2. How does the market of sponsored links operate? User enters a query The auction for the link to appear on the search results page takes place Advertisements of the highest bidders are displayed as sponsored links along the search results If the sponsored link is clicked upon the advertiser pays a provider for bringing the user to her webpage Advertiser submits a bid for slot for the targeted keywords Advertiser pays amount corresponding to a bid of the highest ranked competitor below her Valuation (v) bid (u) bids are weighted by click through rates slots are assigned by weighted bid order Generalized second price auction

3. Sponsored Links in Search auctions Specifics: Specifics: Multiple non-equivalent items are auctioned simultaneously Different definition of internet ad-unit for bidders and ad- platform Large number of bidders competing in large number of actions which cost very little Mean field approximation Markov Decision Process

4. Generalised on-line knapsack problem Decision Maker Unobservable Gain g(i)g(i) )) P. Key et al, working paper cost Value function: Bidding with budgets

5. Infinite time discounted problem

6. Optimal Control Theorem Optimal Control Optimal Control From the point of the optimizing bidder, GSP can be interpreted as a stand-alone standard auction on which the bidder assigns a virtual valuation which is shaded down

7. Existence of Mean Field Equilibrium Existence of Mean Field Equilibrium

8. No budget constraints

9. Conclusions Conclusions Mean field approximation simplifies models for online- bidding making optimization and control tractable MFE exists for on-line single-slot bidding with budgets MFE exists in special cases of online bidding without budgets constraints, but, in general, MFE may not exists Outlook: Modelling beyond mean field