The First- and Second-Price Sealed-Bid Auctions Under Mean–Variance Preferences

Abstract

This paper uses linear mean–variance preferences within the Markowitz mean–variance framework to characterize bidders’ trade-off between return and risk and extends first- and second-price sealed-bid auction models to risky environments. Compared with conventional models that maximize expected payoff or represent risk aversion through expected utility, this framework does not rely on a specific utility function, characterizes bidders’ risk attitudes more directly, and facilitates quantitative analysis of how those attitudes affect equilibrium bidding, optimal reserve prices, seller revenue, and social welfare under the adopted mean–variance criterion. We characterize the equilibrium bidding strategies, optimal reserve prices, and seller’s expected revenue in the two auction formats, together with their rankings; analyze the effects of bidders’ variance aversion and the number of bidders; and trace the new findings to bidders’ variance aversion. For social welfare, we quantify the welfare loss caused by mean–variance preferences, rank social welfare across auction formats and preference types, and show that the welfare-maximizing reserve price lies below the seller-optimal reserve price in each format. These results inform the choice of an appropriate reserve price in practice to reduce the welfare loss caused by bidders’ variance aversion. Finally, we briefly consider three extensions: an asymmetric model, exogenous shocks to bidders’ payoffs, and nonlinear mean–variance preferences.