Social Choice Correspondences Research Articles

A solution for abstract decision problems based on maximum flow value

An abstract decision problem is an ordered pair where the first component is a nonempty and finite set of alternatives from which a society has to make a choice and the second component is an irreflexive relation on that set representing a dominance relation. A crucial problem is to find a reasonable solution that allows to select, for any given abstract decision problem, some of the alternatives. A variety of solutions have been proposed over the years. In this paper we propose a new solution, called maximum flow value set, that naturally stems from the work by Bubboloni and Gori (The flow network method, Social Choice and Welfare 51, pp. 621–656, 2018) and that is based on the concept of maximum flow value in a digraph. We analyze its properties and its relation with other solutions such as the core, the admissible set, the uncovered set, the Copeland set and the generalized stable set. We also show that the maximum flow value set allows to define a new Condorcet social choice correspondence strictly related to the Copeland social choice correspondence and fulfilling lots of desirable properties.

Mechanisms and axiomatics for division problems with single-dipped preferences

AbstractA mechanism allocates one unit of an infinitely divisible commodity among agents reporting a number between zero and one. Nash, Pareto optimal Nash, and strong equilibria are analyzed for the case where the agents have single-dipped preferences. One main result is that when the mechanism satisfies anonymity, monotonicity, the zero–one property, and order preservation, then the Pareto optimal Nash and strong equilibria coincide and result in Pareto optimal allocations that are characterized by so-called maximal coalitions: members of a maximal coalition prefer an equal coalition share over obtaining zero, whereas the outside agents prefer zero over obtaining an equal share from joining the coalition. A second main result is an axiomatic characterization of the associated social choice correspondence as the maximal correspondence satisfying minimal envy Pareto optimality, equal division lower bound, and sharing index order preservation.

Implementation in undominated strategies with applications to auction design, public good provision and matching

This paper considers implementation in undominated strategies by finite mechanisms, where multiple outcomes may be implemented at a single state of the world. We establish a sufficient condition for implementation applicable in a general environment with private values. We apply it to three well-known environments and obtain strikingly permissive results. In the single-object auction, the second-price auction with a reserve price can be outperformed in terms of revenue. In the public good provision problem, the Vickrey–Clarke–Groves mechanism can be outperformed from the viewpoint of a designer who wishes to minimise deficit subject to efficiency. In the two-sided matching environment where preferences on one side of the market are private information, the social choice correspondence that outputs all stable matchings at every preference profile, is implementable.

On a class of strategy-proof social choice correspondences with single-peaked utility functions

We consider the problem of constructing strategy-proof rules that choose sets of alternatives based on the preferences of voters, modelled as Social Choice Correspondences (SCCs) in the literature. We focus on two domain restrictions inspired by Barberà et al. (2001) in the context of single-peaked utility functions. We find that for the narrower domain, the set of tops-only, unanimous, and strategy-proof SCCs coincides with the class of unions of two min–max rules (Moulin, 1980). For the broader domain, the set of SCCs coincides with the class of unions of two ‘adjacent’ min–max rules, meaning the corresponding parameters for the two rules must be either the same or consecutive alternatives.

A Society Can Always Decide How to Decide: A Proof

Infinite regress lies within every democratic procedural choice. If society members try to select an appropriate rule [social choice correspondence (SCC)] entirely endogenously, they will need an appropriate rule to choose such a rule. However, this should also be selected by an appropriate rule to choose a rule to choose a rule, and so on. This paper explores how to solve this infinite regress. A preference profile over the set of alternatives is said to converge if, at a sufficiently high level, every feasible SCC in the menu ultimately results in the same alternative, and hence, further regress has no effective meaning. A menu is said to be convergent if all preference profiles converge under the menu (i.e., infinite regress can “always” be resolved). First, we characterize the convergent menus under a special case. Then, we prove two general possibility theorems: (1) there exists a menu of SCCs that is strongly convergent (i.e., the outcome is uniquely determined); (2) any set of scoring rules can be extended to a superset that is asymptotically convergent for a large society (i.e., the probability of a convergent profile occurring goes to one as the population goes to infinity). Therefore, such a large society can “almost always” resolve the infinite regress by adding multiple SCCs. These theorems are expected to build new ground for SCCs in a distinct way from the axiomatic characterizations of standard social choice theory.

Characterizing the top cycle via strategyproofness

Gibbard and Satterthwaite have shown that the only single‐valued social choice functions (SCFs) that satisfy nonimposition (i.e., the function's range coincides with its codomain) and strategyproofness (i.e., voters are never better off by misrepresenting their preferences) are dictatorships. In this paper, we consider set‐valued social choice correspondences (SCCs) that are strategyproof according to Fishburn's preference extension and, in particular, the top cycle, an attractive SCC that returns the maximal elements of the transitive closure of the weak majority relation. Our main theorem shows that, under mild conditions, the top cycle is the only non‐imposing strategyproof SCC whose outcome only depends on the quantified pairwise comparisons between alternatives. This result effectively turns the Gibbard–Satterthwaite impossibility into a complete characterization of the top cycle by moving from SCFs to SCCs. We also leverage key ideas of the proof of this statement to obtain a more general characterization of strategyproof SCCs.

Self-implementation of social choice correspondences in Nash equilibrium

A social choice correspondence is Nash self-implementable if it can be implemented in Nash equilibrium by a social choice function that selects from it as the game form. We provide a complete characterization of all unanimous and anonymous Nash self-implementable social choice correspondences when there are two agents or two alternatives. For the case of three agents and three alternatives, only the top correspondence is Nash self-implementable. In all other cases, every Nash self-implementable social choice correspondence contains the top correspondence and is contained in the Pareto correspondence. In particular, when the number of alternatives is at least four, every social choice correspondence containing the top correspondence plus the intersection of the Pareto correspondence with a fixed set of alternatives, is self-implementable.

On the Indecisiveness of Kelly-Strategyproof Social Choice Functions

Social choice functions (SCFs) map the preferences of a group of agents over some set of alternatives to a non-empty subset of alternatives. The Gibbard-Satterthwaite theorem has shown that only extremely restrictive SCFs are strategyproof when there are more than two alternatives. For set-valued SCFs, or so-called social choice correspondences, the situation is less clear. There are miscellaneous -- mostly negative -- results using a variety of strategyproofness notions and additional requirements. The simple and intuitive notion of Kelly-strategyproofness has turned out to be particularly compelling because it is weak enough to still allow for positive results. For example, the Pareto rule is strategyproof even when preferences are weak, and a number of attractive SCFs (such as the top cycle, the uncovered set, and the essential set) are strategyproof for strict preferences. In this paper, we show that, for weak preferences, only indecisive SCFs can satisfy strategyproofness. In particular, (i) every strategyproof rank-based SCF violates Pareto-optimality, (ii) every strategyproof support-based SCF (which generalize Fishburn's C2 SCFs) that satisfies Pareto-optimality returns at least one most preferred alternative of every voter, and (iii) every strategyproof non-imposing SCF returns the Condorcet loser in at least one profile. We also discuss the consequences of these results for randomized social choice.

Strategyproof social choice when preferences and outcomes may contain ties

The Gibbard-Satterthwaite theorem implies that all anonymous, Pareto-optimal, and single-valued social choice functions can be strategically manipulated. In this paper, we investigate whether there exist social choice correspondences (SCCs), that satisfy these conditions under various assumptions about how single alternatives are eventually selected from the choice set. These assumptions include even-chance lotteries as well as resolute choice functions and linear tie-breaking orderings unknown to the agents. We show that (i) all anonymous Pareto-optimal SCCs where ties are broken according to some linear tie-breaking ordering or by means of even-chance lotteries are manipulable, and that (ii) all pairwise Pareto-optimal SCCs are manipulable for any deterministic tie-breaking rule. These results are proved by reducing the statements to finite—yet very large—formulas in propositional logic, which are then shown to be unsatisfiable by a computer.

Computational implementation

Following a theoretical analysis of the scope of Nash implementation for a given mechanism, we study the formal framework for computational identification of Nash implementability. We provide computational tools for Nash implementation in finite environments. In particular, we supply Python codes that identify (i) the domain of preferences that allows Nash implementation by a given mechanism, (ii) the maximal domain of preferences that a given mechanism Nash implements Pareto efficiency, (iii) all consistent collections of sets of a given social choice correspondence (SCC), the existence of which is a necessary condition for Nash implementation of this SCC, and (iv) check whether some of the well-known sufficient conditions for Nash implementation hold for a given SCC. Our results exhibit that the computational identification of all collections consistent with an SCC enables the planner to design appealing mechanisms.

Characterising scoring rules by their solution in iteratively undominated strategies

We characterize voting procedures according to the social choice correspondence they implement when voters cast ballots strategically, applying iteratively undominated strategies. In elections with three candidates, the Borda Rule is the unique positional scoring rule that satisfies unanimity (U) (i.e., elects a candidate whenever it is unanimously preferred) and is majoritarian after eliminating a worst candidate (MEW)(i.e., if there is a unanimously disliked candidate, the majority-preferred among the other two is elected). In a larger class of rules, Approval Voting is characterized by a single axiom that implies both U and MEW but is weaker than Condorcet-consistency (CON)—it is the only direct mechanism scoring rule that is majoritarian after eliminating a Pareto-dominated candidate (MEPD)(i.e., if there is a Pareto-dominated candidate, the majority-preferred among the other two is elected); among all finite scoring rules that satisfy MEPD, Approval Voting is the most decisive. However, it fails a desirable monotonicity property: a candidate that is elected for some preference profile, may lose the election once she gains further in popularity. In contrast, the Borda Rule is the unique direct mechanism scoring rule that satisfies U, MEW and monotonicity (MON). There exists no direct mechanism scoring rule that satisfies both MEPD and MON and no finite scoring rule satisfying CON.

Rationalizable implementation of social choice correspondences

A (deterministic) social choice correspondence F, mapping states into outcomes, is rationalizably implementable provided that there exists a mechanism such that the support of its set of rationalizable outcomes coincides with the set of outcomes recommended by F. We provide a necessary condition for rationalizable implementation, called r-monotonicity. This condition, when combined with some other auxiliary conditions, is also sufficient when there are at least three agents.

Rationalizable Implementation of Social Choice Correspondences

A social choice correspondence (SCC) F is implementable in rationalizable strategies provided that there exists a mechanism such that for each state θ, the support of its set of rationalizable outcomes is equal to the socially desirable set F(θ). We find that r -monotonicity is a necessary condition for the rationalizable implementation of a SCC. When there are at least three agents and F satisfies certain auxiliary conditions, r-monotonicity is also sufficient for rationalizable implementation. Finally, we show that a SCC which is never single-valued is rationalizably implementable if and only if it satisfies r-monotonicity.

An incentive mechanism for partner selection from a collaborative network with private information

Partner selection is critical to the success of a Virtual Enterprise. This paper studies the partner selection problem under the virtual organization breeding environment where candidates have private information about their cost to undertake the tasks. A systematic solution is provided to reveal the true value of the private information and then select the optimal partners. This solution mainly comprises two steps: (1) For each possible cost combination, a multi-objective integer nonlinear programming model and an algorithm are designed to obtain the optimal partners. Then, a social choice correspondence, which indicates the optimal partners for any cost combination, is formulated. (2) An incentive mechanism is constructed to implement the social choice correspondence and ensure that candidates will announce their values of cost truthfully. Furthermore, a numerical study is provided to illustrate the effectiveness of the solution and to investigate the relationship between the construction of the collaborative network and the utility to partners. We find that the number of candidates in each period affects the utility of partners as well as the expected surplus of the leading company significantly.

Set and revealed preference axioms for multi-valued choice

We consider choice correspondences that assign a subset to every choice set of alternatives, where the total set of alternatives is an arbitrary finite or infinite set. We focus on the relations between several extensions of the condition of independence of irrelevant alternatives on one hand, and conditions on the revealed preference relation on sets, notably the weak axiom of revealed preference, on the other hand. We also establish the connection between the condition of independence of irrelevant alternatives and so-called strong sets; the latter characterize a social choice correspondence satisfying independence of irrelevant alternatives.

Characterization of the Pareto social choice correspondence

Independent, necessary and sufficient conditions are derived for a social choice correspondence to be the one that selects exactly the Pareto optimal alternatives.

A foundation for Pareto optimality

Can an axiomatic justification be given for the requirement that society picks all and only Pareto optimal alternatives at each profile of individual preferences? Using the framework of fixed-agenda social choice theory, we present a characterization of the Pareto optimal social choice correspondence. We introduce a new independence condition, P-independence. When combined with three natural assumptions, P-independence leads to the conclusion that the social choice set and the Pareto optimal set are the same.

Rationalizable Implementation of Correspondences

A new condition, which we call uniform monotonicity, is shown to be necessary and almost sufficient for rationalizable implementation of correspondences. Uniform monotonicity is much weaker than Maskin monotonicity and reduces to it in the case of functions. Maskin monotonicity, the key condition for Nash implementation, had also been shown to be necessary for rationalizable implementation of social choice functions. Our conclusion is that the conditions for rationalizable implementation are not only starkly different from but also, much weaker than those for Nash implementation when we consider social choice correspondences. Thus, dropping rational expectations significantly expands the class of rules that can be decentralized by communication-based economic institutions.

Balancedness of social choice correspondences

A social choice correspondence satisfies balancedness if, for each pair of alternatives, x and y, and each pair of individuals, i and j , whenever a profile has x adjacent to but just above y for individual i while individual j has y adjacent to but just above x, then only switching x and y in the orderings for both of those two individuals leaves the choice set unchanged. We show how the balancedness condition (satisfied by the Borda, Pareto, and Copeland rules) interacts with other social choice properties, especially tops-only. We also use balancedness to characterize the Borda rule (for a fixed number of voters) within the class of scoring rules.

Double implementation without no-veto-power

We consider the implementation problem with at least three agents. We study double implementability of social choice correspondences in Nash equilibria and undominated Nash equilibria. We prove that “DZ-invariance,” “weak no-veto-power,” and “unanimity” together are sufficient for double implementability. If there is at least one partially honest agent in the sense of Dutta and Sen (2012), then weak no-veto-power and unanimity together are sufficient for double implementability. If there are at least two partially honest agents, then unanimity is sufficient for double implementability. In addition, we show that if there is at least one partially honest agent and unanimity is satisfied, then “LY-condition” is necessary and sufficient for double implementability. From these results, we obtain several positive corollaries.