In this paper we propose a learning model for bidding in multi-round, pay as bid, sealed bid auctions using techniques from partial differential equations and statistical mechanics tools. As an application, we perform a theoretical study of an agent based model. 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 (for reverse auctions), depending on the level of competition. A discontinuous phase transition occurs when half of the bidders win. The purpose of this paper is to introduce this theoretical methodology, and to analyze the dynamics. After establishing the rate equations, we obtain their continuous limit, which is a first-order, non-linear partial differential equation. We study its solutions, we prove the existence of the phase transition, and we explain the qualitative behavior of the solutions observed in the agent-based simulations.

We propose a simple approach to auctions based on statistical mechanics tools and evolutionary game theory to assess the impact of the competition level on the prices. In a sealed-bid, pay-as-bid scheme with several rounds, at each round bidders place their bids following a normal distribution with fixed variance and mean value μ characteristic of each bidder. Bidders learn from round to round, adjusting myopically their μ according to their performance in the round. We study the resulting dynamics using agent-based simulations, and we identify a phase transition depending on the competition level of the auction. Our model is in contrast with the classical literature on auctions which assumes bidders act purely rationally. Despite the simplicity of the model, it is able to explain the increasing and decreasing trends of the outcomes of real auctions, like the wind onshore energy auctions held in Germany from 2017 to 2019. Moreover, we describe a mean field approximation of the agent dynamics, obtaining a partial differential equation for their distribution on the parameter space, which helps to study the dynamics without simulations.