Bribing in Second Price Auctions

Abstract

We analyse the effect of two simple forms of collusion on the efficiency and payoffs of a two-bidder, second-price auction for a single good, where both bidders have private and independent valuations. In the first form, the auction is preceded by one round of bribing, i.e. one bidder can try to bribe the other to stay out of the auction. In the unique and continuous equilibria of this model, bidders with a type below a certain threshold, reveal themselves as the amount they offer depends uniquely on their type. All bidders with a type above the threshold, offer the same amount. This can result in an inefficient auction. Due to the bribing, the payoffs of the bidders increase, while the income of the seller decreases, compared to the regular second-price auction payoffs. In the two-rounds bribing model, both bidders get the opportunity to bribe the other in exchange for her commitment to leave the auction. These bribes are offered in turn and the order is determined exogenously. There exists an efficient equilibrium, in which the first bidder offers her valuation minus her surplus. The second bidder either accepts, or rejects and counteroffers the surplus of the first bidder. Due to full revelation of the first bidders type, the auction remains efficient. In this model, there is always one bidder who excludes herself from the auction. Hence, the payoff of the seller is zero. Furthermore, due to information asymmetry, the first bidder's payoff remains the same as in a regular second-price auction, while the second bidder's payoff increases.