Equilibrium in Sealed High Bid Auctions

There is a voluminous theoretical literature on sealed high-bid auctions (auctions in which bids are sealed and the high bidder pays his bid). See for example, Vickrey (1961), Myerson (1981), Riley and Samuelson (1981), Milgrom and Weber (1982), Matthews (1983), Maskin and Riley (1984), Holt (1980), Cox, Smith and Walker (1988). A critical property on which this literature relies is the existence of equilibrium in which buyers' bidding strategies are monotonic in their types. This enables the analyst to perform comparative statics as the distribution of types changes, and to compare the welfare properties of the high-bid auction with those of other auction institutions. One case in which the existence of monotonic equilibrium is readily established is that in which buyers' types are distributed smoothly, independently and symmetrically. In this case, there exists a symmetric equilibrium one in which all bidders share the same bidding behaviour as a function of their valuations and this bid function is the solution to a first-order differential equation. However, many applications require consideration of asymmetric distributions (see Maskin and Riley (2000)), common values (see Milgrom and Weber (1982)), or discrete distributions. For these cases the issue of equilibrium existence is not as straightforward. Indeed, even in apparently simple cases, monotonic equilibrium may fail to exist (see Examples 1-3). Therefore we attempt here to provide sufficient conditions that are general enough to cover most applications that the literature has considered. In Section 2 we introduce the model. In Section 3, we first show that, under mild assumptions on preferences, equilibrium bidding must be monotonic, if types are distributed independently. With affiliation we introduce the stronger assumption of log super modularity and, in Section 4, consider the case of discrete types. In general there may be no equilibrium under the standard tie-breaking rule (in which the winner is selected at random). However, we are able to establish existence of a monotonic equilibrium of a sealed high-bid auction in which the tying high bidders compete for the item in a secondround Vickrey auction. In Section 5 we extend our existence theorem (under the modified tie-breaking rule) to the case of continuously distributed types. Finally we establish existence of equilibrium under the standard tie-breaking rule by showing that buyers' strategies