Strategy-proof Social Choice Research Articles

A characterization of superdictatorial domains for strategy-proof social choice functions

A domain D is dictatorial iff there exists no surjective, strategy-proof and non-dictatorial social choice function defined over D. A dictatorial domain D is superdictatorial iff every superdomain of D is also dictatorial. The existence of dictatorial but not superdictatorial domains being known, we show that a dictatorial domain D is superdictatorial iff every alternative is ranked at the top by some ordering in D.

The Gibbard–Satterthwaite theorem of social choice theory in an infinite society and LPO (limited principle of omniscience)

This paper is an attempt to examine the main theorems of social choice theory from the viewpoint of constructive mathematics. We examine the Gibbard–Satterthwaite theorem [A.F. Gibbard, Manipulation of voting schemes: a general result, Econometrica 41 (1973) 587–601; M.A. Satterthwaite, Strategyproofness and Arrow’s conditions: existence and correspondence theorems for voting procedures and social welfare functions, Journal of Economic Theory 10 (1975) 187–217] in a society with an infinite number of individuals (infinite society). We will show that the theorem that any coalitionally strategy-proof social choice function may have a dictator or has no dictator in an infinite society is equivalent to LPO (limited principle of omniscience). Therefore, it is non-constructive. A dictator of a social choice function is an individual such that if he strictly prefers an alternative (denoted by x) to another alternative (denoted by y), then the social choice function chooses an alternative other than y. Coalitional strategy-proofness is an extension of the ordinary strategy-proofness. It requires non-manipulability for coalitions of individuals as well as for a single individual.

Strategy-proof allocation of multiple public goods

We characterize the set of strategy-proof social choice functions (SCFs), the outcome of which are multiple public goods. The set of feasible alternatives is a subset of a product set with a finite number of elements. We do not require the SCFs to be ‘onto’, but instead impose the weaker requirement that every element in each category of public goods is attained at some preference profile. Admissible preferences are arbitrary rankings of the goods in the various categories, while a separability restriction concerning preferences among the various categories is assumed. We find that the range of the SCF is uniquely decomposed into a product set in general coarser than the original product set, and that the SCF must be dictatorial in each component of the range. If the range cannot be decomposed at all, the SCF is dictatorial in spite of the separability assumption on preferences, and a form of the Gibbard-Satterthwaite theorem with a restricted preference domain is obtained.

Type two computability of social choice functions and the Gibbard–Satterthwaite theorem in an infinite society

This paper investigates the computability problem of the Gibbard–Satterthwaite theorem [A.F. Gibbard, Manipulation of voting schemes: a general result, Econometrica 41 (1973) 587–601; M.A. Satterthwaite, Strategyproofness and Arrow’s conditions: existence and correspondence theorems for voting procedures and social welfare functions, Journal of Economic Theory 10 (1975) 187–217] of social choice theory in a society with an infinite number of individuals (infinite society) based on Type two computability by Weihrauch [K. Weihrauch, A Simple Introduction to Computable Analysis, Informatik Berichte, vol. 171, second ed., Fern Universität Hagen, Hagen, 1995; K. Weihrauch, Computable Analysis, Springer-Verlag, 2000]. There exists a dictator or there exists no dictator for any coalitionally strategy-proof social choice function in an infinite society. We will show that if there exists a dictator for a social choice function, it is computable in the sense of Type two computability, but if there exists no dictator it is not computable. A dictator of a social choice function is an individual such that if he strictly prefers an alternative (denoted by x) to another alternative (denoted by y), then it does not choose y, and his most preferred alternative is always chosen. Coalitional strategy-proofness is an extension of the ordinary strategy-proofness. It requires non-manipulability by coalitions of individuals as well as by a single individual.

Dominant strategy implementation in economic environments

We study dominant strategy implementation especially in economic environments. We first show that in general environments, strategy-proofness and quasi-strong-non-bossiness together are necessary and sufficient for dominant strategy implementation via the associated direct revelation mechanism. We next prove that in weak separable environments, strategy-proofness is sufficient for dominant strategy implementation, by using an augmented revelation mechanism similar to the one devised by Jackson, Palfrey, and Srivastava [Jackson, M.O., Palfrey, T.R., Srivastava, S., 1994. Undominated Nash implementation in bounded mechanisms. Games Econ. Behav. 6, 474–501]. Moreover, we focus on pure exchange economies without free disposal, and try to construct another augmented revelation mechanism that satisfies balancedness in and out of equilibrium, and which implements all strategy-proof social choice functions in dominant strategy equilibria.

Coalitionally Strategy-Proof Social Choice Correspondences and the Pareto Rule

Coalitionally Strategy-Proof Social Choice Correspondences and the Pareto Rule

On the equivalence of the HEX game theorem and the Duggan–Schwartz theorem for strategy-proof social choice correspondences

Gale [D. Gale, The game of HEX and the Brouwer fixed-point theorem, American Mathematical Monthly 86 (1979) 818–827] has shown that the so called HEX game theorem that any HEX game has one winner is equivalent to the Brouwer fixed point theorem. In this paper we will show that under some assumptions about marking rules of HEX games, the HEX game theorem for a 6 × 6 HEX game is equivalent to the Duggan–Schwartz theorem for strategy-proof social choice correspondences [J. Duggan, T. Schwartz, Strategic manipulability without resoluteness or shared beliefs: Gibbard–Satterthwaite generalized, Social Choice and Welfare 17 (2000) 85–93] that there exists no social choice correspondence which satisfies the conditions of strategy-proofness, non-imposition, residual resoluteness, and has no dictator.

The structure of strategy-proof social choice — Part I: General characterization and possibility results on median spaces

We define a general notion of single-peaked preferences based on abstract betweenness relations. Special cases are the classical example of single-peaked preferences on a line, the separable preferences on the hypercube, the “multi-dimensionally single-peaked” preferences on the product of lines, but also the unrestricted preference domain. Generalizing and unifying the existing literature, we show that a social choice function is strategy-proof on a sufficiently rich domain of generalized single-peaked preferences if and only if it takes the form of voting by issues (“voting by committees”) satisfying a simple condition called the “Intersection Property.” Based on the Intersection Property, we show that the class of preference domains associated with “median spaces” gives rise to the strongest possibility results; in particular, we show that the existence of strategy-proof social choice rules that are non-dictatorial and neutral requires an underlying median space. A space is a median space if, for every triple of elements, there is a fourth element that is between each pair of the triple; numerous examples are given (some well-known, some novel), and the structure of median spaces and the associated preference domains is analysed.

Single-Crossing, Strategic Voting and the Median Choice Rule

This paper studies the strategic foundation of the Representative Voter Theorem (Rothstein in: Pub Choice 72:193–212, 1991), also called the “second version” of the Median Voter Theorem. As a by-product, it also considers the existence of strategy-proof social choice functions over the domain of single-crossing preferences. The main result shows that single-crossing constitutes a domain restriction over the real line that allows not only majority voting equilibria, but also non-manipulable choice rules. In particular, this is true for the median rule, which is found to be group strategic-proof over the full set of alternatives and over every nonempty subset. In addition, the paper also examines the relation between single-crossing and order-restriction. And it uses this relation together with the strategy-proofness of the median rule to prove that the outcome predicted by the Representative Voter Theorem can be implemented in dominant strategies through a simple mechanism. This mechanism is a two-stage voting procedure in which, first, individuals select a representative among themselves, and then the winner chooses a policy to be implemented by the planner.

Dominant Strategy Implementation in Economic Environments

We study dominant strategy implementation especially in economic environments. We first show that in general environments, strategy-proofness and quasi-strong-non-bossiness together are necessary and sufficient for dominant strategy implementation via the associated direct revelation mechanism. We next prove that in weak separable environments, strategy-proofness is sufficient for dominant strategy implementation, by using an augmented revelation mechanism similar to the one devised by Jackson et al. (1994). Moreover, we focus on pure exchange economies without free disposal, and try to construct another augmented revelation mechanism that satisfies balancedness in and out of equilibrium, and which implements all strategy-proof social choice functions in dominant strategy equilibria.

Choosing the level of a public good when agents have an outside option

The work examines strategy-proof social choice functions which select a level or the location of a public good when its consumption is not compulsory. We assume that agents have single-peaked preferences on the consumption of the public good as well as a reservation utility. Agents opt out of the good whenever they get a utility smaller than their reservation utility. We characterize strategy-proof and efficient social choice functions as well as the ones which are anonymous and group strategy-proof. Finally, we observe that for given preferences a Condorcet winner may not exist.

A strategy-proofness characterization of majority rule

A feasible alternative x is a strong Condorcet winner if for every other feasible alternative y there is some majority coalition that prefers x to y. Let \({\cal L}_{C}\) (resp., \(\wp_{C})\) denote the set of all profiles of linear (resp., merely asymmetric) individual preference relations for which a strong Condorcet winner exists. Majority rule is the only non-dictatorial and strategy-proof social choice rule with domain \({\cal L}_{C}\), and majority rule is the only strategy-proof rule with domain \(\wp_{C}\).

Efficient strategy-proof exchange and minimum consumption guarantees

For exchange economies with classical economic preferences, it is shown that any strategy-proof social choice function that selects Pareto optimal outcomes cannot guarantee everyone a consumption bundle bounded away from the origin. This result demonstrates that there is a fundamental conflict between efficiency and distributional goals in exchange economies if the social choice rule is required to be strategy-proof.

A leximin characterization of strategy-proof and non-resolute social choice procedures

We characterize strategy-proof social choice procedures when choice sets need not be singletons. Sets are compared by leximin. For a strategy-proof rule g, there is a positive integer k such that either (i) the choice sets g(r) for all profiles r have the same cardinality k and there is an individual i such that g(r) is the set of alternatives that are the k highest ranking in i's preference ordering, or (ii) all sets of cardinality 1 to k are chosen and there is a coalition L of cardinality k such that g(r) is the union of the tops for the individuals in L. There do not exist any strategy-proof rules such that the choice sets are all of cardinality $k^*$ to k where $1 < k^* < k$ .

Multi-valued strategy-proof social choice rules

In this paper we introduce a new definition of strategy-proofness for multi-valued social choice correspondences. We prove two Gibbard- Satterthwaite type results for strategy-proof social choice correspondences. These results show that allowing multiple outcomes as social choices will not necessarily lead to an escape from the Gibbard-Satterthwaite impossibility theorem.

Preference revelation with a limited number of indifference classes

Abstract. Suppose that g is a strategy-proof social choice rule on the domain of all profiles of complete and transitive binary relations that have exactly m indifference classes. If $m \ge 3$ and the range of g has three or more members, then g is dictatorial. If m = 2, then for any set X of feasible alternatives, there exist non-dictatorial and strategy-proof rules that are sensitive to the preferences of every individual and which have X as range.

An introduction to strategy-proof social choice functions

An introduction to strategy-proof social choice functions

Strategyproof single unit award rules

The problem of allocating a single indivisible unit to one of several agents is considered, where monetary compensations are not allowed, and the unit is not necessarily desirable to each agent. In addition to strategyproofness, three properties of social choice functions are considered: Pareto-optimality, nondictatorship, and nonbossiness. It is shown that these three additional criteria cannot be satisfied simultaneously. However, any two of the additional criteria can be satisfied. We give characterizations of the classes of strategyproof social choice functions satisfying these three pairs of properties.

Existence of a coalitionally strategyproof social choice function: A constructive proof

This paper gives a concrete example of a nondictatorial, coalitionally strategyproof social choice function for countably infinite societies. The function is defined for those profiles such that for each alternative, the coalition that prefers it the most is “describable.” The “describable” coalitions are assumed to form a countable Boolean algebra. The paper discusses oligarchical characteristics of the function, employing a specific interpretation of an infinite society. The discussion clarifies within a single framework a connection between the negative result (the Gibbard-Satterthwaite theorem) for finite societies and the positive result for infinite ones.

On the Shapley–Scarf economy: the case of multiple types of indivisible goods

We study a generalization of Shapley–Scarf’s [Journal of Mathematical Economics 1 (1974) 23–37] economy in which multiple types of indivisible goods are traded. We show that many of the distinctive results from the Shapley–Scarf economy do not carry over to this model, even if agents’ preferences are strict and can be represented by additively separable utility functions. The core may be empty. The strict core, if nonempty, may be multi-valued, and might not coincide with the set of competitive allocations. Furthermore, there is no Pareto efficient, individually rational, and strategy-proof social choice rule. We also show that the core may be empty in the class of economies with a single type of indivisible good and agents consuming multiple units, even if no complementarity exists among the goods.