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.
Recent work, mostly in the CS literature, has led to a more nuanced notion of efficiency. Procaccia and Rosenchein @cite_31 studied the strong welfare factor notion (that they call distortion), and noticed that deterministic mechanisms have unbounded distortion. @cite_21 proposed randomized mechanisms and showed that the strong welfare factor is at most @math , if the consistent cardinal-utility profile is normalized. In contrast, our rank approximation results imply @math -approximate outcomes, but under a stronger restriction on the consistent cardinal utilities. The notion of approximations to scoring rules was studied by Procaccia @cite_24 where he described strongly truthful mechanisms which @math -approximate Borda, but @math -approximate the plurality rule. In contrast, our (non-truthful) mechanism @math -rank approximates any scoring rule, and plurality can be arbitrarily well approximated by a lex-truthful mechanism.
@cite_24 @cite_31 @cite_21
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The Gibbard-Satterthwaite Theorem asserts that any reasonable voting rule cannot be strategyproof. A large body of research in AI deals with circumventing this theorem via computational considerations; the goal is to design voting rules that are computationally hard, in the worst-case, to manipulate. However, recent work indicates that the prominent voting rules are usually easy to manipulate. In this paper, we suggest a new CS-oriented approach to circumventing Gibbard-Satterthwaite, using randomization and approximation. Specifically, we wish to design strategyproof randomized voting rules that are close, in a standard approximation sense, to prominent score-based (deterministic) voting rules. We give tight lower and upper bounds on the approximation ratio achievable via strategyproof randomized rules with respect to positional scoring rules, Copeland, and Maximin. The theoretical guarantees provided by voting have distinguished it as a prominent method of preference aggregation among autonomous agents. However, unlike humans, agents usually assign each candidate an exact utility, whereas an election is resolved based solely on each voter's linear ordering of candidates. In essence, the agents' cardinal (utility-based) preferences are embedded into the space of ordinal preferences. This often gives rise to a distortion in the preferences, and hence in the social welfare of the outcome. In this paper, we formally define and analyze the concept of distortion. We fully characterize the distortion under different restrictions imposed on agents' cardinal preferences; both possibility and strong impossibility results are established. We also tackle some computational aspects of calculating the distortion. Ultimately, we argue that, whenever voting is applied in a multiagent system, distortion must be a pivotal consideration. We adopt a utilitarian perspective on social choice, assuming that agents have (possibly latent) utility functions over some space of alternatives. For many reasons one might consider mechanisms, or social choice functions, that only have access to the ordinal rankings of alternatives by the individual agents rather than their utility functions. In this context, one possible objective for a social choice function is the maximization of (expected) social welfare relative to the information contained in these rankings. We study such optimal social choice functions under three different models, and underscore the important role played by scoring functions. In our worst-case model, no assumptions are made about the underlying distribution and we analyze the worst-case distortion---or degree to which the selected alternative does not maximize social welfare---of optimal social choice functions. In our average-case model, we derive optimal functions under neutral (or impartial culture) distributional models. Finally, a very general learning-theoretic model allows for the computation of optimal social choice functions (i.e., that maximize expected social welfare) under arbitrary, sampleable distributions. In the latter case, we provide both algorithms and sample complexity results for the class of scoring functions, and further validate the approach empirically.