Mean Field Equilibria of Dynamic Auctions with Learning

Abstract

We study learning in a dynamic setting where identical copies of a good are sold over time through a sequence of second-price auctions. Each agent in the market has an unknown independent private valuation that determines the distribution of the reward she obtains from the good; for example, in sponsored search settings, advertisers may initially be unsure of the value of a click. Though the induced dynamic game is complex, we simplify analysis of the market using an approximation methodology known as mean field equilibrium (MFE). The methodology assumes that agents optimize only with respect to long-run average estimates of the distribution of other players' bids. We show a remarkable fact: In a mean field equilibrium, the agent has an optimal strategy where she bids truthfully according to a conjoint valuation. The conjoint valuation is the sum of her current expected valuation, together with an overbid amount that is exactly the expected marginal benefit of one additional observation about her true private valuation. Under mild conditions on the model, we show that an MFE exists, and that it is a good approximation to a rational agent’s behavior as the number of agents increases. Formally, if every agent except one follows the MFE strategy, then the remaining agent’s loss on playing the MFE strategy converges to zero as the number of agents in the market increases. We conclude by discussing the implications of the auction format and design on the auctioneer’s revenue. In particular, we establish the revenue equivalence of standard auctions in dynamic mean field settings, and discuss optimal selection of reserve prices in dynamic auctions. This paper was accepted by Assaf Zeevi, stochastic models and simulation.