MaPSS: Maths Postgraduate Seminar Series: Klinger

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.