primarily to bined 2-card turn/river. However, it is not clear what the ap-
advanced tactics on the turn. There is a smaller effect on the propriate bet size should be for this composite round. In any
flop strategy, and the strategy for the first betting round may case, the solutions for these models (over a full range of possi-
have no significant changes, since it incorporates all the out- ble bet sizes), all turned out to be substantially different from
comes of two future betting rounds. We use this particular their 3-round counterparts, and the method was therefore re-
abstraction to define an appropriate strategy for play in the jected.
first round, and thus call it a preflop model (see Figure 2).
Some rules allow unlimited raises when only two players are
involved. However, occasions with more than three legitimate raises
4.3 Composition of preflop and postflop models
are relatively rare, and do not greatly alter an optimal strategy.
Technically, a player may fold even though there is no outstand- Although the nodes of an imperfect information game tree are
ing bet. This is logically dominated by not folding, and therefore not independent in general, some decomposition is possible.
does not occur in an optimal strategy, and is almost never seen in For example, the sub-trees resulting from different preflop
practice. betting sequences can no longer have nodes that belong to the
same information set. 3 The sub-trees for our postflop models
can be computed in isolation, provided that the appropriate
preconditions are given as input. Unfortunately, knowing the
correct conditional probabilities would normally entail solv-
ing the whole game, so there would be no advantage to the
For simple postflop models, we dispense with the prior
probabilities. For the postflop models used in PsOpti0 and
PsOpti1, we simply ignore the implications of the preflop
betting actions, and assume a uniform distribution over all
possible hands for each player. Different postflop solutions
were computed for initial pot sizes of two, four, six, and eight
bets (corresponding to preflop sequences with zero, one, two,
or three raises, but ignoring which player initially made each
raise). In PsOpti1, the four postflop solutions are simply ap-
pended to the Selby preflop strategy (Figure 2). Although
these simplifying assumptions are technically wrong, the re-
sulting play is still surprisingly effective. Figure 2: Composition of PsOpti1 and PsOpti2.
A better way to compose postflop models is to estimate
the conditional probabilities, using the solution to a preflop
model. With a tractable preflop model, we have a means of 17 equivalence classes for each player would be sufficient for
estimating an appropriate strategy at the root, and thereby de- each betting round. However, since a mixed strategy may be
termine the consequent probability distributions. indicated for optimal play in some cases, we would like to
In PsOpti2, a 3-round preflop model was designed and group hands that have a similar probability distribution over
solved. The resulting pseudo-optimal strategy for the pre- action plans.
flop (which was significantly different from the Selby strat- One obvious but rather crude bucketing function is to group
egy) was used to determine the corresponding distribution of all hands according to strength (ie. its rank with respect to all
hands for each player in each context. This provided the nec- possible hands, or the probability of currently being in the
essary input parameters for each of the seven preflop betting lead). This can be improved by considering the roll-out of all
sequences that carry over to the flop stage. Since each of future cards, giving an (unweighted) estimate of the chance
these postflop models has been given (an approximation of) of winning the hand.
the perfect recall knowledge of the full game, they are fully However, this is only a one-dimensional view of hand
compatible with each other, and are properly integrated un- 
types, in what can be considered to be an -dimensional
der the umbrella of the preflop model (Figure 2). In theory, space of strategies, with a vast number of different ways
this should be equivalent to computing the much larger tree, to classify them. A superior practical method would be to
but it is limited by the accuracy and appropriateness of the project the set of all hands onto a two-dimensional space con-
proposed preflop betting model. sisting of (roll-out) hand strength and hand potential (sim-
ilar to the hand assessment used in Poki, [Billings et al.,
4.4 Abstraction by bucketing 2002]). Clusters in the resulting scattergram suggest reason-
The most important method of abstraction for the computa- able groups of hands to be treated similarly.
tion of our pseudo-optimal strategies is called bucketing. This We eventually settled on a simple compromise. With
is an extension of the natural and intuitive concept that has available buckets, we allocate   to roll-out hand strength.
been applied many times in previous research (e.g. [Sklansky The number of hand types in each class is not uniform; the
and Malmuth, 1994 ] [Takusagawa, 2000 ] [Shi and Littman, classes for the strongest hands are smaller than those for
2001]). The set of all possible hands is partitioned into equiv- mediocre and weak hands, allowing for better discrimination
alence classes (also called buckets or bins). A many-to-one of the smaller fractions of hands that should be raised or re-
mapping function determines which hands will be grouped raised.
together. Ideally, the hands should be grouped according to One special bucket is designated for hands that are low in
strategic similarity, meaning that they can all be played in a strength but have high potential, such as good draws to a flush
similar manner without much loss in EV. or straight. This plays an important role in identifying good
If every hand was played with a particular pure strategy hands to use for bluffing (known as semi-bluffs in [Sklansky
(ie. only one of the available choices), then a perfect mapping and Malmuth, 1994 ]). Comparing postflop solutions that use
function would group all hands that follow the same plan, and six strength buckets to solutions with five strength plus one
3 high-potential bucket, we see that most bluffs in the latter are
To see this, each decision node of the tree can be labeled with
all the cards known to that player, and the full path that led to that taken from the special bucket, which is sometimes played in
node. Nodes with identical labels differ only in the hidden informa- the same way as the strongest bucket. This confirmed our
tion, and are therefore in the same information set. Since the betting expectations that the high-potential bucket would improve the
history is different for these sub-trees, none of the nodes are inter- selection of hands for various betting tactics, and increase the
dependent. overall EV.
5 Experiments
5.1 Testing against computer players
A series of matches between computer programs was con-
ducted, with the results shown in Table 1. Win rates are mea-
sured in small bets per hand (sb/h). Each match was run for at
least 20,000 games (and over 100,000 games in some cases).
The variance per game depends greatly on the styles of the
two players involved, but is typically +/- 6 sb. The standard
deviation for each match outcome is not shown, but is nor-
Figure 3: Transition probabilities (six buckets per player). mally less than +/- 0.03 sb/h.
The bot players were:
PsOpti2, composed of a hand-crafted 3-round preflop
The number of buckets that can be used in conjunction with model, providing conditional probability distributions to each
a 3-round model is very small, typically six or seven for each of seven 3-round postflop models (Figure 2). All models in
player (ie. 36 or 49 pairs of bucket assignments). Obviously this prototype used six buckets per player per round.
this results in a very coarse-grained abstract game, but it may PsOpti1, composed of four 3-round postflop models un-
not be substantially different from the number of distinctions der the naive uniform distribution assumption, with 7 buck-
an average human player might make. Regardless, it is the ets per player per round. Selbys optimal solution for preflop
best we can currently do given the computational constraints Holdem is used to play the preflop ( [Selby, 1999]).
of this approach. PsOpti0, composed of a single 3-round postflop model,
wrongly assuming uniform distributions and an initial pot size
The final thing needed to sever the abstract game from the of two bets, with seven buckets per player per round. This
underlying real game tree are the transition probabilities. The program used an always-call policy for the preflop betting
chance node between the flop and turn represents a particular round.
card being dealt from the remaining stock of 45 cards. In the Poki, the University of Alberta poker program. This older
abstract game, there are no cards, only buckets. The effect of version of Poki was not designed to play the 2-player game,
the turn card in the abstract game is to dictate the probability and can be defeated rather easily, but is a useful benchmark.
of moving from one pair of buckets on the flop to any pair of Anti-Poki, a rule-based program designed to beat Poki by
buckets on the turn. Thus the collection of chance nodes in
the game tree is represented by an   to   tran-
exploiting its weaknesses and vulnerabilities in the 2-player
game. Any specific counter-strategy can be even more vul-
sition network as shown in Figure 3. For postflop models, nerable to adaptive players.
this can be estimated by walking the entire tree, enumerating
Aadapti, a relatively simple adaptive player, capable of
all transitions for a small number of characteristic flops. For

   
slowly learning and exploiting persistent patterns in play.
preflop models,  the full enumeration is more expensive (en-
compassing all  
 
  possible flops), so it Always Call, a very weak benchmark strategy.
Always Raise, a very weak benchmark strategy.
is estimated either by sampling, or by (parallel) enumeration
of a truncated tree. It is important to understand that a game-theoretic optimal
player is, in principle, not designed to win. Its purpose is to
For a 3-round postflop model, we can comfortably solve not lose. An implicit assumption is that the opponent is also
abstract games with up to seven buckets for each player in playing optimally, and nothing can be gained by observing
each round. Changing the distribution of buckets, such as six the opponent for patterns or weaknesses.
for the flop, seven for the turn, and eight for the river, does In a simple game like RoShamBo (also known as Rock-
not appear to significantly affect the quality of the solutions, Paper-Scissors), playing the optimal strategy ensures a break-
better or worse. even result, regardless of what the opponent does, and is
The final linear programming solution produces a large ta- therefore insufficient to defeat weak opponents, or to win a
ble of mixed strategies (probabilities for fold, call, or raise) tournament ( [Billings, 2000]). Poker is more complex, and
for every reachable scenario in the abstract game. To use this, in theory an optimal player can win, but only if the oppo-
the poker-playing program looks for the corresponding situa- nent makes dominated errors. Any time a player makes any
tion based on the same hand strength and potential measures, choice that is part of a randomized mixed strategy of some
and randomly selects an action from the mixed strategy. game-theoretic optimal policy, that decision is not dominated.
In other words, it is possible to play in a highly sub-optimal
The large LP computations typically take less than a day manner, but still break even against an optimal player, be-
(using CPLEX with the barrier method), and use up to two cause those choices are not strictly dominated.
Gigabytes of RAM. Larger problems will exceed available Since the pseudo-optimal strategies do no opponent model-
memory, which is common for large LP systems. Certain ing, there is no guarantee that they will be especially effective
LP techniques such as constraint generation could potentially against very bad or highly predictable players. They must rely
extend the range of solvable instances considerably, but this only on these fundamental strategic errors, and the margin of
would probably only allow the use of one or two additional victory might be relatively modest as a result.
buckets per player.
No. Program 1 2 3 4 5 6 7 8
1 PsOpti1 X +0.090 +0.091 +0.251 +0.156 +0.047 +0.546 +0.635
2 PsOpti2 -0.090 X +0.069 +0.118 +0.054 +0.045 +0.505 +0.319
3 PsOpti0 -0.091 -0.069 X +0.163 +0.135 +0.001 +0.418 +0.118
4 Aadapti -0.251 -0.118 -0.163 X +0.178 +0.550 +0.905 +2.615
5 Anti-Poki -0.156 -0.054 -0.135 -0.178 X +0.385 +0.143 +0.541
6 Poki -0.047 -0.045 -0.001 -0.550 -0.385 X +0.537 +2.285
7 Always Call -0.546 -0.505 -0.418 -0.905 -0.143 -0.537 X =0.000
8 Always Raise -0.635 -0.319 -0.118 -2.615 -0.541 -2.285 =0.000 X

Table 1: Computer vs computer matches (sb/h).

The critical question is whether such errors are common in Player Hands Posn 1 Posn 2 sb/h
practice. There is no definitive answer to this question yet, Master-1 early 1147 -0.324 +0.360 +0.017
but preliminary evidence suggests that dominated errors oc- Master-1 late 2880 -0.054 +0.396 +0.170
cur often enough to gain a measurable EV advantage over Experienced-1 803 +0.175 +0.002 +0.088
weaker players, but may not be very common in the play of Experienced-2 1001 -0.166 -0.168 -0.167
very good players. Experienced-3 1378 +0.119 -0.016 +0.052
The first tests of the viability of pseudo-optimal solutions Experienced-4 1086 +0.042 -0.039 +0.002
Intermediate-1 2448 +0.031 +0.203 +0.117
were done with PsOpti0 playing postflop Holdem, where
Novice-1 1277 -0.159 -0.154 -0.156
both players agree to simply call in the preflop (thereby
matching the exact pre-conditions for the postflop solution). All Opponents 15125 -0.015
In those preliminary tests, a poker master (the first author) Table 2: Human vs PsOpti2 matches.
played more than 2000 hands, and was unable to defeat the
pseudo-optimal strategy. In contrast, Poki had been beaten Player Hands Posn 1 Posn 2 sb/h
consistently at a rate of over 0.8 sb/h (which is more than
thecount 7030 -0.006 +0.103 +0.048
would be lost by simply folding every hand). Master-1 2872 +0.141 +0.314 +0.228
Using the same no-bet preflop policy, PsOpti0 was able to Master-2 569 -0.007 +0.035 +0.014
defeat Poki at a rate of +0.144 sb/h (compared to +0.001 sb/h Master-3 425 +0.047 +0.373 +0.209
for the full game including preflop), and defeated Aadapti at Experienced-1 4078 -0.058 +0.164 +0.053
+0.410 sb/h (compared to +0.163 sb/h for the full game). Experienced-2 511 +0.152 +0.369 +0.260
All of the pseudo-optimal players play substantially bet- Experienced-3 2303 -0.252 +0.128 -0.062
ter than any previously existing computer programs. Even Experienced-5 610 -0.250 -0.229 -0.239
PsOpti0, which is not designed to play the full game, earns Intermediate-1 16288 -0.145 +0.048 -0.049
enough from the postflop betting rounds to offset the EV Intermediate-2 478 -0.182 +0.402 +0.110
losses from the preflop round (where it never raises good Novice-1 5045 -0.222 -0.010 -0.116
hands, nor folds bad ones). Novice-2 485 -0.255 -0.139 -0.197
It is suspicious that PsOpti1 outperformed PsOpti2, which Novice-3 1017 -0.369 -0.051 -0.210
in principle should be a better approximation. Subsequent Novice-4 583 -0.053 -0.384 -0.219
analysis of the play of PsOpti2 revealed some programming Novice-5 425 -0.571 -0.296 -0.433
errors, and also suggested that the bucket assignments for the All Opponents 46479 -0.057
preflop model were flawed. This may have resulted in an in-
Table 3: Human vs PsOpti1 matches.
accurate pseudo-optimal preflop strategy, and consequent im-
balances in the prior distributions used as input for the post-
flop models. We expect that this will be rectified in future rized in Table 2 and Table 3. (Master-1 is the first author,
versions, and that the PsOpti2 design will surpass PsOpti1 in Experienced-1 is the third author).
performance. In most cases, the relatively short length of the match
leaves a high degree of uncertainty in the outcome, limit-
5.2 Testing against human players ing how much can be safely concluded. Nevertheless, some
While these results are encouraging, none of the non-pseudo- players did appear to have a definite edge, while others were
optimal computer opponents are better than intermediate clearly losing.
strength at 2-player Texas Holdem. Therefore, matches were A number of interesting observations were made over the
conducted against human opponents. course of these games. It was obvious that most people had a
More than 100 participants volunteered to play against the lot of difficulty learning and adjusting to the computers style
pseudo-optimal players on our public web applet (www.cs. of play. In poker, knowing the basic approach of the oppo-, including many expe- nent is essential, since it will dictate how to properly handle
rienced players, a few masters, and one world-class player. many situations that arise. Some players wrongly attributed
The programs provided some fun opposition, and ended up intelligence where none was present. After losing a 1000-
with a winning record overall. The results are summa- game match, one experienced player commented the bot has
PsOpti1, playing more than 7000 hands over the course of
thecount (+0.046) several days. The graph in Figure 4 shows the progression of
the match.
The pseudo-optimal player started with some good fortune,
but lost at a rate of about -0.2 sb/h over the next 2000 hands.
Then there was a sudden reversal, following a series of for-
200 tuitous outcomes for the program. Although thecount is
Small Bets Won

renown for his mental toughness, an uncommon run of bad

luck can be very frustrating even for the most experienced
players. Mr. Rao believes he played below his best level dur-
ing that stage, which contributed to a dramatic drop where he
lost 300 sb in less than 400 hands. Mr. Rao resumed play the
following day, but was unable to recover the losses, slipping
-200 further to -200 sb after 3700 hands. At this point he stopped
play and did a careful reassessment.
0 1000 2000 3000 4000 5000 6000 7000 It was clear that his normal style for maximizing income
Hands Played against typical human opponents was not effective against the
Figure 4: Progress of the thecount vs PsOpti1 pseudo-optimal player. Whereas human players would nor-
mally succumb to a lot of pressure from aggressive betting,
the bot was willing to call all the way to the showdown with
me figured out now, indicating that its opponent model was as little as a Jack or Queen high card. That kind of play would
accurate, when in fact the pseudo-optimal player is oblivious be folly against most opponents, but is appropriate against an
and does no modeling at all. extremely aggressive opponent. Most human players fail to
It was also evident that these programs do considerably make the necessary adjustment under these atypical condi-
better in practice than might be expected, due to the emo- tions, but the program has no sense of fear.
tional frailty of their human opponents. Many players com- Mr. Rao changed his approach to be less aggressive, with
mented that playing against the pseudo-optimal opponent was immediate rewards, as shown by the +600 sb increase over
an exasperating experience. The bot routinely makes uncon- the next 1100 hands (some of which he credited to a good run
ventional plays that confuse and confound humans. Invari- of cards). Mr. Rao was able to utilize his knowledge that the
ably, some of these bizarre plays happen to coincide with a computer player did not do any opponent modeling. Knowing
lucky escape, and several of these bad beats in quick succes- this allows a human player to systematically probe for weak-
sion will often cause strong emotional reactions (sometimes nesses, without any fear of being punished for playing in a
referred to as going on tilt). The level of play generally methodical and highly predictable manner, since an oblivious
goes down sharply in these circumstances. opponent does not exploit those patterns and biases.
This suggests that a perfect game-theoretic optimal poker Although he enjoyed much more success in the match from
player could perhaps beat even the best humans in the long that point forward, there were still some adventures, such as
run, because any EV lost in moments of weakness would the sharp decline at 5400 hands. Poker is a game of very high
never be regained. However, the win rate for such a program variance, especially between two opponents with sharp styles,
could still be quite small, giving it only a slight advantage. as can be seen by the dramatic swings over the course of this
Thus it would be unable to exert its superiority convincingly match. Although 7000 games may seem like a lot, Mr. Raos
over the short term, such as the few hundred hands of one victory in this match was still not statistically conclusive.
session, or over the course of a world championship tourna- We now believe that a human poker master can eventu-
ment. Since even the best human players are known to have ally gain a sizable advantage over these pseudo-optimal pro-
biases and weaknesses, opponent modeling will almost cer- totypes (perhaps +0.20 sb/h or more is sustainable). However,
tainly be necessary to produce a program that surpasses all it requires a good understanding of the design of the program
human players. and its resulting weaknesses. That knowledge is difficult to
learn during normal play, due to the good information hiding
5.3 Testing against a world-class player provided by an appropriate mixture of plans and tactics. This
The elite poker expert was Gautam Rao, who is known as cloud of confusion is a natural barrier to opponent learning.
thecount or CountDracula in the world of popular online It would be even more difficult to learn against an adaptive
poker rooms. Mr. Rao is the #1 all-time winner in the history program with good opponent modeling, since any methodical
of the oldest online game, by an enormous margin over all testing by the human would be easily exploited. This is in
other players, both in total earnings and in dollar-per-hand stark contrast to typical human opponents, who can often be
rate. His particular specialty is in short-handed games with accurately modeled after only a small number of hands.
five or fewer players. He is recognized as one of the best
players in the world in these games, and is also exceptional
6 Conclusions and Future Work
at 2-player Holdem. Like many top-flight players, he has a The pseudo-optimal players presented in this paper are the
dynamic ultra-aggressive style. first complete approximations of a game-theoretic optimal
Mr. Rao agreed to play an exhibition match against strategy for a full-scale variation of real poker.
Several abstraction techniques were explored, resulting in [Billings, 2000] D. Billings. The first international roshambo
the reasonably accurate representation of a large imperfect programming competition. International Computer
information game tree having    
nodes with a small Games Association Journal, 23(1):38, 4250, 2000.
collection of models of size   
. Despite these massive [Koller and Megiddo, 1992 ] D. Koller and N. Megiddo. The
reductions and simplifications, the resulting programs play complexity of two-person zero-sum games in extensive
respectably. For the first time ever, computer programs are form. Games and Economic Beh., 4(4):528552, 1992.
not completely outclassed by strong human opposition in the
game of 2-player Texas Holdem. [Koller and Pfeffer, 1997 ] D. Koller and A. Pfeffer. Repre-
Useful abstractions included betting tree reductions, trun- sentations and solutions for game-theoretic problems. Ar-
cation of betting rounds combined with EV leaf nodes, and tificial Intelligence, pages 167215, 1997.
bypassing betting rounds. A 3-round model anchored at [Koller et al., 1994] D. Koller, N. Megiddo, and B. von Sten-
the root provided a pseudo-optimal strategy for the preflop gel. Fast algorithms for finding randomized strategies in
round, which in turn provided the proper contextual informa- game trees. STOC, pages 750759, 1994.
tion needed to determine conditional probabilities for post- [Kuhn, 1950] H. W. Kuhn. A simplified two-person poker.
flop models. The most powerful abstractions for reducing the Contributions to the Theory of Games, 1:97103, 1950.
problem size were based on bucketing, a method for parti-
tioning all possible holdings according to strategic similarity. [Nash, 1950 ] J. Nash. Equilibrium points in n-person games.
Although these methods exploit the particular structure of the National Academy of Sciences, 36:4849, 1950.
Texas Holdem game tree, the principles are general enough [Sakaguchi and Sakai, 1992 ] M. Sakaguchi and S. Sakai.
to be applied to a wide variety of imperfect information do- Solutions of some three-person stud and draw poker.
mains. Mathematics Japonica, pages 11471160, 1992.
Many refinements and improvements will be made to the
[Selby, 1999 ] A. Selby. Optimal heads-up preflop poker.
basic techniques in the coming months. Further testing will
also continue, since accurate assessment in a high variance 1999.
domain is always difficult. [Shi and Littman, 2001 ] J. Shi and M. Littman. Abstrac-
The next stage of the research will be to apply these tech- tion models for game theoretic poker. In Computers and
niques to obtain approximations of Nash equilibria for -  Games, pages 333345. Springer-Verlag, 2001.
player Texas Holdem. This promises to be a challenging ex- [Sklansky and Malmuth, 1994 ] D. Sklansky and M. Mal-
tension, since multi-player games have many properties that muth. Texas Holdem for the Advanced Player. Two Plus
do not exist in the 2-player game. Two Publishing, 2nd edition, 1994.
Finally, having reasonable approximations of optimal
strategies does not lessen the importance of good oppo- [Takusagawa, 2000] K. Takusagawa. Nash equilibrium of
nent modeling. Learning against an adaptive adversary in a Texas Holdem poker, 2000. Undergraduate thesis, Com-
stochastic game is a challenging problem, and there will be puter Science, Stanford University.
many ideas to explore in combining the two different forms [von Neumann and Morgenstern, 1944 ] J. von Neumann and
of information. That will likely be the key difference between O. Morgenstern. Theory of Games and Economic Behav-
a program that can compete with the best, and a program that ior. Princeton University Press, 1944.
surpasses all human players.
Quoting thecount:
You have a very strong program. Once you add
opponent modeling to it, it will kill everyone.
The authors would like to thank Gautam Rao, Sridhar
Mutyala, and the other poker players for donating their valu-
able time. We also wish to thank Daphne Koller, Michael
Littman, Matthew Ginsberg, Rich Sutton, David McAllester,
Mason Malmuth, and David Sklansky for their valuable in-
sights in past discussions.
This research was supported in part by grants from the Nat-
ural Sciences and Engineering Research Council of Canada
(NSERC), the Alberta Informatics Circle of Research Excel-
lence (iCORE), and an Izaak Walton Killam Memorial post-
graduate scholarship.

[Billings et al., 2002 ] D. Billings, A. Davidson, J. Schaeffer,
and D. Szafron. The challenge of poker. Artificial Intelli-
gence, 134(12):201240, 2002.