Large Auctions

We posit a standard model of an asymmetric double auction with interdependent values in which each trader observes a private signal about a hidden state before submitting a bid or ask price for a unit demand or supply. The state and signals are one-dimensional, traders’ signals are independent conditional on the state, and their distributions have the strict monotone likelihood ratio property. The model encompasses auctions by allowing sellers to be non-strategic. We study a version in which there are n replicates of each type of trader, with each replicate observing a signal drawn independently from the same conditional distribution as the original trader of that type, and all traders of the same type using the same strategy. The limit economy with a countable set of traders has a unique Walrasian equilibrium, whose clearing price reveals the state. If this equilibrium is totally monotone in that each buyer’s (resp. seller’s) probability of trading decreases (resp. increases) with the state, then the limit auction has a monotone equilibrium yielding the Walrasian price as the clearing price.<br><br>We present four asymptotic results as n grows large: (1) a sequence of monotone strategies comprises epsilon-equilibria if and only if limit points are monotone equilibria of the limit auction; (2) for a sequence of monotone strategy profiles converging to a monotone equilibrium, the Strong Law of Large Numbers for prices holds, in that the sequence of price functions converges a.s. to the price function of the limit equilibrium; (3) if the effect of the state on traders’ valuations is symmetric (around the equilibrium) then large but finite auctions have monotone equilibria whose outcomes approximate the Walrasian equilibrium outcome when bidders are restricted to sufficiently fine bid-grids; and (4) the same conclusion holds true without the symmetry assumption when we discretize the state space as well. Total monotonicity seems to be crucial: an example has a Walrasian equilibrium that is not the outcome of a Nash equilibrium of an auction.