Sharp Thresholds for Monotone Non Boolean Functions and Social Choice Theory

A key fact in the theory of Boolean functions $f : \{0,1\}^n \to \{0,1\}$ is that they often undergo sharp thresholds. For example: if the function $f : \{0,1\}^n \to \{0,1\}$ is monotone and symmetric under a transitive action with \E_p[f] = \eps and \E_q[f] = 1-\eps then $q-p \to 0$ as $n \to \infty$. Here \E_p denotes the product probability measure on $\{0,1\}^n$ where each coordinate takes the value $1$ independently with probability $p$. The fact that symmetric functions undergo sharp thresholds is important in the study of random graphs and constraint satisfaction problems as well as in social choice.In this paper we prove sharp thresholds for monotone functions taking values in an arbitrary finite sets. We also provide examples of applications of the results to social choice and to random graph problems. Among the applications is an analog for Condorcet's jury theorem and an indeterminacy result for a large class of social choice functions

A stability result for the cube edge isoperimetric inequality

We prove the following stability version of the edge isoperimetric inequality for the cube: any subset of the cube with average boundary degree within $K$ of the minimum possible is $\varepsilon$-close to a union of $L$ disjoint cubes, where $L \leq L(K,\varepsilon )$ is independent of the dimension. This extends a stability result of Ellis, and can viewed as a dimension-free version of Friedgut's junta theorem.Comment: 12 page

Probabilistic and analytic aspects of Boolean functions

This thesis will focus on the study of Boolean functions. In point of fact, they can be represented with a Fourier expansion and many of the definitions and results for these functions can be rewritten in terms of the Fourier coefficients. The definition of Boolean functions is simple, this implies that they have a natural interpretation and hence have applications in many areas of scientific research. Specifically, in this thesis we will see applications in Social Choice Theory, Theoretical Computer Science and Combinatorics. For the first area, we will see and prove with Fourier analysis Arrow's theorem and KKL theorem in order to show that it is not possible to define a perfect voting election system from an ethical standpoint. Additionally, we will translate the proof presented by Arrow for his own theorem in terms of mathematical language. The work for the second application will follow the steps to prove Sensitivity Conjecture which, although it has remained unsolved for 30 years, Huang has presented a brilliant short proof in a paper published at Annals of Mathematics very recently (2019). For the last area we will present the strange phenomena of thresholds in Random Graph properties and we will show Margulis-Russo Formula to study this event in terms of Boolean functions Fourier analysis

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