Bertrand-Edgeworth Auction with Multiple Asymmetric Bidders

Abstract

Motivated by supply competitions in the service sector, we consider a version of the Bertrand-Edgeworth game where capacitated suppliers compete in prices to serve a deterministic demand and a price cap is imposed exogenously. We characterize the equilibrium structure for games with multiple suppliers that are asymmetric in cost and capacity. A pure-strategy equilibrium exists only in special cases, otherwise a mixed-strategy equilibrium prevails. For a mixed-strategy equilibrium, we show there exists an anchoring supplier whose pricing range covers those of the other suppliers, and we pin down the lower and upper bounds of that price range. We derive a canonical closed-form solution for a set of games, and for other more general cases, we demonstrate a numerical scheme for equilibrium calculation. In contrast to a well-known intuition established in the existing literature, we identify a new equilibrium pricing structure in games with N >= 3 suppliers who have similar costs but different capacities: A low-capacity supplier's price distribution is more concentrated than those of its high-capacity competitors, by spanning a smaller interval. We demonstrate an interesting strategy for a buyer with a competitive supply base: Subsidizing a cost-competitive incumbent may reduce the total procurement cost.