1. Better automated abstraction techniques for imperfect information games, with application to Texas Hold’em poker Andrew Gilpin and Tuomas Sandholm Carnegie Mellon University Computer Science Department

2. Games and information Perfect information games: agents have complete knowledge about the state of the world –E.g.: Chess, Go, Checkers Imperfect information games: agents face uncertainty about the state of the world –E.g.: Robot facing adversaries in an uncertain, stochastic environment Almost any economic situation in which the other participants have private information (e.g. valuations, quality information) Almost any card game in which the other players’ cards are hidden –Presents several challenges for AI Imperfect information Risk assessment and management Speculation and counter-speculation Signaling and interpreting signals –Misrepresentation (bluffing…)

3. Nash equilibrium In multiagent systems, an agent’s outcome depends on the actions of the other agents an agent’s optimal strategy depends on the strategies of the other agents Game theory defines how agents should play Definition: A Nash equilibrium is a strategy for each agent such that no agent benefits by deviating to another strategy

4. Finding an equilibrium In two-person zero-sum games, –If opponent plays a non-equilibrium strategy, that can only help me –Nash equilibria are interchangeable, so there is no equilibrium selection problem –Equilibrium can be found using LP Any extensive form game (of perfect recall) can be converted into a matrix game –Create one pure strategy in the matrix game for every possible pure contingency plan in the sequential game –Leads to exponential blowup in #strategies Sequence form: More compact representation based on sequences of moves rather than pure strategies [Romanovskii 62, Koller & Megiddo 92, von Stengel 96] –Two-person zero-sum games can be solved in time polynomial in size of game tree –Doesn’t scale to Rhode Island Hold’em (3.1 billion nodes) or Texas Hold’em (10 18 nodes)

5. Finding an equilibrium In two-person zero-sum games, –If opponent plays a non-equilibrium strategy, that can only help me –Nash equilibria are interchangeable, so there is no equilibrium selection problem –Equilibrium can be found using LP Any extensive form game (of perfect recall) can be converted into a matrix game –Create one pure strategy in the matrix game for every possible pure contingency plan in the sequential game (set product of actions at information sets) –Leads to exponential blowup in #strategies even in the reduced normal form Sequence form: More compact representation based on sequences of moves rather than pure strategies [Romanovskii 62, Koller & Megiddo 92, von Stengel 96] –Two-person zero-sum games can be solved in time polynomial in size of game tree –Not enough to solve Rhode Island Hold’em (3.1 billion nodes) or Texas Hold’em (10 18 nodes)

6. Automated abstraction [EC-06] Automatic method for performing abstractions in a broad class of sequential games of imperfect information Equilibrium-preserving game transformation –certain information sets are merged, and –certain nodes within an information set are collapsed GameShrink identifies and applies all those transformations –Õ(n 2 ) time n = #nodes in the signal tree. In poker, these are possible card deals Run-time tends to be highly sublinear in the size of the game tree Solved Rhode Island Hold’em –Largest poker game solved to date by over four orders of magnitude Also developed approximate (lossy) version of GameShrink –Uses a similarity metric on nodes in the signal tree (e.g., |#wins 1 - #wins 2 | + |#losses 1 - #losses 2 |) and a similarity threshold

7. Illustration of our approach Nash equilibrium Original game Abstracted game Abstraction Nash equilibrium Compute Nash

8. Texas Hold’em poker

17. Tackling Texas Hold’em: High-level view Two-person game tree has ~10 18 leaves –Too large to run lossless GameShrink –Even after that, LP would be too large Already too large when we applied this to first two rounds We split the 4 betting rounds into two phases –Phase I (first 3 rounds) solved offline using new approximate version of GameShrink followed by LP –Phase II (last 2 rounds): abstractions computed offline real-time equilibrium computation using updated hand probabilities and anytime LP

18. Optimized approximate abstractions Original version of GameShrink is “greedy” when used as an approximation algorithm => lopsided abstractions GS2 instead finds an abstraction via clustering & IP For round 1 in signal tree, use k-means clustering –Similarity metric is win probability (ties count as half a win) Computed based on rollout For each round 2..3 of signal tree: –For each group i of hands (children of a parent at round – 1): use k-means clustering to split group i into k i abstract “states” for each value of k i, compute expected error (considering hand probs) –IP decides how many children different parents (from round – 1) may have: Decide k i ’s to minimize this expected error, subject to Σ i k i ≤ K round K round is set based on acceptable LP size Solving this IP is fast in practice

19. Optimized approximate abstractions Original version of GameShrink yielded lopsided abstractions when used as an approximation algorithm Now we instead find an abstraction via clustering: For each level of the tree (starting from root): –For each group of hands (children of a parent at the previous betting round): use k-means clustering to split group i into k i abstract “states” –win probability as the similarity metric (ties count as half a win) for each value of k i, compute expected error (considering hand probs) –Use integer programming to find an abstraction (split of K into k i ’s) that minimizes this expected error, subject to a constraint on the total number of states, K, at that level (=size of the resulting LP in the zero-sum case) Solving this class of integer programs is quite easy in practice

20. Phase I (first three rounds) Allowed 15, 225, and 900 abstracted states in rounds 1, 2, and 3, respectively Optimizing the approximate abstraction took 3 days on 4 CPUs LP took 7 days and 80 GB using CPLEX’s barrier method

21. Phase I (first three rounds) Optimized abstraction –Round 1 There are 1,326 hands, of which 169 are strategically different We allowed 15 abstract states –Round 2 There are 25,989,600 distinct possible hands –GameShrink (in lossless mode for Phase I) determined there are ~10 6 strategically different hands (still too large to solve) Allowed 225 abstract states –Round 3 There are 1,221,511,200 distinct possible hands Allowed 900 abstract states Optimizing the approximate abstraction took 3 days on 4 CPUs LP took 7 days and 80 GB using CPLEX’s barrier method

22. Mitigating effect of round-based abstraction (i.e., having 2 phases) For leaves of Phase I, GS1 & Sparbot assumed rollout Can do better by estimating the amount of betting that occurs in later rounds –Incorporate this info in LP for Phase I For each possible hand strength and in each possible betting situation, we store the probability of each possible action –Mine history of how betting has gone in later rounds –from 100,000’s of hands that Sparbot played

23. Example of betting in 4th round Player 1 has bet. Player 2 to fold, call, or raise

24. Mitigating effect of round-based abstraction (i.e., having 2 phases) For leaves in the first phase, we could assume no betting in the later rounds Ignores implied odds Can do better by estimating the amount of betting that occurs in later rounds –Incorporate this information into the LP for the first phase For each possible hand strength and in each possible betting situation, we store the probability of each possible action –Mine the betting history in the later rounds from hundreds of thousands of played hands

25. Example of betting in fourth round Player 1 has bet. Player 2 to fold, call, or raise

26. Phase II (rounds 3 and 4) Just as in GS1, except now with the optimized approximate abstraction algorithm

27. Phase II (last two rounds) Abstractions computed offline –Betting history doesn’t matter => ( ) situations –Simple suit isomorphisms at the root of Phase II halves this –For each such setting, we use clustering+IP to generate an abstraction with 10 and 100 strategically different hands in the last two rounds, respectively Real-time equilibrium computation (using LP) –So that our strategies are specific to particular hand (too many to precompute) –Updated hand probabilities from Phase I equilibrium using betting histories and community card history: s i is player i’s strategy, h is an information set –Conditional choice of primal vs. dual simplex Achieve anytime capability for the player that is us –Dealing with running off the equilibrium path 52 4

28. Precompute several databases db5 : possible wins and losses (for a single player) for every combination of two hole cards and three community cards (25,989,600 entries) –Used by GameShrink for quickly comparing the similarity of two hands db223 : possible wins and losses (for both players) for every combination of pairs of two hole cards and three community cards based on a roll-out of the remaining cards (14,047,378,800 entries) –Used for computing payoffs of the Phase I game to speed up the LP creation handval : concise encoding of a 7-card hand rank used for fast comparisons of hands (133,784,560 entries) –Used in several places, including in the construction of db5 and db223 Colexicographical ordering used to compute indices into the databases allowing for very fast lookups

29. Experimental results GS1: Game theory-based player, old version of automated abstraction, no strategy simulation in later rounds [Gilpin & Sandholm AAAI-06] Sparbot: Game theory-based player, manual abstraction [Billings et al 2003] Vexbot: Opponent modeling, miximax search with statistical sampling [Billings et al 2004] OpponentSeries wonWin rate (small bets per 100) GS138 of 50+3.12 Sparbot28 of 50+0.43 Vexbot32 of 50-0.62

30. Measuring the contribution of new techniques OpponentSeries wonWin rate (small bets per 100) GS2 without improved abstraction and without estimated payoffs 48 of 50+2.87 GS2 without improved abstraction 35 of 50+2.73 GS2 without estimated payoffs 44 of 50+0.72

31. Summary Main ideas –Optimized approximate state-space abstraction based on clustering and IP –Computing leaf payoffs for truncated game based on statistics Automatically generated competitive Texas Hold’em bot –First phase (rounds 1, 2 & 3): automated abstraction & LP solved offline, using statistical data to compute payoffs at end of round 3 –Second phase (rounds 3 & 4): abstraction precomputed automatically; LP solved in real-time using updated hand probabilities and anytime Techniques are applicable to many sequential games of imperfect information, not just poker

32. Ongoing research Provable approximation, e.g., ex ante, ex post Other types of abstraction More scalable equilibrium-finding algorithms Tournament poker [e.g. Miltersen & Sørensen 07] No-limit poker More than two players –May need to use alternative solution concept Thank you