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.

We study the Maximum Weighted Matching problem in a partial information setting where the agents' utilities for being matched to other agents are hidden and the mechanism 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, and look to design algorithms that approximate the true optimum matching using only the preference orderings for each agent (induced by the hidden weights) as input. If no restrictions are placed on the weights, then one cannot hope to do better than the simple greedy algorithm, which yields a half optimal matching. Perhaps surprisingly, we show that by imposing a little structure on the weights, we can improve upon the trivial algorithm significantly: we design a 1.6-approximation algorithm for instances where the hidden weights obey the metric inequality. Our algorithm is obtained using a simple but powerful framework that allows us to combine greedy and random techniques in unconventional ways. These results are the first non-trivial ordinal approximation algorithms for such problems, and indicate that we can design robust matchings even when we are agnostic to the precise agent utilities.

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We study the efficiency of allocations in large markets with a network structure where every seller owns an edge in a graph and every buyer desires a path connecting some nodes. While it is known that stable allocations in such settings can be very inefficient, the exact properties of equilibria in markets with multiple sellers are not fully understood even in single-source singlesink networks. In this work, we show that for a large class of natural buyer demand functions, we are guaranteed the existence of an equilibrium with several desirable properties. The crucial insight that we gain into the equilibrium structure allows us to obtain tight bounds on efficiency in terms of the various parameters governing the market, especially the number of monopolies M . All of our efficiency results extend to markets without the network structure.While it is known that monopolies can cause large inefficiencies in general, our main results for single-source single-sink networks indicate that for several natural demand functions the efficiency only drops linearly with M . For example, for concave demand we prove that the efficiency loss is at most a factor 1 + M 2 from the optimum, for demand with monotone hazard rate it is at most 1+M , and for polynomial demand the efficiency decreases logarithmically with M . In contrast to previous work that showed that monopolies may adversely affect welfare, our main contribution is showing that monopolies may not be as 'evil' as they are made out to be; the loss in efficiency is bounded in many natural markets. Finally, we consider more general, multiple-source networks and show that in the absence of monopolies, mild assumptions on the network topology guarantee an equilibrium that maximizes social welfare.

The increasingly tight coupling between humans and system operations in domains ranging from intelligent infrastructure to e-commerce has led to a challenging new class of problems founded on a well-established area of research: incentive design. There is a clear need for a new tool kit for designing mechanisms that help coordinate self-interested parties while avoiding unexpected outcomes in the face of information asymmetries, exogenous uncertainties from dynamic environments, and resource constraints. This article provides a perspective on the current state of the art in incentive design from three core communities—economics, control theory, and machine learning—and highlights interesting avenues for future research at the interface of these domains.