On the convergence of iterative voting: how restrictive should restricted dynamics be?

We study convergence properties of iterative voting procedures. Such procedures are defined by a voting rule and a (restricted) iterative process, where at each step one agent can modify his vote towards a better outcome for himself. It is already known that if the iteration dynamics (the manner in which voters are allowed to modify their votes) are unrestricted, then the voting process may not converge. For most common voting rules this may be observed even under the best response dynamics limitation. It is therefore important to investigate whether and which natural restrictions on the dynamics of iterative voting procedures can guarantee convergence. To this end, we provide two general conditions on the dynamics based on iterative myopic improvements, each of which is sufficient for convergence. We then identify several classes of voting rules (including Positional Scoring Rules, Maximin, Copeland and Bucklin), along with their corresponding iterative processes, for which at least one of these conditions hold

Reaching Consensus Under a Deadline

Committee decisions are complicated by a deadline, e.g., the next start of a budget, or the beginning of a semester. In committee hiring decisions, it may be that if no candidate is supported by a strong majority, the default is to hire no one - an option that may cost dearly. As a result, committee members might prefer to agree on a reasonable, if not necessarily the best, candidate, to avoid unfilled positions. In this paper, we propose a model for the above scenario - Consensus Under a Deadline (CUD)- based on a time-bounded iterative voting process. We provide convergence guarantees and an analysis of the quality of the final decision. An extensive experimental study demonstrates more subtle features of CUDs, e.g., the difference between two simple types of committee member behavior, lazy vs.~proactive voters. Finally, a user study examines the differences between the behavior of rational voting bots and real voters, concluding that it may often be best to have bots play on the voters' behalf

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

Strategic Behavior is Bliss: Iterative Voting Improves Social Welfare

Recent work in iterative voting has defined the additive dynamic price of anarchy (ADPoA) as the difference in social welfare between the truthful and worst-case equilibrium profiles resulting from repeated strategic manipulations. While iterative plurality has been shown to only return alternatives with at most one less initial votes than the truthful winner, it is less understood how agents' welfare changes in equilibrium. To this end, we differentiate agents' utility from their manipulation mechanism and determine iterative plurality's ADPoA in the worst- and average-cases. We first prove that the worst-case ADPoA is linear in the number of agents. To overcome this negative result, we study the average-case ADPoA and prove that equilibrium winners have a constant order welfare advantage over the truthful winner in expectation. Our positive results illustrate the prospect for social welfare to increase due to strategic manipulation.Comment: 21 pages, 5 figures, in NeurIPS 202

Convergence of Multi-Issue Iterative Voting under Uncertainty

We study the effect of strategic behavior in iterative voting for multiple issues under uncertainty. We introduce a model synthesizing simultaneous multi-issue voting with Meir, Lev, and Rosenschein (2014)'s local dominance theory and determine its convergence properties. After demonstrating that local dominance improvement dynamics may fail to converge, we present two sufficient model refinements that guarantee convergence from any initial vote profile for binary issues: constraining agents to have O-legal preferences and endowing agents with less uncertainty about issues they are modifying than others. Our empirical studies demonstrate that although cycles are common when agents have no uncertainty, introducing uncertainty makes convergence almost guaranteed in practice.Comment: 19 pages, 4 figure

Collective decision-making under the influence of bribers and temporal constraints

Jo estudio la connexió entre la corrupció i les característiques estructurals dels parlaments: nombre de seients, el nombre de partits representats, i regles de decisió adoptades. Amb l'aplicació d'enfocaments analítics i computacionals, a més de simulacions, mostro que el nombre mitjà de diputats que han de ser subornats disminueix a mesura que el nombre de partits augmenta, de manera que el suborn se sent encoratjat per un nombre cada vegada més gran de parts. També investigo dues formes en que pot afectar el temps a la presa de decisions. En primer lloc, suggereixo un procediment de votació iteratiu en el que el retard en prendre una decisió és costós. Amb dos electors, dues opcions i un ordre de votació fix, demostro que en l’únic equilibri perfecte en subjocs, l’elector que vota primer, obté la seva opció preferida a l'inici del procediment. Si l'ordre s'inverteix en algun moment, identifico la condició sota la qual el votant que vota segon pot obtenir la seva opció preferida al principi. En segon lloc, proposo un altre procediment de votació iterativa, permetent que els votants canvien els seus vots, però ara amb una data límit: una etapa que, si no s'ha pres una decisió, els resultats de la votació són pitjors. Mostro que (i) si hi ha temps suficient perquè tots els votants canviïn el seu vot, es prendrà una decisió, i (ii) si hi ha una alternativa preferida per la majoria dels votants, aquesta alternativa serà finalment triada. Afegeixo un estudi experimental que indica que fins i tot amb menys temps del necessari per a què cada votant pugui canviar el seu vot, els electors estaran d'acord amb una decisió de totes maneres.Estudio la conexión entre la corrupción y las características estructurales de los parlamentos: número de asientos, el número de partidos representados, y reglas de decisión adoptadas. Con la aplicación de enfoques analíticos y computacionales, además de simulaciones, muestro que el número medio de diputados que deben ser sobornados disminuye a medida que el número de partidos aumenta, por lo que el soborno se siente alentado por un número cada vez mayor de partes. También investigo dos formas en que puede afectar el tiempo en la toma de decisiones. En primer lugar, sugiero un procedimiento de votación iterativo en el que el retraso en tomar una decisión es costoso. Con dos electores, dos opciones y un orden de votación fijo, demuestro que en el único equilibrio perfecto en subjuegos, el elector que vota primero obtiene su opción preferida al inicio del procedimiento. Si el orden se invierte en algún momento, identifico la condición bajo la cual el votante que vota segundo puede obtener su opción preferida al principio. En segundo lugar, propongo otro procedimiento de votación iterativa, permitiendo que los votantes cambian sus votos, pero ahora con una fecha límite: una etapa que, si no se ha tomado una decisión, los resultados de la votación son peores. Muestro que (i) si hay tiempo suficiente para que todos los votantes cambien su voto, se tomará una decisión, y (ii) si hay una alternativa preferida por la mayoría de los votantes, esta alternativa será finalmente elegida. Añado un estudio experimental que indica que los electores estarán de acuerdo con una decisión aunque no haya tiempo sufficiente para que cada votante pueda cambiar su voto.I study the connection between corruption and structural characteristics of parliaments: number of seats, the number of parties represented, and decision rules adopted. Applying analytical and computational approaches, and running simulations, I show that the average number of deputies needed to be bribed decreases as the number of parties increases, so bribery is encouraged by a growing number of parties. I also investigate two ways in which time may affect decision-making. First, I suggest an iterative voting procedure in which delay to reach a decision is costly. For two voters and two options, with a fixed voting order, I prove that in the unique subgame perfect equilibrium the voter who votes first obtains his most preferred option at the beginning of the procedure. If the fixed order is reversed once at some stage, I identify the condition under which the voter initially voting the second obtains this most preferred option, also at the beginning. Second, I propose another iterative voting procedure, allowing voters to change their votes, but now with a deadline: a stage such that, if no decision has been taken by then, the worst outcome results. I show that (i) if there is enough time for all the voters to change their vote, a decision will be taken, and (ii) if there is an alternative preferred by a majority of the voters, this alternative will be finally chosen. I add an experimental study indicating that even with less time necessary for every voter to change his vote, the voters will agree with a decision anyway