Approximate Equilibrium and Incentivizing Social Coordination

We study techniques to incentivize self-interested agents to form socially desirable solutions in scenarios where they benefit from mutual coordination. Towards this end, we consider coordination games where agents have different intrinsic preferences but they stand to gain if others choose the same strategy as them. For non-trivial versions of our game, stable solutions like Nash Equilibrium may not exist, or may be socially inefficient even when they do exist. This motivates us to focus on designing efficient algorithms to compute (almost) stable solutions like Approximate Equilibrium that can be realized if agents are provided some additional incentives. Our results apply in many settings like adoption of new products, project selection, and group formation, where a central authority can direct agents towards a strategy but agents may defect if they have better alternatives. We show that for any given instance, we can either compute a high quality approximate equilibrium or a near-optimal solution that can be stabilized by providing small payments to some players. We then generalize our model to encompass situations where player relationships may exhibit complementarities and present an algorithm to compute an Approximate Equilibrium whose stability factor is linear in the degree of complementarity. Our results imply that a little influence is necessary in order to ensure that selfish players coordinate and form socially efficient solutions.Comment: A preliminary version of this work will appear in AAAI-14: Twenty-Eighth Conference on Artificial Intelligenc

Price Competition in Networked Markets: How do monopolies impact social welfare?

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 single-sink 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.Comment: To appear in Proceedings of WINE 2015: 11th Conference on Web and Internet Economic

A test score based approach to stochastic submodular optimization

We study the canonical problem of maximizing a stochastic submodular function subject to a cardinality constraint, where the goal is to select a subset from a ground set of items with uncertain individual perfor- mances to maximize their expected group value. Although near-optimal algorithms have been proposed for this problem, practical concerns regarding scalability, compatibility with distributed implementation, and expensive oracle queries persist in large-scale applications. Motivated by online platforms that rely on indi- vidual item scores for content recommendation and team selection, we study a special class of algorithms that select items based solely on individual performance measures known as test scores. The central contribution of this work is a novel and systematic framework for designing test score based algorithms for a broad class of naturally occurring utility functions. We introduce a new scoring mechanism that we refer to as replication test scores and prove that as long as the objective function satisfies a diminishing returns condition, one can leverage these scores to compute solutions that are within a constant factor of the optimum. We then extend these scoring mechanisms to the more general stochastic submodular welfare maximization problem, where the goal is to partition items into groups to maximize the sum of the expected group values. For this more difficult problem, we show that replication test scores can be used to develop an algorithm that approximates the optimum solution up to a logarithmic factor. The techniques presented in this work bridge the gap between the rigorous theoretical work on submodular optimization and simple, scalable heuristics that are useful in certain domains. In particular, our results establish that in many applications involving the selection and assignment of items, one can design algorithms that are intuitive and practically relevant with only a small loss in performance compared to the state-of-the-art approaches

Blind, Greedy, and Random: Algorithms for Matching and Clustering Using Only Ordinal Information

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