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

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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
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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 choosing
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