Kinetic theory of active particles meets auction theory

In this paper we study Nash equilibria in auctions from the kinetic theory of active particles point of view. We propose a simple learning rule for agents to update their bidding strategies based on their previous successes and failures, in first-price auctions with two bidders. Then, we formally derive the corresponding kinetic equations which describe the evolution over time of the distribution of agents on the bidding strategies. We show that the stationary solution of the equation corresponds to the symmetric Nash equilibrium of the auction, and we prove the convergence to this stationary solution when time goes to infinity. We also introduce a more general learning rule that only depends on the income of agents, and we apply to both first- and second-price auctions. We show that agents learn the Nash equilibrium in first- and second-price auctions with these rules. We present agent-based simulations of the models, and we discuss several open problems.