On Monotone Strategy Equilibria in Simultaneous Auctions for Complementary Goods

We propose a general theoretical model of bidding in simultaneous first price auctions and explore existence and properties of equilibrium in this model. Due to the combinatorial nature of the simultaneous bidding problem, standard notions of coordinatewise monotonicity may fail dramatically in this setting – for instance, a strict increase in all elements of a bidder’s private type can induce a strict decrease in all elements of that bidder’s corresponding best-response bid. Building on the methodology in Athey (2001), McAdams (2003) and Reny (2011), we introduce a notion of cross-object complementarity – weak quasisupermodularity – and a novel partial order on bidder types under which each bidder’s interim best-reply correspondence is monotone. We then apply this result to establish existence of pure strategy Bayes-Nash equilibrium in simultaneous first-price auctions for complementary goods, which are also monotone with respect to the proposed partial order. Finally, we present several numeric examples exploring strategic behavior, revenue and efficiency in simultaneous first-price auctions.