Bertrand-Edgeworth Auction with Multiple Asymmetric Bidders
Shanshan Hu, Roman Kapuscinski, William Lovejoy
We define Bertrand-Edgeworth (B-E) auctions as a modiﬁcation of Bertrand-Edgeworth games where demand is inelastic and a price cap is set exogenously. B-E auctions are motivated by the discriminatory procurement auctions used in some wholesale electricity markets. We characterize the equilibrium structure for B-E auctions with multiple asymmetric suppliers (bidders). A pure-strategy equilibrium will not exist except in special cases, but when it does, it features active suppliers bidding uniformly at the competitive price. In most cases a mixed-strategy equilibrium prevails, for which there always exits an anchoring supplier whose random pricing interval covers other suppliers’ mixed bidding ranges. We show that the interval can be uniquely determined, where the upper bound maximizes the anchoring supplier’s expected payoff as a residual claimant (the highest bidder being admitted), and the lower bound provides her the same expected payoff as a base supplier (the lowest bidder). For the subset of the games where all active bidders reach the lower bound, closed-form equilibrium solutions can be uniquely derived. For the general case, based on a numerical algorithm that we propose, we numerically illustrate that in an oligopoly a weak (low-capacity) bidder does not necessarily price more aggressively (stochastically lower), as is the case for a duopoly (Kreps and Scheinkman (1983)). Instead, the weak supplier’s distribution is more concentrated, that is it spans a smaller interval. We show that most of the derived properties also hold when demand is price-sensitive. The model generalizes some results for multiple known versions of B-E models.
Hu, Shanshan, Roman Kapuscinski, and William Lovejoy (2008), "Bertrand-Edgeworth Auction with Multiple Asymmetric Bidders," Under Submission.