Games and also the evaluation of the potential that plays a crucial role in theĭetermination of the preferred Nash equilibrium, and defines the Boltzmannĭistribution towards which these systems evolve for suitable types of dynamical The concept of decomposition helps the identification of potential That can be decomposed into elementary matrices (games) and classified intoįour types. In these systems the interactions are characterized by matrices Represent a wide scale of biological objects, human individuals, or even theirĪssociations. Multi-player games where players with a finite number of strategies may Pair interactions and evolutionary rules on the macroscopic behavior of
Game theory provides a general mathematical background to study the effect of
Analysis of the general properties indicates similar types of interactions at larger number of strategies for the symmetric matrix games. The main features of the four classes of games are discussed separately and we illustrate some characteristic strategy distributions on a square lattice in the low noise limit if logit rule controls the strategy evolution. In the absence of cyclic components the game is a potential game and the potential matrix is evaluated. Six of 16 basis games describe coordination-type interactions among the strategy pairs and three basis games span the parameter space of the cyclic components that are analogous to the rock-paper-scissors games. The games with self-dependent (cross-dependent) payoffs are characterized by matrices consisting of uniform rows (columns). Among these basis games four classes can be distinguished as it is already found for the three-strategy games. The symmetric four-strategy games are decomposed into a linear combination of 16 basis games represented by orthogonal matrices. We show that for a range of fitness values of the mutants, the Comet and Comet-swarm graphs have fixation probability strictly larger than the fixation probability of the Star graph, for fixed population size and at the limit of large populations, respectively. In this work we present the Comet and Comet-swarm families of undirected graphs. In the regime of undirected graphs, the strongest amplifier known has been the Star graph, and the existence of undirected graphs with stronger amplification properties has remained open for over a decade. In this work we focus on symmetric population structures represented as undirected graphs. Amplifiers of natural selection are population structures which increase the fixation probability of advantageous mutants, as compared to the baseline case of well-mixed populations. The fixation probability is a fundamental quantity of natural selection, and known to depend on the population structure. The fixation probability is the probability that a new mutant introduced in a homogeneous population eventually takes over the entire population.
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