Auctions have emerged globally as a crucial mechanism for acquiring renewable energy, enhancing both transparency and the expansion of clean energy initiatives. However, they differ to other auctions, and no theoretical results are available to analyze them. In this work we compute the Nash equilibrium bidding strategy for bidders participating in those auctions. By simulating the auction with bidders following this strategy and comparing the outcomes to actual results, we can gain valuable insights into market development. As an application, we analyze the German solar photo-voltaic auctions.

We study renewable energy auctions in the presence of de-risking instruments using an agent-based model. A sealed-bid, pay-as-bid scheme with several rounds of auctions is considered. A certain fraction of the auctioned volume is guaranteed, and each bidder has then two costs, presenting two bids depending on whether the contract will be covered by the guarantee or not. We study the resulting dynamics, and identify a sharp phase transition depending on the competition level of the auction. The guarantee impacts throughout the program and contributes to a significant decrease of the final prices when the auction is competitive enough. However, in not competitive auctions, its impact becomes negligible, and does not depend on the guaranteed volume fraction. A novel differentiated ceiling price mechanism applied only to bids benefiting from the guarantee is then introduced and studied. In particular, a new phase transition appears when the auction is not competitive, showing that relatively low prices can still be obtained when the guaranteed volume is high enough. We use the German wind on-shore auction datasets to study the differentiated ceiling price and its impact. JEL Classification: C53, C54, C57, C73, D44, Q21

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 correspond 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-price and second-price auctions with these rules. We present agent based simulations of the models, and we discuss several open problems.

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.