Welfare Maximization and Truthfulness in Mechanism Design with Ordinal Preferences

We study mechanism design problems in the {\em 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~\cite{Gibbard73,Satterthwaite75} force one to move to randomized mechanisms, one needs a more nuanced notion of truthfulness. We propose {\em rank approximation} as a metric for measuring the quality of an outcome, which allows us to evaluate mechanisms based on worst-case performance, and {\em 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 {\em 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 {\em (one-sided) matching markets}, and its generalizations, {\em matroid} and {\em scheduling} markets.