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Introduction to Mixed Strategy Games: Understanding Non-Strictly Determined Matrix Games

Explore strategies, probabilities, and expected values in non-strictly determined games. Learn how mixed strategies impact gameplay decisions in scenarios like the penny matching game. Dive into matrix calculations and the fundamental theorem of game theory. Discover the solution to a 2x2 non-strictly determined matrix game.

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Introduction to Mixed Strategy Games: Understanding Non-Strictly Determined Matrix Games

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  1. 9.2 Mixed Strategy Games In this section, we look at non-strictly determined games. For these type of games the payoff matrix has no saddle points.

  2. Non-strictly determined matrix games • A strategy consisting of possible moves and a probability distribution (collection of weights) which corresponds to how frequently each move is to be played. A player would only use a mixed strategy when she is indifferent between several pure strategies, and when keeping the opponent guessing is desirable - that is, when the opponent can benefit from knowing the next move.

  3. Penny matching game • Two players R and C each have a penny , and they simultaneously choose to show the side of the coin of their choice. (H = heads, T = tails) If the pennies match, R wins ( C loses) 1cent. If the pennies do not match, R loses (C wins) 1 cent. In terms of a game matrix , we have

  4. Because there is no saddle point for the penny-matching game, there is no pure strategy for R. We will assign probabilities corresponding to the likelihood that R will choose row 1 or row 2. Similarly, probabilities will be found for C’s likelihood of choosing column one or column 2. The strategy will be to choose the row with a probability that will yield the largest expected value. • Determination of Strategy

  5. Expected Value of a Matrix Game For R • For the matrix game • and strategies • for R and C, respectively, the expected value of the game for R is given by

  6. Fundamental Theorem of Game Theory (The number v is the value of the game. If v = 0, the game is said to be fair. ) For every m x n matrix game , M , there exists strategies and for R and C , respectively, and a unique number v such that for every strategy Q of C and for every strategy P of R.

  7. Solution to a 2 x 2 Non-strictly Determined Matrix Game • For the non-strictly determined game • the optimal strategies and and the value of the game are given by • = • = • where

  8. Original problem : • 1. Identify a,b,c,d : a = 1, b = -1, c= -1, d = 1. • 2. Find D: (a+d)-(b+c)=4 (not zero) • 3. The value of the game is v = • v = 0 so it is a fair game • 4. Find = = [ 0.5 , 0.5] • 5. Find = = • 6. Find the expected value of game: E(P,Q)=PMQ=0

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