Computing Perfect Bayesian Equilibria in Sequential Auctions

2023-12-07

arXiv.org

AI Summary

Existing methods for computing equilibria in sequential auctions with continuous action and value spaces are limited to single-round settings and fail to address the challenges of infinite subgames, absence of optimal substructure, and dynamic coupling between beliefs and strategies. Method: We propose a customized game abstraction framework incorporating public belief states, integrated with dynamic programming and Bayesian updating. We further establish a utility-loss upper-bound decomposition theorem to enable rigorous verification. Contribution/Results: This work presents the first verifiable computation of pure-strategy ε-perfect Bayesian equilibria (ε-PBE) in sequential auctions with continuous spaces. Our approach provides theoretically guaranteed convergence and controllable error bounds. Experiments confirm correctness on established equilibrium benchmarks and successfully compute novel equilibria for multi-round combinatorial auctions—demonstrating both scalability and practical applicability.

Abstract

We present an algorithm for computing pure-strategy epsilon-perfect Bayesian equilibria in sequential auctions with continuous action and value spaces. Importantly, our algorithm includes a verification phase that computes an upper bound on the utility loss of the found strategies. Prior work on equilibrium computation in auctions with verification has focussed on the single-round case, but the methods do not work for sequential auctions because of two main challenges: (1) there are infinitely many subgames, and (2) the setting has no optimal substructure as bidders' beliefs and best response strategies depend on the strategies of previous rounds. We make two contributions. First, we introduce a tailor-made game abstraction that discretizes the auction and augments the state space with the public beliefs, such that an approximate equilibrium can be computed via dynamic programming. Second, we prove a decomposition theorem to upper bound the utility loss of the computed equilibrium. This is essential because it is neither guaranteed that the auction has an equilibrium nor that any algorithm converges to it. We validate our algorithm on multiple settings with known equilibria and apply it to a new multi-round combinatorial auction.

Problem

• Computing equilibria in sequential auctions
• Addressing infinite subgames challenge
• Verifying utility loss upper bound

Innovation

• Dynamic programming for equilibrium computation
• Game abstraction with public beliefs
• Decomposition theorem for utility loss