Common priors under endogenous uncertainty

For a fixed game and a type structure that admits a common prior, Action Independence states that the conditional beliefs induced by the common prior do not depend on the players' own strategies. It has been conjectured that Action Independence can be behaviorally characterized by means of a suitable no-betting condition (Dekel and Siniscalchi, 2015), but whether this is indeed the case remains an open problem. In this paper, we prove this conjecture true by focusing on strategy-invariant bets, which are bets that cannot be manipulated by the players. In particular, first, we show that at least one of the common priors satisfies Action Independence if and only if there exists no mutually acceptable strategy-invariant bet among the players. Second, we show that, all common priors satisfy Action Independence if and only if there exists no mutually acceptable strategy-invariant bet among the players and an outside observer. These results give us a deeper understanding of existing foundations of solution concepts using only epistemic conditions that are expressed in terms of type structures and are therefore elicitable.