In a strategic game, a strategy of player i is justifiable if it is a best response to some possible (mixed) strategy (example 1.98) of the other players, that is,
si is justifiable ⇔ si ∊ Bi (∑-i)
where ∑-i is the set of mixed strategies of the opposing players (example 1.110). Let B1i denote the set of justifiable strategies of player i. Then
B1i = Bi(∑-i
A strategy of player i is rationalizable if it is justifiable using a belief that assigns positive probability only to strategies of j ≠ i that are justifiable, if these strategies are justified using beliefs that assign positive probability only to justifiable strategies of i, and so on. To formalize this definition, define the sequence of justifiable strategies
Bni = Bi(Bn-1-1
The set of rationalizable strategies for player i is Ri ∩∞n=0 Bni. That is, the set of rationalizable strategies is those that are left after iteratively discarding unjustified strategies. Show that when S is compact and ⊁i is continuous, there are rationalizable strategies for every game, that is, Ri∅.