Question

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1. Consider the following two-player game in strategic form: T4,5 3,0 0,2 M 5,2 2, 1,0 B0,02,84,2 (a) What strategies are rationalizable? (b) What strategies survive the iterative elimination of strictly dominant strategies? (c) What strategies are ruled out by the assumption of rationality alone (i.e, without the assumption of common knowledge)? (d) Find all pure-strategy nash equilibria.

Answer 1

(a) To find out the rationalizable strategies , we need to eliminate those strategies which are never the best responses . Player 1 assumes player 2 is rational and player 2 assumes player 1 is rational . In the above question , there is no strategy for any player which is never the best response . T is the best response for C , M is the best response for l and B is the best response for R and you can find for the player 2 as well . Thus all the strategies are rationalizable .

(b) In the iterative elimination , we eliminate the strictly dominated strategies one by one . For player 2 , there is no strategy which is strictly dominated any strategy and for the player 1 as well . You can see in this way , for player 2 in strategy l and R , pay offs for player 2 by choosing L are ( 5 , 2 , 0) and for R (2,0,2) so L does not strictly dominate R . check for others ( C  and R ) and ( L and C) and for the combinations of player 1 strategies as well .

(c) No strategies ruled out by the assumption of rationality .

(d) Pure strategy equilibrium : Those strategies which are best response of the other's player best response . In other words , no player has the incentive to deviate , if i say (T,l) is the pure strategy equilibrium , but player 1 has the incentive to deviate and choose m instead because 5 > 4 . Thus you can check all the strategies in this way and you will find there is only one pure strategy equilibrium and that is ( M , L )