Rationalizable Strategic Behavior - Bernheim, B. Douglas (Econometrica, 1984)
This paper examines the nature of rational choice in strategic games.Although there are many reasons why an agent might select a Nash equilibrium strategy in a particular game, rationality alone does not require him to do so.A natural extension of widely accepted axioms for rational choice under uncertainty to strategic environments generates an alternative class of strategies, labelled "rationalizable."It is argued that no rationalizable strategy can be discarded on the basis of rationality alone, and that all rationally justifiable strategies are members of the rationalizable set.The properties of rationalizable strategies are studied, and refinements are considered.PROOF OF PROPOSITION 5.5: Since under the specified conditionsf(-) is a contraction mapping, Nash equilibrium is unique.We need only show P(G) = N*(G).Since P(G) is the intersection of an infinite sequence of compact, nested sets, it is compact.Consequently, we can define di = max d(s,, s,).s,,s, E-P,(G) Assume without loss of generality that d, > di Vi > 1.If point rationalizable strategies are not unique, then di > 0. Let s' and sj' be the strategies for which d(s'1, sj') = dl.There must exist t', t' e P(G)such that fl(t') = sj, and fl(t") = sj', with g,(t') = v1(t") (the first component doesn't effect fl( )).Now d(t', t") < ( d2 ) < d1(I-1)1/2. (i=2 J Further, d(f(t'), f(t")) > d(s', sj') = dl.So d(f(t), ft"))> d(t, t"I(I 1 ,/2