Algorithms for Social Choice Functions

This is an advance summary of a forthcoming article in the Oxford Research Encyclopedia of Politics. Please check back later for the full article.Narrowly understood, social choice theory is a specialized branch of applied logic and mathematics that analyzes abstract objects called preference aggregation functions, social welfare functions, and social choice functions. But more broadly, social choice theory identifies, analyzes, and evaluates rules that may be used to make collective decisions. So understood, social choice is a subfield of the social sciences that examines what may be called “voting rules” of various sorts. While social choice theory typically assumes a finite set of alternatives over which voter preferences are unrestricted, the spatial model of social choice assumes that policy alternatives can be represented by points in a space of one or more dimensions, and that voters have preferences that are plausibly shaped by this spatial structure.Social choice theory has considerable relevance for the study of legislative (as well as electoral) institutions. The concepts and tools of social choice theory make possible formal descriptions of legislative institutions such as bicameralism, parliamentary voting procedures, effects of decision rules (e.g., supramajority vs. simple majority rule and executive veto rules), sincere vs. strategic voting by legislators, agenda control, and other parliamentary maneuvers. Spatial models of social choice further enrich this analysis and raise additional questions regarding policy stability and change. Spatial models are used increasingly to guide empirical research on legislative institutions and processes.

This paper introduces a new axiom for choice in preference profiles and tournaments, called composition-consistency. A social choice function is composition-consistent if it is non-sensitive to the cloning of one or several outcomes. The key feature of the composition consistency property is an operation concept called multiple composition product of profiles. The paper provides a brief overview of some social choice functions studied in the literature. Concerning the tournament solutions, it is proved that the Top Cycle, the Slater and the Copeland solutions are not composition-consistent, whereas the Banks, Uncovered Set, TEQ, Minimal Covering Set are composition-consistent. Moreover, we define the composition-consistent hull of a solution ф as the smallest composition-consistent solution containing ф. The composition-consistent hulls of the Top cycle and Copeland solutions are specified, and we give some hints about the location of the hull of the Slater set. Concerning social choice functions, it is shown that Kemeny, Borda and Minimax social choice functions are not composition-consistent, whereas the Paretian one is composition-consistent. Moreover, we prove that the latter is the composition-consistent hull of the Borda and Minimax functions.

We introduce a logic specifically designed to support reasoning about social choice functions. The logic includes operators to capture strategic ability, and operators to capture agent preferences. We establish a correspondence between formulae in the logic and properties of social choice functions, and show that the logic is expressively complete with respect to social choice functions, i.e., that every social choice function can be characterised as a formula of the logic. We prove that the logic is decidable, and give a complete axiomatization. To demonstrate the value of the logic, we show in particular how it can be applied to the problem of determining whether a social choice function is strategy-proof.

We study a mechanism design problem under the assumption that renegotiation cannot be prevented. We investigate what kind of equilibria of which mechanisms are renegotiation-proof under a variety of renegotiation procedures, and which social choice functions can be implemented in a way that is renegotiation-proof. In complete information environments, we show that the set of ex post renegotiation-proof implementable social choice functions contains all ex post efficient allocations when there at least three agents, but only budget balanced Groves allocations when there are two agents. In incomplete information environments with correlated beliefs and at least three agents, every ex post efficient social choice function can be implemented in the presence of ex post renegotiation, but with independent private values only social choice functions that are given by budget balanced “Groves in expectations” mechanisms are implementable in such a way. We further show that the requirement of interim renegotiation-proofness does not impose additional restrictions on implementable social choice functions under complete information, but is likely to impose additional restrictions under incomplete information.

Implementation with Extensive Form Games: One Round of Signaling Is Not Enough

A social choice correspondence is self-implementable in strong equilibrium if it is implementable in strong equilibrium by a social choice function selecting from the correspondence itself as a game form. We characterize all social choice correspondences implementable this way by an anonymous social choice function satisfying no veto power, given that the number of agents is large relative to the number of alternatives. It turns out that these are exactly the social choice correspondences resulting from feasible elimination procedures as introduced in Peleg (Econometrica 46:153–161, 1978).

Perhaps the most influential result in social choice theory is Arrow’s impossibility theorem, which states that a seemingly modest set of desiderata cannot be satisfied when aggregating preferences [1]. While Arrow’s theorem might appear rather negative, it can also be interpreted in a positive way by identifying what can be achieved in preference aggregation.In this talk, I present a number of variations of Arrow’s theorem–such as those due to Mas-Colell and Sonnenschein [8] and Blau and Deb [2]–in their choice-theoretic version. The critical condition in all these theorems is the assumption of a rationalizing binary relation or equivalent notions of choice-consistency. The bulk of my presentation contains three escape routes from these results. The first one is to ignore consistency with respect to a variable set of alternatives altogether and require consistency with respect to a variable electorate instead. As Smith [12] and Young [14] have famously shown, this essentially characterizes the class of scoring rules, which contains plurality and Borda’s rule. For the second escape route, we factorize choice-consistency into two parts, contraction-consistency and expansions-consistency [11]. While even the mildest dose of the former has severe consequences on the possibility of choice, varying degrees of the latter characterize a number of appealing social choice functions, namely the top cycle, the uncovered set, and the Banks set [3,9,4]. Finally, I suggest to redefine choice-consistency by making reference to the set of chosen alternatives rather than individual chosen alternatives [6]. It turns out that the resulting condition is a weakening of transitive rationalizability and can be used to characterize the minimal covering set and the bipartisan set. Based on a two decades-old conjecture due to Schwartz [10], the tournament equilibrium set can be characterized by the same condition or, alternatively, by a weak expansion-consistency condition from the second escape route. Whether Schwartz’s conjecture actually holds remains a challenging combinatorial problem as well as one of the enigmatic open problems of social choice theory.Throughout the presentation I will discuss the algorithmic aspects of all considered social choice functions. While some of the mentioned functions can be easily computed, other ones do not admit an efficient algorithm unless P equals NP [13,5,7].

<p>One important issue in mechanism design theory is to model agents’ behaviors under uncertainty. The classical approach assumes that agents hold commonly known probability assessments towards uncertainty, which has been challenged by economists in many fields. My thesis adopts alternative methods to model agents’ behaviors. The new findings contribute to understanding how the mechanism designer can benefit from agents’ uncertainty aversion and how she should respond to the lack of information on agents’ probability assessments.</p> <p>Chapter 1 of this thesis allows the mechanism designer to introduce ambiguity to the mechanism. Instead of informing agents of the precise payment rule that she commits to, the mechanism designer can tell agents multiple payment rules that she may have committed to. The multiple payment rules are called ambiguous transfers. As agents do not know which rule is chosen by the designer, they are assumed to make decisions based on the worst-case scenario. Under this assumption, this chapter characterizes when the mechanism designer can obtain the first-best outcomes by introducing ambiguous transfers. Compared to the standard approach where the payment rule is unambiguous, first-best mechanism design becomes possible under a broader information structure. Hence, there are cases when the mechanism designer can benefit from introducing ambiguity.</p> <p>Chapter 2 assumes that the mechanism designer does not know agents’ probability assessments about others’ private information. The mechanisms designed to implement the social choice function thus should not depend on the probability assessments, which are called robust mechanisms. Different from the existing robust mechanism design literature where agents are always assumed to act non-cooperatively, this chapter allows them to communicate and form coalitions. This chapter provides necessary and almost sufficient conditions for robustly implementing a social choice function as an equilibrium that is immune to all coalitional deviations. As there are social choice functions that are only implementable with coalitional structures, this chapter provides insights on when agents should be allowed to communicate. As an extension, when the mechanism designer has no information on which coalitions can be formed, this chapter also provides conditions for robust implementation under all coalition patterns.</p> <p>Chapter 3 assumes that agents are not probabilistic about others’ private information. Instead, when they hold ambiguous assessments about others’ information, they make decisions based on the worst-case belief. This chapter provides necessary and almost sufficient conditions on when a social choice goal is implementable under such a behavioral assumption. As there are social choice goals that are only implementable under ambiguous assessments, this chapter provides insights on what information structure is desirable to the mechanism designer.</p>

We propose a new functional form characterization of binary nonmanipulable social choice functions on a universal domain and an arbitrary, possibly infinite, set of agents. In order to achieve this, we considered the more general case of two-valued social choice functions and describe the structure of the family consisting of groups of agents having no power to determine the values of a nonmanipulable social choice function. With the help of such a structure, we introduce a class of functions that we call powerless revealing social choice functions and show that the binary nonmanipulable social choice functions are the powerless revealing ones.

Boundedness of the range of a strategy-proof social choice function

A promising direction in computational social choice is to address research problems using computer-aided proving techniques. In particular with SAT solvers, this approach has been shown to be viab...

Social choice functions: A tool for ranking variables involved in action plans against road noise

We consider classes of non-manipulable social choice functions with range of cardinality at most two within a set of at least two alternatives. We provide the functional form for each of the classes we consider. This functional form is a characterization that explicitly describes how a social choice function of that particular class selects the collective choice corresponding to a profile. We provide a unified formulation of these characterizations using the new concept of “character”. The choice of the character, depending on the class of social choice functions, gives the functional form of all social choice functions of the class.

Two-person ex post implementation

A significant chapter of the short history of formal philosophy is related with the notion and the theory of the so-called “Social Welfare Functions (SWFs)”, as a substantial component of the “social choice theory”. One of the main uses of SWFs is aimed, indeed, at representing coherent patterns (effectively, algebraic structures of relations) of individual and collective choices/preferences, with respect to a fixed ranking of alternative social/economical states. Indeed, the SWF theory is originally inspired by Samuelson’s pioneering work on the foundations of mathematical economic analysis. It uses explicitly Gibbs’ thermodynamics of ensembles “at equilibrium” based on statistical mechanics as the physical paradigm for the mathematical theory of economic systems. In both theories, indeed, the differences and the relationships among individuals are systematically considered as irrelevant. On the contrary, in the mathematical theory of “Social Choice Functions” (SCFs) developed by Amartya Sen, the interpersonal comparison and the real-time information exchanges among different social actors and their environments—different—ethical values and constraints, included—play an essential role. This means that the inspiring physical paradigm is no longer “gas” but “fluid thermodynamics” of interacting systems passing through different “phases” of fast “dissolution/aggregation of coherent behaviors”, and then staying persistently in far from equilibrium conditions. These processes are systematically studied by the quantum field theory (QFT) of “dissipative systems”, at the basis of the physics of condensed matter, modeled by the “algebra doubling” of coalgebras. This coalgebraic modeling is highly significant for making computationally effective Sen’s SCF theory, because both based on a dynamic and not statistical weighing of the variables for interacting systems, respectively in the physical and in the social realms.