Type: Seminar

Distribution: World

Expiry: 30 Jun 2018

CalTitle1: Using mean field games to study cost-per-click advertising behaviour

Auth: gwiggins@10.19.115.108 (gwig6180) in SMS-WASM

In online sponsored search mechanism, advertisements are sold based on a cost-per-click model and where each ad unit sold is measured per click. The pricing of each click is determined by the multi-keyword sponsored search auction mechanism which involves multiple generalised second price auctions running simultaneously for each query. In a dynamic game setting, a continuum of advertisers participate in a sequence of multi-keyword sponsored search auctions, and their bidding behaviour can be analysed as a non-cooperative game of incomplete and imperfect information. Each advertiser has a private valuation that is modelled by a stationary stochastic process, and the motion of cost state is driven by the optimal drift, which can be derived from the ex-post Bayesian Nash equilibrium bids generated by the static version of the game. Though the induced dynamic game is complex, we can simplify the analysis of the market using an approximation methodology known as mean field games, to study a specific example. The methodology assumes that advertisers optimise only with respect to long run average estimates of the distribution of other advertisers’ bids. Closed-form analytic solutions do not exist; however, I developed a numerical method for computing both stationary and time-varying equilibria. The problem can be broken down into a system of coupled PDEs, where an individual advertiser’s bidding choices can be analysed by solving Hamilton-Jacobi-Bellman equations, and the evolution of joint distribution of costs and valuations can be characterised by Fokker-Planck equations. I also show that a mean-field equilibrium exists, and that it is a good approximation to the rational advertisers’ behaviour when the number of advertisers is large. This was then followed by computing the hypothetical best response via solving a mixed-integer nonlinear problem to produce optimal bids.

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