The Price of Stability for First Price Auction

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 $⊊$ Bayes Correlated Equilibrium $⊊$ Bayes Coarse Correlated Equilibrium} $\bullet$ Bayes Nash Equilibrium: For independent valuations, the tight PoS is $1-1\mathrm{/}{e}^2\approx 0.8647$, matching the counterpart Price of Anarchy (PoA) bound \cite{JL22}. For correlated valuations, the tight $\text{PoS}$ is $1-1\mathrm{/}e\approx 0.6321$, 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 $1=100\mathrm{\%}$, i.e., no efficiency degradation, different from the counterpart PoA bound $1-1\mathrm{/}e\approx 0.6321$ \cite{ST13,S14}. This result indicates that First Price Auctions can be fully efficient when we allow the more general equilibrium concepts.