Online Ascending Auctions for Gradually Expiring Items

In this paper we consider online auction mechanisms for the allocation of M items that are identical to each other except for the fact that they have different expiration times, and each item must be allocated before it expires. Players arrive at different times, and wish to buy one item before their deadline. The main difficulty is that players act "selfishly" and may mis-report their values, deadlines, or arrival times. We begin by showing that the usual notion of truthfulness (where players follow a single dominant strategy) cannot be used in this case, since any (deterministic) truthful auction cannot obtain better than an M-approximation of the social welfare. Therefore, instead of designing auctions in which players should follow a single strategy, we design two auctions that perform well under a wide class of selfish, "semi-myopic", strategies. For every combination of such strategies, the auction is associated with a different algorithm, and so we have a family of "semi-myopic" algorithms. We show that any algorithm in this family obtains a 3-approximation, and by this conclude that our auctions will perform well under any choice of such semi-myopic behaviors. We next turn to provide a game-theoretic justification for acting in such a semi-myopic way. We suggest a new notion of "Set-Nash" equilibrium, where we cannot pin-point a single best-response strategy, but rather only a set of possible best-response strategies. We show that our auctions have a Set-Nash equilibrium which is all semi-myopic, hence guarantees a 3-approximation. We believe that this notion is of independent interest

Sum of Us: Strategyproof Selection from the Selectors

We consider directed graphs over a set of n agents, where an edge (i,j) is taken to mean that agent i supports or trusts agent j. Given such a graph and an integer k\leq n, we wish to select a subset of k agents that maximizes the sum of indegrees, i.e., a subset of k most popular or most trusted agents. At the same time we assume that each individual agent is only interested in being selected, and may misreport its outgoing edges to this end. This problem formulation captures realistic scenarios where agents choose among themselves, which can be found in the context of Internet search, social networks like Twitter, or reputation systems like Epinions. Our goal is to design mechanisms without payments that map each graph to a k-subset of agents to be selected and satisfy the following two constraints: strategyproofness, i.e., agents cannot benefit from misreporting their outgoing edges, and approximate optimality, i.e., the sum of indegrees of the selected subset of agents is always close to optimal. Our first main result is a surprising impossibility: for k \in {1,...,n-1}, no deterministic strategyproof mechanism can provide a finite approximation ratio. Our second main result is a randomized strategyproof mechanism with an approximation ratio that is bounded from above by four for any value of k, and approaches one as k grows

Core-competitive Auctions

One of the major drawbacks of the celebrated VCG auction is its low (or zero) revenue even when the agents have high value for the goods and a {\em competitive} outcome could have generated a significant revenue. A competitive outcome is one for which it is impossible for the seller and a subset of buyers to `block' the auction by defecting and negotiating an outcome with higher payoffs for themselves. This corresponds to the well-known concept of {\em core} in cooperative game theory. In particular, VCG revenue is known to be not competitive when the goods being sold have complementarities. A bottleneck here is an impossibility result showing that there is no auction that simultaneously achieves competitive prices (a core outcome) and incentive-compatibility. In this paper we try to overcome the above impossibility result by asking the following natural question: is it possible to design an incentive-compatible auction whose revenue is comparable (even if less) to a competitive outcome? Towards this, we define a notion of {\em core-competitive} auctions. We say that an incentive-compatible auction is $\alpha$-core-competitive if its revenue is at least $1/\alpha$ fraction of the minimum revenue of a core-outcome. We study the Text-and-Image setting. In this setting, there is an ad slot which can be filled with either a single image ad or $k$ text ads. We design an $O(\ln \ln k)$ core-competitive randomized auction and an $O(\sqrt{\ln(k)})$ competitive deterministic auction for the Text-and-Image setting. We also show that both factors are tight

Optimal Competitive Auctions

We study the design of truthful auctions for selling identical items in unlimited supply (e.g., digital goods) to n unit demand buyers. This classic problem stands out from profit-maximizing auction design literature as it requires no probabilistic assumptions on buyers' valuations and employs the framework of competitive analysis. Our objective is to optimize the worst-case performance of an auction, measured by the ratio between a given benchmark and revenue generated by the auction. We establish a sufficient and necessary condition that characterizes competitive ratios for all monotone benchmarks. The characterization identifies the worst-case distribution of instances and reveals intrinsic relations between competitive ratios and benchmarks in the competitive analysis. With the characterization at hand, we show optimal competitive auctions for two natural benchmarks. The most well-studied benchmark $\mathcal{F}^{(2)}(\cdot)$ measures the envy-free optimal revenue where at least two buyers win. Goldberg et al. [13] showed a sequence of lower bounds on the competitive ratio for each number of buyers n. They conjectured that all these bounds are tight. We show that optimal competitive auctions match these bounds. Thus, we confirm the conjecture and settle a central open problem in the design of digital goods auctions. As one more application we examine another economically meaningful benchmark, which measures the optimal revenue across all limited-supply Vickrey auctions. We identify the optimal competitive ratios to be $(\frac{n}{n-1})^{n-1}-1$ for each number of buyers n, that is $e-1$ as $n$ approaches infinity