There are 4 main categories of games depending on whether the game ...

#### Heads-up no-limit Texas hold’em
Common variant of poker playe...

In 1944 John von Neumann and Oskar Morgenstern co-authored a book e...

This is a massive number. Chess has approximately $10^{47}$ decisio...

**Counterfactual Regret Minimization (CFR)** is an algorithm by whi...

**mbb/g** - milli-big blinds per game. In poker, blinds are forced ...

In heads-up poken one player will post the big blind and their oppo...

### Nash Equilibrium
Let’s say you were playing rock, paper scisso...

This figure provides a great high level picture of DeepStack. As th...

**Huber Loss** is a loss function that is less sensitive to outlier...

DeepStack: Expert-Level Artiﬁcial Intelligence in

Heads-Up No-Limit Poker

Matej Morav

ˇ

c

´

ık

♠,♥,†

, Martin Schmid

♠,♥,†

, Neil Burch

♠

, Viliam Lis

´

y

♠,♣

,

Dustin Morrill

♠

, Nolan Bard

♠

, Trevor Davis

♠

,

Kevin Waugh

♠

, Michael Johanson

♠

, Michael Bowling

♠,∗

♠

Department of Computing Science, University of Alberta,

Edmonton, Alberta, T6G2E8, Canada

♥

Department of Applied Mathematics, Charles University,

Prague, Czech Republic

♣

Department of Computer Science, FEE, Czech Technical University,

Prague, Czech Republic

†

These authors contributed equally to this work and are listed in alphabetical order.

∗

To whom correspondence should be addressed; E-mail: bowling@cs.ualberta.ca

Artiﬁcial intelligence has seen several breakthroughs in recent years, with

games often serving as milestones. A common feature of these games is that

players have perfect information. Poker is the quintessential game of imperfect

information, and a longstanding challenge problem in artiﬁcial intelligence.

We introduce DeepStack, an algorithm for imperfect information settings. It

combines recursive reasoning to handle information asymmetry, decomposi-

tion to focus computation on the relevant decision, and a form of intuition that

is automatically learned from self-play using deep learning. In a study involv-

ing 44,000 hands of poker, DeepStack defeated with statistical signiﬁcance pro-

fessional poker players in heads-up no-limit Texas hold’em. The approach is

theoretically sound and is shown to produce more difﬁcult to exploit strategies

than prior approaches.

1

Games have long served as benchmarks and marked milestones of progress in artiﬁcial

intelligence (AI). In the last two decades, computer programs have reached a performance that

1

This is the author’s version of the work. It is posted here by permission of the AAAS for personal use, not for

redistribution. The deﬁnitive version was published in Science, (March 02, 2017), doi: 10.1126/science.aam6960.

1

exceeds expert human players in many games, e.g., backgammon (1), checkers (2), chess (3),

Jeopardy! (4), Atari video games (5), and go (6). These successes all involve games with

information symmetry, where all players have identical information about the current state of

the game. This property of perfect information is also at the heart of the algorithms that enabled

these successes, e.g., local search during play (7, 8).

The founder of modern game theory and computing pioneer, von Neumann, envisioned

reasoning in games without perfect information. “Real life is not like that. Real life consists of

blufﬁng, of little tactics of deception, of asking yourself what is the other man going to think

I mean to do. And that is what games are about in my theory.” (9) One game that fascinated

von Neumann was poker, where players are dealt private cards and take turns making bets or

blufﬁng on holding the strongest hand, calling opponents’ bets, or folding and giving up on the

hand and the bets already added to the pot. Poker is a game of imperfect information, where

players’ private cards give them asymmetric information about the state of game.

Heads-up no-limit Texas hold’em (HUNL) is a two-player version of poker in which two

cards are initially dealt face-down to each player, and additional cards are dealt face-up in

three subsequent rounds. No limit is placed on the size of the bets although there is an overall

limit to the total amount wagered in each game (10). AI techniques have previously shown

success in the simpler game of heads-up limit Texas hold’em, where all bets are of a ﬁxed size

resulting in just under 10

14

decision points (11). By comparison, computers have exceeded

expert human performance in go (6), a perfect information game with approximately 10

170

decision points (12). The imperfect information game HUNL is comparable in size to go, with

the number of decision points exceeding 10

160

(13).

Imperfect information games require more complex reasoning than similarly sized perfect

information games. The correct decision at a particular moment depends upon the probability

distribution over private information that the opponent holds, which is revealed through their

past actions. However, how our opponent’s actions reveal that information depends upon their

knowledge of our private information and how our actions reveal it. This kind of recursive

reasoning is why one cannot easily reason about game situations in isolation, which is at the

heart of heuristic search methods for perfect information games. Competitive AI approaches

in imperfect information games typically reason about the entire game and produce a complete

strategy prior to play (14–16). Counterfactual regret minimization (CFR) (14,17,18) is one such

technique that uses self-play to do recursive reasoning through adapting its strategy against itself

over successive iterations. If the game is too large to be solved directly, the common response

is to solve a smaller, abstracted game. To play the original game, one translates situations and

actions from the original game to the abstract game.

Although this approach makes it feasible for programs to reason in a game like HUNL, it

does so by squeezing HUNL’s 10

160

situations down to the order of 10

14

abstract situations.

Likely as a result of this loss of information, such programs are behind expert human play. In

2015, the computer program Claudico lost to a team of professional poker players by a margin

of 91 mbb/g (19), which is a “huge margin of victory” (20). Furthermore, it has been recently

shown that abstraction-based programs from the Annual Computer Poker Competition have

2

massive ﬂaws (21). Four such programs (including top programs from the 2016 competition)

were evaluated using a local best-response technique that produces an approximate lower-bound

on how much a strategy can lose. All four abstraction-based programs are beatable by over

3,000 mbb/g, which is four times as large as simply folding each game.

DeepStack takes a fundamentally different approach. It continues to use the recursive rea-

soning of CFR to handle information asymmetry. However, it does not compute and store a

complete strategy prior to play and so has no need for explicit abstraction. Instead it consid-

ers each particular situation as it arises during play, but not in isolation. It avoids reasoning

about the entire remainder of the game by substituting the computation beyond a certain depth

with a fast approximate estimate. This estimate can be thought of as DeepStack’s intuition:

a gut feeling of the value of holding any possible private cards in any possible poker situa-

tion. Finally, DeepStack’s intuition, much like human intuition, needs to be trained. We train it

with deep learning (22) using examples generated from random poker situations. We show that

DeepStack is theoretically sound, produces strategies substantially more difﬁcult to exploit than

abstraction-based techniques, and defeats professional poker players at HUNL with statistical

signiﬁcance.

DeepStack

DeepStack is a general-purpose algorithm for a large class of sequential imperfect information

games. For clarity, we will describe its operation in the game of HUNL. The state of a poker

game can be split into the players’ private information, hands of two cards dealt face down, and

the public state, consisting of the cards laying face up on the table and the sequence of betting

actions made by the players. Possible sequences of public states in the game form a public tree

with every public state having an associated public subtree (Fig. 1).

A player’s strategy deﬁnes a probability distribution over valid actions for each decision

point, where a decision point is the combination of the public state and the hand for the acting

player. Given a player’s strategy, for any public state one can compute the player’s range,

which is the probability distribution over the player’s possible hands given that the public state

is reached.

Fixing both players’ strategies, the utility for a particular player at a terminal public state,

where the game has ended, is a bilinear function of both players’ ranges using a payoff matrix

determined by the rules of the game. The expected utility for a player at any other public

state, including the initial state, is the expected utility over reachable terminal states given the

players’ ﬁxed strategies. A best-response strategy is one that maximizes a player’s expected

utility against an opponent strategy. In two-player zero-sum games, like HUNL, a solution

or Nash equilibrium strategy (23) maximizes the expected utility when playing against a best-

response opponent strategy. The exploitability of a strategy is the difference in expected utility

against its best-response opponent and the expected utility under a Nash equilibrium.

The DeepStack algorithm seeks to compute and play a low-exploitability strategy for the

3

PRE-FLOP

FLOP

Raise

Raise

Fold

Call

Call

Fold

Call

Fold

Call

Call

Fold

Bet

Check

Check

Raise

Check

TURN

-50

-100

100

100

-200

Fold

1

22

1

1

2

2

1

Figure 1: A portion of the public tree in HUNL. Nodes represent public states, whereas edges

represent actions: red and turquoise showing player betting actions, and green representing

public cards revealed by chance. The game ends at terminal nodes, shown as a chip with an

associated value. For terminal nodes where no player folded, the player whose private cards

form a stronger poker hand receives the value of the state.

game, i.e., solve for an approximate Nash equilibrium. DeepStack computes this strategy dur-

ing play only for the states of the public tree that actually arise. Although computed during

play, DeepStack’s strategy is static, albeit stochastic, because it is the result of a deterministic

computation that produces a probability distribution over the available actions.

The DeepStack algorithm (Fig. 2) is composed of three ingredients: a sound local strategy

computation for the current public state, depth-limited lookahead using a learned value func-

tion to avoid reasoning to the end of the game, and a restricted set of lookahead actions. At a

conceptual level these three ingredients describe heuristic search, which is responsible for many

of AI’s successes in perfect information games. Until DeepStack, no theoretically sound appli-

cation of heuristic search was known in imperfect information games. The heart of heuristic

search methods is the idea of “continual re-searching”, where a sound local search procedure is

invoked whenever the agent must act without retaining any memory of how or why it acted to

reach the current state. At the heart of DeepStack is continual re-solving, a sound local strategy

computation which only needs minimal memory of how and why it acted to reach the current

public state.

Continual re-solving. Suppose we have taken actions according to a particular solution strat-

egy but then in some public state forget this strategy. Can we reconstruct a solution strategy

for the subtree without having to solve the entire game again? We can, through the process of

4

re-solving (17). We need to know both our range at the public state and a vector of expected

values achieved by the opponent under the previous solution for each opponent hand (24). With

these values, we can reconstruct a strategy for only the remainder of the game, which does not

increase our overall exploitability. Each value in the opponent’s vector is a counterfactual value,

a conditional “what-if” value that gives the expected value if the opponent reaches the public

state with a particular hand. The CFR algorithm also uses counterfactual values, and if we use

CFR as our solver, it is easy to compute the vector of opponent counterfactual values at any

public state.

Re-solving, however, begins with a strategy, whereas our goal is to avoid ever maintaining

a strategy for the entire game. We get around this by doing continual re-solving: reconstructing

a strategy by re-solving every time we need to act; never using the strategy beyond our next

action. To be able to re-solve at any public state, we need only keep track of our own range

and a suitable vector of opponent counterfactual values. These values must be an upper bound

on the value the opponent can achieve with each hand in the current public state, while being

no larger than the value the opponent could achieve had they deviated from reaching the public

state. This is an important relaxation of the counterfactual values typically used in re-solving,

with a proof of sufﬁciency included in our proof of Theorem 1 below (10).

At the start of the game, our range is uniform and the opponent counterfactual values are

initialized to the value of being dealt each private hand. When it is our turn to act we re-solve the

subtree at the current public state using the stored range and opponent values, and act according

to the computed strategy, discarding the strategy before we act again. After each action, either

by a player or chance dealing cards, we update our range and opponent counterfactual values

according to the following rules: (i) Own action: replace the opponent counterfactual values

with those computed in the re-solved strategy for our chosen action. Update our own range using

the computed strategy and Bayes’ rule. (ii) Chance action: replace the opponent counterfactual

values with those computed for this chance action from the last re-solve. Update our own range

by zeroing hands in the range that are impossible given new public cards. (iii) Opponent action:

no change to our range or the opponent values are required.

These updates ensure the opponent counterfactual values satisfy our sufﬁcient conditions,

and the whole procedure produces arbitrarily close approximations of a Nash equilibrium (see

Theorem 1). Notice that continual re-solving never keeps track of the opponent’s range, instead

only keeping track of their counterfactual values. Furthermore, it never requires knowledge of

the opponent’s action to update these values, which is an important difference from traditional

re-solving. Both will prove key to making this algorithm efﬁcient and avoiding any need for the

translation step required with action abstraction methods (25, 26).

Continual re-solving is theoretically sound, but by itself impractical. While it does not ever

maintain a complete strategy, re-solving itself is intractable except near the end of the game.

In order to make continual re-solving practical, we need to limit the depth and breadth of the

re-solved subtree.

5

Limited depth lookahead via intuition. As in heuristic search for perfect information games,

we would like to limit the depth of the subtree we have to reason about when re-solving. How-

ever, in imperfect information games we cannot simply replace a subtree with a heuristic or

precomputed value. The counterfactual values at a public state are not ﬁxed, but depend on how

players play to reach the public state, i.e., the players’ ranges (17). When using an iterative

algorithm, such as CFR, to re-solve, these ranges change on each iteration of the solver.

DeepStack overcomes this challenge by replacing subtrees beyond a certain depth with a

learned counterfactual value function that approximates the resulting values if that public state

were to be solved with the current iteration’s ranges. The inputs to this function are the ranges

for both players, as well as the pot size and public cards, which are sufﬁcient to specify the

public state. The outputs are a vector for each player containing the counterfactual values of

holding each hand in that situation. In other words, the input is itself a description of a poker

game: the probability distribution of being dealt individual private hands, the stakes of the

game, and any public cards revealed; the output is an estimate of how valuable holding certain

cards would be in such a game. The value function is a sort of intuition, a fast estimate of the

value of ﬁnding oneself in an arbitrary poker situation. With a depth limit of four actions, this

approach reduces the size of the game for re-solving from 10

160

decision points at the start of

the game down to no more than 10

17

decision points. DeepStack uses a deep neural network as

its learned value function, which we describe later.

Sound reasoning. DeepStack’s depth-limited continual re-solving is sound. If DeepStack’s

intuition is “good” and “enough” computation is used in each re-solving step, then DeepStack

plays an arbitrarily close approximation to a Nash equilibrium.

Theorem 1 If the values returned by the value function used when the depth limit is reached

have error less than , and T iterations of CFR are used to re-solve, then the resulting strategy’s

exploitability is less than k

1

+ k

2

/

√

T , where k

1

and k

2

are game-speciﬁc constants. For the

proof, see (10).

Sparse lookahead trees. The ﬁnal ingredient in DeepStack is the reduction in the number of

actions considered so as to construct a sparse lookahead tree. DeepStack builds the lookahead

tree using only the actions fold (if valid), call, 2 or 3 bet actions, and all-in. This step voids

the soundness property of Theorem 1, but it allows DeepStack to play at conventional human

speeds. With sparse and depth-limited lookahead trees, the re-solved games have approximately

10

7

decision points, and are solved in under ﬁve seconds using a single NVIDIA GeForce GTX

1080 graphics card. We also use the sparse and depth-limited lookahead solver from the start of

the game to compute the opponent counterfactual values used to initialize DeepStack’s continual

re-solving.

Relationship to heuristic search in perfect information games. There are three key chal-

lenges that DeepStack overcomes to incorporate heuristic search ideas in imperfect information

6

C

Sampled poker

situations

BA

Agent's possible actions

Lookahead tree

Current public state

Agent’s range

Opponent counterfactual values

Neural net [see B]

Action history

Public tree

Subtree

Values

Ranges

Figure 2: DeepStack overview. (A) DeepStack reasons in the public tree always producing

action probabilities for all cards it can hold in a public state. It maintains two vectors while

it plays: its own range and its opponent’s counterfactual values. As the game proceeds, its

own range is updated via Bayes’ rule using its computed action probabilities after it takes an

action. Opponent counterfactual values are updated as discussed under “Continual re-solving”.

To compute action probabilities when it must act, it performs a re-solve using its range and the

opponent counterfactual values. To make the re-solve tractable it restricts the available actions

of the players and lookahead is limited to the end of the round. During the re-solve, counterfac-

tual values for public states beyond its lookahead are approximated using DeepStack’s learned

evaluation function. (B) The evaluation function is represented with a neural network that takes

the public state and ranges from the current iteration as input and outputs counterfactual values

for both players (Fig. 3). (C) The neural network is trained prior to play by generating ran-

dom poker situations (pot size, board cards, and ranges) and solving them to produce training

examples. Complete pseudocode can be found in Algorithm S1 (10).

games. First, sound re-solving of public states cannot be done without knowledge of how and

why the players acted to reach the public state. Instead, two additional vectors, the agent’s range

and opponent counterfactual values, must be maintained to be used in re-solving. Second, re-

solving is an iterative process that traverses the lookahead tree multiple times instead of just

once. Each iteration requires querying the evaluation function again with different ranges for

every public state beyond the depth limit. Third, the evaluation function needed when the depth

limit is reached is conceptually more complicated than in the perfect information setting. Rather

than returning a single value given a single state in the game, the counterfactual value function

needs to return a vector of values given the public state and the players’ ranges. Because of this

complexity, to learn such a value function we use deep learning, which has also been successful

at learning complex evaluation functions in perfect information games (6).

Relationship to abstraction-based approaches. Although DeepStack uses ideas from ab-

straction, it is fundamentally different from abstraction-based approaches. DeepStack restricts

the number of actions in its lookahead trees, much like action abstraction (25, 26). However,

7

each re-solve in DeepStack starts from the actual public state and so it always perfectly un-

derstands the current situation. The algorithm also never needs to use the opponent’s actual

action to obtain correct ranges or opponent counterfactual values, thereby avoiding translation

of opponent bets. We used hand clustering as inputs to our counterfactual value functions, much

like explicit card abstraction approaches (27, 28). However, our clustering is used to estimate

counterfactual values at the end of a lookahead tree rather than limiting what information the

player has about their cards when acting. We later show that these differences result in a strategy

substantially more difﬁcult to exploit.

Deep Counterfactual Value Networks

Deep neural networks have proven to be powerful models and are responsible for major ad-

vances in image and speech recognition (29, 30), automated generation of music (31), and

game-playing (5, 6). DeepStack uses deep neural networks with a tailor-made architecture, as

the value function for its depth-limited lookahead (Fig. 3). Two separate networks are trained:

one estimates the counterfactual values after the ﬁrst three public cards are dealt (ﬂop network),

the other after dealing the fourth public card (turn network). An auxiliary network for values

before any public cards are dealt is used to speed up the re-solving for early actions (10).

Architecture. DeepStack uses a standard feedforward network with seven fully connected

hidden layers each with 500 nodes and parametric rectiﬁed linear units (32) for the output. This

architecture is embedded in an outer network that forces the counterfactual values to satisfy

the zero-sum property. The outer computation takes the estimated counterfactual values, and

computes a weighted sum using the two players’ input ranges resulting in separate estimates of

the game value. These two values should sum to zero, but may not. Half the actual sum is then

subtracted from the two players’ estimated counterfactual values. This entire computation is

differentiable and can be trained with gradient descent. The network’s inputs are the pot size as

a fraction of the players’ total stacks and an encoding of the players’ ranges as a function of the

public cards. The ranges are encoded by clustering hands into 1,000 buckets, as in traditional

abstraction methods (27, 28, 33), and input as a vector of probabilities over the buckets. The

output of the network are vectors of counterfactual values for each player and hand, interpreted

as fractions of the pot size.

Training. The turn network was trained by solving 10 million randomly generated poker turn

games. These turn games used randomly generated ranges, public cards, and a random pot

size (10). The target counterfactual values for each training game were generated by solving

the game with players’ actions restricted to fold, call, a pot-sized bet, and an all-in bet, but no

card abstraction. The ﬂop network was trained similarly with 1 million randomly generated

ﬂop games. However, the target counterfactual values were computed using our depth-limited

8

Input

Bucket

ranges

7 Hidden Layers

• fully connected

• linear, PReLU

Output

Bucket

values

FEEDFORWARD

NEURAL NET

ZERO-SUM

NEURAL NET

Output

Counterfactual

values

CARD

COUNTERFACTUAL

VALUES

Zero-sum

Error

BUCKETING

(INVERSE)

BUCKETING

CARD

RANGES

500

500

500

500

500

500

500

1000

1

P2

P1

P1 P2

1326

P1

P2

Pot Public

1326

22100

1

1000

P2

P1

1000

P2

P1

Figure 3: Deep counterfactual value network. The inputs to the network are the pot size,

public cards, and the player ranges, which are ﬁrst processed into hand clusters. The output

from the seven fully connected hidden layers is post-processed to guarantee the values satisfy

the zero-sum constraint, and then mapped back into a vector of counterfactual values.

solving procedure and our trained turn network. The networks were trained using the Adam

gradient descent optimization procedure (34) with a Huber loss (35).

Evaluating DeepStack

We evaluated DeepStack by playing it against a pool of professional poker players recruited

by the International Federation of Poker (36). Thirty-three players from 17 countries were

recruited. Each was asked to complete a 3,000 game match over a period of four weeks between

November 7th and December 12th, 2016. Cash incentives were given to the top three performers

($5,000, $2,500, and $1,250 CAD).

Evaluating performance in HUNL is challenging because of the large variance in per-game

outcomes owing to randomly dealt cards and stochastic choices made by the players. The better

player may lose in a short match simply because they were dealt weaker hands or their rare

bluffs were made at inopportune times. As seen in the Claudico match (20), even 80,000 games

may not be enough to statistically signiﬁcantly separate players whose skill differs by a consid-

erable margin. We evaluate performance using AIVAT (37), a provably unbiased low-variance

technique for evaluating performance in imperfect information games based on carefully con-

structed control variates. AIVAT requires an estimated value of holding each hand in each public

9

Participant

Hands played

1250

1000

500

250

750

50

0

-500

-250

DeepStack win rate (mbb/g)

1000

2000

3000

0

5 10 15 20 25 30

Figure 4: Performance of professional poker players against DeepStack. Performance esti-

mated with AIVAT along with a 95% conﬁdence interval. The solid bars at the bottom show the

number of games the participant completed.

state, and then uses the expected value changes that occur due to chance actions and actions of

players with known strategies (i.e., DeepStack) to compute the control variate. DeepStack’s

own value function estimate is perfectly suited for AIVAT. Indeed, when used with AIVAT we

get an unbiased performance estimate with an impressive 85% reduction in standard deviation.

Thanks to this technique, we can show statistical signiﬁcance (38) in matches with as few as

3,000 games.

In total 44,852 games were played by the thirty-three players with 11 players completing the

requested 3,000 games. Over all games played, DeepStack won 492 mbb/g. This is over 4 stan-

dard deviations away from zero, and so, highly signiﬁcant. Note that professional poker players

consider 50 mbb/g a sizable margin. Using AIVAT to evaluate performance, we see DeepStack

was overall a bit lucky, with its estimated performance actually 486 mbb/g. However, as a lower

variance estimate, this margin is over 20 standard deviations from zero.

The performance of individual participants measured with AIVAT is summarized in Fig-

ure 4. Amongst those players that completed the requested 3,000 games, DeepStack is esti-

mated to be winning by 394 mbb/g, and individually beating 10 out of 11 such players by a

statistically signiﬁcant margin. Only for the best performing player, still estimated to be losing

by 70 mbb/g, is the result not statistically signiﬁcant. More details on the participants and their

results are presented in (10).

10

Table 1: Exploitability bounds from Local Best Response. For all listed programs, the value

reported is the largest estimated exploitability when applying LBR using a variety of different

action sets. Table S2 gives a more complete presentation of these results (10). ‡: LBR was

unable to identify a positive lower bound for DeepStack’s exploitability.

Program LBR (mbb/g)

Hyperborean (2014) 4675

Slumbot (2016) 4020

Act1 (2016) 3302

Always Fold 750

DeepStack 0 ‡

Exploitability

The main goal of DeepStack is to approximate Nash equilibrium play, i.e., minimize exploitabil-

ity. While the exact exploitability of a HUNL poker strategy is intractable to compute, the re-

cent local best-response technique (LBR) can provide a lower bound on a strategy’s exploitabil-

ity (21) given full access to its action probabilities. LBR uses the action probabilities to compute

the strategy’s range at any public state. Using this range it chooses its response action from a

ﬁxed set using the assumption that no more bets will be placed for the remainder of the game.

Thus it best-responds locally to the opponent’s actions, providing a lower bound on their overall

exploitability. As already noted, abstraction-based programs from the Annual Computer Poker

Competition are highly exploitable by LBR: four times more exploitable than folding every

game (Table 1). However, even under a variety of settings, LBR fails to exploit DeepStack at all

— itself losing by over 350 mbb/g to DeepStack (10). Either a more sophisticated lookahead is

required to identify DeepStack’s weaknesses or it is substantially less exploitable.

Discussion

DeepStack defeated professional poker players at HUNL with statistical signiﬁcance (39), a

game that is similarly sized to go, but with the added complexity of imperfect information.

It achieves this goal with little domain knowledge and no training from expert human games.

The implications go beyond being a milestone for artiﬁcial intelligence. DeepStack represents

a paradigm shift in approximating solutions to large, sequential imperfect information games.

Abstraction and ofﬂine computation of complete strategies has been the dominant approach for

almost 20 years (33,40,41). DeepStack allows computation to be focused on speciﬁc situations

that arise when making decisions and the use of automatically trained value functions. These

are two of the core principles that have powered successes in perfect information games, albeit

conceptually simpler to implement in those settings. As a result, the gap between the largest

11

perfect and imperfect information games to have been mastered is mostly closed.

With many real world problems involving information asymmetry, DeepStack also has im-

plications for seeing powerful AI applied more in settings that do not ﬁt the perfect information

assumption. The abstraction paradigm for handling imperfect information has shown promise

in applications like defending strategic resources (42) and robust decision making as needed

for medical treatment recommendations (43). DeepStack’s continual re-solving paradigm will

hopefully open up many more possibilities.

References and Notes

1. G. Tesauro, Communications of the ACM 38, 58 (1995).

2. J. Schaeffer, R. Lake, P. Lu, M. Bryant, AI Magazine 17, 21 (1996).

3. M. Campbell, A. J. Hoane Jr., F. Hsu, Artiﬁcial Intelligence 134, 57 (2002).

4. D. Ferrucci, IBM Journal of Research and Development 56, 1:1 (2012).

5. V. Mnih, et al., Nature 518, 529 (2015).

6. D. Silver, et al., Nature 529, 484 (2016).

7. A. L. Samuel, IBM Journal of Research and Development 3, 210 (1959).

8. L. Kocsis, C. Szepesv

´

ari, Proceedings of the Seventeenth European Conference on Machine

Learning (2006), pp. 282–293.

9. J. Bronowski, The ascent of man, Documentary (1973). Episode 13.

10. See Supplementary Materials.

11. In 2008, Polaris defeated a team of professional poker players in heads-up limit Texas

hold’em (44). In 2015, Cepheus essentially solved the game (18).

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12

16. End-game solving (17, 45, 46) is one exception to computation occurring prior to play.

When the game nears the end, a new computation is invoked over the remainder of the

game. Thus, the program need not store this part of the strategy or can use a ﬁner-grained

abstraction aimed to improve the solution quality. We discuss this as re-solving when we

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big-blind is one thousandth of the forced big blind bet amount that starts the game. This

normalizes performance for the number of games played and the size of stakes. For com-

parison, a win rate of 50 mbb/g is considered a sizable margin by professional players and

750 mbb/g is the rate that would be lost if a player folded each game. The poker community

commonly uses big blinds per one hundred games (bb/100) to measure win rates, where 10

mbb/g equals 1 bb/100.

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have only been applied in small games (48).

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strategy. In practice, though, using expected values from the previous solution in self-play

often works as well or better than best-response values (10).

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13

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38. Statistical signiﬁcance where noted was established using a two-tailed Student’s t-test at

the 0.05 level with N ≥ 3000.

39. Subsequent to our study, the computer program Libratus, developed at CMU by Tuomas

Sandholm and Noam Brown, defeated a team of four professional heads-up poker spe-

cialists in a HUNL competition held January 11-30, 2017. Libratus has been described as

using “nested endgame solving” (49), a technique with similarities to continual re-solving,

but developed independently. Libratus employs this re-solving when close to the end of the

game rather than at every decision, while using an abstraction-based approach earlier in the

game.

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ligence (2016), pp. 544–550.

14

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44. J. Rehmeyer, N. Fox, R. Rico, Wired 16.12, 186 (2008).

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Autonomous Agents and Multi-Agent Systems (2015), pp. 37–45.

46. M. Moravcik, M. Schmid, K. Ha, M. Hlad

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ık, S. J. Gaukrodger, Proceedings of the Thirtieth

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ﬁcial Intelligence (2016), pp. 360–367.

48. J. Heinrich, D. Silver, arXiv preprint arXiv:1603.01121 (2016).

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Imperfect Information Games (2017).

50. M. Zinkevich, M. Littman, Journal of the International Computer Games Association 29,

166 (2006). News item.

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ternational Joint Conference on Artiﬁcial Intelligence (2015), pp. 645–652.

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telligence (2014), pp. 682–690.

Acknowledgements. The hand histories of all games played in the human study as well as

those used to generate the LBR results against DeepStack are in (10). We would like to thank

the IFP, all of the professional players who committed valuable time to play against DeepStack,

the anonymous reviewers’ invaluable feedback and suggestions, and R. Holte, A. Brown, and

K. Bla

ˇ

zkov

´

a for comments on early drafts of this article. We especially would like to thank

IBM for their support of this research through an IBM faculty grant. The research was also

supported by Alberta Innovates through the Alberta Machine Intelligence Institute, the Natural

Sciences and Engineering Research Council of Canada, and Charles University (GAUK) Grant

no. 391715. This work was only possible thanks to computing resources provided by Compute

Canada and Calcul Qu

´

ebec. MM and MS are on leave from IBM Prague. MJ serves as a

Research Scientist, and MB as a Contributing Brain Trust Member, at Cogitai, Inc. DM owns

200 shares of Gamehost Inc.

15

**Huber Loss** is a loss function that is less sensitive to outliers in data than the squared error loss.
In 1944 John von Neumann and Oskar Morgenstern co-authored a book entitled **Theory of Games and Economic Behavior**.
![Theory of Games and Economic Behavior cover](https://upload.wikimedia.org/wikipedia/en/1/1e/VNM-TGEB.jpg)
This was a groundbreaking book that would go on to essentially kickstart the field of game theory. John Nash would go on to greatly expand and explore the field of Game Theory.
This is a massive number. Chess has approximately $10^{47}$ decision points. The estimated number of atoms in the universe is $10^{80}$
In heads-up poken one player will post the big blind and their opponent will post the small blind. If your strategy is to always fold, you will be loosing at a rate of 750 *mbb/g* (you are loosing *1.5* big blinds every 2 games - big blind + small blind).
**mbb/g** - milli-big blinds per game. In poker, blinds are forced bets posted by players on each round. Usually there is a small blind and a big blind (twice the small blind). If the big blind is \$20 and you are averaging 500 *mbb/g*, your expected winnings are \$10 per hand.
There are 4 main categories of games depending on whether the game is deterministic or not and on whether or not it contains perfect information.
Deterministic games contain no element of chance and the outcome is solely determined by the skill and tactics of the players (e.g. chess). The distinction between perfect or imperfect information games depends upon whether the competitors have all the information relating to the game state. In this sense, Checkers is a perfect information game (both players can see everything that is going on the board) whereas Battleships is not.
![games categories](http://i.imgur.com/59biDnO.png)
This figure provides a great high level picture of DeepStack. As the figure indicates all that DeepStack keeps track of is:
- **Agent’s Range** - The probability distribution over the agent’s possible hands taking into account the current state of the game.
- ** Opponent’s counterfactual values ** - Set of expected values for each hand the opponent might have given the current state.
#### Heads-up no-limit Texas hold’em
Common variant of poker played between two players with no limits on bets. In contrast, *heads-up **limit** Texas hold’em* "was essentially weakly solved” in 2015 ([paper](http://ai.cs.unibas.ch/_files/teaching/fs15/ki/material/ki02-poker.pdf))
Multi-player poker is still a considerably harder challenge than 1 vs. 1.
### Nash Equilibrium
Let’s say you were playing rock, paper scissors with an opponent. The strategy that is Game Theory Optimal and therefore achieves a Nash equilibrium is to play each one of your 3 available actions with the same frequency ($1/3$). With 2 perfect players playing, no participant can gain by making a change to this strategy.
It is worth noting that if your player is not a rational / perfect player, the Game Theory Optimal strategy might not be the one that yields the best results. If you are playing against an opponent that plays rock half of the times and scissors the other half, then the best strategy is to always play rock.
**Counterfactual Regret Minimization (CFR)** is an algorithm by which an agent learns how to play a game by repeatedly playing against itself. The idea is that you initialize your strategy with a uniform distribution over the actions available to you. You then simulate playing the game against yourself repeatedly and each time, you revisit your strategy and update it.
You improve your strategy by minimizing the **regret** for each action at each decision point. Regret is how much better you would have done overall (over all the games you’ve played) if at a certain decision point, you had always chosen a specific action available to you at the time (vs using the probability distribution over the possible actions provided by your strategy). You then go back and recompute your strategy with probabilities proportional to the regret for each action.
[Here is a good resource](http://modelai.gettysburg.edu/2013/cfr/) in case you want to learn more about CFR.