Bilateral Collusion in Second-Price Sealed-Bid Auctions Under Heterogeneous Risk Aversion

Second-price sealed-bid auctions with collusion are well studied, but the role of bidders’ risk aversion remains unclear. We study a two-bidder auction with heterogeneous constant relative risk aversion under the variable-bribe and fixed-bribe mechanisms, characterize the corresponding collusive equilibria, and examine effects on collusion and the seller’s expected revenue. Under the variable-bribe mechanism, we obtain an equilibrium in which lower proposer values separate and higher values pool. Greater proposer risk aversion increases the bribe, the responder’s acceptance cutoff, the probability of successful collusion, and the probability of misallocation, while reducing the seller’s expected revenue. Responder risk aversion does not affect the equilibrium. Under the fixed-bribe mechanism, primitive sufficient conditions establish a regular interior cutoff equilibrium. Within this regular interior equilibrium class, greater responder risk aversion raises the responder’s acceptance cutoff and the probability of misallocation, but lowers the proposer’s offer cutoff and the seller’s expected revenue. Greater proposer risk aversion lowers both the proposer’s offer cutoff and the responder’s acceptance cutoff, while its effects on the probability of successful collusion and the seller’s expected revenue can be positive or negative. Finally, we briefly discuss four extensions: variable-bribe collusive equilibria with positive initial wealth or a reserve price, and fixed-bribe collusive equilibria with general concave utility or an endogenously determined amount.