求解一道博弈证明题

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收藏 2012-12-04

Prove the following lemma: Let G be a finite strategic game, and let G` be a strategic game obtained by eliminating a weakly dominated action in G. If a* is a NE(nash equilibrium) of G`, then it is also a NE(nash equilibrium) of G.

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vesperw

2012-12-4 12:07:32

Let Ai be the set of pure strategies of player i in game G. Ai` be the set of pure strategies of game G`. Note that Ai \Ai`=/=empty for some i since some weakly dominated strategies are eliminated from G to G`.

Now a* is a NE in G`, by definition , for all i and ai in Ai`, ui(ai*, a-i*)>= ui (ai, a-i*). For any si in Ai \ Ai`, since si is weakly dominated, there exists some bi in Ai` such that ui(bi, a-i*)>=ui(si,a-i*). But then ui(ai*, a-i*)>=ui(si,a-i*). Thus ai* is still a best response to a-i* in the original game G. This argument is true for all other players, so a* is a NE in G