Equilibria in a Japanese-English auction with discrete bid levels for the wallet game

We consider the set-up of a Japanese-English auction with exogenously fixed discrete bid levels for the wallet game with two bidders. We prove that bidding twice the signal - the equilibrium strategy with continuous bid levels - is never an equilibrium in this set up. We show that partition equilibria exist that may be separating or pooling. We illustrate some separating and pooling equilibria with two and three discrete bid levels; we also compare the revenues of the seller from these equilibria and thereby find the optimal bid levels in these cases.