[Paper Review] Beyond Equilibria: Mechanisms for Repeated Combinatorial Auctions

Brendan Lucier|ArXiv.org|Sep 30, 2009

Auction Theory and Applications17 references3 citations

TL;DR

This paper presents a mechanism for repeated combinatorial auctions that achieves an $O(\sqrt{m})$ approximation to optimal social welfare when bidders use regret-minimizing or best-response strategies. It introduces a black-box reduction for monotone, loser-independent algorithms, ensuring approximate efficiency without dominant-strategy truthfulness, leveraging learning-based bidder behavior models.

ABSTRACT

We study the design of mechanisms in combinatorial auction domains. We focus on settings where the auction is repeated, motivated by auctions for licenses or advertising space. We consider models of agent behaviour in which they either apply common learning techniques to minimize the regret of their bidding strategies, or apply short-sighted best-response strategies. We ask: when can a black-box approximation algorithm for the base auction problem be converted into a mechanism that approximately preserves the original algorithm's approximation factor on average over many iterations? We present a general reduction for a broad class of algorithms when agents minimize external regret. We also present a new mechanism for the combinatorial auction problem that attains an $O(\sqrt{m})$ approximation on average when agents apply best-response dynamics.

Motivation & Objective

• To design mechanisms for repeated combinatorial auctions that maintain approximation guarantees under non-equilibrium bidder behavior.
• To decouple computational efficiency from incentive compatibility by avoiding dominant-strategy truthfulness.
• To analyze mechanisms under two realistic behavioral models: regret-minimization and myopic best-response dynamics.
• To provide a black-box conversion of approximation algorithms into mechanisms that preserve their performance on average over repeated rounds.
• To achieve an $O(\sqrt{m})$ approximation to optimal social welfare under both behavioral models.

Proposed method

• Proposes a mechanism based on monotone, loser-independent approximation algorithms for combinatorial auctions.
• Applies a black-box reduction to convert such algorithms into mechanisms that preserve their approximation factor under external regret minimization.
• Introduces a new mechanism, $\mathcal{M}_{CA}$, that achieves $O(\sqrt{m})$ welfare approximation under best-response dynamics.
• Uses a threshold-based bidding strategy where agents bid at least half their true value for a set if it is beneficial.
• Employs probabilistic concentration bounds (e.g., Chernoff bound) to show high-probability convergence to the approximation factor.
• Analyzes welfare outcomes by comparing social welfare under actual bids to welfare under hypothetical truthful bids and empty bids.

Experimental results

Research questions

• RQ1 Can a black-box approximation algorithm be converted into a mechanism that preserves its approximation factor under regret-minimizing bidder behavior?
• RQ2 What approximation ratio can be achieved in repeated combinatorial auctions when bidders use myopic best-response strategies?
• RQ3 How does the structure of the allocation algorithm (e.g., monotonicity, loser-independence) affect mechanism design in repeated settings?
• RQ4 To what extent can computational efficiency be preserved without requiring dominant-strategy truthfulness in repeated auctions?
• RQ5 Can mechanisms achieve constant-factor welfare approximation under realistic, non-equilibrium behavioral models?

Key findings

• The mechanism achieves an $O(\sqrt{m})$ approximation to the optimal social welfare on average over repeated auction rounds under regret-minimizing behavior.
• Under best-response dynamics, the mechanism attains an $O(\sqrt{m})$ approximation to the optimal welfare with high probability over the random order of bidder updates.
• The mechanism ensures that no bidder can gain utility by deviating from a bid of at least half their true value for a set, under the given behavioral model.
• The analysis shows that for any allocation of sets of size at most $\sqrt{m}$, the mechanism achieves an $O(\sqrt{m})$ approximation in expectation.
• The mechanism maintains $O(\sqrt{m})$ approximation for allocations of sets of size at least $\sqrt{m}$, by bounding the loss due to non-truthful bidding.
• With high probability $1 - 2ne^{-T\epsilon^2/n}$, the mechanism achieves a $\left(\frac{1}{O(\sqrt{m})} - \epsilon\right)$ approximation to the optimal welfare after $T$ rounds.