14 research outputs found
A Smooth Transition from Powerlessness to Absolute Power
We study the phase transition of the coalitional manipulation problem for
generalized scoring rules. Previously it has been shown that, under some
conditions on the distribution of votes, if the number of manipulators is
, where is the number of voters, then the probability that a
random profile is manipulable by the coalition goes to zero as the number of
voters goes to infinity, whereas if the number of manipulators is
, then the probability that a random profile is manipulable
goes to one. Here we consider the critical window, where a coalition has size
, and we show that as goes from zero to infinity, the limiting
probability that a random profile is manipulable goes from zero to one in a
smooth fashion, i.e., there is a smooth phase transition between the two
regimes. This result analytically validates recent empirical results, and
suggests that deciding the coalitional manipulation problem may be of limited
computational hardness in practice.Comment: 22 pages; v2 contains minor changes and corrections; v3 contains
minor changes after comments of reviewer
Generalized polymorphisms
We find all functions and
satisfying the following
identity for all matrices : Our results
generalize work of Dokow and Holzman (2010), which considered the case , and of Chase, Filmus, Minzer, Mossel and Saurabh (2022),
which considered the case .Comment: 19 page
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
Voting with Coarse Beliefs
The classic Gibbard-Satterthwaite theorem says that every strategy-proof
voting rule with at least three possible candidates must be dictatorial.
Similar impossibility results hold even if we consider a weaker notion of
strategy-proofness where voters believe that the other voters' preferences are
i.i.d.~(independent and identically distributed). In this paper, we take a
bounded-rationality approach to this problem and consider a setting where
voters have "coarse" beliefs (a notion that has gained popularity in the
behavioral economics literature). In particular, we construct good voting rules
that satisfy a notion of strategy-proofness with respect to coarse
i.i.d.~beliefs, thus circumventing the above impossibility results
The Probability of Intransitivity in Dice and Close Elections
We study the phenomenon of intransitivity in models of dice and voting.
First, we follow a recent thread of research for -sided dice with pairwise
ordering induced by the probability, relative to , that a throw from one
die is higher than the other. We build on a recent result of Polymath showing
that three dice with i.i.d. faces drawn from the uniform distribution on
and conditioned on the average of faces equal to are
intransitive with asymptotic probability . We show that if dice faces are
drawn from a non-uniform continuous mean zero distribution conditioned on the
average of faces equal to , then three dice are transitive with high
probability. We also extend our results to stationary Gaussian dice, whose
faces, for example, can be the fractional Brownian increments with Hurst index
.
Second, we pose an analogous model in the context of Condorcet voting. We
consider voters who rank alternatives independently and uniformly at
random. The winner between each two alternatives is decided by a majority vote
based on the preferences. We show that in this model, if all pairwise elections
are close to tied, then the asymptotic probability of obtaining any tournament
on the alternatives is equal to , which markedly differs
from known results in the model without conditioning. We also explore the
Condorcet voting model where methods other than simple majority are used for
pairwise elections. We investigate some natural definitions of "close to tied"
for general functions and exhibit an example where the distribution over
tournaments is not uniform under those definitions.Comment: Ver3: 45 pages, additional details and clarifications; Ver2: 43
pages, additional co-author, major revision; Ver1: 23 page