mixed-strategies - Ecos3012 - Strategic Behavior Semeter 1,...

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Unformatted text preview: Ecos3012 - Strategic Behavior Semeter 1, 20011 Lecture Note on Mixed Strategies Note. Complement the reading of the Chapter titled “Mixed Strategy Nash Equilibrium” from Watson’s text with this lecture note. 1 Mixed Strategies We have so far dealt with a number of solution concepts: IEDS, Rationalizability and Nash Equilibrium. Even with the most general of them – Nash Equilibrium — an equilibrium may fail to exist. Indeed, one need only consider a simple example such as the one shown below: L R T 2,-1-1,1 B-1,2 2,-1 Table 1: No Nash equilibrium in pure straegies (Partly) to overcome the issue of existence of an equilibrium, we enlarge the strategy space of a player. In the above instance, instead of assuming that P1 can choose either T or B, we may entertain the possibility that she may randomize between these two strategies. Thus, a strategy for P1 now becomes a number p that lies between 0 and 1 (inclusive) and it represents the probability with which P1 plays the strategy T. Correspondingly, 1- p is the probability with which she would then play the strategy B. Similarly we may enlarge the strategy space of P2 by saying that she chooses a number q which denotes the probability of her choosing L. Thus, we have transformed the strategy set of P1 from { T,B } to the interval [0 , 1] and that of P2 from { L,R } to [0 , 1]. The original strategies { T,B } are referred to as pure strategies of P1 whereas ( p, 1- p ), which gives the probability with which each of the pure strategies are played is said to be a mixed strategy. More generally, if P1 has m pure strategies labeled s 1 ,s 2 ,...,s m then a mixed strategy of P1 is a probability distribution p = ( p 1 ,...,p m ), an m-dimensional vector where p i denotes the probability that P1 plays strategy s i . Similarly, if P2 has n pure strategies labeled t 1 ,...,t n , then q = ( q 1 ,...,q n ) denotes a typical mixed strategy of P2. Observe that when p i * = 1 and p i = 0 for all i 6 = i * , it means that P1 plays the strategy s i * with probability 1, which is the same as saying P1 plays the pure strategy s i * . So, a pure strategy is nothing but a particular mixed strategy. Having enlarged the strategy spaces, we will now address in sequence the issue of extending the payoffs from pure strategies to mixed strategies, computing the best responses & reaction functions and finally using those to figure out the Nash equilibria. 1 2 Payoff from Mixed Strategies Our interest here is to figure out U i ( p , q ), the payoff that player i gets when the mixed strategy ( p , q ) is played. In order to do this, we first recall that a player satisfies the Expected Utility Hypothesis. Next, as usual u 1 ( s i ,t j ) denote the payoff that P1 gets when the pure strategy profile is played out. So, if P2 were to play the mixed strategy it is as if P1 is choosingstrategy profile is played out....
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This note was uploaded on 08/20/2011 for the course ECON 101 taught by Professor Etw during the Spring '11 term at Università di Bologna.

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mixed-strategies - Ecos3012 - Strategic Behavior Semeter 1,...

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