  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. 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. We compare equilibrium
characterization and equilibrium payoffs with the continuous variant of all-pay
auctions.
