On the Distortion Value of the Elections with Abstention

In Spatial Voting Theory, distortion is a measure of how good the winner is. It is proved that no deterministic voting mechanism can guarantee a distortion better than $3$, even for simple metrics such as a line. In this study, we wish to answer the following question: how does the distortion value change if we allow less motivated agents to abstain from the election? We consider an election with two candidates and suggest an abstention model, which is a more general form of the abstention model proposed by Kirchgassner. We define the concepts of the expected winner and the expected distortion to evaluate the distortion of an election in our model. Our results fully characterize the distortion value and provide a rather complete picture of the model.Comment: Revised version of the paper appeared in AAAI-1

Sequential Deliberation for Social Choice

In large scale collective decision making, social choice is a normative study of how one ought to design a protocol for reaching consensus. However, in instances where the underlying decision space is too large or complex for ordinal voting, standard voting methods of social choice may be impractical. How then can we design a mechanism - preferably decentralized, simple, scalable, and not requiring any special knowledge of the decision space - to reach consensus? We propose sequential deliberation as a natural solution to this problem. In this iterative method, successive pairs of agents bargain over the decision space using the previous decision as a disagreement alternative. We describe the general method and analyze the quality of its outcome when the space of preferences define a median graph. We show that sequential deliberation finds a 1.208- approximation to the optimal social cost on such graphs, coming very close to this value with only a small constant number of agents sampled from the population. We also show lower bounds on simpler classes of mechanisms to justify our design choices. We further show that sequential deliberation is ex-post Pareto efficient and has truthful reporting as an equilibrium of the induced extensive form game. We finally show that for general metric spaces, the second moment of of the distribution of social cost of the outcomes produced by sequential deliberation is also bounded

Bad cycles in iterative Approval Voting

This article is about synchronized iterative voting in the context of Approval Voting. Assuming that, before an election, successive polls occur to which voters react strategically, we shall exhibit examples showing the possibility of cycles with strong negative properties (in particular, non election of an existing Condorcet winner, or possible election of a candidate strongly rejected by a majority of the electorate). We also show that such cycles persist if only a proportion of the voters adjust their ballot at each iteration and if their strategy changes when close ties occur

Bad cycles and chaos in iterative Approval Voting

We consider synchronized iterative voting in the Approval Voting system. We give examples with a Condorcet winner where voters apply simple, sincere, consistent strategies but where cycles appear that can prevent the election of the Condorcet winner, or that can even lead to the election of a ''consensual loser'', rejected in all circumstances by a majority of voters. We conduct numerical experiments to determine how rare such cycles are. It turns out that when voters apply Laslier's Leader Rule they are quite uncommon, and we prove that they cannot happen when voters' preferences are modeled by a one-dimensional culture. However a slight variation of the Leader Rule accounting for possible draws in voter's preferences witnesses much more bad cycle, especially in a one-dimensional culture.Then we introduce a continuous-space model in which we show that these cycles are stable under perturbation. Last, we consider models of voters behavior featuring a competition between strategic behavior and reluctance to vote for candidates that are ranked low in their preferences. We show that in some cases, this leads to chaotic behavior, with fractal attractors and positive entropy.Comment: v2: added a numerical study of rarity of bad cycles and equilibriums, and a case of chaotic Continuous Polling Dynamics. Many other improvements throughout the tex

Acyclic Games and Iterative Voting

We consider iterative voting models and position them within the general framework of acyclic games and game forms. More specifically, we classify convergence results based on the underlying assumptions on the agent scheduler (the order of players) and the action scheduler (which better-reply is played). Our main technical result is providing a complete picture of conditions for acyclicity in several variations of Plurality voting. In particular, we show that (a) under the traditional lexicographic tie-breaking, the game converges for any order of players under a weak restriction on voters' actions; and (b) Plurality with randomized tie-breaking is not guaranteed to converge under arbitrary agent schedulers, but from any initial state there is \emph{some} path of better-replies to a Nash equilibrium. We thus show a first separation between restricted-acyclicity and weak-acyclicity of game forms, thereby settling an open question from [Kukushkin, IJGT 2011]. In addition, we refute another conjecture regarding strongly-acyclic voting rules.Comment: some of the results appeared in preliminary versions of this paper: Convergence to Equilibrium of Plurality Voting, Meir et al., AAAI 2010; Strong and Weak Acyclicity in Iterative Voting, Meir, COMSOC 201

On the Welfare of Cardinal Voting Mechanisms

A voting mechanism is a method for preference aggregation that takes as input preferences over alternatives from voters, and selects an alternative, or a distribution over alternatives. While preferences of voters are generally assumed to be cardinal utility functions that map each alternative to a real value, mechanisms typically studied assume coarser inputs, such as rankings of the alternatives (called ordinal mechanisms). We study cardinal mechanisms, that take as input the cardinal utilities of the voters, with the objective of minimizing the distortion - the worst-case ratio of the best social welfare to that obtained by the mechanism. For truthful cardinal mechanisms with m alternatives and n voters, we show bounds of Theta(mn), Omega(m), and Omega(sqrt{m}) for deterministic, unanimous, and randomized mechanisms respectively. This shows, somewhat surprisingly, that even mechanisms that allow cardinal inputs have large distortion. There exist ordinal (and hence, cardinal) mechanisms with distortion O(sqrt{m log m}), and hence our lower bound for randomized mechanisms is nearly tight. In an effort to close this gap, we give a class of truthful cardinal mechanisms that we call randomized hyperspherical mechanisms that have O(sqrt{m log m}) distortion. These are interesting because they violate two properties - localization and non-perversity - that characterize truthful ordinal mechanisms, demonstrating non-trivial mechanisms that differ significantly from ordinal mechanisms. Given the strong lower bounds for truthful mechanisms, we then consider approximately truthful mechanisms. We give a mechanism that is delta-truthful given delta in (0,1), and has distortion close to 1. Finally, we consider the simple mechanism that selects the alternative that maximizes social welfare. This mechanism is not truthful, and we study the distortion at equilibria for the voters (equivalent to the Price of Anarchy, or PoA). While in general, the PoA is unbounded, we show that for equilibria obtained from natural dynamics, the PoA is close to 1. Thus relaxing the notion of truthfulness in both cases allows us to obtain near-optimal distortion

Iterative Voting, Control and Sentiment Analysis

In multi-agent systems agents often need to take a collective decision based on the preferences of individuals. A voting rule is used to decide which decision to take, mapping the agents' preferences over the possible candidate decisions into a winning decision for the collection of agents. In these kind of scenarios acting strategically can be seen in two opposite way. On one hand it may be desirable that agents do not have any incentive to act strategically. That is, to misreport their preferences in order to influence the result of the voting rule in their favor or acting on the structure of the election to change the outcome. On the other hand manipulation can be used to improve the quality of the outcome by enlarging the consensus of the winner. These two different scenarios are studied in this thesis. The first one by modeling and describing a natural form of control named ``replacement control'' and characterizing for several voting rules its computational complexity. The second scenario is studied in the form of iterative voting frameworks where individuals are allowed to change their preferences to change the outcome of the election. Computational social choice techniques can be used in very different scenarios. This work reports a first attempt to introduce the use of voting procedures in the field of sentiment analysis. In this area computer scientists extract the opinion of the community about a specific item. This opinion is extracted aggregating the opinion expressed by each individual which leaves a text in a blog or social network about the given item. We studied and proposed a new aggregation method which can improve performances of sentiment analysis, this new technique is a new variance of a well-known voting rule called Borda

Strategic Voting and Social Networks

With the ever increasing ubiquity of social networks in our everyday lives, comes an increasing urgency for us to understand their impact on human behavior. Social networks quantify the ways in which we communicate with each other, and therefore shape the flow of information through the community. It is this same flow of information that we utilize to make sound, strategic decisions. This thesis focuses on one particular type of decisions: voting. When a community engages in voting, it is soliciting the opinions of its members, who present it in the form of a ballot. The community may then choose a course of action based on the submitted ballots. Individual voters, however, are under no obligation to submit sincere ballots that accurately reflects their opinions; they may instead submit a strategic ballot in hopes of affecting the election's outcome to their advantage. This thesis examines the interplay between social network structure and strategic voting behavior. In particular, we will explore how social network structure affects the flow of information through a population, and thereby affect the strategic behavior of voters, and ultimately, the outcomes of elections. We will begin by considering how network structure affects information propagation. This work builds upon the rich body of literature called opinion dynamics by proposing a model for skeptical agents --- agents that distrust other agents for holding opinions that differ too wildly from their own. We show that network structure is one of several factors that affects the degree of penetration that radical opinions can achieve through the community. Next, we propose a model for strategic voting in social networks, where voters are self-interested and rational, but may only use the limited information available through their social network contacts to formulate strategic ballots. In particular, we study the ``Echo Chamber Effect'', the tendency for humans to favor connections with similar people, and show that it leads to the election of less suitable candidates. We also extend this voter model by using boundedly-rational heuristics to scale up our simulations to larger populations. We propose a general framework for voting agents embedded in social networks, and show that our heuristic models can demonstrate a variation of the ``Micromega Law'' which relates the popularity of smaller parties to the size of the population. Finally, we examine another avenue for strategic behavior: choosing when to cast your vote. We propose a type of voting mechanism called ``Sticker Voting'', where voters cast ballots by placing stickers on their favored alternatives, thereby publicly and irrevocably declaring their support. We present a complete analysis of several simple instances of the Sticker Voting game and discuss how our results reflect human voting behavior