A symbiosis between cellular automata and genetic algorithms

Cellular automata are systems which use a rule to describe the evolution of a population in a discrete lattice, while genetic algorithms are procedures designed to find solutions to optimization problems inspired by the process of natural selection. In this paper, we introduce an original implementation of a cellular automaton whose rules use a fitness function to select for each cell the best mate to reproduce and a crossover operator to determine the resulting offspring. This new system, with a proper definition, can be both a cellular automaton and a genetic algorithm. We show that in our system the Conway’s Game of Life can be easily implemented and, consequently, it is capable of universal computing. Moreover two generalizations of the Game of Life are created and also implemented with it. Finally, we use our system for studying and implementing the prisoner’s dilemma and rock-paper-scissors games, showing very interesting behaviors and configurations (e.g., gliders) inside these games

Competitive intransitivity, population interaction structure, and strategy coexistence

Sherpa Romeo green journal. Permission to archive accepted author manuscriptIntransitive competition occurs when competing strategies cannot be listed in a hierarchy, but rather form loops – as in the game Rock-Paper-Scissors. Due to its cyclic competitive replacement, competitive intransitivity promotes strategy coexistence, both in Rock-Paper-Scissors and in higher-richness communities. Previous work has shown that this intransitivity-mediated coexistence is strongly influenced by spatially explicit interactions, compared to when populations are well mixed. Here, we extend and broaden this line of research and examine the impact on coexistence of intransitive competition taking place on a continuum of small-world networks linking spatial lattices and regular random graphs. We use simulations to show that the positive effect of competitive intransitivity on strategy coexistence holds when competition occurs on networks toward the spatial end of the continuum. However, in networks that are sufficiently disordered, increasingly violent fluctuations in strategy frequencies can lead to extinctions and the prevalence of monocultures. We further show that the degree of disorder that leads to the transition between these two regimes is positively dependent on population size; indeed for very large populations, intransitivity-mediated strategy coexistence may even be possible in regular graphs with completely random connections. Our results emphasize the importance of interaction structure in determining strategy dynamics and diversity

Evolutionary games on graphs

Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in non-equilibrium statistical physics. This review gives a tutorial-type overview of the field for physicists. The first three sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fourth section surveys the topological complications implied by non-mean-field-type social network structures in general. The last three sections discuss in detail the dynamic behavior of three prominent classes of models: the Prisoner's Dilemma, the Rock-Scissors-Paper game, and Competing Associations. The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.Comment: Review, final version, 133 pages, 65 figure

Limiting Behavior of Stochastic Processes Involving Martingale Structures

A stochastic process is given by a family of random variables indexed by elements of a set. We have considered stochastic processes of three different types, each involving an associated martingale structure. Martingale is a sequence of random variables for which the conditional expectation at a certain time point given the entire past is given by the present value of the sequence. Martingales possess nice theoretical properties with wide applicability. We have exploited martingale tools and techniques to derive the limiting results related to the stochastic processes. The processes we have considered are given below. An evolutionary urn scheme based on the rock-paper-scissors game The random multiplicative cascade model for intermittent processes A set-indexed partial sum process with dependent increments The evolutionary urn scheme based on the rock-paper-scissors game is known to model species interactions in ecological systems. Therefore its limiting behavior is of interest to ecologists to understand the long term species composition of a certain ecological system. We have considered a generalization of the process to accommodate more than three species. Simulations in the general set up suggest interesting phenomena that are counter-intuitive when compared to the three-player case. The second chapter of this thesis is motivated by data sets with variable intermittency, which makes it diffcult to use standard modeling tools. It has been observed that a special class of multiplicative models, namely the Random Multiplicative Cascade models reproduce some characteristics of the data. We have derived theoretical results related to the multiplicative cascade models under a missing data set up. We have applied the method to the daily stock volume data of Tesla. Also, we have proposed a change point detection method for intermittent time series. This can possibly be extended to spatial processes as well. The last chapter of the thesis is related to a set indexed partial sum process, with martingale differences as its increments. We have derived the weak limit of the system under the Lindeberg type condition and the metric entropy integrability condition. In spite of a common martingale structure underlying each of these three processes, they are fundamentally different. Therefore the methods to derive the limiting properties are unique to each process. For example, in the case of the rock-paper-scissors urn scheme, the key idea behind the derivation of almost sure limit is noticing a connection between sub-martingale structures within the game and a well known convergence theorem of polynomial sequence. However, for the random multiplicative cascade model, the main challenge lies in deriving asymptotic theory on a tree structure. In the third chapter, we have used probabilistic tools and techniques like generic chaining, symmetrization, and truncation to derive weak limit of the set indexed partial sum process