Learning, Mean Field Approximations, and Phase Transitions in Auction Models

In this paper, we study an agent-based model for multi-round, pay as bid, sealed bid reverse auctions using techniques from partial differential equations and statistical mechanics tools. We assume that in each round a fixed fraction of bidders is awarded, and bidders learn from round to round using simple microscopic rules, adjusting myopically their bid according to their performance. Agent-based simulations show that bidders coordinate in the sense that they tend to bid the same value in the long-time limit. Moreover, this common value is the true cost or the ceiling price of the auction, depending on the level of competition. A discontinuous phase transition occurs when half of the bidders win. We establish the corresponding rate equations, and we obtain a system of ordinary differential equations describing the dynamics. Finally, we derive formally the kinetic equations modeling the dynamics, and we study the asymptotic behavior of solutions of the corresponding first-order, nonlinear partial differential equation satisfied by the distribution of agents.