All-pay auctions with reserve price and bid cap

I characterize equilibria in an all-pay auction with a reserve price and a bid cap. I consider the cases of two and three players. I show that equilibrium bidding is characterized by atoms at 0, the reserve price, and the bid cap, as well as continuous bidding above the bid cap. If the valuations are high enough, the range for continuous bidding shrinks completely. I show that for some parameter ranges there exist multiple equilibria. Under three players, I show that there exist equilibria with the following features: the player with the non-top valuation can have a positive rent; the players with completely different valuations can actively compete for the single prize; the player with positive payoff can have different payoff in different equilibria.