On the existence of a "complete" possibility structure: Adam Brandenburger
Fix a game between two players, Ann and Bob. A very basic idea in game theory is that ofcommon beliefofrationality. Ann and Bob are both rational, Ann believes Bob is rational, Bob believes Ann is rational, Ann believes Bob believes she (Ann) is rational, and so on indefinitely. Now, under the usual definition, a rational player is one who chooses a strategy that is optimal, given his or her belief about the other players' strategies. So, to talk about the rationality of Ann and Bob, we have to talk about what each believes (about the other's strategy). To talk about Ann's belief in Bob's rationality, we have to talk about what Ann believes about what Bob believes (about her strategy). And so on.