Understanding Some Recent Existence Results for Discontinuous Games∗

We introduce a new condition, weak better-reply security, and show that every compact, locally convex, metric, quasiconcave and weakly better-reply secure game has a Nash equilibrium. This result is established using simple generalizations of classical ideas. Furthermore, we show that, when players’ action spaces are metric and locally convex, it implies the existence results of Reny (1999) and Carmona (2009) and that it is equivalent to a recent result of Barelli and Soza (2009). Our general existence result also implies a new existence result for weakly upper reciprocally semicontinuous and weakly payoff secure games that satisfy a strong quasiconcavity property. Journal of Economic Literature Classification Numbers: C72