A two-sided reputation result with long-run players

Cripps et al. (2005) conjectured that in an infinitely repeated game with two equally patient players, if there is positive probability that the players could be Stackelberg types, then equilibrium behavior would resemble a war of attrition, i.e., a two-sided reputation result would hold. In this note we show that this conjecture is indeed true for a wide set of stage games for which the one-sided reputation result of Atakan and Ekmekci (2008) holds.