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Berkeley ECON 101A - Lecture Notes on Mixed Strategy Equilibria

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Economics 101A Lecture Notes on Mixed Strategy EquilibriaSteve GoldmanApril 19, 2001Abstract. This note describes the construction of a mixed strategyequilibrium in a simple 2x2 game.1. Mixed Strategy EquilibriaConsider a two player game with a payo® matrixBA2641 21 a11; b11a12; b122 a21; b21a22; b22375where players A amd B may each play actions (1) or (2). Thus, if A chooosesaction (i) and B chooses (j) then A will receive an outcome aijand B will receive bij:Suppose that A plays the mixed strategy of choosing (1) with probability ® and (2)with probability (1 ¡ ®), and B plays (1) with probability ¯ and (2) with probability(1 ¡ ¯) . Then the probabilities of the outcomes are described by the matrix:BA2641 21 ®¯ ®(1 ¡ ¯)2 (1 ¡ ®)¯ (1 ¡ ®)(1 ¡ ¯)375The expected payo® to player A is then¦A(®; ¯) = ®¯a11+ ®(1 ¡ ¯)a12+ (1 ¡ ®)¯a21+ (1 ¡ ®)(1 ¡ ¯)a22and to player B¦B(®; ¯) = ®¯b11+ ®(1 ¡ ¯)b12+ (1 ¡ ®)¯b21+ (1 ¡ ®)(1 ¡ ¯)b22An equilibrium describes a pair of probabilities (®¤; ¯¤) such that¦A(®¤; ¯¤) ¸ ¦A(®; ¯¤) for all ® 2 [0; 1]1Economics 101A Lecture Notes on Mixed Strategy Equilibria 2and¦B(®¤; ¯¤) ¸ ¦A(®¤; ¯) for all ¯ 2 [0; 1]:Now if B chooses probability ¯¤then the best that A can do is as follows:if the derivative of his payo® with respect to ® is positive then he should make® = 1 and if it is negative then ® = 0. If the payo® is insensitive to ®; then any valueof ® between 0 and 1 will do equally well.So if¯¤a11+ (1 ¡ ¯¤)a12¡ ¯¤a21¡ (1 ¡ ¯¤)a22> 0 then ® = 1,¯¤a11+ (1 ¡ ¯¤)a12¡ ¯¤a21¡ (1 ¡ ¯¤)a22< 0 then ® = 0.Similarly for B if®¤b11¡ ®¤b12+ (1 ¡ ®¤)b21¡ (1 ¡ ®¤)b22> 0 then ¯ = 1,®¤b11¡ ®¤b12+ (1 ¡ ®¤)b21¡ (1 ¡ ®¤)b22< 0 then ¯ = 0.Equating the derivative of A's payo® function w.r.t. ® to zero lets us determinethat value for ¯ at which A is indi®erent about her choice for ® and we ¯nd¯¤=a22¡ a12(a11¡ a12¡ a21+ a22)Similarly, there is a critical value for ® where B will be indi®erent to the choiceof ¯:®¤=b22¡ b21(b11¡ b12¡ b21+ b22)If both ®¤and ¯¤lie in the interval [0; 1] then they consitute a mixed strategyequilibrium. If either lies outside of that interval then that player will choose an actionwith certainty, e.g. if ¯¤< 0 then A will choose ® = 1 if (a11¡ a12¡ a21+ a22) > 0or ® = 0 if (a11¡ a12¡ a21+ a22) < 0. If ¯¤> 1 then the reverse conditions are true.In the event that (a11¡ a12¡ a21+ a22) = 0, then ® = 1 if a12¡ a22> 0 and ® = 0 ifa12¡ a22< 0. Similar conditions can be easily stated for ®¤and the choice of ¯.Examples:. Coordination GameP layer 2Ball Game MovieP layer 1 Ball Game 2; 1 0:5; 0:5Movie 0; 0 1; 2Player 1's payo® = ®¯2 + ®(1 ¡ ¯)0:5 + (1 ¡ ®)(1 ¡ ¯)Economics 101A Lecture Notes on Mixed Strategy Equilibria 3and Player 2's payo® = ®¯ + ®(1 ¡ ¯)0:5 + (1 ¡ ®)(1 ¡ ¯)2Setting the derivatives equal to 0 and solving:¯2 + (1 ¡ ¯)0:5 ¡ (1 ¡ ¯) = 0 and ® ¡ ®0:5 ¡ (1 ¡ ®)2 = 0so® = 0:8; ¯ = 0:2and the payo®s are 0:8 for each player. Note here that there are three equilibria - twoare pure strategy (one at the ball game and the other at the movie) and the third ismixed.Stackelberg Game:P layer 2H LP layer 1 H 0; 0 1800; 900L 900; 1800 1600; 1600There are two equilirbia at (L; H) and (H; L). But there is also a mixed strategyequilibrium at ® = ¯ =211which yields pro¯ts of 1472.7272... each or join pro¯ts of2945.45...


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