We study the problem of an advertising agent who needs to intelligently distribute her budget across a sequence of online keyword bidding auctions. We assume the closing price of each auction is governed by the same unknown distribution, and study the problem of making provably optimal bidding decisions. Learning the distribution is done under censored observations, i.e. the closing price of an auction is revealed only if the bid we place is above it. We consider three algorithms, namely ε-First, Greedy Product- Limit (GPL) and LuekerLearn, respectively, and we show that these algorithms provably achieve Hannan-consistency. In particular, we show that the regret bound of ε-First is at most O(T^2/3) with high probability. For the other two algorithms, we first prove that, by using a censored data distribution estimator proposed by Zeng [19], the empirical distribution of the closing market price converges in probability to its true distribution with a O(1/εt) rate, where t is the number of updates. Based on this result, we prove that both GPL and Lueker Learn achieve O(√T) regret bound with high probability. This in fact provides an affirmative answer to the research question raised in [1]. We also evaluate the abovementioned algorithms using real bidding data, and show that although GPL achieves the best performance on average (up to 90% of the optimal solution), its long running time may limit its suitability in practice. By contrast, LuekerLearn and ε-First proposed in this paper achieve up to 85% of the optimal, but with an exponential reduction in computational complexity (a saving up to 95%, compared to GPL).

@en

Following previous work on payment redistribution in static mechanisms, we develop the theory of redistribution in online mechanisms (e.g., [2, 10, 8]). In static mechanisms, redistribution is important as it increases social welfare in scenarios with no residual claimant. Many online scenarios also do not have a natural residual claimant, and redistribution there is equally important. In this work, we adopt a fundamental online mechanism design model where a single expiring item is allocated in each of T periods. Agents with unit demand are present in the market between their arrival and departure periods, which are private information along with the value an agent attributes to the item. For this model, we derive a number of properties characterizing redistribution in online mechanisms (including revenue monotonicity properties, and quantifying the effect an agent can have on the total revenue). We then design two redistribution functions. The first one generalizes the static redistribution proposed by Cavallo [2] making redistribution after the departure of the last agent. For this redistribution function we provide theoretical worst-case guarantees. The second function is truly "online" making redistribution to each agent on her departure. The performance of both functions is evaluated using numerical simulations.@en