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Counterfactual Regret Minimization and Domination in Extensive-Form Games Richard Gibson University of Alberta Edmonton, Alberta, Canada Overview Counterfactual Regret Minimization (CFR) Overview Counterfactual Regret Minimization (CFR)

  1. Counterfactual Regret Minimization and Domination in Extensive-Form Games Richard Gibson University of Alberta Edmonton, Alberta, Canada

  2. Overview Counterfactual Regret Minimization (CFR)

  3. Overview Counterfactual Regret Minimization (CFR) Provably solves for Nash equilibrium 2-Player Zero-Sum Extensive-Form Games

  4. Overview Counterfactual Regret Minimization (CFR) Provably solves for Seems to work well... Nash equilibrium 2-Player Zero-Sum Extensive-Form Extensive-Form Games, any Games Number of players

  5. Overview Counterfactual Regret Minimization (CFR) Provably solves for Seems to work well... Nash equilibrium 2-Player Zero-Sum Extensive-Form Extensive-Form Games, any Games Number of players Question : Why do CFR strategies work well in extensive-form games outside of the 2-player zero-sum case?

  6. Extensive-Form Games C QJ QK 0.5 0.5 1 1 c b c b 2 2 2 2 c b f c c b f c (1,-1) 1 (1,-1) (2,-2) (-1,1) -1 1 (1,-1) (-2,2) f c f c (-1,1) (2,-2) (-1,1) (-2,2)

  7. Extensive-Form Games C QJ QK 0.5 0.5 1 1 c b c b 2 2 2 2 c b f c c b f c (1,-1) 1 (1,-1) (2,-2) (-1,1) -1 1 (1,-1) (-2,2) f c f c (-1,1) (2,-2) (-1,1) (-2,2) Information sets group states that are indistinguishable to the player.

  8. Extensive-Form Games C QJ QK 0.5 0.5 1 1 0.4 0.6 0.4 0.6 2 2 2 2 0.7 0.3 0 1 0.9 0.1 1 0 (1,-1) 1 (1,-1) (2,-2) (-1,1) 1 (1,-1) (-2,2) 0.2 0.8 0.2 0.8 (-1,1) (2,-2) (-1,1) (-2,2) A strategy profile σ = ( σ 1 , σ 2 ) assigns a probability distribution over actions at each information set. Example: Probability player 1 checks is σ 1 ( Q?, c ) = 0.4 .

  9. Extensive-Form Games C QJ QK 0.5 0.5 1 1 0.4 0.6 0.4 0.6 2 2 2 2 0.7 0.3 0 1 0.9 0.1 1 0 (1,-1) 1 (1,-1) (2,-2) (-1,1) 1 (1,-1) (-2,2) 0.2 0.8 0.2 0.8 (-1,1) (2,-2) (-1,1) (-2,2) u i ( σ ) is the expected utility for player i , assuming players play according to σ .

  10. Counterfactual Regret Minimization (CFR) [Zinkevich et al ., NIPS 2007] ● CFR is an iterative algorithm that generates strategy profiles ( σ 1 , σ 2 , ... , σ T ) over many iterations T . ● Final output of CFR: σ AVG = Average( σ 1 , σ 2 , ... , σ T ). ● For 2-player zero-sum games, σ AVG is an ϵ -Nash equilbrium, with ϵ → 0 as T → ∞: AVG , σ 2 AVG ) ≥ max u 1 ( σ 1 * , σ 2 AVG ) - ϵ u 1 ( σ 1 * σ 1 AVG , σ 2 AVG ) ≥ max u 2 ( σ 1 AVG , σ 2 * ) - ϵ u 2 ( σ 1 σ 2 *

  11. Counterfactual Regret Minimization (CFR) ● Outside of 2-player zero-sum games, σ AVG is not necessarily an approximate Nash equilibrium [Abou Risk and Szafron, AAMAS 2010] . – A player may gain by deviating from σ AVG . ● In these games, a Nash equilibrium might not be the most appropriate solution concept anyways. ● On the other hand, σ AVG performs very well in practice...

  12. Annual Computer Poker Competition 3-Player Limit Hold'em - 2009 3-Player Limit Hold'em - 2011 Agent Instant Run-off: Round 0 Agent Instant Run-off: Round 0 Hyperborean-Eqm 319 ± 2 Sartre3p 243 ± 20 Hyperborean-BR 299 ± 2 Hyperborean-3p-limit-iro 204 ± 20 akuma 151 ± 2 LittleRock 113 ± 19 dpp 171 ± 2 AAIMontybot 96 ± 44 CMURingLimit -37 ± 2 dcubot3plr 77 ± 19 dcu3pl -63 ± 2 OwnBot -4 ± 30 Bluechip -548 ± 2 Bnold3 -91 ± 22 Entropy -108 ± 36 3-Player Limit Hold'em - 2010 player.zeta.3p -530 ± 33 Agent Instant Run-off: Round 0 Hyperborean.iro 144 ± 32 dcu3pl.tbr 98 ± 30 LittleRock 65 ± 35 Arnold3 -135 ± 39 Bender -172 ± 16

  13. Counterfactual Regret Minimization (CFR) ● In games with more than 2-players, σ AVG is a “good” strategy. Why? ● What properties make a strategy good in games with more than 2-players? ● We know what a bad strategy is...

  14. Domination C QJ QK 0.5 0.5 1 1 c b c b 2 2 2 2 c b f c c b f c (1,-1) 1 (1,-1) (2,-2) (-1,1) -1 1 (1,-1) (-2,2) f c f c (-1,1) (2,-2) (-1,1) (-2,2)

  15. Domination C QJ QK 0.5 0.5 1 1 c b c b 2 2 2 2 c b 0 1 c b f c (1,-1) 1 (1,-1) (2,-2) (-1,1) -1 1 (1,-1) (-2,2) f c f c (-1,1) (2,-2) (-1,1) (-2,2) J, c that always calls with the Jack when faced Consider any player 2 strategy σ 2 with a bet.

  16. Domination C QK QJ 0.5 0.5 1 1 c b c b 2 2 2 2 c b 0 1 c b f c (1,-1) 1 (1,-1) (2, -2 ) (-1,1) -1 1 (1,-1) (-2,2) f c f c (-1,1) (2,-2) (-1,1) (-2,2) J, c ) = ... + 0.5 σ 1 ( Q?, b )1 ( -2 ) + ... u 2 ( σ 1 , σ 2

  17. Domination C QK QJ 0.5 0.5 1 1 c b c b 2 2 2 2 c b 1 0 c b f c (1,-1) 1 (1,-1) (2,-2) (-1,1) -1 1 (1,-1) (-2,2) f c f c (-1,1) (2,-2) (-1,1) (-2,2) J, c ) = ... + 0.5 σ 1 ( Q?, b )1 ( -2 ) + ... u 2 ( σ 1 , σ 2 J, f . Now consider the same player 2 strategy, except always folds the J. Call it σ 2

  18. Domination C QK QJ 0.5 0.5 1 1 c b c b 2 2 2 2 c b 1 0 c b f c (1,-1) 1 (1, -1 ) (2,-2) (-1,1) -1 1 (1,-1) (-2,2) f c f c (-1,1) (2,-2) (-1,1) (-2,2) J, c ) = ... + 0.5 σ 1 ( Q?, b )1 ( -2 ) + ... u 2 ( σ 1 , σ 2 J, f ) = '' + 0.5 σ 1 ( Q?, b )1 ( -1 ) + '' u 2 ( σ 1 , σ 2

  19. Domination C QK QJ 0.5 0.5 1 1 c b c b 2 2 2 2 c b 1 0 c b f c (1,-1) 1 (1, -1 ) (2,-2) (-1,1) -1 1 (1,-1) (-2,2) f c f c (-1,1) (2,-2) (-1,1) (-2,2) J, c ) = ... + 0.5 σ 1 ( Q?, b )1 ( -2 ) + ... u 2 ( σ 1 , σ 2 J, f ) = '' + 0.5 σ 1 ( Q?, b )1 ( -1 ) + '' u 2 ( σ 1 , σ 2 J, c ) ≤ u 2 ( σ 1 , σ 2 J, f ) for all σ 1 . u 2 ( σ 1 , σ 2 J, c ) < u 2 ( σ 1 , σ 2 J, f ) if σ 1 ( Q?, b ) > 0 u 2 ( σ 1 , σ 2

  20. Domination C QJ QK 0.5 0.5 1 1 c b c b 2 2 2 2 c b f c c b f c (1,-1) 1 (1,-1) (2,-2) (-1,1) -1 1 (1,-1) (-2,2) f c f c (-1,1) (2,-2) (-1,1) (-2,2) σ 2 is a dominated strategy if there exists σ 2 ' such that u 2 ( σ 1 , σ 2 , σ 3 , ... ) ≤ u 2 ( σ 1 , σ 2 ' , σ 3 , ... ) for all σ 1 , σ 3 , ... u 2 ( σ 1 , σ 2 , σ 3 , ... ) < u 2 ( σ 1 , σ 2 ' , σ 3 , ... ) for some σ 1 , σ 3 , ...

  21. Domination C QJ QK 0.5 0.5 1 1 c b c b 2 2 2 2 c b f c c b f c (1,-1) 1 (1,-1) (2,-2) (-1,1) -1 1 (1,-1) (-2,2) f c f c (-1,1) (2,-2) (-1,1) (-2,2) J, c is dominated by σ 2 J, f σ 2 K, f is dominated by σ 2 K, c σ 2

  22. Domination C QJ QK 0.5 0.5 1 1 c b c b 2 2 2 2 c b f c c b f c (1,-1) 1 (1,-1) (2,-2) (-1,1) -1 1 (1,-1) (-2,2) f c f c (-1,1) (2,-2) (-1,1) (-2,2) Define a dominated action to be an action such that any strategy that always plays that action is dominated (assuming that player plays to reach that action).

  23. Domination C QJ QK 0.5 0.5 1 1 c b c b 2 2 2 2 c b f c b c (1,-1) 1 (1,-1) (-1,1) -1 1 (-2,2) f c f c (-1,1) (2,-2) (-1,1) (-2,2)

  24. Domination C QJ QK 0.5 0.5 1 1 0 1 0 1 2 2 2 2 c b f c b c (1,-1) 1 (1,-1) (-1,1) -1 1 (-2,2) f c f c (-1,1) (2,-2) (-1,1) (-2,2) b that always bets. Consider the player 1 strategy σ 1

  25. Domination C QJ QK 0.5 0.5 1 1 0 1 0 1 2 2 2 2 c b 1 c b 1 (1,-1) 1 ( 1 ,-1) (-1,1) -1 1 ( -2 ,2) f c f c (-1,1) (2,-2) (-1,1) (-2,2) b , σ 2 ) = 0.5 ( 1 )( 1 )( 1 ) + 0.5 ( 1 )( 1 )( -2 ) = -0.5 u 1 ( σ 1

  26. Domination C QJ QK 0.5 0.5 1 1 1 0 1 0 2 2 2 2 c b f c b c (1,-1) 1 (1,-1) (-1,1) -1 1 (-2,2) 0 1 0 1 (-1,1) (2,-2) (-1,1) (-2,2) b , σ 2 ) = 0.5 ( 1 )( 1 )( 1 ) + 0.5 ( 1 )( 1 )( -2 ) = -0.5 u 1 ( σ 1 cc that checks then calls. Now consider the player 1 strategy σ 1

  27. Domination C QJ QK 0.5 0.5 1 1 1 0 1 0 2 2 2 2 1 0 f 0 1 c ( 1 ,-1) 1 (1,-1) (-1,1) -1 1 (-2,2) 0 1 0 1 (-1,1) (2,-2) (-1,1) ( -2 ,2) b , σ 2 ) = 0.5 ( 1 )( 1 )( 1 ) + 0.5 ( 1 )( 1 )( -2 ) = -0.5 u 1 ( σ 1 cc , σ 2 J c ,K b ) = 0.5 ( 1 )( 1 )( 1 ) + 0.5 ( 1 )( 1 )( 1 )( -2 ) = -0.5 u 1 ( σ 1

  28. Domination C QJ QK 0.5 0.5 1 1 1 0 1 0 2 2 2 2 0 1 f 1 0 c (1,-1) 1 (1,-1) ( -1 ,1) -1 1 (-2,2) 0 1 0 1 (-1,1) ( 2 ,-2) (-1,1) (-2,2) b , σ 2 ) = 0.5 ( 1 )( 1 )( 1 ) + 0.5 ( 1 )( 1 )( -2 ) = -0.5 u 1 ( σ 1 cc , σ 2 J c ,K b ) = 0.5 ( 1 )( 1 )( 1 ) + 0.5 ( 1 )( 1 )( 1 )( -2 ) = -0.5 u 1 ( σ 1 cc , σ 2 J b ,K c ) = 0.5 ( 1 )( 1 )( 1 )( 2 ) + 0.5 ( 1 )( 1 )( -1 ) = +0.5 u 1 ( σ 1

  29. Domination C QJ QK 0.5 0.5 1 1 c b c b 2 2 2 2 c b f c b c (1,-1) 1 (1,-1) (-1,1) -1 1 (-2,2) f c f c (-1,1) (2,-2) (-1,1) (-2,2) σ 1 is an iteratively dominated strategy if there exists σ 1 ' such that u 1 ( σ 1 , σ 2 , σ 3 , ... ) ≤ u 1 ( σ 1 ' , σ 2 , σ 3 , ... ) for all non-iteratively dominated σ 2 , σ 3 , ... u 1 ( σ 1 , σ 2 , σ 3 , ... ) < u 1 ( σ 1 ' , σ 2 , σ 3 , ... ) for some non-iteratively dominated σ 2 , σ 3 , ...

  30. Domination C QJ QK 0.5 0.5 1 1 c b c b 2 2 2 2 c b f c b c (1,-1) 1 (1,-1) (-1,1) -1 1 (-2,2) f c f c (-1,1) (2,-2) (-1,1) (-2,2) b is iteratively dominated by σ 1 cc σ 1

  31. Domination C QJ QK 0.5 0.5 1 1 c b c b 2 2 2 2 c b f c b c (1,-1) 1 (1,-1) (-1,1) -1 1 (-2,2) f c f c (-1,1) (2,-2) (-1,1) (-2,2) Define an iteratively dominated action to be an action such that any strategy that always plays that action is iteratively dominated (assuming that player plays to reach that action).

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