Strategy-proof Social Choice Functions Research Articles

A Note on Binary Strategy-Proof Social Choice Functions

Let Φn be the set of the binary strategy-proof social choice functions referred to a group of n voters who are allowed to declare indifference between the alternatives. We provide a recursive way to obtain the set Φn+1 from the set Φn. Computing the cardinalities |Φn| presents difficulties as the computation of the Dedekind numbers. The latter give the analogous number of social choice functions when only strict preferences are admitted. A comparison is given for the known values. Based on our results, we present a graphical description of the binary strategy-proof social choice functions in the case of three voters.

The structure of two-valued coalitional strategy-proof social choice functions

The paper investigates the structure of coalitional strategy-proof social choice functions – CSP scfs, for short – whose range is a subset of cardinality two of an arbitrary set A of alternatives. The study is conducted in the case where the voters/agents are allowed to express indifference among elements of A, and the domain of the scfs consists of preference profiles P=(Pv)v∈V over a society V of arbitrary cardinality. A representation formula for the two-valued CSP scfs is obtained that provides the structure of such functions.

Binary strategy-proof social choice functions with indifference

Binary strategy-proof social choice functions with indifference

The Structure of Two-Valued Strategy-Proof Social Choice Functions With Indifference

We give a structure theorem for all coalitionally strategy-proof social choice functions whose range is a subset of cardinality two of a given larger set of alternatives. We provide this in the case where the voters/agents are allowed to express indifference and the domain consists of profiles of preferences over a society of arbitrary cardinality. The theorem, that takes the form of a representation formula, can be used to construct all functions under consideration.

On single-peaked domains and min–max rules

We consider social choice problems where the admissible set of preferences of each agent is single-peaked. First, we show that if all the agents have the same admissible set of preferences, then every unanimous and strategy-proof social choice function (SCF) is tops-only. Next, we consider situations where different agents have different admissible sets of single-peaked preferences. We show by means of an example that unanimous and strategy-proof SCFs need not be tops-only in this situation, and consequently provide a sufficient condition on the admissible sets of preferences of the agents so that unanimity and strategy-proofness guarantee tops-onlyness. Finally, we characterize all domains on which (i) every unanimous and strategy-proof SCF is a min–max rule, and (ii) every min–max rule is strategy-proof. As an application of our result, we obtain a characterization of the unanimous and strategy-proof social choice functions on maximal single-peaked domains (Moulin in Public Choice 35(4):437–455. https://doi.org/10.1007/BF00128122 , 1980; Weymark in SERIEs 2(4):529–550. https://doi.org/10.1007/s13209-011-0064-5 , 2011), minimally rich single-peaked domains (Peters et al. in J Math Econ 52:123–127. https://doi.org/10.1016/j.jmateco.2014.03.008 . http://www.sciencedirect.com/science/article/pii/S0304406814000470 , 2014), maximal regular single-crossing domains (Saporiti in Theor Econ 4(2):127–163, 2009, J Econ Theory 154:216–228. https://doi.org/10.1016/j.jet.2014.09.006 . http://www.sciencedirect.com/science/article/pii/S0022053114001276 , 2009), and distance based single-peaked domains.

Condorcet domains, median graphs and the single-crossing property

Condorcet domains are sets of linear orders with the property that, whenever the preferences of all voters of a society belong to this set, their majority relation has no cycles. We observe that, without loss of generality, every such domain can be assumed to be closed in the sense that it contains the majority relation of every profile with an odd number of voters whose preferences belong to this domain. We show that every closed Condorcet domain can be endowed with the structure of a median graph and that, conversely, every median graph is associated with a closed Condorcet domain (in general, not uniquely). Condorcet domains that have linear graphs (chains) associated with them are exactly the preference domains with the classical single-crossing property. As a corollary, we obtain that a domain with the so-called ‘representative voter property’ is either a single-crossing domain or a very special domain containing exactly four different preference orders whose associated median graph is a 4-cycle. Maximality of a Condorcet domain imposes additional restrictions on the associated median graph. We prove that among all trees only (some) chains can be associated graphs of maximal Condorcet domains, and we characterize those single-crossing domains which are maximal Condorcet domains. Finally, using the characterization of Nehring and Puppe (J Econ Theory 135:269–305, 2007) of monotone Arrovian aggregation, we show that any closed Condorcet domain admits a rich class of strategy-proof social choice functions.

On the ultrafilter representation of coalitionally strategy-proof social choice functions

By means of a simple new characterization of ultrafilters, we elementarily prove, in the case of finitely many alternatives and arbitrarily large societies, that every coalitionally strategy-proof social choice function with at least three alternatives in its range is given by an ultrafilter. This provides an alternate and simple proof of results in Mihara (Soc Choice Welf 17:393–402, 2000). In case there are only two alternatives in the range of a coalitionally strategy-proof social choice function, we describe its structure, supplementing the work of Barbera et al. (Int J Game Theory 41:791–808, 2012).

The decomposition of strategy-proof random social choice functions on dichotomous domains

A feature of strategy-proof and efficient random social choice functions (RSCFs) defined over several important domains is that they are fixed probability distributions over deterministic strategy-proof and efficient social choice functions. We call such domains deterministic extreme point (DEP) domains. Examples of DEP domains are the domain of all strict preferences and the domain of single-peaked preferences. We show that the dichotomous domain introduced in Bogomolnaia et al. (2005) is not a DEP domain. We find a necessary condition for a strategy-proof RSCF to be written as a fixed probability distribution of deterministic strategy proof social choice functions. We show that this condition is compatible with efficiency. We also show that the condition is sufficient for decomposability in a special case.

Finding Strategyproof Social Choice Functions via SAT Solving

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 viable not only for proving classic impossibility theorems such as Arrow's Theorem but also for finding new impossibilities in the context of preference extensions. In this paper, we demonstrate that these computer-aided techniques can also be applied to improve our understanding of strategyproof irresolute social choice functions. These functions, however, requires a more evolved encoding as otherwise the search space rapidly becomes much too large. Our contribution is two-fold: We present an efficient encoding for translating such problems to SAT and leverage this encoding to prove new results about strategyproofness with respect to Kelly's and Fishburn's preference extensions. For example, we show that no Pareto-optimal majoritarian social choice function satisfies Fishburn-strategyproofness. Furthermore, we explain how human-readable proofs of such results can be extracted from minimal unsatisfiable cores of the corresponding SAT formulas.

Finding strategyproof social choice functions via SAT solving

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...

Affine Maximizers in Domains with Selfish Valuations

We consider the domain of selfish and continuous preferences over a “rich” allocation space and show that onto, strategyproof and allocation non-bossy social choice functions are affine maximizers. Roberts [1979] proves this result for a finite set of alternatives and an unrestricted valuation space. In this article, we show that in a subdomain of the unrestricted valuations with the additional assumption of allocation non-bossiness , using the richness of the allocations, the strategyproof social choice functions can be shown to be affine maximizers. We provide an example to show that allocation non-bossiness is indeed critical for this result. This work shows that an affine maximizer result needs a certain amount of richness split across valuations and allocations.

Strategy-proof social choice on multiple and multi-dimensional single-peaked domains

We generalize the traditional concept of single-peaked preference domains in two ways. First, we introduce the concept of a multiple single-peaked domain, where the set of alternatives is equipped with several underlying orderings with respect to which a preference can be single-peaked, and we argue that these domains are appropriate to represent preferences over political parties. Second, we define a domain of multi-dimensional single-peaked preferences based on the condition of value-restricted preferences by Sen [34]. We provide complete characterizations of the strategy-proof social choice functions on both multiple and multi-dimensional single-peaked domains. For both domains, we also identify the subclasses consisting of all anonymous strategy-proof social choice functions.

Strategy proofness and Pareto efficiency in quasilinear exchange economies

In this paper, we revisit a longstanding question on the structure of strategy-proof and Pareto-efficient social choice functions (SCFs) in classical exchange economies (Hurwicz 1972). Using techniques developed by Myerson in the context of auction design, we show that in a specific quasilinear domain, every Pareto-efficient and strategy-proof SCF that satisfies non-bossiness and a mild continuity property is dictatorial. The result holds for an arbitrary number of agents, but the two-person version does not require either the non-bossiness or the continuity assumptions. It also follows that the dictatorship conclusion holds on any superset of this domain. We also provide a minimum consumption guarantee result in the spirit of Serizawa and Weymark (2003).

Strategy-proof voting for multiple public goods

In a voting model where the set of feasible alternatives is a subset of a product set A = A1×⋯×Am of m finite categories, we characterize the set of all strategy-proof social choice functions for three different types of preference domains over A, namely for the domains of additive, completely separable, and weakly separable preferences over A.

Another direct proof for the Gibbard–Satterthwaite Theorem

We prove the following result which is equivalent to the Gibbard–Satterthwaite Theorem: when there are at least 3 alternatives, for any unanimous and strategy-proof social choice function, at any given profile if an individual’s top ranked alternative differs from the social choice, then she can not change the social choice at that profile by changing her ranking. Hence, proving it yields a new proof for the Gibbard–Satterthwaite Theorem.

An extreme point characterization of random strategy-proof social choice functions: The two alternative case

We show that every strategy-proof random social choice function is a convex combination of strategy-proof deterministic social choice functions in a two-alternative voting model. This completely characterizes all strategy-proof random social choice functions in this setting.

Group strategy-proof social choice functions with binary ranges and arbitrary domains: characterization results

We define different concepts of group strategy-proofness for social choice functions. We discuss the connections between the defined concepts under different assumptions on their domains of definition. We characterize the social choice functions that satisfy each one of them and whose ranges consist of two alternatives, in terms of two types of basic properties. Finally, we obtain the functional form of all rules satisfying our strongest version of group strategy-proofness.

A unified approach to strategy-proofness for single-peaked preferences

This article establishes versions of Moulin’s (Public Choice 35:437–455, 1980) characterizations of various classes of strategy-proof social choice functions when the domain consists of all profiles of single-peaked preferences on an arbitrary subset of the real line. Two results are established that show that the median of 2n + 1 numbers can be expressed using a combination of minimization and maximization operations applied to subsets of these numbers when either these subsets or the numbers themselves are restricted in a particular way. These results are used to show how Moulin’s characterizations of generalized median social choice functions can be obtained as corollaries of his characterization of min–max social choice functions.

On strategy-proof social choice under categorization

I consider social choice problems such that (i) the set of alternatives can be partitioned into categories based on a prominent and objective feature and (ii) agents have strict preferences over the alternatives. Main results are characterizations of the structure of the strategy-proof social choice functions. I prove that each social choice function is strategy-proof if and only if it is decomposable into “small” strategy-proof social choice functions; one of them chooses one category and each of the others chooses one alternative from a category.

Algorithmic verification of feasibility for generalized median voter schemes on compact ranges

AbstractBarberá, Massó and Serizawa (1998) provided full characterization for class of strategy-proof social choice functions for societies where the set of alternatives is any full dimensional compact subset of a Euclidean space and all voters have generalized single-peaked preferences. They proved that this class is composed by generalized median voter schemes satisfying an additional condition, called the “intersection property”. But according to their results in order to understand whether any generalized median voter scheme satisfies intersection property for given set of alternatives or not it was necessary to check all the alternatives from the set of unfeasible alternatives – addition of the set of feasible alternatives to minimal Cartesian product range, containing this set. So the number of alternatives to be checked, was infinite. In this paper it is proved, that it is enough to check finite number of alternatives from the set of unfeasible alternatives and constructive algorithm to determine alternatives that should be checked is provided.