Rationality and Common Belief in Rationality in Brandenburger & Dekel (1987)

Question

One of the fundamental results in epistemic game theory is that the solution concept of correlated rationalizability gives exactly those action profiles that are compatible with rationality and common belief in rationality. A precise statement and formulation of this result is given in

Tan, Tommy Chin-Chiu, and Sérgio Ribeiro da Costa Werlang. "The Bayesian foundations of solution concepts of games." Journal of Economic Theory 45.2 (1988): 370-391.

as Theorem 5.2 and Theorem 5.3. An alternative reference often cited for this result (at least in the context of finite games, Tan & Werlang allow for compact metric action spaces) is

Brandenburger, Adam, and Eddie Dekel. "Rationalizability and correlated equilibria." Econometrica: Journal of the Econometric Society (1987): 1391-1402.

For example, the survey on epitemic game theory in the fourth volume of the handbook of game theory credits Brandenburger & Dekel for this result (online version, see Theorem 1 there). I have actually seen many such references but was not able to locate the result in their paper. That paper contains 4 propositions and none of them corresponds to this result. The authors actually credit Tan & Werlang and write "Tan and Werlang (1984) and Bernheim (1985) provide formal proofs of the equivalence between rationalizability and common knowledge of rationality." (Tan & Werlang 1984 is the working paper version).

What am I missing that everyone else gets?

Answer 1

The concept that Brandenburger and Dekel (1987) call an "a posteriori equilibrium" is roughly the same as what Dekel and Siniscalchi call an "epistemic type structure for a complete information game" in which all types are rational and there is common belief in rationality. Therefore, Brandenburger and Dekel's Proposition 2.1, together with the remark that immediately follows the proof of Propoistion 2.1, is roughly the same as Theorem 1 in Dekel and Siniscalchi.