The Price of Stability for First Price Auction

TL;DR

This paper determines the Price of Stability for First Price Auctions across various equilibrium concepts, revealing that efficiency can be optimal or near-optimal depending on the equilibrium type and valuation correlation.

Contribution

It provides tight bounds on the Price of Stability for all studied equilibrium concepts in First Price Auctions, including new insights for Bayesian Coarse Correlated Equilibrium.

Findings

PoS for Bayes Nash Equilibrium with independent valuations is approximately 0.8647.

PoS for Bayes Nash Equilibrium with correlated valuations is approximately 0.6321.

PoS for Bayesian Coarse Correlated Equilibrium is always 1, indicating full efficiency.

Abstract

This paper establishes the Price of Stability (PoS) for First Price Auctions, for all equilibrium concepts that have been studied in the literature: Bayes Nash Equilibrium $\subsetneq$ Bayes Correlated Equilibrium $\subsetneq$ Bayes Coarse Correlated Equilibrium} $\bullet$ Bayes Nash Equilibrium: For independent valuations, the tight PoS is $1 - 1/ e^{2} \approx 0.8647$, matching the counterpart Price of Anarchy (PoA) bound \cite{JL22}. For correlated valuations, the tight $\PoS$ is $1 - 1 / e \approx 0.6321$, matching the counterpart PoA bound \cite{ST13,S14}. This result indicates that, in the worst cases, efficiency degradation depends not on different selections among Bayes Nash Equilibria. $\bullet$ Bayesian Coarse Correlated Equilibrium: For independent or correlated valuations, the tight PoS is always $1 = 100\%$, i.e., no efficiency degradation, different from the counterpart PoA bound $1 - 1 / e \approx 0.6321$ \cite{ST13,S14}. This result indicates that First Price Auctions can be fully efficient when we allow the more general equilibrium concepts.