Martin Starnberger

On Multiple Keyword Sponsored Search Auctions with Budgets

We study multiple keyword sponsored search auctions with budgets. Each keyword has multiple ad slots with a click-through rate. The bidders have additive valuations, which are linear in the click-through rates, and budgets, which are restricting their overall payments. Additionally, the number of slots per keyword assigned to a bidder is bounded.n We show the following results: (1) We give the first mechanism for multiple keywords, where click-through rates differ among slots. Our mechanism is incentive compatible in expectation, individually rational in expectation, and Pareto optimal. (2) We study the combinatorial setting, where each bidder is only interested in a subset of the keywords. We give an incentive compatible, individually rational, Pareto-optimal, and deterministic mechanism for identical click-through rates. (3) We give an impossibility result for incentive compatible, individually rational, Pareto-optimal, and deterministic mechanisms for bidders with diminishing marginal valuations.

Valuation Compressions in VCG-Based Combinatorial Auctions

The focus of classic mechanism design has been on truthful direct-revelation mechanisms. In the context of combinatorial auctions the truthful direct-revelation mechanism that maximizes social welfare is the VCG mechanism. For many valuation spaces computing the allocation and payments of the VCG mechanism, however, is a computationally hard problem. We thus study the performance of the VCG mechanism when bidders are forced to choose bids from a subspace of the valuation space for which the VCG outcome can be computed efficiently. We prove improved upper bounds on the welfare loss for restrictions to additive bids and upper and lower bounds for restrictions to non-additive bids. These bounds show that the welfare loss increases in expressiveness. All our bounds apply to equilibrium concepts that can be computed in polynomial time as well as to learning outcomes.

Auctions for Heterogeneous Items and Budget Limits

We study individual rational, Pareto-optimal, and incentive compatible mechanisms for auctions with heterogeneous items and budget limits. We consider settings with multiunit demand and additive valuations. For single-dimensional valuations we prove a positive result for randomized mechanisms, and a negative result for deterministic mechanisms. While the positive result allows for private budgets, the negative result is for public budgets. For multidimensional valuations and public budgets we prove an impossibility result that applies to deterministic and randomized mechanisms. Taken together this shows the power of randomization in certain settings with heterogeneous items, but it also shows its limitations.

Auctions with heterogeneous items and budget limits

We study individual rational, Pareto optimal, and incentive compatible mechanisms for auctions with heterogeneous items and budget limits. For multi-dimensional valuations we show that there can be no deterministic mechanism with these properties for divisible items. We use this to show that there can also be no randomized mechanism that achieves this for either divisible or indivisible items. For single-dimensional valuations we show that there can be no deterministic mechanism with these properties for indivisible items, but that there is a randomized mechanism that achieves this for either divisible or indivisible items. The impossibility results hold for public budgets, while the mechanism allows private budgets, which is in both cases the harder variant to show. While all positive results are polynomial-time algorithms, all negative results hold independent of complexity considerations.

Limiting Price Discrimination when Selling Products with Positive Network Externalities

Assume a seller wants to sell a digital product in a social network where a buyer’s valuation of the item has positive network externalities from her neighbors that already have the item. The goal of the seller is to maximize his revenue. Previous work on this problem [7] studies the case where clients are offered the item in sequence and have to pay personalized prices. This is highly infeasible in large scale networks such as the Facebook graph: (1) Offering items to the clients one after the other consumes a large amount of time, and (2) price-discrimination of clients could appear unfair to them and result in negative client reaction or could conflict with legal requirements.

Welfare Maximization with Friends-of-Friends Network Externalities

Online social networks allow the collection of large amounts of data about the influence between users connected by a friendship-like relationship. When distributing items among agents forming a social network, this information allows us to exploit network externalities that each agent receives from his neighbors that get the same item. In this paper we consider Friends-of-Friends (2-hop) network externalities, i.e., externalities that not only depend on the neighbors that get the same item but also on neighbors of neighbors. For these externalities we study a setting where multiple different items are assigned to unit-demand agents. Specifically, we study the problem of welfare maximization under different types of externality functions. Let n be the number of agents and m be the number of items. Our contributions are the following: (1) We show that welfare maximization is APX-hard; we show that even for step functions with 2-hop (and also with 1-hop) externalities it is NP-hard to approximate social welfare better than (1-1/e). (2) On the positive side we present (i) an O(sqrt n)-approximation algorithm for general concave externality functions, n(ii) an O(log m)-approximation algorithm for linear externality functions, and (iii) an (1-1/e)frac{1}{6}-approximation algorithm for 2-hop step function externalities. We also improve the result from [6] for 1-hop step function externalities by giving a (1-1/e)/2-approximation algorithm.

Combinatorial Auctions with Conflict-Based Externalities

Combinatorial auctions CA are a well-studied area in algorithmic mechanism design. However, contrary to the standard model, empirical studies suggest that a bidders valuation often does not depend solely on the goods assigned to him. For instance, in adwords auctions an advertiser might not want his ads to be displayed next to his competitors ads. In this paper, we propose and analyze several natural graph-theoretic models that incorporate such negative externalities, in which bidders form a directed conflict graph with maximum out-degree

Combinatorial auctions with bidding constraints and network externalities

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