Truthful Mechanisms for Matching and Clustering in an Ordinal World

We study truthful mechanisms for matching and related problems in a partial information setting, where the agents' true utilities are hidden, and the algorithm only has access to ordinal preference information. Our model is motivated by the fact that in many settings, agents cannot express the numerical values of their utility for different outcomes, but are still able to rank the outcomes in their order of preference. Specifically, we study problems where the ground truth exists in the form of a weighted graph of agent utilities, but the algorithm can only elicit the agents' private information in the form of a preference ordering for each agent induced by the underlying weights. Against this backdrop, we design truthful algorithms to approximate the true optimum solution with respect to the hidden weights. Our techniques yield universally truthful algorithms for a number of graph problems: a 1.76-approximation algorithm for Max-Weight Matching, 2-approximation algorithm for Max k-matching, a 6-approximation algorithm for Densest k-subgraph, and a 2-approximation algorithm for Max Traveling Salesman as long as the hidden weights constitute a metric. We also provide improved approximation algorithms for such problems when the agents are not able to lie about their preferences. Our results are the first non-trivial truthful approximation algorithms for these problems, and indicate that in many situations, we can design robust algorithms even when the agents may lie and only provide ordinal information instead of precise utilities.Comment: To appear in the Proceedings of WINE 201

On the Manipulability of Maximum Vertex-Weighted Bipartite bb-matching Mechanisms

In this paper, we study the Maximum Vertex-weighted $b$-Matching (MVbM) problem on bipartite graphs in a new game-theoretical environment. In contrast to other game-theoretical settings, we consider the case in which the value of the tasks is public and common to every agent so that the private information of every agent consists of edges connecting them to the set of tasks. In this framework, we study three mechanisms. Two of these mechanisms, namely \MB and \MD, are optimal but not truthful, while the third one, \MG, is truthful but sub-optimal. Albeit these mechanisms are induced by known algorithms, we show \MB and \MD are the best possible mechanisms in terms of Price of Anarchy and Price of Stability, while \MG is the best truthful mechanism in terms of approximated ratio. Furthermore, we characterize the Nash Equilibria of \MB and \MD and retrieve sets of conditions under which \MB acts as a truthful mechanism, which highlights the differences between \MB and \MD. Finally, we extend our results to the case in which agents' capacity is part of their private information.Comment: 10 pages, 0 figure

Combinatorial Secretary Problems with Ordinal Information

The secretary problem is a classic model for online decision making. Recently, combinatorial extensions such as matroid or matching secretary problems have become an important tool to study algorithmic problems in dynamic markets. Here the decision maker must know the numerical value of each arriving element, which can be a demanding informational assumption. In this paper, we initiate the study of combinatorial secretary problems with ordinal information, in which the decision maker only needs to be aware of a preference order consistent with the values of arrived elements. The goal is to design online algorithms with small competitive ratios. For a variety of combinatorial problems, such as bipartite matching, general packing LPs, and independent set with bounded local independence number, we design new algorithms that obtain constant competitive ratios. For the matroid secretary problem, we observe that many existing algorithms for special matroid structures maintain their competitive ratios even in the ordinal model. In these cases, the restriction to ordinal information does not represent any additional obstacle. Moreover, we show that ordinal variants of the submodular matroid secretary problems can be solved using algorithms for the linear versions by extending [Feldman and Zenklusen, 2015]. In contrast, we provide a lower bound of Omega(sqrt(n)/log(n)) for algorithms that are oblivious to the matroid structure, where n is the total number of elements. This contrasts an upper bound of O(log n) in the cardinal model, and it shows that the technique of thresholding is not sufficient for good algorithms in the ordinal model