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

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

Maximum Skew-Symmetric Flows and Matchings

The maximum integer skew-symmetric flow problem (MSFP) generalizes both the maximum flow and maximum matching problems. It was introduced by Tutte in terms of self-conjugate flows in antisymmetrical digraphs. He showed that for these objects there are natural analogs of classical theoretical results on usual network flows, such as the flow decomposition, augmenting path, and max-flow min-cut theorems. We give unified and shorter proofs for those theoretical results. We then extend to MSFP the shortest augmenting path method of Edmonds and Karp and the blocking flow method of Dinits, obtaining algorithms with similar time bounds in general case. Moreover, in the cases of unit arc capacities and unit ``node capacities'' the blocking skew-symmetric flow algorithm has time bounds similar to those established in Even and Tarjan (1975) and Karzanov (1973) for Dinits' algorithm. In particular, this implies an algorithm for finding a maximum matching in a nonbipartite graph in $O(\sqrt{n}m)$ time, which matches the time bound for the algorithm of Micali and Vazirani. Finally, extending a clique compression technique of Feder and Motwani to particular skew-symmetric graphs, we speed up the implied maximum matching algorithm to run in $O(\sqrt{n}m\log(n^2/m)/\log{n})$ time, improving the best known bound for dense nonbipartite graphs. Also other theoretical and algorithmic results on skew-symmetric flows and their applications are presented.Comment: 35 pages, 3 figures, to appear in Mathematical Programming, minor stylistic corrections and shortenings to the original versio