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Mean Field Equilibria of Dynamic Auctions with Learning

49 Pages Posted: 31 Mar 2011 Last revised: 10 Nov 2014

Krishnamurthy Iyer

Cornell University - Operations Research & Industrial Engineering

Ramesh Johari

Stanford University

Mukund Sundararajan

Google Inc.

Date Written: February 26, 2012


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 which 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 to 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 a dynamic version of the revenue equivalence theorem, and discuss optimal selection of reserve prices in dynamic auctions.

Keywords: mean field equilibrium, dynamic auction markets, conjoint valuation

Suggested Citation

Iyer, Krishnamurthy and Johari, Ramesh and Sundararajan, Mukund, Mean Field Equilibria of Dynamic Auctions with Learning (February 26, 2012). Available at SSRN: or

Krishnamurthy Iyer (Contact Author)

Cornell University - Operations Research & Industrial Engineering ( email )

225 Rhodes Hall
Ithaca, NY 14853
United States

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Ramesh Johari

Stanford University ( email )

473 Via Ortega
Stanford, CA 94305-9025
United States

Mukund Sundararajan

Google Inc. ( email )

1600 Amphitheatre Parkway
Second Floor
Mountain View, CA 94043
United States

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