We show that in an n-player m-action strategic form game, we can obtain an approximate equilibrium by sampling any mixed-action equilibrium a small number of times. We study three notions of equilibrium: Nash, correlated and coarse correlated. For each one of them we obtain upper and lower bounds on the asymptotic (where max(m, n) → ∞) worst-case number of samples required for the empirical frequency of 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 include a substantial improvement over the previously known upper bounds on the existence of a small-support approximate equilibrium in games with many players. For all three notions of equilibrium, we show the existence of approximate equilibrium with support size polylogarithmic in n and m, whereas the best previously known results were polynomial in n [8,6,7].

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 [16,11,15].

New ways of doing things often get started through the actions of a few innovators, then diffuse rapidly as more and more people come into contact with prior adopters in their social network. Much of the literature focuses on the speed of diffusion as a function of the network topology. In practice, the topology may not be known with any precision, and it is constantly in flux as links are formed and severed. Here, we establish an upper bound on the expected waiting time until a given proportion of the population has adopted that holds independently of the network structure. Kreindler and Young (2014) demonstrated such a bound for regular networks when agents choose between two options: the innovation and the status quo. Our bound holds for directed and undirected networks of arbitrary size and degree distribution, and for multiple competing innovations with different payoffs.

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 ≥ 0} and {W (s) : s ≥ 0}. Let τ t := min{s ≥ 0 : B(s) = t}. Then X t = W (τ t ) is a Cauchy process, and K := {(a, X t + at) : a, t ∈ [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|).

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 neighborhood 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.