The Frequency of Convergent Games under Best-Response Dynamics

We calculate the frequency of games with a unique pure strategy Nash equilibrium in the ensemble of n-player, m-strategy normal-form games. To obtain the ensemble, we generate payoff matrices at random. Games with a unique pure strategy Nash equilibrium converge to the Nash equilibrium. We then consider a wider class of games that converge under a best-response dynamic, in which each player chooses their optimal pure strategy successively. We show that the frequency of convergent games with a given number of pure Nash equilibria goes to zero as the number of players or the number of strategies goes to infinity. In the 2-player case, we show that for large games with at least 10 strategies, convergent games with multiple pure strategy Nash equilibria are more likely than games with a unique Nash equilibrium. Our novel approach uses an n-partite graph to describe games.

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 independent and identically distributed 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 pure Nash equilibria and show that, when the probability of ties is small, a best-response dynamics reaches a pure Nash equilibrium with a probability that quickly approaches one as the number of players grows. We show that various phase transitions depend only on a single parameter of the model, that is, the probability of having ties.

Improving parity games in practice

Parity games are infinite-round two-player games played on directed graphs whose nodes are labeled with priorities. The winner of a play is determined by the smallest priority (even or odd) that is encountered infinitely often along the play. In the last two decades, several algorithms for solving parity games have been proposed and implemented in , a platform written in OCaml. includes the Zielonka’s recursive algorithm (, for short) which is known to be the best performing one over random games. Notably, several attempts have been carried out with the aim of improving the performance of in , but with small advances in practice. In this work, we deeply revisit the implementation of by dealing with the use of specific data structures and programming languages such as Scala, Java, C++, and Go. Our empirical evaluation shows that these choices are successful, gaining up to three orders of magnitude in running time over the classic version of the algorithm implemented in .

Pengaruh Permainan Acak Geometri Terhadap Perkembangan Kecerdasan Logika-Matematika Anak

Based on the results of observations in the field on the development of mathematical logical intelligence in students has not developed optimally so it is necessary to apply the geometry random game media. This study aims to determine the effect of random geometry on the development of mathematical-logic intelligence in children at Nasyithatun Nisa Kindergarten, Teluk Kiambang, Tempuling District, Indragiri Hilir Regency. The sample used in this study was 13 students consisting of one class. The data collection techniques used are test, observation and documentation. The data analysis technique used the Prerequisite test and Hypothesis test using the SPSS 17. The research hypothesis was that the activity of using geometric random games had an influence on the development of mathematical-logic intelligence in group B children at Nasyithatun Nisa Kindergarten. This can be seen from the results of data analysis on the comparison of the pretest and posttest classes obtained by tcount = 50,229 and Sig. (2-tailed) = 0,000. because of Sig. (2-tailed) = 0,000 <0,05, it can be concluded that there is a significant effect after using geometric random play in learning. So that means Ho is rejected and Ha is accepted, which means that in this study there is the influence of random geometry games before and after treatment. The effect of random geometry on the development of logic-mathematical intelligence in children at Nasyithatun Nisa Kindergarten was 82,005%.

On Equilibrium Properties of the Replicator–Mutator Equation in Deterministic and Random Games

In this paper, we study the number of equilibria of the replicator–mutator dynamics for both deterministic and random multi-player two-strategy evolutionary games. For deterministic games, using Descartes’ rule of signs, we provide a formula to compute the number of equilibria in multi-player games via the number of change of signs in the coefficients of a polynomial. For two-player social dilemmas (namely the Prisoner’s Dilemma, Snow Drift, Stag Hunt and Harmony), we characterize (stable) equilibrium points and analytically calculate the probability of having a certain number of equilibria when the payoff entries are uniformly distributed. For multi-player random games whose pay-offs are independently distributed according to a normal distribution, by employing techniques from random polynomial theory, we compute the expected or average number of internal equilibria. In addition, we perform extensive simulations by sampling and averaging over a large number of possible payoff matrices to compare with and illustrate analytical results. Numerical simulations also suggest several interesting behaviours of the average number of equilibria when the number of players is sufficiently large or when the mutation is sufficiently small. In general, we observe that introducing mutation results in a larger average number of internal equilibria than when mutation is absent, implying that mutation leads to larger behavioural diversity in dynamical systems. Interestingly, this number is largest when mutation is rare rather than when it is frequent.