On monotone strategy equilibria in simultaneous auctions for complementary goods

We explore existence and properties of equilibrium when N ≥ 2 bidders compete for L ≥ 2 objects via simultaneous but separate auctions. Bidders have private combinatorial valuations over all sets of objects they could win, and objects are complements in the sense that these valuations are supermodular in the set of objects won. We provide a novel partial order on typesunder which best replies are monotone, and demonstrate that Bayesian Nash equilibria which are monotone with respect to this partial order exist on any finite bid lattice. We apply this result to show existence of monotone Bayesian Nash equilibria in continuous bid spaces when a single global bidder competesfor L objects against many local bidders who bid for single objects only. Wethen consider monotone equilibrium with endogenous tiebreaking building onJackson, Simon, Swinkels and Zame (2002), and demonstrate that these existin general. These existence results apply to many auction formats, including first-price, second-price, and all-pay.