Computing Perfect Bayesian Equilibria in Sequential Auctions with Verification

TL;DR

This paper develops a principled method for computing pure-strategy $\varepsilon$-Perfect Bayesian Equilibria in sequential auctions with continuous action and value spaces, augmented by a verification phase that upper-bounds the utility loss relative to the unabstracted game. It introduces a belief-augmented dynamic program and a finite abstraction that discretizes bidder types and public beliefs, enabling backward induction to obtain equilibrium candidates by solving finite stage auctions via iterated best responses. A decomposition theorem guarantees that the computed equilibria in the abstracted game yield bounded utility loss in the original auction, with a practical verification procedure to compute $\varepsilon$ using only finite checks on vertices and a finite belief set. The approach reproduces known PBEs in several sequential settings (e.g., sequential sales with/without ascending reserves) and yields new insights in multi-round combinatorial auctions, including non-trivial shading patterns, while remaining computationally scalable due to parallelizable Monte Carlo and pattern-search methods. Overall, the work provides a verifiable, scalable framework that complements RL approaches for equilibrium computation in complex, multi-round auction environments.

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.

Figures (38)

• Figure 1: Bidding strategies for the first round of the first-price sequential sales auction with 3 bidders, 2 goods, and reserve prices $r_1=0,r_2=0.5$. Green denotes the analytical solution and red our found strategies. Types are plotted on the x-axis, bids on the y-axis.
• Figure 2: Bidding strategies in the first round of the sequential LLG auction. Green denotes truthful bidding, blue the strategy of the local bidder and red the strategy of the global bidder. Types are plotted on the x-axis, bids on the y-axis.
• Figure 3: Bidding strategies for first round of the second price sequential sales auction with 3 bidders and 2 goods. Green denotes the theoretical prediction and red our found strategies.
• Figure 4: Bidding strategies for second round of the second price sequential sales auction with 3 bidders and 2 goods. Green denotes the theoretical prediction and red our found strategies.
• Figure 5: Bidding strategies for first round of the second price sequential sales auction with 3 bidders and 2 goods. Green denotes the theoretical prediction and red our found strategies.

Theorems & Definitions (29)

• Example 1: Two-round FPSB auction
• Definition 1: $\varepsilon$-PBE
• Definition 2: PBS
• Definition 3: PBS Transition Function
• Definition 4: Consistent Transition
• Definition 5: Stage Auction
• Definition 6: Bellman Utility Update
• Proposition 1
• Proposition 2
• Definition 7: Abstracted Belief-Based Sequential Auction