Bribing in first-price auctions

I study a symmetric 2-bidder IPV first-price auction prior to which one bidder can offer his rival a bribe in exchange for the latterʼs abstention. I focus on pure and undominated strategies, and on continuous monotonic equilibria—equilibria in which the bribing function is continuous and nondecreasing. When types are distributed continuously on the unit interval, such an equilibrium, if it at all exists, is necessarily trivial—its bribing function is identically zero. I provide a sufficient condition for its existence and sufficient conditions for its nonexistence. When the minimum type is strictly positive, a non-trivial equilibrium may exist, but it must be pooling. I provide a sufficient condition for the existence of such an equilibrium. When types are distributed continuously on the unit interval and dominated strategies are allowed, a non-trivial non-pooling equilibrium exists, at least under the uniform prior.