Class Of Social Choice Functions Research Articles

On the manipulability of a class of social choice functions: plurality kth rules

In this paper we introduce the plurality kth social choice function selecting an alternative, which is ranked kth in the social ranking following the number of top positions of alternatives in the individual ranking of voters. As special case the plurality 1st is the same as the well-known plurality rule. Concerning individual manipulability, we show that the larger k the more preference profiles are individually manipulable. We also provide maximal non-manipulable domains for the plurality kth rules. These results imply analogous statements on the single non-transferable vote rule. We propose a decomposition of social choice functions based on plurality kth rules, which we apply for determining non-manipulable subdomains for arbitrary social choice functions. We further show that with the exception of the plurality rule all other plurality kth rules are group manipulable, i.e. coordinated misrepresentation of individual rankings are beneficial for each group member, with an appropriately selected tie-breaking rule on the set of all profiles.

Strongly Rationalizable Virtual Implementation in Social Environments With Evidence

This article investigates rationalizable implementation in social environments where agents can provide hard evidence on the private information they possess. Specifically, we study necessary and sufficient conditions for virtual implementation using a notion of Rationalizability that captures forward-induction thinking. Our results non-trivially strengthen those from previous contributions to Rationalizable Implementation with dynamic mechanisms. In particular, we prove that, under a mild economic condition, dynamic mechanisms that rely on evidence can strategically distinguish all agents' information types endowed with different type-dependent messages. Thus, in a large class of social environments, if agents' evidence technologies are responsive, i.e., different types are endowed with different evidence, preference interdependence no longer plays a role in determining which class of Social Choice Functions can be rationalizably virtually implemented.

Mechanism Design with Farsighted Agents

Agents are farsighted when they consider the ultimate results to which their own actions may lead to. We re-examine the classical questions of implementation theory under complete information in a setting with transfers where farsighted coalitions are regarded as fundamental behavioral units and the equilibrium outcomes of their interactions are predicted via the stability notion of the largest consistent set. The designer's exercise consists of designing a rights structure, which formalizes the idea of power distribution in society. His or her challenge lies in designing a rights structure in which the equilibrium behavior of agents always coincides with the recommendation given by a social choice function. We show that (Maskin) monotonicity fully identifies the class of social choice functions that are implementable by a rights structure. We also discuss how this result changes when social choice correspondences are considered.

Symmetric mechanism design

Designers of economic mechanisms can often benefit by using discriminatory mechanisms which treat agents asymmetrically. This paper studies the extent to which a policy prohibiting biased mechanisms is effective in achieving fair outcomes. Our main result is a characterization of the class of social choice functions that can be implemented by symmetric mechanisms. When the solution concept used is Bayes–Nash equilibrium, symmetry is typically not very restrictive and discriminatory social choice functions can be implemented by symmetric mechanisms. Our characterization in this case is based on a ‘revelation principle’ type of result, where we show that a social choice function can be symmetrically implemented if and only if a particular kind of (indirect) symmetric mechanism implements it. When implementation in dominant strategies is considered, only symmetric social choice functions can be implemented by symmetric mechanisms. We illustrate our results in environments of voting with private values, voting with a common value, and assignment of indivisible goods.

Sharp Thresholds for Monotone Non-Boolean Functions and Social Choice Theory

A key fact in the theory of Boolean functions f: {0, 1}n → {0, 1} is that they often undergo sharp thresholds. For example, if the function f: {0, 1}n → {0, 1} is monotone and symmetric under a transitive action with Ep[f] = ε and Eq[f] = 1−ε, then q − p → 0 as n → ∞. Here Ep denotes the product probability measure on {0, 1}n where each coordinate takes the value 1 independently with probability p. The fact that symmetric functions undergo sharp thresholds is important in the study of random graphs and constraint satisfaction problems as well as in social choice. In this paper we prove sharp thresholds for monotone functions taking values in an arbitrary finite set. We also provide examples of applications of the results to social choice and to random graph problems. Among the applications is an analog for Condorcet’s Jury Theorem and an indeterminacy result for a large class of social choice functions.

In Praise of Manipulation

Many theorists believe that the manipulation of voting procedures is a serious problem. Accordingly, much of social choice theory examines the conditions under which strategy-proofness can be ensured, and what kind of procedures do a better job of preventing manipulation. This article argues that democrats should not be worried about manipulation. Two arguments against manipulation are examined: first, the ‘sincerity argument’, according to which manipulation should be rejected because it displays a form of insincere behaviour. This article distinguishes between sincere and non-sincere manipulation and shows that a familiar class of social choice functions is immune to insincere manipulation. Secondly, the ‘transparency’ argument against manipulation is discussed and it is argued that (sincere or insincere) manipulation may indeed lead to non-transparency of the decision-making process, but that, from a democratic perspective, such non-transparency is often a virtue rather than a vice.

In Praise of Manipulation

Many theorists believe that the manipulation of voting procedures is a serious problem. Accordingly, much of social choice theory examines the conditions under which strategy proofness can be ensured, and what kind of procedures do a better job of preventing manipulation. In this paper we argue that democrats should not be worried about manipulation. Two arguments against manipulation are examined. First we discuss the 'sincerity argument', according to which manipulation should be rejected because it displays a form of insincere behaviour. We argue however that a distinction can be made between sincere and non-sincere manipulation and we show that a familiar class of social choice functions is immune to insincere manipulation. We then discuss the 'transparency' argument against manipulation. We argue that (sincere or insincere) manipulation may indeed lead to non-transparency of the decision making process, but that, from a democratic perspective, such non-transparency is often a virtue rather than a vice.

Ordinally Bayesian Incentive Compatible Voting Rules

We study strategic voting after weakening the notion of strategy-proofness to Ordinal Bayesian Incentive Compatibility (OBIC). Under OBIC, truth-telling is required to maximize the expected utility of every voter, expected utility being computed with respect to the voter's prior beliefs and under the assumption that everybody else is also telling the truth. We show that for a special type of priors, i.e., the uniform priors, there exists a large class of social choice functions that are OBIC. However, for priors that are generic in the set of independent beliefs, a social choice function is OBIC only if it is dictatorial. This result underlines the robustness of the Gibbard–Satterthwaite Theorem.

On Some Recent Results in Incomplete Information Implementation

This paper provides a review of some recent results on implementation in incomplete information environments. This literature is concerned with characterizing the class of social choice functions that can be implemented as equilibrium outcomes of appropriately constructed games. Different game forms (normal and extensive forms) and different equilibrium concepts (Nash equilibrium, sequential equilibrium, etc.) lead to different characterizations. The discussion focuses on the key difficulties that arise in implementing social choice functions in incomplete information environments and describes some of the alternative characterizations.

A Note on Abreu-Matsushima Mechanisms

IN A STIMULATING RECENT PAPER, Abreu and Matsushima (1992) (hereafter A-M) show how a class of social choice functions can be virtually implemented in iteratively undominated strategies. This work has several important features: the mechanisms used are finite and not too difficult to understand, and so less objectionable in this regard than many in the literature; the class of social choice functions implemented is large; and the solution concept-Nash equilibrium determined uniquely by iterative elimination of strongly dominated strategies-is relatively uncontroversial. The point of this note is to argue that the mechanisms used by A-M unfortunately tend to generate games in which the iterative removal of strongly dominated strategies sometimes is indeed (or ought to be) controversial.2 We proceed by first examining an example of a related but simpler implementation problem in which the argument is easily exposed, then indicating how the argument applies generally in the A-M setup. Consider the following much-discussed two-player coordination game: