On the Welfare of Cardinal Voting Mechanisms

On the Welfare of Cardinal Voting Mechanisms Umang Bhaskar1 Tata Institute of Fundamental Research, Mumbai, India Abheek Ghosh The University of Texas at Austin, TX, USA Abstract A voting mechanism is a method for preference aggregation that takes as input preferences over alternatives from voters, and selects an alternative, or a distribution over alternatives. While preferences of voters are generally assumed to be cardinal utility functions that map each alternative to a real value, mechanisms typically studied assume coarser inputs, such as rankings of the alternatives (called ordinal mechanisms). We study cardinal mechanisms, that take as input the cardinal utilities of the voters, with the objective of minimizing the distortion – the worst-case ratio of the best social welfare to that obtained by the mechanism. For truthful cardinal mechanisms with m alternatives and n voters, we show bounds of Θ(mn), √ Ω(m), and Ω( m) for deterministic, unanimous, and randomized mechanisms respectively. This shows, somewhat surprisingly, that even mechanisms that allow cardinal inputs have large dis√ tortion. There exist ordinal (and hence, cardinal) mechanisms with distortion O( m log m), and hence our lower bound for randomized mechanisms is nearly tight. In an effort to close this gap, we give a class of truthful cardinal mechanisms that we call randomized hyperspher√ ical mechanisms that have O( m log m) distortion. These are interesting because they violate two properties – localization and non-perversity – that characterize truthful ordinal mechanisms, demonstrating non-trivial mechanisms that differ significantly from ordinal mechanisms. Given the strong lower bounds for truthful mechanisms, we then consider approximately truthful mechanisms. We give a mechanism that is δ-truthful given δ ∈ (0, 1), and has distortion close to 1. Finally, we consider the simple mechanism that selects the alternative that maximizes social welfare. This mechanism is not truthful, and we study the distortion at equilibria for the voters (equivalent to the Price of Anarchy, or PoA). While in general, the PoA is unbounded, we show that for equilibria obtained from natural dynamics, the PoA is close to 1. Thus relaxing the notion of truthfulness in both cases allows us to obtain near-optimal distortion. 2012 ACM Subject Classification Theory of computation → Algorithmic game theory Keywords and phrases computational social choice, voting rules, cardinal mechanisms, price of anarchy, distortion Digital Object Identifier 10.4230/LIPIcs.FSTTCS.2018.27 1 Introduction A society or a group of people may have different views and preferences but want to make a collective decision that will impact the entire group. For example, the people of India may have conflicting opinions on which party should win the Lok Sabha elections, and who should be the Prime Minister. This is the problem of preference aggregation, and the methods 1 Research funded in part by a Ramanujan fellowship. © Umang Bhaskar and Abheek Ghosh; licensed under Creative Commons License CC-BY 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Editors: Sumit Ganguly and Paritosh Pandya; Article No. 27; pp. 27:1–27:22 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 27:2 On the Welfare of Cardinal Voting Mechanisms of achieving this aggregation are called voting mechanisms – functions that map the given preferences of voters over a set of alternatives to a single alternative or a distribution over alternatives, without money being exchanged. Central to the question of preference aggregation is the question of how preferences are perceived by the voters, and how they are expressed to the mechanism. In classical social choice theory, particularly when a voting mechanism is randomized, i.e., can output a distribution over the set of alternatives, voter preferences are assumed to be von NeumannMorgenstern utility functions, that map each alternative to a real-valued utility. We assume that the total utility each voter has for the alternatives is 1. This is the unit-sum assumption, though other normalizations such as unit-range are also studied. A voter then prefers distributions over the alternatives that maximize her expected utility. These utility functions may be latent and hidden from the mechanism but are required for the voter to rationally compare distributions or lotteries over the set of outcomes. In contrast to these cardinal utility functions of the voters, frequently the mechanisms studied in the literature, and used in practice, have coarser inputs, such as a ranking of the alternatives, or simply a vote for the alternative with the highest utility (called ordinal and plurality voting respectively). If utility functions are cardinal, then an understanding of cardinal voting mechanisms, where voters give as input their utility functions, seems fundamental to understanding the problem of preference aggregation. Though less popular than other voting mechanisms owing to their complexity, cardinal voting mechanisms find use in many areas. For example, they are motivated by automated agents in recommender systems that use exact numeric values for making decisions, and hence naturally have easily expressible cardinal utilities. The use of these automated agents in a movie recommendation system is described by Ghosh et al. [14] (cf. [24]). Hillinger further argues for the use of cardinal voting mechanisms, especially since they do not artificially restrict the freedom of expression of voters [17]. Given an input format for voter preferences, how then should the mechanism choose an alternative? A widely studied property is incentive compatibility or truthfulness – a voter should maximize her expected utility by truthfully expressing her preferences, irrespective of the votes of others. Truthful mechanisms are desirable since voters need not strategize or seek information on the behavior of other voters. Other properties for mechanisms that are studied include Pareto-efficiency and polynomial-time computation. A natural objective for the voting mechanism, given the cardinal utilities of voters, is to maximize the social welfare or the total utility of the voters. This has been a mechanism objective in a number of recent papers (e.g., [7, 13]). The objective of social welfare assumes that the utilities allow for interpersonal comparison: that a unit of voter 1’s utility is equivalent to a unit of agent 2’s utility. Such comparisons may not be generally applicable, but even then, aggregate utility (or disutility) is frequently used as a quantitative measure, e.g., man-hours required for a project, or total time spent in traffic. The motivation for studying social welfare from classical economic theory, as well as further uses in modern recommendation systems, is also described by Boutilier (...truncated)