Simple collusive agreements in one-shot first-price auctions

The strategies of the colluding parties must be mixed since agreeing to submit a bid equal to the reservation price of the seller provides the incentive for that bidder to cheat on the designated winner. We deal with a complete information environment with arbitrary number of bidders. We characterize the sensible equilibrium outcome where the two bidders with the highest valuations collude. We show that the equilibrium outcome is unique and that the probability of collusion exceeds 1/2.