Unanimous and strategy-proof probabilistic rules for single-peaked preference profiles on graphs

Finitely many agents have preferences on a finite set of alternatives, single-peaked with respect to a connected graph with these alternatives as vertices. A probabilistic rule assigns to each preference profile a probability distribution over the alternatives. First, all unanimous and strategy-proof probabilistic rules are characterized when the graph is a tree. These rules are uniquely determined by their outcomes at those preference profiles at which all peaks are on leaves of the tree and, thus, extend the known case of a line graph. Second, it is shown that every unanimous and strategy-proof probabilistic rule is random dictatorial if and only if the graph has no leaves. Finally, the two results are combined to obtain a general characterization for every connected graph by using its block tree representation

On the structure of division rules

We consider the problem of dividing one unit of an infinitely divisible object among a finite number of agents. We provide a characterization of all single-peaked domains on which the uniform rule is the unique division rule satisfying efficiency, strategy-proofness, and equal treatment of equals (ETE). We also provide a class of division rules satisfying these properties on the remaining single-peaked domains. Next, we consider non single-peaked domains and provide a characterization of all such domains on which the uniform rule satisfies efficiency, strategy-proofness, and ETE. We also show that under some mild richness conditions the uniform rule is the unique rule satisfying the mentioned properties on these domains. Finally, we provide a class of division rules satisfying efficiency, strategy-proofness, and ETE on the remaining non single-peaked domains. We conclude the paper by providing a wide range of applications to justify the usefulness of our results

On the equivalence of strategy-proofness and upper contour strategy-proofness for randomized social choice functions

We consider a weaker notion of strategy-proofness called upper contour strategy-proofness (UCSP) and investigate its relation with strategy-proofness (SP) for random social choice functions (RSCFs). Apart from providing a simpler way to check whether a given RSCF is SP or not, UCSP is useful in modeling the incentive structures for certain behavioral agents. We show that SP is equivalent to UCSP and elementary monotonicity on any domain satisfying the upper contour no restoration (UCNR) property. To analyze UCSP on multi-dimensional domains, we consider some block structure over the preferences. We show that SP is equivalent to UCSP and block monotonicity on domains satisfying the block restricted upper contour preservation property. Next, we analyze the relation between SP and UCSP under unanimity and show that SP becomes equivalent to UCSP and multi-swap monotonicity on any domain satisfying the multi-swap UCNR property. Finally, we show that if there are two agents, then under unanimity, UCSP alone becomes equivalent to SP on any domain satisfying the swap UCNR property. We provide applications of our results on the unrestricted, single-peaked, single-crossing, single-dipped, hybrid, and multi-dimensional domains such as lexicographically separable domains with one component ordering and domains under committee formation

On the structure of division rules

We consider the problem of dividing one unit of an infinitely divisible object among a finite number of agents. We provide a characterization of all single-peaked domains on which the uniform rule is the unique division rule satisfying efficiency, strategy-proofness, and equal treatment of equals (ETE). We also provide a class of division rules satisfying these properties on the remaining single-peaked domains. Next, we consider non single-peaked domains and provide a characterization of all such domains on which the uniform rule satisfies efficiency, strategy-proofness, and ETE. We also show that under some mild richness conditions the uniform rule is the unique rule satisfying the mentioned properties on these domains. Finally, we provide a class of division rules satisfying efficiency, strategy-proofness, and ETE on the remaining non single-peaked domains. We conclude the paper by providing a wide range of applications to justify the usefulness of our results

A Unified Characterization of Randomized Strategy-proof Rules

This paper presents a unified characterization of the unanimous and strategy-proof random rules on a class of domains that are based on some prior ordering over the alternatives. It identifies a condition called top-richness so that, if a domain satisfies top-richness, then an RSCF on it is unanimous and strategy-proof if and only if it is a convex combination of tops-restricted min-max rules. Well-known domains like single-crossing, single-peaked, single-dipped etc. satisfy top-richness. This paper also provides a characterization of the random min-max domains. Furthermore, it offers a characterization of the tops-only and strategy-proof random rules on top-rich domains satisfying top-connectedness. Finally, it presents a characterization of the unanimous (tops-only) and group strategy-proof random rules on those domains

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

Formation of committees under constraints through random voting rules

We consider the problem of choosing a committee from a set of available candidates through a randomized social choice function when there are bounds on the size (the number of members) of the committee to be formed. We show that for any (non-vacuous) restriction on the size of the committee, a random social choice function (RSCF) is onto and strategy-proof if and only if it is a range-restricted random dictatorial rule. Next, we consider the situation where an “undesirable committee” can be chosen with positive probability only if everyone in the society wants it as his best committee. We call this property strong unanimity. We characterize all strongly unanimous and strategy-proof RSCFs when there is exactly one undesirable committee. A common situation where a single committee is undesirable is one where the null committee is not allowed to be formed. We further show that there is no RSCF satisfying strong unanimity and strategy-proofness when there are more than one undesirable committees. Finally, we extend all our results when strategy-proofness is strengthened with group strategy-proofness

Random social choice functions for single-peaked domains on trees

Finitely many agents have single-peaked preferences on a finite set of alternatives structured as a tree. Under a richness condition on the domain we characterize all unanimous and strategy-proof random social choice functions. These functions are uniquely determined by the values they assign to preference profiles where all peaks are on leafs of the tree

Nash Welfare Guarantees for Fair and Efficient Coverage

We study coverage problems in which, for a set of agents and a given threshold $T$, the goal is to select $T$ subsets (of the agents) that, while satisfying combinatorial constraints, achieve fair and efficient coverage among the agents. In this setting, the valuation of each agent is equated to the number of selected subsets that contain it, plus one. The current work utilizes the Nash social welfare function to quantify the extent of fairness and collective efficiency. We develop a polynomial-time $\left(18 + o(1) \right)$-approximation algorithm for maximizing Nash social welfare in coverage instances. Our algorithm applies to all instances wherein, for the underlying combinatorial constraints, there exists an FPTAS for weight maximization. We complement the algorithmic result by proving that Nash social welfare maximization is APX-hard in coverage instances.Comment: 19 page