Prove that abstaining is not a weakly dominated strategy.
�12
1 2
1 2
1 2
7 8
1 8
e
μ
3.5 Appendix: Strict and Weak Dominance CONSIDER A GAME WITH n players: 1, 2, . . ., n. Let Si denote player i’s strategy set, and read as “strategy is a member of ” Let be composed of (n � 1)-tuples of strategies for the n � 1 players other than player i. Finally, let Vi( , ) be the payoff of player i when his strategy is , and the other players use � ( , . . ., , , . . . ). Then we have the following definitions:
1. Strategy strictly dominates if and only if
In other words, yields a strictly higher payoff than , regardless of the strategies used by the other n � 1 players.
2. is the dominant strategy if and only if
That is, strictly dominates every other strategy for player i.
3. Strategy weakly dominates if and only if
That is, yields at least as high a payoff as for all strategies of the other players and yields a strictly higher payoff for some strategies of the other players.
3.6 Appendix: Rationalizability (Advanced) WE BEGAN THIS CHAPTER with the statement that a rational player would not use a strictly dominated strategy. Since rationality means choosing what is best, given expectations of other players’ strategies, and since the existence of a strictly dominated strategy implies the existence of another strategy that gives a strictly higher payoff regardless of what the other players’ strategies are, it logically follows that a rational player will not use a strictly dominated strategy. If, in addition, all players believe that all players are rational, then each player believes that no other player will use any strictly dominated strategies. This logic led us to eliminate strictly dominated strategies from
si¿s–i
Vi(si–, s�i) 7 Vi(s¿i, s�i) for some s�i � S�i. Vi(si–, s�i) � Vi(s¿i, s�i) for all s�i � S�i, and
si¿s–i
s–i
Vi(si–, s�i) 7 Vi(si, s�i) for all s�i � S�i, for all si si–.
s–i
si¿s–i
Vi(si–, s�i) 7 Vi(si¿, s�i) for all s�i � S�i.
si¿s–i
sn¿si¿�1si¿�1s1¿s�i¿ si¿s�i¿si¿
S�iSi.si¿s¿i � Si
84 CHAPTER 3: ELIMINATING THE IMPOSSIBLE: SOLVING A GAME WHEN RATIONALITY IS COMMON KNOWLEDGE
12. Derive all of the rationalizable strategies for the game shown in FIGURE PR3.12.
FIGURE PR3.12
0,4 1,1
3,2
1,1
0,0
2,2Player 1
Player 2
a
b
c
2,3
1,4
0,0
x y z