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 The Game of Morra: Payoff Matrix [1,1] [1,2] [2,1] [2,2] [1,1][1,2][2,1][2,2] AB x i – probability that row i is selected by row player y j – relative frequency with which column j is selected by column player – X and Y are stochastic vectors y 1 y 2 y 3 y 4 y = [] x1x2x3x4x1x2x3x4 x =

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 c 1, c 2, c 3, c 4 rounds c 1 + c 2 +c 3 +c 4 = N, where N is total number of rounds Record of the game –In c 1 /2 rounds, A played [1,2] and B played [1,1]: A losing 2 francs –In c 1 /2 rounds, A played [2,1] and B played [1,1]: A winning 3 francs –In c 4 /2 rounds, A played [1,2] and B played [2,2]: A winning 3 francs –In c 4 /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 : ( c 1 – c 4 )/2 francs What if the roles of A and B are swapped?

8 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 a 11 a ij ……. a mn Possible moves of the row player 12i..m12i..m Possible moves of the column player 1 2 … j … n The resulting payoff of the row player

9 Problem Statement Given the payoff matrix A = [a ij ], –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 LP Formulation of Matrix Games x i – probability that row i is selected by row player y j – 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 or

11 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., or

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

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 Gambling: Payoff Matrix and LP Solution 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? [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 Bluffing, Underbidding: Pure Strategies A’s strategies 1.Pass; if B bets, pass again 2.Pass; if B bets, bet again 3.Bet 3x3x3 pure strategies x 1 x 2 x 3 – strategy for A instructing him to follow line x j when holding j B’s strategies 1.Pass no matter what A did 2.If A passes, pass; if A bets, bet 3.If A passes, bet; if A bets, pass 4.Bet no matter what A did 4x4x4 pure strategies y 1 y 2 y 3 – strategy for B Payoff matrix size: 27x64! 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 Payoff matrix size: 8x4!

18 Bluffing, Underbidding: Payoff Matrix and LP Solution /6 -1/ /6 -1/3 -1/ /6 -1/6 1/6 1/ / / /6 -1/3 0 -1/ /6 -1/6 -1/6 -1/ /2 1/3 -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 Holding 1:BLUFF A is allowed to bet 1/6 th times! B is allowed to bet 1/3 rd times! A is allowed to pass 1/2 times! Holding 3:UNDERBID

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20 LP Formulation of Matrix Games: Identity (15.1) min y xAy = min j  i m a ij x i –It is trivial that min y xAy <= min j  i m a ij x i –Now, we show min y xAy >= min j  i m a ij x i –Let t = min j  i m a ij x i, thus we have xAy =  j n y j ( i m a ij x i ) >=  j n y j t = t