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Tuomas Sandholm, Andrew Gilpin Lossless Abstraction of Imperfect Information Games Presentation : B94902042 趙峻甫 B94902064 蔡旻光 B94902066 駱家淮 B95902095 李政緯 B95902098 周于荃

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Outline Introduction of Rhode Island Hold’em and Lossless abstraction Extensive form game Ordered game isomorphic abstraction transformation /Game shrink Proof of correctness Time complexity analysis

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Rhode Island Hold’em Taxes-Hold’em(Poker) is a large, well-structured, zero-sum multiplayer game of imperfect information Two player Texas-Hold’em has a game tree of O( ) nodes, too big to be solved using conventional Linear Programming techniques. Rhode Island Hold’em is a reduced game from commonly played Taxes-Hold’em, with limited bet, less cards dealt to each player. The author is aim to create a Taxes-Hold’em AI, but begin with a reduced form of the original game.

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Rhode Island Hold’em Each player pays an ante of 5 chips. Each player is dealt a single card, placed face down. This is the player's hole card. After receiving the hole cards, the players take part in one betting round. Each player may check, or bet if no bets have been placed. If a bet has been placed, then the player may 1. fold (thus forfeiting the game) 2. call (adding chips to the pot equal to the last player's bet 3. raise (calling the current bet and making an additional bet). In Rhode Island Hold ‘em, the players are limited to 3 raises per betting round. In this betting round, the bets are 10 chips.

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Rhode Island Hold’em After the betting round, a community card is dealt face up. This is called the flop. Another betting round take places at this point, with bets equal to 20 chips. Another community card is dealt face up. This is called the turn card. Another (and final) betting round takes place at this point, with bets equal to 20 chips. If neither player folds, then the showdown takes place. Both players turn over their cards. The player who has the best 3- card poker hand takes the pot. In the event of a draw, the pot is split evenly.

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Ranking of Rhode Island Hold’em

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Games With Ordered Signal A game can be explained by the tuple as follow: Γ = I = ｛ 1, …, n ｝ is a finite set of players. G =, = ( ), is a finite collection of finite directed trees with nodes and edges. Let denote the leaf nodes of and let denote the outgoing neighbors of v ∈. is the stage game for round j. L =,, indicates which player acts (choose an outgoing edge),at each internal node in round j. Θ is a finite set of signals.

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Games With Ordered Signal κ =, γ = are vectors of nonnegative integers, where and denotes the number of public and private signals ( per player ), respectively revealed in round j. Each signal θ ∈ Θ may only be revealed once. In each round, every player receives the same number of private signals. Then we require.

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Games With Ordered Signal Some important notation to describe signals is the public information revealed in round j. is all the public information revealed up through round j. is the private information revealed to player i ∈ I in round j. is all the private information revealed to player i ∈ I up through round j. is with replaced with. is said to be legal if no signals are repeated.

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Games With Ordered Signal p is a probability distribution over Θ, with p( θ ) > 0 for all θ ∈ Θ. Signals are drawn from Θ according to p, so if X is the set of signals already revealed, then is a partial ordering of subsets of Θ and is defined for at least those pairs required by u. ω ： → ｛ over, continue ｝, is a mapping of terminal nodes within a stage game. Clearly, we need ω (z)=over for all z ∈. And we define,,.

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Games With Ordered Signal u = is a utility function such that for every j,,for i ∈ I. For every, at least one of the following two conditions holds. (A) Utility is signal independent: for all legal,. (B) is define for all legal signals ( ), and ( ) through round j and a player’s utility is increasing in her private signals, every else equal:

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