## Invited Session Thu.1.MA 043

#### Thursday, 10:30 - 12:00 h, Room: MA 043

**Cluster 8: Game theory** [...]

### Mean-field approaches to large scale dynamic auctions and mechanisms

**Chair: Gabriel Weintraub and Santiago Balseiro**

**Thursday, 10:30 - 10:55 h, Room: MA 043, Talk 1**

**Krishnamurthy Iyer**

Mean field equilibria of dynamic auctions with learning

**Coauthors: Ramesh Johari, Mukund Sundararajan**

**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 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. We conclude by establishing a dynamic version of the revenue equivalence theorem.

**Thursday, 11:00 - 11:25 h, Room: MA 043, Talk 2**

**Santiago Balseiro**

Auctions for online display advertising exchanges: Approximations and design

**Coauthors: Omar Besbes, Gabriel Weintraub**

**Abstract:**

We study the competitive landscape that arises in Ad Exchanges and the implications for publishers' decisions. Advertisers join these markets with a pre-specified budget and participate in multiple auctions over the length of a campaign. They bid on online ad placements based on specific viewer information. We introduce the notion of a Fluid Mean Field Equilibrium (FMFE) to study the advertisers' dynamic bidding strategies. This concept is based on a mean field approximation to relax the informational requirements of advertisers, together with a fluid approximation to approximate the complex dynamics of the advertisers' stochastic control problems. We derive a closed-form characterization of the bidding strategies under a FMFE, and of the resulting landscape. Using this characterization, we study the auction design problem from the publisher's perspective, and analyze the impact of three design levers: (1) the reserve price; (2) the supply of impressions to the Exchange versus an alternative channel; and (3) the disclosure of viewers' information. Our results provide novel insights with regard to the description and design of such markets.

**Thursday, 11:30 - 11:55 h, Room: MA 043, Talk 3**

**Alexandre Proutiere**

Optimal bidding strategies and equilibria in repeated auctions with budget constraints

**Coauthors: Ramki Gummadi, Peter Key**

**Abstract:**

How should agents bid in repeated sequential auctions when they are budget constrained? A motivating

example is that of sponsored search auctions, where advertisers bid in a sequence of generalized second

price (GSP) auctions. These auctions have many idiosyncratic

features that distinguish them from other models of sequential auctions. (1) Each bidder competes in a large

number of auctions, where each auction is worth very little. (2) The total bidder population is large, which

means it is unrealistic to assume that the bidders could possibly optimize their strategy by modeling specific

opponents. (3) The presence of a virtually unlimited supply of these auctions means bidders are necessarily

expense constrained.

Motivated by these three factors, we first frame the generic problem as a discounted Markov Decision Process and provide a structural characterization of the associated value function and the optimal bidding strategy, which specifies the extent to which agents underbid from their true valuation due to budget constraints. We then show the existence of Mean Field Equilibria for both the repeated second price and GSP auctions with a large number of bidders.