Truthful Mechanisms for Matching and Clustering in an Ordinal World

Abstract

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.

Ordinal Approximation for Social Choice, Matching, and Facility Location Problems Given Candidate Positions

“…While we leave the analysis of randomized algorithms which know the location of the facilities to future work, and consider the worst-case candidate locations, it is worth pointing out that our deterministic algorithm achieves a distortion of 3, which is also the best known distortion bound for any randomized mechanism which only knows the ordinal preferences of the agents. Similarly, another common goal is to form truthful mechanisms with small distortion for matching and social choice, as in [6,13,21]; we focus on general mechanisms in this paper in order to understand the limitations of knowing only certain kinds of ordinal information, and leave the goal of forming truthful mechanisms for future work.…”

“…The distortion of matching in a metric space has received far less attention than social choice questions. [5][6][7] analyzed maximum-weight metric matching; the maximization objective makes this problem far easier, and even choosing a uniformly random matching yields a distortion of a small constant. This is very different from our goal of computing a minimum-cost matching, for which no ordinal approximations better than O(n) are known.…”

“…These observations have recently led to a large body of work using the utilitarian approach, in which we assume that some latent numerical costs or utilities exist, but we only know the ordinal preferences of the agents, not their underlying numerical costs. See for example [3,4,11,16,21,23,29] for the social choice setting, [1,[5][6][7] for matching and other graph problems, and [13] for facility location. These works focus on measuring the distortion of various algorithms: a measure of how well an algorithm behaves when using only ordinal information, as compared to the optimum algorithm which has access to the true underlying numerical information.…”

Tradeoffs Between Information and Ordinal Approximation for Bipartite Matching

“…On the other hand, eliciting ordinal information (i.e., the preference ordering of each agent over the outcomes) is often much more reasonable. Because of this, there has been a lot of recent work on ordinal approximation algorithms: these are algorithms which only use ordinal preference information as their input, and yet return a solution provably close to the optimum one (e.g., [3][4][5][9][10][11][12]17]). In other words, these are algorithms which only use limited ordinal information, and yet can compete in the quality of solution produced with omniscient algorithms which know the true (possibly latent) numerical utility information.…”

“…Different from two-sided model, α does not equal to the number of agent pairs we are able to match by greedy algorithm in total ordering model. |X | x∈X (w(P (x)) − w(D(x))) ≤ β − 1 (5) ∀x ∈ X , w(P (x)) ≤ w(λ(x)), so it is obvious that w(OP T (S)) ≤ x∈X w(λ(x)).…”