How many candidates are needed to make elections hard to manipulate?

In multiagent settings where the agents have different preferences, preference aggregation is a central issue. Voting is a general method for preference aggregation, but seminal results have shown that all general voting protocols are manipulable. One could try to avoid manipulation by using voting protocols where determining a beneficial manipulation is hard computationally. The complexity of manipulating realistic elections where the number of candidates is a small constant was recently studied (Conitzer 2002), but the emphasis was on the question of whether or not a protocol becomes hard to manipulate for some constant number of candidates. That work, in many cases, left open the question: How many candidates are needed to make elections hard to manipulate? This is a crucial question when comparing the relative manipulability of different voting protocols. In this paper we answer that question for the voting protocols of the earlier study: plurality, Borda, STV, Copeland, maximin, regular cup, and randomized cup. We also answer that question for two voting protocols for which no results on the complexity of manipulation have been derived before: veto and plurality with runoff. It turns out that the voting protocols under study become hard to manipulate at 3 candidates, 4 candidates, 7 candidates, or never

Social choice of convex risk measures through Arrovian aggregation of variational preferences

It is known that a combination of the Maccheroni-Marinacci-Rustichini (2006) axiomatisation of variational preferences with the Föllmer-Schied (2002,2004) representation theorem for concave monetary utility functionals provides an (individual) decision-theoretic foundation for convex risk measures. The present paper is devoted to collective decision making with regard to convex risk measures and addresses the existence problem for non-dictatorial aggregation functions of convex risk measures - in the guise of variational preferences - satisfying Arrow-type rationality axioms (weak universality, systematicity, Pareto principle). We prove an impossibility result for finite electorates, viz. a variational analogue of Arrow's impossibility theorem. For infinite electorates, the possibility of rational aggregation of variational preferences (i.e. convex risk measures) depends on a uniform continuity condition for the variational preference profiles: We shall prove variational analogues of both Campbell's impossibility theorem and Fishburn's possibility theorem. Methodologically, we adopt the model-theoretic approach to aggregation theory inspired by Lauwers-Van Liedekerke (1995). In an appendix, we apply the Dietrich-List (2010) analysis of logical aggregation based on majority voting to the problem of variational preference aggregation. The fruit is a possibility theorem, but at the cost of considerable and - at least at first sight - rather unnatural restrictions on the domain of the variational preference aggregator.variational preference representation, convex risk measure, multiple priors preferences, Arrow-type preference aggregation, judgment aggregation, abstract aggregation theory, model theory, first-order predicate logic, ultrafilter, ultraproduct

Preference Integration and Optimization of Multistage Weighted Voting System Based on Ordinal Preference

Multistage voting is a common voting form through which the winners are selected. By virtue of weighted multistage voting rules, in this paper, we establish a weighted voting model by analyzing the correlation between individual preference and group preference. The weights of voters in each voting stage are adjusted through preference deviation degrees between individual preferences and group preference, and the ranking among candidates in each stage is determined according to weighted Borda function value. Examples are given to verify our model, which shows that weighted information aggregation model can mine more useful information from different individual preferences of voters to quicken the aggregation of group preference

Approximate Judgement Aggregation

In this paper we analyze judgement aggregation problems in which a group of agents independently votes on a set of complex propositions that has some interdependency constraint between them (e.g., transitivity when describing preferences). We consider the issue of judgement aggregation from the perspective of approximation. That is, we generalize the previous results by studying approximate judgement aggregation. We relax the main two constraints assumed in the current literature, Consistency and Independence and consider mechanisms that only approximately satisfy these constraints, that is, satisfy them up to a small portion of the inputs. The main question we raise is whether the relaxation of these notions significantly alters the class of satisfying aggregation mechanisms. The recent works for preference aggregation of Kalai, Mossel, and Keller fit into this framework. The main result of this paper is that, as in the case of preference aggregation, in the case of a subclass of a natural class of aggregation problems termed `truth-functional agendas', the set of satisfying aggregation mechanisms does not extend non-trivially when relaxing the constraints. Our proof techniques involve Boolean Fourier transform and analysis of voter influences for voting protocols. The question we raise for Approximate Aggregation can be stated in terms of Property Testing. For instance, as a corollary from our result we get a generalization of the classic result for property testing of linearity of Boolean functions.judgement aggregation, truth-functional agendas, computational social choice, computational judgement aggregation, approximate aggregation, inconsistency index, dependency index

A Mechanism for Participatory Budgeting With Funding Constraints and Project Interactions

Participatory budgeting (PB) has been widely adopted and has attracted significant research efforts; however, there is a lack of mechanisms for PB which elicit project interactions, such as substitution and complementarity, from voters. Also, the outcomes of PB in practice are subject to various minimum/maximum funding constraints on 'types' of projects. There is an insufficient understanding of how these funding constraints affect PB's strategic and computational complexities. We propose a novel preference elicitation scheme for PB which allows voters to express how their utilities from projects within 'groups' interact. We consider preference aggregation done under minimum and maximum funding constraints on 'types' of projects, where a project can have multiple type labels as long as this classification can be defined by a 1-laminar structure (henceforth called 1-laminar funding constraints). Overall, we extend the Knapsack voting model of Goel et al. in two ways - enriching the preference elicitation scheme to include project interactions and generalizing the preference aggregation scheme to include 1-laminar funding constraints. We show that the strategyproofness results of Goel et al. for Knapsack voting continue to hold under 1-laminar funding constraints. Although project interactions often break the strategyproofness, we study a special case of vote profiles where truthful voting is a Nash equilibrium under substitution project interactions. We then turn to the study of the computational complexity of preference aggregation. Social welfare maximization under project interactions is NP-hard. As a workaround for practical instances, we give a fixed parameter tractable (FPT) algorithm for social welfare maximization with respect to the maximum number of projects in a group

Universal Voting Protocol Tweaks to Make Manipulation Hard

Voting is a general method for preference aggregation in multiagent settings, but seminal results have shown that all (nondictatorial) voting protocols are manipulable. One could try to avoid manipulation by using voting protocols where determining a beneficial manipulation is hard computationally. A number of recent papers study the complexity of manipulating existing protocols. This paper is the first work to take the next step of designing new protocols that are especially hard to manipulate. Rather than designing these new protocols from scratch, we instead show how to tweak existing protocols to make manipulation hard, while leaving much of the original nature of the protocol intact. The tweak studied consists of adding one elimination preround to the election. Surprisingly, this extremely simple and universal tweak makes typical protocols hard to manipulate! The protocols become NP-hard, #P-hard, or PSPACE-hard to manipulate, depending on whether the schedule of the preround is determined before the votes are collected, after the votes are collected, or the scheduling and the vote collecting are interleaved, respectively. We prove general sufficient conditions on the protocols for this tweak to introduce the hardness, and show that the most common voting protocols satisfy those conditions. These are the first results in voting settings where manipulation is in a higher complexity class than NP (presuming PSPACE $\neq$ NP)

Majority Dynamics and Aggregation of Information in Social Networks

Consider n individuals who, by popular vote, choose among q >= 2 alternatives, one of which is "better" than the others. Assume that each individual votes independently at random, and that the probability of voting for the better alternative is larger than the probability of voting for any other. It follows from the law of large numbers that a plurality vote among the n individuals would result in the correct outcome, with probability approaching one exponentially quickly as n tends to infinity. Our interest in this paper is in a variant of the process above where, after forming their initial opinions, the voters update their decisions based on some interaction with their neighbors in a social network. Our main example is "majority dynamics", in which each voter adopts the most popular opinion among its friends. The interaction repeats for some number of rounds and is then followed by a population-wide plurality vote. The question we tackle is that of "efficient aggregation of information": in which cases is the better alternative chosen with probability approaching one as n tends to infinity? Conversely, for which sequences of growing graphs does aggregation fail, so that the wrong alternative gets chosen with probability bounded away from zero? We construct a family of examples in which interaction prevents efficient aggregation of information, and give a condition on the social network which ensures that aggregation occurs. For the case of majority dynamics we also investigate the question of unanimity in the limit. In particular, if the voters' social network is an expander graph, we show that if the initial population is sufficiently biased towards a particular alternative then that alternative will eventually become the unanimous preference of the entire population.Comment: 22 page

Computational aspects of voting: a literature survey

Preference aggregation is a topic of study in different fields such as philosophy, mathematics, economics and political science. Recently, computational aspects of preference aggregation have gained especial attention and “computational politics” has emerged as a marked line of research in computer science with a clear concentration on voting protocols. The field of voting systems, rooted in social choice theory, has expanded notably in both depth and breadth in the last few decades. A significant amount of this growth comes from studies concerning the computational aspects of voting systems. This thesis comprehensively reviews the work on voting systems (from a computing perspective) by listing, classifying and comparing the results obtained by different researchers in the field. This survey covers a wide range of new and historical results yet provides a profound commentary on related work as individual studies and in relation to other related work and to the field in general. The deliverables serve as an overview where students and novice researchers in the field can start and also as a depository that can be referred to when searching for specific results. A comprehensive literature survey of the computational aspects of voting is a task that has not been undertaken yet and is initially realized here. Part of this research was dedicated to creating a web-depository that contains material and references related to the topic based on the survey. The purpose was to create a dynamic version of the survey that can be updated with latest findings and as an online practical reference