Pure Nash Equilibria and Best-Response Dynamics in Random Games

In finite games mixed Nash equilibria always exist, but pure equilibria may fail to exist. To assess the relevance of this nonexistence, we consider games where the payoffs are drawn at random. In particular, we focus on games where a large number of players can each choose one of two possible strategies, and the payoffs are i.i.d. with the possibility of ties. We provide asymptotic results about the random number of pure Nash equilibria, such as fast growth and a central limit theorem, with bounds for the approximation error. Moreover, by using a new link between percolation models and game theory, we describe in detail the geometry of Nash equilibria and show that, when the probability of ties is small, a best-response dynamics reaches a Nash equilibrium with a probability that quickly approaches one as the number of players grows. We show that a multitude of phase transitions depend only on a single parameter of the model, that is, the probability of having ties.Comment: 29 pages, 7 figure

Quantum Tomography under Prior Information

We provide a detailed analysis of the question: how many measurement settings or outcomes are needed in order to identify a quantum system which is constrained by prior information? We show that if the prior information restricts the system to a set of lower dimensionality, then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the system. Conversely, we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the Minkowski dimension of the set. We apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints. In particular, we show that 4d-4 measurement outcomes (POVM elements) is enough in order to identify all pure states in a d-dimensional Hilbert space, and that the minimal number is at most 2 log_2(d) smaller than this upper bound.Comment: v3: There was a mistake in the derived finer upper bound in Theorem 3. The corrected upper bound is +1 to the earlier versio

Prevalence and risk factors for autism spectrum disorder in epilepsy: a systematic review and meta-analysis

AIM: To assess the prevalence and risk factors for autism spectrum disorder (ASD) in epilepsy, and to better understand the relationship and comorbidity between these disorders. METHOD: PsychINFO and PubMed were searched for articles published in the past 15 years that examined the prevalence of ASD in individuals with epilepsy. RESULTS: A total of 19 studies were found with a pooled ASD prevalence of 6.3% in epilepsy. When divided by type, the risks of ASD for general epilepsy, infantile spasms, focal seizures, and Dravet syndrome were 4.7%, 19.9%, 41.9%, and 47.4% respectively. Studies with populations under 18 years showed a 13.2 times greater risk of ASD than study populations over 18 years, and samples with most (>50%) individuals with intellectual disability showed a greater risk 4.9 times higher than study populations with a minority of individuals with intellectual disability. The main risk factors for ASD reported in the 19 studies included presence of intellectual disability, sex, age, and symptomatic aetiology of epilepsy. INTERPRETATION: Current research supports a high prevalence of ASD in epilepsy. This study helps to define the clinical profile of patients with epilepsy who are at risk for ASD, which may help clinicians in early screening and diagnosis of ASD in this population