The Price of Stability for First Price Auction

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.