We consider ordinal approximation algorithms for a broad class of utility maximization problems for multi-agent systems. In these problems, agents have utilities for connecting to each other, and the goal is to compute a maximum-utility solution subject to a set of constraints. We represent these as a class of graph optimization problems, including matching, spanning tree problems, TSP, maximum weight planar subgraph, and many others. We study these problems in the ordinal setting: latent numerical utilities exist, but we only have access to ordinal preference information, i.e., every agent specifies an ordering over the other agents by preference. We prove that for the large class of graph problems we identify, ordinal information is enough to compute solutions which are close to optimal, thus demonstrating there is no need to know the underlying numerical utilities. For example, for problems in this class with bounded degree @math a simple ordinal greedy algorithm always produces a ( @math )-approximation; we also quantify how the quality of ordinal approximation depends on the sparsity of the resulting graphs. In particular, our results imply that ordinal information is enough to obtain a 2-approximation for Maximum Spanning Tree; a 4-approximation for Max Weight Planar Subgraph; a 2-approximation for Max-TSP; and a 2-approximation for various Matching problems.
Historically, it has been common to approach problems in the ordinal setting with a normative view by designing mechanisms which satisfy axiomatic properties, like stability or truthfulness. These axiomatic properties are useful in many applications, but do not provide a quantitative measure of the quality of a solution. The notion of distortion and the implicit utilitarian framework were first introduced by @cite_5 in the context of voting to provide such a measure. Since then the distortion, or approximation factor, of various ordinal utility maximization mechanisms has been studied, particularly for matchings @cite_22 @cite_16 @cite_8 @cite_2 @cite_10 and social choice @cite_18 @cite_20 @cite_1 @cite_0 @cite_4 @cite_6 @cite_3 @cite_17 @cite_15 .
@cite_18 @cite_4 @cite_22 @cite_8 @cite_1 @cite_6 @cite_3 @cite_0 @cite_2 @cite_5 @cite_15 @cite_16 @cite_10 @cite_20 @cite_17
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We study mechanism design problems in the ordinal setting wherein the preferences of agents are described by orderings over outcomes, as opposed to specific numerical values associated with them. This setting is relevant when agents can compare outcomes, but aren't able to evaluate precise utilities for them. Such a situation arises in diverse contexts including voting and matching markets. Our paper addresses two issues that arise in ordinal mechanism design. To design social welfare maximizing mechanisms, one needs to be able to quantitatively measure the welfare of an outcome which is not clear in the ordinal setting. Second, since the impossibility results of Gibbard and Satterthwaite Gibbard73,Satterthwaite75 force one to move to randomized mechanisms, one needs a more nuanced notion of truthfulness. We propose rank approximation as a metric for measuring the quality of an outcome, which allows us to evaluate mechanisms based on worst-case performance, and lex-truthfulness as a notion of truthfulness for randomized ordinal mechanisms. Lex-truthfulness is stronger than notions studied in the literature, and yet flexible enough to admit a rich class of mechanisms circumventing classical impossibility results . We demonstrate the usefulness of the above notions by devising lex-truthful mechanisms achieving good rank-approximation factors, both in the general ordinal setting, as well as structured settings such as (one-sided) matching markets , and its generalizations, matroid and scheduling markets. We consider a task of scheduling with a common deadline on a single machine. Every player reports to a scheduler the length of his job and the scheduler needs to finish as many jobs as possible by the deadline. For this simple problem, there is a truthful mechanism that achieves maximum welfare in dominant strategies. The new aspect of our work is that in our setting players are uncertain about their own job lengths, and hence are incapable of providing truthful reports (in the strict sense of the word). For a probabilistic model for uncertainty we show that even with relatively little uncertainty, no mechanism can guarantee a constant fraction of the maximum welfare. To remedy this situation, we introduce a new measure of economic efficiency, based on a notion of a fair share of a player, and design mechanisms that are Ω(1)-fair. In addition to its intrinsic appeal, our notion of fairness implies good approximation of maximum welfare in several cases of interest. In our mechanisms the machine is sometimes left idle even though there are jobs that want to use it. We show that this unfavorable aspect is unavoidable, unless one gives up other favorable aspects (e.g., give up Ω(1)-fairness). We also consider a qualitative approach to uncertainty as an alternative to the probabilistic quantitative model. In the qualitative approach we break away from solution concepts such as dominant strategies (they are no longer well defined), and instead suggest an axiomatic approach, which amounts to listing desirable properties for mechanisms. We provide a mechanism that satisfies these properties. We study social choice rules under the utilitarian distortion framework, with an additional metric assumption on the agents' costs over the alternatives. In this approach, these costs are given by an underlying metric on the set of all agents plus alternatives. Social choice rules have access to only the ordinal preferences of agents but not the latent cardinal costs that induce them. Distortion is then defined as the ratio between the social cost (typically the sum of agent costs) of the alternative chosen by the mechanism at hand, and that of the optimal alternative. This model was introduced by (2015) who conjectured that Ranked Pairs, the well-known social choice rule, achieves a distortion of at most 3. We disprove this conjecture by constructing a sequence of instances which shows that the worst-case distortion of Ranked Pairs is at least 5. Our lower bound on the worst case distortion of Ranked Pairs matches a previously known upper bound for the Copeland rule, proving that in the worst case, the (simpler) Copeland rule is at least as good as Ranked Pairs. And as long as we are limited to rules, we demonstrate that randomization cannot help achieve an expected worst-case distortion of less than 3. Using the concept of approximate majorization within the distortion framework, we prove that Copeland and Randomized Dictatorship achieve low constant factor fairness-ratios (5 and 3 respectively); this approximates essentially all fairness measures, and is a considerable generalization of the work of (2015) who prove similar results for objectives such as the sum and median of agent costs. In addition to all of the above, we outline several interesting directions for further research in this space. We examine the quality of social choice mechanisms using a utilitarian view, in which all of the agents have costs for each of the possible alternatives. While these underlying costs determine what the optimal alternative is, they may be unknown to the social choice mechanism; instead the mechanism must decide on a good alternative based only on the ordinal preferences of the agents which are induced by the underlying costs. Due to its limited information, such a social choice mechanism cannot simply select the alternative that minimizes the total social cost (or minimizes some other objective function). Thus, we seek to bound the distortion: the worst-case ratio between the social cost of the alternative selected and the optimal alternative. Distortion measures how good a mechanism is at approximating the alternative with minimum social cost, while using only ordinal preference information. The underlying costs can be arbitrary, implicit, and unknown; our only assumption is that the agent costs form a metric space, which is a natural assumption in many settings. We quantify the distortion of many well-known social choice mechanisms. We show that for both total social cost and median agent cost, many positional scoring rules have large distortion, while on the other hand Copeland and similar mechanisms perform optimally or near-optimally, always obtaining a distortion of at most 5. We also give lower bounds on the distortion that could be obtained by any deterministic social choice mechanism, and extend our results on median agent cost to more general objective functions.