We consider the class of two-player symmetric n x n games with the total bandwagon property (TBP) introduced by Kandori and Rob (1998). We show that a game has TBP if and only if the game has 2^n - 1 symmetric Nash equilibria. We extend this result to bimatrix games by introducing the generalized TBP. This sheds light on the (wrong) conjecture of Quint and Shubik (1997) that any n x n bimatrix game has at most 2^n - 1 Nash equilibria. As for an equilibrium selection criterion, I show the existence of a ½-dominant equilibrium for two subclasses of games with TBP: (i) supermodular games; (ii) potential games. As an application, we consider the minimum-effort game, which does not satisfy TBP, but is a limit case of TBP. (author's abstract) / Series: Department of Economics Working Paper Series