Iterative elimination of dominated strategies A strategy S; of player i is strictly dominated if there is another strategy available to i which yields strictly higher payoffs regardless of the strategies chosen by the rivals of i. more formally, a strategy of player i is strictly dominated if there exists another strategy si in Si such that for al possible S i, Ti (si, S-i >Tisi, S-i. this also means that strategy s; strictly dominates strategy si On the basis of iterative elimination of strictly dominated strategies, We can predict the outcomes of the game In many games there will be no strictly dominated strategies
Iterative elimination of dominated strategies • A strategy si of player i is strictly dominated if there is another strategy available to i which yields strictly higher payoffs regardless of the strategies chosen by the rivals of i. More formally, a strategy of player i is strictly dominated if there exists another strategy si ’ in Si such that for all possible s-i , πi (si ’, s-i )> πi (si ’, s-i ). This also means that strategy si ’ strictly dominates strategy si . • On the basis of iterative elimination of strictly dominated strategies, we can predict the outcomes of the game. • In many games there will be no strictly dominated strategies
C 4.3 5.1 62 R R2|2,1 3.4 3.6 R33,0 9.6 2.8 Normal form of a strategic game with no strictly dominated strategies
4,3 5,1 6,2 2,1 3,4 3,6 3,0 9,6 2,8 R1 R2 R3 C1 C2 C3 Normal form of a strategic game with no strictly dominated strategies
R 8.10 100.9 D|7,6 6.5 Rationality and strictly dominated strategies R U|2,0 35 4,4 D|03 52 No strictly dominated strategies
8,10 -100,9 7,6 6,5 2,0 3,5 0,3 2,1 Rationality and strictly dominated strategies 4,4 5,2 No strictly dominated strategies U D L R U D L M R
Rationalizable strategies(1) In a game-theoretic situation, a player's payoffs depends on what her rivals do When doing what to do, a player will have to make a conjecture or prediction about what she thinks her rival will do. on the basis of this prediction or belief about what the rival will do, a rational player will then choose her payoff-maximizing strategy. The approach inherent in iteratively eliminating strictly dominated strategies was to identify rivals strategies that would not be rational for them to play (because they are strictly dominated)and therefore would not be reasonable predictions
Rationalizable strategies(1) • In a game-theoretic situation, a player’s payoffs depends on what her rivals do. • When doing what to do, a player will have to make a conjecture or prediction about what she thinks her rival will do. On the basis of this prediction or belief about what the rival will do, a rational player will then choose her payoff-maximizing strategy. • The approach inherent in iteratively eliminating strictly dominated strategies was to identify rivals’ strategies that would not be rational for them to play (because they are strictly dominated) and therefore would not be reasonable predictions
Rationalizable strategies(2) For the same reason, a rational player should only play a best response. The strategy si is a best response for player i to s- i if TT ( Si, S i 2 T(si, s i for all Si in S. This simply says that there is some strategy profile of is rivals for which S is the best choice. If player i believed that her rivals were going to play S-, then si would be her best choice. It seems reasonable that rational players would not play a strategy that is never a best response. A player could always increase her payoff by playing a best response Rationalizable strategies are justifiable on the basis that a player's conjecture or belief about what her rivals wi conjecture is that the rival will always play a best o do is reasonable-where reasonable means that the response and would not use a strategy that is never a best response
Rationalizable strategies(2) • For the same reason, a rational player should only play a best response. The strategy si is a best response for player i to s-i if πi (si , s-i )≥ πi (si ’ , s-i ) for all si ’ in Si . This simply says that there is some strategy profile of i’s rivals for which si is the best choice. If player i believed that her rivals were going to play s-I , then si would be her best choice. It seems reasonable that rational players would not play a strategy that is never a best response. A player could always increase her payoff by playing a best response. • Rationalizable strategies are justifiable on the basis that a player’s conjecture or belief about what her rivals will do is reasonable-where reasonable means that the conjecture is that the rival will always play a best response and would not use a strategy that is never a best response