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Matrix Games

Matrix Games. Mahesh Arumugam Borzoo Bonakdarpour Ali Ebnenasir CSE 960: Selected Topics in Algorithms and Complexity Instructor: Dr. Torng. Outline. Basic concepts Problem statement LP Formulation of Matrix Games Minimax Theorem Gambling Bluffing and Underbidding. Basic Concepts.

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Matrix Games

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  1. Matrix Games Mahesh Arumugam Borzoo Bonakdarpour Ali Ebnenasir CSE 960: Selected Topics in Algorithms and Complexity Instructor: Dr. Torng

  2. Outline • Basic concepts • Problem statement • LP Formulation of Matrix Games • Minimax Theorem • Gambling • Bluffing and Underbidding

  3. Basic Concepts • Game: A description of strategic interaction between rationale parties based on a set of rules • Rules: Constraints on the set of actions that each party can take and the players’ interest • Finite Game: Set of actions of each player is finite • Two-Player Game: There exist only two players [OR94] Osborne and Rubinstein, A Course in Game Theory, MIT press, 1994.

  4. Example:The Game of Morra • Rule: • Each player hides one or two francs, and • Tries to guess how many francs the other player has hidden • Payoff: • If only one player guesses correctly • he wins the total amount of hidden money • Otherwise, the result is a draw

  5. The Game of Morra: Pure Strategies • Possible courses of action for each player • Hide one, guess one  [1, 1] • Hide one, guess two  [1, 2] • Hide two, guess one  [2, 1] • Hide two, guess two  [2, 2] • Pure strategy: a course of action • Denoted [x,y]; i.e., hide x, guess y

  6. y1 y2 y3 y4 y = [ ] x1 x2 x3 x4 x = The Game of Morra: Payoff Matrix A B [1,1] [1,2] [2,1] [2,2] 0 2 -3 0 [1,1] [1,2] -2 0 0 3 [2,1] 3 0 0 -4 [2,2] 0 -3 4 0 • xi – probability that row i is selected by row player • yj – relative frequency with which column j is selected • by column player • X and Y are stochastic vectors

  7. The Game of Morra - Cont’d • A only plays [1,2] or [2,1] with probability 0.5 • B plays • [1,1] , [1,2], [2,1], [2,2] in c1, c2, c3, c4 rounds • c1+ c2+c3 +c4 = N, where N is total number of rounds • Record of the game • In c1/2 rounds, A played [1,2] and B played [1,1]: A losing 2 francs • In c1/2 rounds, A played [2,1] and B played [1,1]: A winning 3 francs • In c4/2 rounds, A played [1,2] and B played [2,2]: A winning 3 francs • In c4/2 rounds, A played [2,1] and B played [2,2]: A losing 4 francs • Other rounds, result in a draw • Total winning of A : (c1 – c4)/2 francs What if the roles of A and B are swapped?

  8. The resulting payoff of the row player Basic Concepts - Cont’d • Round: a course of actions in which each player moves once • Payoff: the value gained by a player in a round • The Payoff Matrix defines a game for two players • Zero-sum game: The sum of the average payoffs of the two players is 0 Possible moves of the column player Possible moves of the row player 1 2 … j … n ……. a11 1 2 i . . m ……. aij ……. amn

  9. Problem Statement Given the payoff matrix A = [aij ], • identify a mixture of moves of the row player where the average payoff per round is optimal no matter what moves the column player takes

  10. or LP Formulation of Matrix Games xi – probability that row i is selected by row player yj – relative frequency with which column j is selected by column player • X and Y are stochastic vectors • Average payoff to the row player in each round

  11. s.t., or LP Formulation of Matrix Games - Cont’d • If row player adopts the strategy specified by stochastic vector x, he is assured to win = • The objective is to maximize this payoff s.t.,

  12. D s.t., s.t., LP Formulation of Matrix Games - Cont’d • What is the dual of this problem? P • What does this problem formalize? Column player’s optimal strategy and the value he is assured to win if he adopts such a strategy!

  13. Minimax Theorem For every m n matrix A there is a stochastic row vector x* of length m and a stochastic column vector y* of length n such that min x*Ay = max xAy* with the minimum taken over all stochastic column vectors y of length n and maximum taken over all stochastic row vectors x of length m. Value of game In a game, v = min x*Ay = max xAy*is called the value of that game. What are the implications of this theorem?

  14. Ready for Gambling?!! • As long as a player adopts an optimal strategy, the player can reveal it to the opponent • Example: (The Game of Morra) • column player announces his/her guess • row player announces his/her guess either independent of the opponent or adjust his/her guess based on the extra information • Additional pure strategies for row player • Hide 1, make the same guess  [1, S] • Hide 1, make a different guess  [1, D] • Hide 2, make the same guess  [2, S] • Hide 2, make a different guess  [2, D]

  15. Consider the optimal solution x=[0, 56/99, 40/99, 0, 0, 2/99, 0, 1/99] y=[28/99, 30/99, 21/99, 20/99] Game value = 4/99 row player is assured to win at least this amount on the average column player is assured to lose no more than this amount on the average Do you think this game is fair? What does this suggest? Gambling:Payoff Matrix and LP Solution [1,1] [1,2] [2,1] [2,2] Revealing the guess does not hurt the prospects for the column player!!

  16. How about Bluffing or Underbidding? • Are bluffing or underbidding rational strategies? • Example: (Game invented by H. W. Kuhn) • 2 players, deck of cards numbered 1, 2, or 3 • Each player bets or passes in every round • Play terminates when • Bet is answered by bet; payoff 2 to player holding higher card • Pass is answered by pass; payoff 1 to player holding higher card • Bet is answered by pass; payoff 1 to the player who bets

  17. A’s strategies Pass; if B bets, pass again Pass; if B bets, bet again Bet 3x3x3 pure strategies x1x2x3 – strategy for A instructing him to follow line xj when holding j B’s strategies Pass no matter what A did If A passes, pass; if A bets, bet If A passes, bet; if A bets, pass Bet no matter what A did 4x4x4 pure strategies y1y2y3 – strategy for B Bluffing, Underbidding: Pure Strategies Payoff matrix size: 27x64! Payoff matrix size: 8x4! Holding 1: A – refrain line 2; B – refrain lines 2 and 4; Holding 3: A – refrain line 1; B – refrain lines 1, 2 and 3; Holding 2: choose to pass in the first round; lines 1 or 2

  18. 114 124 314 324 11 2 0 0 -1/6 -1/6 113 0 1/6 -1/3 -1/6 122 -1/6 -1/6 1/6 1/6 123 -1/6 0 0 1/6 312 1/6 -1/3 0 -1/2 313 1/6 -1/6 -1/6 -1/2 322 0 -1/2 1/3 -1/6 323 0 -1/3 1/6 -1/6 Consider the optimal solution A: [1/3, 0, 0, 1/2, 1/6, 0, 0, 0] B: [2/3, 0, 0, 1/3] Game Value = -1/18 Bluffing, Underbidding: Payoff Matrix and LP Solution Holding 1: BLUFF A is allowed to bet 1/6th times! B is allowed to bet 1/3rd times! Holding 3: UNDERBID A is allowed to pass 1/2 times!

  19. Thank U!

  20. LP Formulation of Matrix Games: Identity (15.1) miny xAy = minjimaij xi • It is trivial that miny xAy <= minjimaij xi • Now, we show miny xAy >= minjimaij xi • Let t = minjimaij xi , thus we have xAy = jn yj (imaij xi) >= jn yj t = t

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