Bid optimization in online advertisement auctions

We consider the problem of online keyword advertising auctions among multiple bidders with limited budgets, and propose a bidding heuristic to optimize the utility for bidders by equalizing the return-on-investment for each bidder across all keywords. We show that natural auction mechanisms combined with this heuristic can experience chaotic cycling (as is the case with many current advertisement auction systems), and therefore propose a modified class of mechanisms with small random perturbations. This perturbation is reminiscent of the small time-dependent perturbations employed in the dynamical systems literature to convert many types of chaos into attracting motions. We show that our perturbed mechanism provably converges in the case of first-price auctions and experimentally converges in the case of second-price auctions. Moreover, we show that our bidder-optimal system does not decrease the revenue of the auctioneer in the sense that it converges to the unique market equilibrium in the case of first-price auctions. In the case of second-price auctions, we conjecture that it converges to the non-unique “supplyaware” market equilibrium. We also observe that our perturbed auction scheme is useful in a broader context: In general, it can allow bidders to “share” a particular item, leading to stable allocations and pricing for the bidders, and improved revenue for the auctioneer.