Discrete two player all-pay auction with complete information

We study discrete two player all-pay auction with complete information. We provide full characterization of mixed strategy Nash equilibria and show that they constitute a subset of Nash equilibria of discrete General Lotto game. We show that equilibria are not unique in general but they are interchangeable and sets of equilibrium strategies are convex. We also show that equilibrium payoffs are unique, unless valuation of at least one of the players is an even integer number. If equilibrium payoffs are not unique, continuum of equilibrium payoffs are possible.