arXiv subjects

Ron Peretz

Publications and source records attributed to Ron Peretz.

13 recordsLinked to original sources

arXiv2026

Entropy Bounds for Local Coordination and Graph Amenability

We study local pure coordination games on finite graphs. In these games, each vertex must choose one of two symmetric actions using only local information, and the cost is the average disagreement across edges. Hutchcroft, Rospuskova, and Tamuz showed that if such local coordination can be achieved with low cost, then the underlying graph must be amenable, or hyperfinite, but their quantitative bound has a square root loss. We improve this loss in the unbiased binary setting. The main idea is to associate with each player's local output a probability measure that records, along an ordered list of information sources, the mutual information with that output. For binary outputs, two players who usually agree have nearby associated measures, with a bound given by the binary entropy of their disagreement probability. Combining this estimate with a grand coupling theorem yields an improved amenability bound of order $\varepsilon\log(1/\varepsilon)$, where $\varepsilon$ is the average disagreement. We also show that the square root loss in the earlier general theorem is essentially unavoidable for non-binary coordination profiles. Thus the binary assumption is not merely technical: it is what makes the improved entropy bound possible.

cs.GT↗

arXiv2025

The edge-averaging process on graphs with random initial opinions

In several settings (e.g., sensor networks and social networks), nodes of a graph are equipped with initial opinions, and the goal is to estimate the average of these opinions using local operations. A natural algorithm to achieve this is the edge-averaging process, where edges are repeatedly selected at random (according to independent Poisson clocks) and the opinions on the nodes of each selected edge are replaced by their average. The effectiveness of this algorithm is determined by its convergence rate. It is known that on a finite graph of $n$ nodes, the opinions reach approximate consensus in polynomial time. We prove that the convergence is much faster when the initial opinions are disordered (independent identically distributed): the time to reach approximate consensus is $O (\log^2n)$, and this bound is sharp. For infinite graphs, we show that for every $p\geq 1$, if the initial opinions are in $L^p$, then the opinion at each vertex converges to the mean in $L^p$, and if $p>4$, then almost sure convergence holds as well.

math.PR↗

arXiv2022

The Asynchronous DeGroot Dynamics

We analyze the asynchronous version of the DeGroot dynamics: In a connected graph $G$ with $n$ nodes, each node has an initial opinion in $[0,1]$ and an independent Poisson clock. When a clock at a node $v$ rings, the opinion at $v$ is replaced by the average opinion of its neighbors. It is well known that the opinions converge to a consensus. We show that the expected time $\mathbb E(\tau_\varepsilon)$ to reach $\varepsilon$-consensus is poly$(n)$ in undirected graphs and in Eulerian digraphs, but for some digraphs of bounded degree it is exponential. Our main result is that in undirected graphs and Eulerian digraphs, if the degrees are uniformly bounded and the initial opinions are i.i.d., then $\mathbb E(\tau_\varepsilon)=\text{polylog}(n)$ for every fixed $\varepsilon>0$. We give sharp estimates for the variance of the limiting consensus opinion, which measures the ability to aggregate information (``wisdom of the crowd''). We also prove generalizations to non-reversible Markov chains and infinite graphs. New results of independent interest on fragmentation processes and coupled random walks are crucial to our analysis.

math.PR↗

arXiv2021

Dynamical noise sensitivity for the voter model

We study noise sensitivity of the consensus opinion of the voter model on finite graphs, with respect to noise affecting the initial opinions and noise affecting the dynamics. We prove that the final opinion is stable with respect to small perturbations of the initial configuration, and is sensitive to perturbations of the dynamics governing the evolution of the process. Our proofs rely on the duality relationship between the voter model and coalescing random walks, and on a precise description of this evolution when we have coupled dynamics.

math.PR↗

arXiv2021

Granular DeGroot Dynamics -- a Model for Robust Naive Learning in Social Networks

We study a model of opinion exchange in social networks where a state of the world is realized and every agent receives a zero-mean noisy signal of the realized state. It is known from [Golub and Jackson 2010] that under DeGroot dynamics [DeGroot 1974] agents reach a consensus that is close to the state of the world when the network is large. The DeGroot dynamics, however, is highly non-robust and the presence of a single ``stubborn agent'' that does not adhere to the updating rule can sway the public consensus to any other value. We introduce a variant of DeGroot dynamics that we call \emph{ $\frac{1}{m}$-DeGroot}. $\frac{1}{m}$-DeGroot dynamics approximates standard DeGroot dynamics to the nearest rational number with $m$ as its denominator and like the DeGroot dynamics it is Markovian and stationary. We show that in contrast to standard DeGroot dynamics, $\frac{1}{m}$-DeGroot dynamics is highly robust both to the presence of stubborn agents and to certain types of misspecifications.

math.PR↗

arXiv2020

The Lipschitz Constant of Perturbed Anonymous Games

The worst-case Lipschitz constant of an $n$-player $k$-action $δ$-perturbed game, $λ(n,k,δ)$, is given an explicit probabilistic description. In the case of $k\geq 3$, $λ(n,k,δ)$ is identified with the passage probability of a certain symmetric random walk on $\mathbb Z$. In the case of $k=2$ and $n$ even, $λ(n,2,δ)$ is identified with the probability that two two i.i.d.\ Binomial random variables are equal. The remaining case, $k=2$ and $n$ odd, is bounded through the adjacent (even) values of $n$. Our characterisation implies a sharp closed form asymptotic estimate of $λ(n,k,δ)$ as $δn /k\to\infty$.

cs.GT↗

arXiv2017

Stable Secretaries

We define and study a new variant of the secretary problem. Whereas in the classic setting multiple secretaries compete for a single position, we study the case where the secretaries arrive one at a time and are assigned, in an on-line fashion, to one of multiple positions. Secretaries are ranked according to talent, as in the original formulation, and in addition positions are ranked according to attractiveness. To evaluate an online matching mechanism, we use the notion of blocking pairs from stable matching theory: our goal is to maximize the number of positions (or secretaries) that do not take part in a blocking pair. This is compared with a stable matching in which no blocking pair exists. We consider the case where secretaries arrive randomly, as well as that of an adversarial arrival order, and provide corresponding upper and lower bounds.

cs.GT↗

arXiv2015

Effective Martingales with Restricted Wagers

The classic model of computable randomness considers martingales that take real or rational values. Recent work by Bienvenu et al. (2012) and Teutsch (2014) shows that fundamental features of the classic model change when the martingales take integer values. We compare the prediction power of martingales whose wagers belong to three different subsets of rational numbers: (a) all rational numbers, (b) rational numbers excluding a punctured neighbourhood of 0, and (c) integers. We also consider three different success criteria: (i) accumulating an infinite amount of money, (ii) consuming an infinite amount of money, and (iii) making the accumulated capital oscillate. The nine combinations of (a)--(c) and (i)--(iii) define nine notions of computable randomness. We provide a complete characterization of the relations between these notions, and show that they form five linearly ordered classes. Our results solve outstanding questions raised in Bienvenu et al. (2012), Teutsch (2014), and Chalcraft et al. (2012), and strengthen existing results.

math.LO↗

arXiv2014

Empirical Distribution of Equilibrium Play and Its Testing Application

We show that in any $n$-player $m$-action normal-form game, we can obtain an approximate equilibrium by sampling any mixed-action equilibrium a small number of times. We study three types of equilibria: Nash, correlated and coarse correlated. For each one of them we obtain upper and lower bounds on the number of samples required for the empirical distribution over the sampled action profiles to form an approximate equilibrium with probability close to one. These bounds imply that using a small number of samples we can test whether or not players are playing according to an approximate equilibrium, even in games where $n$ and $m$ are large. In addition, our results substantially improve previously known upper bounds on the support size of approximate equilibria in games with many players. In particular, for all the three types of equilibria we show the existence of approximate equilibrium with support size polylogarithmic in $n$ and $m$, whereas the previously best-known upper bounds were polynomial in $n$.

cs.GT↗

arXiv2014

How to Gamble Against All Odds

A decision maker observes the evolving state of the world while constantly trying to predict the next state given the history of past states. The ability to benefit from such predictions depends not only on the ability to recognize patters in history, but also on the range of actions available to the decision maker. We assume there are two possible states of the world. The decision maker is a gambler who has to bet a certain amount of money on the bits of an announced binary sequence of states. If he makes a correct prediction he wins his wager, otherwise he loses it. We compare the power of betting strategies (aka martingales) whose wagers take values in different sets of reals. A martingale whose wagers take values in a set $A$ is called an $A$-martingale. A set of reals $B$ anticipates a set $A$, if for every $A$-martingale there is a countable set of $B$-martingales, such that on every binary sequence on which the $A$-martingale gains an infinite amount at least one of the $B$-martingales gains an infinite amount, too. We show that for two important classes of pairs of sets $A$ and $B$, $B$ anticipates $A$ if and only if the closure of $B$ contains $rA$, for some positive $r$. One class is when $A$ is bounded and $B$ is bounded away from zero; the other class is when $B$ is well ordered (has no left-accumulation points). Our results generalize several recent results in algorithmic randomness and answer a question posed by Chalcraft et al. (2012).

cs.GT↗

arXiv2013

Small-Support Approximate Correlated Equilibria

We prove the existence of approximate correlated equilibrium of support size polylogarithmic in the number of players and the number of actions per player. In particular, using the probabilistic method, we show that there exists a multiset of polylogarithmic size such that the uniform distribution over this multiset forms an approximate correlated equilibrium. Along similar lines, we establish the existence of approximate coarse correlated equilibrium with logarithmic support. We complement these results by considering the computational complexity of determining small-support approximate equilibria. We show that random sampling can be used to efficiently determine an approximate coarse correlated equilibrium with logarithmic support. But, such a tight result does not hold for correlated equilibrium, i.e., sampling might generate an approximate correlated equilibrium of support size Ω(m) where m is the number of actions per player. Finally, we show that finding an exact correlated equilibrium with smallest possible support is NP-hard under Cook reductions, even in the case of two-player zero-sum games.

cs.GT↗

arXiv2013

Approximate Nash Equilibria via Sampling

We prove that in a normal form n-player game with m actions for each player, there exists an approximate Nash equilibrium where each player randomizes uniformly among a set of O(log(m) + log(n)) pure strategies. This result induces an $N^{\log \log N}$ algorithm for computing an approximate Nash equilibrium in games where the number of actions is polynomial in the number of players (m=poly(n)), where $N=nm^n$ is the size of the game (the input size). In addition, we establish an inverse connection between the entropy of Nash equilibria in the game, and the time it takes to find such an approximate Nash equilibrium using the random sampling algorithm.

cs.GT↗

arXiv2012

Hunter, Cauchy Rabbit, and Optimal Kakeya Sets

A planar set that contains a unit segment in every direction is called a Kakeya set. We relate these sets to a game of pursuit on a cycle $\Z_n$. A hunter and a rabbit move on the nodes of $\Z_n$ without seeing each other. At each step, the hunter moves to a neighbouring vertex or stays in place, while the rabbit is free to jump to any node. Adler et al (2003) provide strategies for hunter and rabbit that are optimal up to constant factors and achieve probability of capture in the first $n$ steps of order $1/\log n$. We show these strategies yield a Kakeya set consisting of $4n$ triangles with minimal area, (up to constant), namely $Θ(1/\log n)$. As far as we know, this is the first non-iterative construction of a boundary-optimal Kakeya set. Considering the continuum analog of the game yields a construction of a random Kakeya set from two independent standard Brownian motions $\{B(s): s \ge 0\}$ and $\{W(s): s \ge 0\}$. Let $τ_t:=\min\{s \ge 0: B(s)=t\}$. Then $X_t=W(τ_t)$ is a Cauchy process, and $K:=\{(a,X_t+at) : a,t \in [0,1]\}$ is a Kakeya set of zero area. The area of the $ε$-neighborhood of $K$ is as small as possible, i.e., almost surely of order $Θ(1/|\log ε|)$.

math.PR↗