Sequential Auctions, Price Trends, and Risk Preferences

We analyze sequential Dutch and Vickrey auctions where risk averse, or risk preferring, bidders may have heterogeneous risk exposures. We derive and characterize a pure strategy equilibrium of both auctions for arbitrary number of identical objects. A sufficient, and to certain extent necessary, condition for this result is that bidders' marginal utilities are log-submodular in income and type. We then show that when bidders are risk averse (preferring), the equilibrium price sequences should be downward (upward) drifting, and in each period the conditional expected revenue is higher (lower) in the Dutch than in the Vickrey sequential auctions. In particular, the "declining price anomaly" is perfectly consistent with nonincreasing absolute risk aversion when bidders have exposures to background risk.