Optimal Voting Mechanisms on Generalized Single-Peaked Domains

Stochastic mechanisms in settings without monetary transfers: The regular case

We consider the design of mechanisms for locating facilities on an interval. There are multiple agents on the interval, each receiving a utility determined by their distances to the facilities. The objectives considered are maximization of social welfare (sum of utilities) and egalitarian welfare (minimum utility). Agents can misreport their locations, and so we require the mechanisms to be strategyproof—no agent should be able to benefit from misreporting; subject to strategyproofness, we attempt to design mechanisms that are approximately optimal (have small worst-case approximation ratios). The novelty of our work is the consideration of models in which single-dipped and single-peaked preferences exist simultaneously. We consider two models. In the first model, there is a single facility, and agents may disagree about its nature: some agents prefer to be near the facility, while others prefer to be far from it. In the second model, there are two facilities: a desirable facility that all agents want near, and an undesirable facility that all agents want far. We design a variety of approximately optimal strategyproof mechanisms for both models, and prove several lower bounds as well. For the social welfare objective, we provide best-possible deterministic strategyproof mechanisms in the first model and the second model. We then provide improved randomized strategyproof mechanisms for each model, as well as a non-tight lower bound on the worst-case approximation ratio attainable by such mechanisms for the first model. For the egalitarian welfare objective, we provide a lower bound on randomized strategyproof mechanisms for the first model, as well as an optimal (non-approximate) strategyproof mechanism for the second model. All of our mechanisms are also group strategyproof: no coalition of agents can unanimously benefit from misreporting.

We ask whether the absence of information about other voters’ preferences allows optimal voting to be interpreted as sincere.We start by classifying voting mechanisms as simple and complex according to the number of message types voters can use to elect alternatives. We show that while in simple voting mechanisms the elimination of information about other voters’ preferences allows optimal voting to be interpreted as sincere, this is no longer always true for complex ones. In complex voting mechanisms, voters’ optimal strategy may vary with the size of the electorate. Therefore, in order to interpret optimal voting as sincere for complex voting mechanisms, we describe the optimal voting strategy when voters not only have no information but also have no pivotal power, i.e., as the size of the electorate tends to infinity.

A budget-limited mechanism for category-aware crowdsourcing of multiple-choice tasks

For agents it can be advantageous to vote insincerely in order to change the outcome of an election. This behavior is called manipulation. The Gibbard-Satterthwaite theorem states that in principle every non-trivial voting rule with at least three candidates is susceptible to manipulation. Since the seminal paper by Bartholdi, Tovey, and Trick in 1989, (coalitional) manipulation has been shown $$\mathrm{NP}$$ -hard for many voting rules. However, under single-peaked preferences – one of the most influential domain restrictions – the complexity of manipulation often drops from $$\mathrm{NP}$$ -hard to $$\mathrm{P}$$ . In this paper, we investigate the complexity of manipulation for the k-approval and veto families of voting rules in nearly single-peaked elections, exploring the limits where the manipulation problem turns from $$\mathrm{P}$$ to $$\mathrm{NP}$$ -hard. Compared to the classical notion of single-peakedness, notions of nearly single-peakedness are more robust and thus more likely to appear in real-world data sets.

Single-peaked preferences over multidimensional binary alternatives

Anonymous social choice function for a large atomless population maps cross-section distributions of preferences into outcomes. Because any one individual is too insignificant to a¤ect these distributions, every anonymous social choice function is individually strategy-proof. However, not every anonymous social choice function is group strategy-proof. If the set of outcomes is linearly ordered and participants have single-peaked preferences, an anonymous social choice function is group strategy-proof if and only if it can be implemented by a mechanism involving binary votes between neighbouring outcomes with nondecreasing thresholds for moving higher up. Such a mechanism can be interpreted as a version of Moulin's (1980) generalized median-voter mechanism for a large population.

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

Group strategy-proof probabilistic voting with single-peaked preferences

Voting mechanisms often exhibit a large multiplicity of equilibria. In many of these equilibria, coordination failures among voters arise where some (coalition of) voters could have induced an outcome that they all prefer to the equilibrium outcome had they agreed on a common strategy. In order to prevent these failures, we study voting mechanisms requiring that every Nash equilibrium is coalition-proof ([1]) so that, in equilibrium, no coalition of voters has some mutually beneficial deviation. The described mechanisms achieve coalitional implementation, that is they implement social choice rules in both Nash and in coalition-proof equilibria. Our approach assumes complete information and is hence well-suited for voting mechanisms for small committees. We consider a committee of n voters selecting one out of k alternatives. The main focus of the paper is on the case with k > n which is a natural environment for a small committee. The first part of the paper considers the simultaneous veto (SV) mechanisms. In each of these mechanisms, each voter has the right to select a list of alternatives to veto, and the outcome is selected randomly from the nonvetoed alternatives. Each such equilibrium admits a pure strategy equilibrium for each preference profile assuming that voters' preferences over lotteries satisfy the mild assumption of stochastic dominance. In equilibrium, no pair of voters veto the same alternative. As we show, for each specification of the veto rights, each of these mechanisms coalitionally implements a Veto by random priority rule (VRP) introduced by [6].

We develop a novel geometric approach to mechanism design using an important result in convex analysis: the duality between a closed convex set and its support function. By deriving the support function for the set of feasible interim values we extend the wellknown Maskin-Riley-Matthews-Border conditions for reduced-form auctions to social choice environments. We next refine the support function to include incentive constraints using a geometric characterization of incentive compatibility. Borrowing results from majorization theory that date back to the work of Hardy, Littlewood, and Polya (1929) we elucidate the ironing procedure introduced by Myerson (1981) and Mussa and Rosen (1978). The inclusion of Bayesian and dominant strategy incentive constraints result in the same support function, which establishes equivalence between these implementation concepts. Using Hotelling's lemma we next derive the optimal mechanism for any social choice problem and any linear objective, including revenue and surplus maximization. We extend the approach to include general concave objectives by providing a fixed-point condition characterizing the optimal mechanism. We generalize reduced-form implementation to environments with multi-dimensional, correlated types, non-linear utilities, and interdependent values. When value interdependencies are linear we are able to include incentive constraints into the support function and provide a condition when the second-best allocation is ex post incentive compatible.

We develop a novel geometric approach to mechanism design using an important result in convex analysis: the duality between a closed convex set and its support function. By deriving the support function for the set of feasible interim values we extend the well known Maskin--Riley--Matthews--Border conditions for reduced-form auctions to social choice environments. We next refine the support function to include incentive constraints using a geometric characterization of incentive compatibility. Borrowing results from majorization theory that date back to the work of Hardy, Littlewood, and P\'olya (1929) we elucidate the ironing procedure introduced by Myerson (1981) and Mussa and Rosen (1978). The inclusion of Bayesian and dominant strategy incentive constraints result in the same support function, which establishes equivalence between these implementation concepts. Using Hotelling's lemma we next derive the optimal mechanism for any social choice problem and any linear objective, including revenue and surplus maximization. We extend the approach to include general concave objectives by providing a fixed-point condition characterizing the optimal mechanism. We generalize reduced-form implementation to environments with multi-dimensional, correlated types, non-linear utilities, and interdependent values. When value interdependencies are linear we are able to include incentive constraints into the support function and provide a condition when the second-best allocation is ex post incentive compatible.

Single-Peaked preferences play an important role in the social choice literature. In this paper, we provide necessary and sufficient conditions for observed behaviour to be consistent with a mixture model of single-peaked preferences for a given ordering of the alternatives. These conditions can be tested in time polynomial in the number of choice alternatives. In addition, algorithms are provided which identify the underlying ordering of choice alternatives if ordering is unknown. These algorithms also run in polynomial time, providing an efficient test for the mixture model of single-peaked preferences.

In this paper we establish the link between strategy-proofness and unanimity in a domain of private good economies with single-peaked preferences. We introduce a new condition and we prove that if this new property together with the requirement of citizen sovereignty are held, a social choice function satisfies strategy-proofness if and only if it is unanimous. As an implication, we show that strategy-proofness and Maskin monotonicity become equivalent. We also give applications to implementation literature: We provide a full characterization for standard Nash implementation and partially honest Nash implementation and we determine a certain equivalence among these theories.

Optimal mechanisms for agents with multi-dimensional preferences are generally complex. This complexity makes them challenging to solve for and impractical to run. In a typical mechanism design approach, a model is posited and then the optimal mechanism is designed for the model. Successful mechanism design gives mechanisms that one could at least imagine running. By this measure, multi-dimensional mechanism design has had only limited success. In this paper we take the opposite approach, which we term reverse mechanism design. We start by hypothesizing the optimality of a particular form of mechanism that is simple and reasonable to run, then we solve for sufficient conditions for the mechanism to be optimal (among all mechanisms). This paper has two main contributions. The first is in codifying the method of virtual values from single-dimensional auction theory and extending it to agents with multidimensional preferences. The second is in applying this method to two paradigmatic classes of multi-dimensional preferences. The first class is unit-demand preferences (e.g., a homebuyer who wishes to buy at most one house); for this class we give sufficient conditions under which posting a uniform price for each item is optimal. This result generalizes one of Alaei et al. [2013] for a consumer with values uniform on interval [0; 1], and contrasts with an example of Thanassoulis [2004] for a consumer with values uniform on interval [5; 6] where uniform pricing is not optimal. The second class is additive preferences, for this class we give sufficient conditions under which posting a price for the grand bundle is optimal. This result generalizes a recent result of Hart and Nisan [2012] and relates to work of Armstrong [1999]. Similarly to an approach of Alaei et al. [2013], these results for single-agent pricing problems can be generalized naturally to multi-agent auction problems.