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Verifying Approximate Equilibrium in Auctions

Fabian R. Pieroth, Tuomas Sandholm

TL;DR

This work develops a data-driven framework to verify how close a bidding strategy profile in auctions is to an equilibrium, even when equilibria are unknown or non-truthful. By sampling from agents' priors and bids, and by evaluating potential gains from deviating within a finite strategy subset, the authors derive PAC-style guarantees that translate empirical utility differences into bounds on the true incentive gaps. The method extends prior work to interdependent priors and strategic bidding, using concepts like dispersion and pseudo-dimension to control discretization and sampling errors; it also provides explicit bounds for a range of auction formats, including multi-unit and combinatorial auctions. The approach enables practical equilibrium verification for designers and economists, highlighting both the potential and current scalability limits for complex auction settings, and pointing to future work on exploiting structure to achieve polynomial rather than exponential scaling.

Abstract

In practice, most auction mechanisms are not strategy-proof, so equilibrium analysis is required to predict bidding behavior. In many auctions, though, an exact equilibrium is not known and one would like to understand whether -- manually or computationally generated -- bidding strategies constitute an approximate equilibrium. We develop a framework and methods for estimating the distance of a strategy profile from equilibrium, based on samples from the prior and either bidding strategies or sample bids. We estimate an agent's utility gain from deviating to strategies from a constructed finite subset of the strategy space. We use PAC-learning to give error bounds, both for independent and interdependent prior distributions. The primary challenge is that one may miss large utility gains by considering only a finite subset of the strategy space. Our work differs from prior research in two critical ways. First, we explore the impact of bidding strategies on altering opponents' perceived prior distributions -- instead of assuming the other agents to bid truthfully. Second, we delve into reasoning with interdependent priors, where the type of one agent may imply a distinct distribution for other agents. Our main contribution lies in establishing sufficient conditions for strategy profiles and a closeness criterion for conditional distributions to ensure that utility gains estimated through our finite subset closely approximate the maximum gains. To our knowledge, ours is the first method to verify approximate equilibrium in any auctions beyond single-item ones. Also, ours is the first sample-based method for approximate equilibrium verification.

Verifying Approximate Equilibrium in Auctions

TL;DR

This work develops a data-driven framework to verify how close a bidding strategy profile in auctions is to an equilibrium, even when equilibria are unknown or non-truthful. By sampling from agents' priors and bids, and by evaluating potential gains from deviating within a finite strategy subset, the authors derive PAC-style guarantees that translate empirical utility differences into bounds on the true incentive gaps. The method extends prior work to interdependent priors and strategic bidding, using concepts like dispersion and pseudo-dimension to control discretization and sampling errors; it also provides explicit bounds for a range of auction formats, including multi-unit and combinatorial auctions. The approach enables practical equilibrium verification for designers and economists, highlighting both the potential and current scalability limits for complex auction settings, and pointing to future work on exploiting structure to achieve polynomial rather than exponential scaling.

Abstract

In practice, most auction mechanisms are not strategy-proof, so equilibrium analysis is required to predict bidding behavior. In many auctions, though, an exact equilibrium is not known and one would like to understand whether -- manually or computationally generated -- bidding strategies constitute an approximate equilibrium. We develop a framework and methods for estimating the distance of a strategy profile from equilibrium, based on samples from the prior and either bidding strategies or sample bids. We estimate an agent's utility gain from deviating to strategies from a constructed finite subset of the strategy space. We use PAC-learning to give error bounds, both for independent and interdependent prior distributions. The primary challenge is that one may miss large utility gains by considering only a finite subset of the strategy space. Our work differs from prior research in two critical ways. First, we explore the impact of bidding strategies on altering opponents' perceived prior distributions -- instead of assuming the other agents to bid truthfully. Second, we delve into reasoning with interdependent priors, where the type of one agent may imply a distinct distribution for other agents. Our main contribution lies in establishing sufficient conditions for strategy profiles and a closeness criterion for conditional distributions to ensure that utility gains estimated through our finite subset closely approximate the maximum gains. To our knowledge, ours is the first method to verify approximate equilibrium in any auctions beyond single-item ones. Also, ours is the first sample-based method for approximate equilibrium verification.
Paper Structure (31 sections, 44 theorems, 91 equations, 1 figure, 1 table)

This paper contains 31 sections, 44 theorems, 91 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related research
  4. Preliminaries
  5. The model
  6. Concentration bounds from learning theory
  7. Challenges for sampling-based equilibrium verification
  8. Sufficient properties for strategies to be verifiable
  9. Verifying approximate equilibrium under independent priors
  10. A sampling-based bound on the ex interim utility loss via grid search
  11. Verifying approximate equilibrium under interdependent priors
  12. Bounding best-response utility differences with constant best-responses
  13. A sampling-based bound on the ex ante utility loss via finite precision step functions
  14. Guarantees on dispersion and pseudo-dimension for four mechanisms
  15. Dispersion guarantees under strategic bidding
  16. ...and 16 more sections

Key Result

Theorem 3.3

Let $\Phi$ be a distribution over $\mathcal{X}$ and $\mathcal{F} \subset \{f: \mathcal{X} \rightarrow [-1, 1]\;|\;f \text{ measurable.} \}$. Set $d=\text{Pdim}(\mathcal{F})$, then, with probability $1-\delta$ over a draw $x^{(1)}, \dots, x^{(N)} \sim \Phi$, for all $f \in \mathcal{F}$, it holds that

Figures (1)

  • Figure 1: A beta-distributed density function of agent $i$'s valuations (left) and the corresponding bidding density functions under different strategies $\beta_i$ (right).

Theorems & Definitions (77)

  • Definition 3.1: $\varepsilon$-Bayesian Nash equilibrium
  • Definition 3.2: DBLP:books/daglib/0034861
  • Theorem 3.3: DBLP:books/daglib/0034861
  • Example 4.1
  • Definition 4.2: balcanDispersionDataDrivenAlgorithm2018
  • Definition 4.3: Bi-Lipschitz function, verineExpressivityBiLipschitzNormalizing2023
  • Theorem 4.4
  • Theorem 5.1
  • Theorem 5.3
  • Definition 6.1: Total variation, gibbsChoosingBoundingProbability2002
  • ...and 67 more