Social diversity and promotion of cooperation in the spatial prisoner's dilemma game

The diversity in wealth and social status is present not only among humans, but throughout the animal world. We account for this observation by generating random variables that determ ine the social diversity of players engaging in the prisoner's dilemma game. Here the term social diversity is used to address extrinsic factors that determine the mapping of game pay offs to individual fitness. These factors may increase or decrease the fitness of a player depending on its location on the spatial grid. We consider different distributions of extrin sic factors that determine the social diversity of players, and find that the power-law distribution enables the best promotion of cooperation. The facilitation of the cooperative str ategy relies mostly on the inhomogeneous social state of players, resulting in the formation of cooperative clusters which are ruled by socially high-ranking players that are able to prevail against the defectors even when there is a large temptation to defect. To confirm this, we also study the impact of spatially correlated social diversity and find that coopera tion deteriorates as the spatial correlation length increases. Our results suggest that the distribution of wealth and social status might have played a crucial role by the evolution of cooperation amongst egoistic individuals.Comment: 5 two-column pages, 5 figure

Heteroclinic Chaos, Chaotic Itinerancy and Neutral Attractors in Symmetrical Replicator Equations with Mutations

A replicator equation with mutation processes is numerically studied. Without any mutations, two characteristics of the replicator dynamics are known: an exponential divergence of the dominance period, and hierarchical orderings of the attractors. A mutation introduces some new aspects: the emergence of structurally stable attractors, and chaotic itinerant behavior. In addition, it is reported that a neutral attractor can exist in the mutataion rate -> +0 region.Comment: 4 pages, 9 figure

Evolutionary Stability of Ecological Hierarchy

A self-similar hierarchical solution that is both dynamically and evolutionarily stable is found to the multi dimensional Lotka-Volterra equation with a single chain of prey-predator relations. This gives a simple and natural explanation to the key features of hierarchical ecosystems, such as its ubiquity, pyramidal population distribution, and higher aggressiveness among higher trophic levels. pacs{87.23.Kg, 89.75.Da, 05.45.-a} keywords{Lotka-Volterra equation, Trophic pyramid, Self-similarity}Comment: 4 Pages RevTeX4, 1 Fig, 1 Table, shortened by publishers reques

Strictly Dominated Strategies in the Replicator-Mutator Dynamics

The replicator-mutator dynamics is a set of differential equations frequently used in biological and socioeconomic contexts to model evolutionary processes subject to mutation, error or experimentation. The replicator-mutator dynamics generalizes the widely used replicator dynamics, which appears in this framework as the extreme case where replication is perfectly precise. This paper studies the influence of strictly dominated strategies on the location of the rest points of the replicator-mutator dynamics, at the limit where the mutation terms become arbitrarily small. It can be proved that such limit rest points for small mutation are Nash equilibria, so strictly dominated strategies do not occur at limit stationary points. However, we show through a simple case how strictly dominated strategies can have an influence on the location of the limit rest points for small mutation. Consequently, the characterization of the limit rest points of the replicator-mutator dynamics cannot in general proceed safely by readily eliminating strictly dominated strategiesJCyL (GREX251-2009 and VA006B09), Ministry of Science and Innovation (TIN2008-06464-C03-02, DPI2010-16920 and CSD2010-00034), Ministry of Education (grant JC2009-00263

Evolution of emotions on networks leads to the evolution of cooperation in social dilemmas

We show that the resolution of social dilemmas in random graphs and scale-free networks is facilitated by imitating not the strategy of better-performing players but, rather, their emotions. We assume sympathy and envy to be the two emotions that determine the strategy of each player in any given interaction, and we define them as the probabilities of cooperating with players having a lower and a higher payoff, respectively. Starting with a population where all possible combinations of the two emotions are available, the evolutionary process leads to a spontaneous fixation to a single emotional profile that is eventually adopted by all players. However, this emotional profile depends not only on the payoffs but also on the heterogeneity of the interaction network. Homogeneous networks, such as lattices and regular random graphs, lead to fixations that are characterized by high sympathy and high envy, while heterogeneous networks lead to low or modest sympathy but also low envy. Our results thus suggest that public emotions and the propensity to cooperate at large depend, and are in fact determined by, the properties of the interaction network

Evolutionary prisoner's dilemma game on hierarchical lattices

An evolutionary prisoner's dilemma (PD) game is studied with players located on a hierarchical structure of layered square lattices. The players can follow two strategies [D (defector) and C (cooperator)] and their income comes from PD games with the ``neighbors.'' The adoption of one of the neighboring strategies is allowed with a probability dependent on the payoff difference. Monte Carlo simulations are performed to study how the measure of cooperation is affected by the number of hierarchical levels (Q) and by the temptation to defect. According to the simulations the highest frequency of cooperation can be observed at the top level if the number of hierarchical levels is low (Q<4). For larger Q, however, the highest frequency of cooperators occurs in the middle layers. The four-level hierarchical structure provides the highest average (total) income for the whole community.Comment: appendix adde

On Phase Transitions to Cooperation in the Prisoner's Dilemma

Game theory formalizes certain interactions between physical particles or between living beings in biology, sociology, and economics, and quantifies the outcomes by payoffs. The prisoner's dilemma (PD) describes situations in which it is profitable if everybody cooperates rather than defects (free-rides or cheats), but as cooperation is risky and defection is tempting, the expected outcome is defection. Nevertheless, some biological and social mechanisms can support cooperation by effectively transforming the payoffs. Here, we study the related phase transitions, which can be of first order (discontinous) or of second order (continuous), implying a variety of different routes to cooperation. After classifying the transitions into cases of equilibrium displacement, equilibrium selection, and equilibrium creation, we show that a transition to cooperation may take place even if the stationary states and the eigenvalues of the replicator equation for the PD stay unchanged. Our example is based on adaptive group pressure, which makes the payoffs dependent on the endogeneous dynamics in the population. The resulting bistability can invert the expected outcome in favor of cooperation.Comment: For related work see http://www.soms.ethz.ch

Random replicators with high-order interactions

We use tools of the equilibrium statistical mechanics of disordered systems to study analytically the statistical properties of an ecosystem composed of N species interacting via random, Gaussian interactions of order p >= 2, and deterministic self-interactions u <= 0. We show that for nonzero u the effect of increasing the order of the interactions is to make the system more cooperative, in the sense that the fraction of extinct species is greatly reduced. Furthermore, we find that for p > 2 there is a threshold value which gives a lower bound to the concentration of the surviving species, preventing then the existence of rare species and, consequently, increasing the robustness of the ecosystem to external perturbations.Comment: 7 pages, 4 Postscript figure

Networking Effects on Cooperation in Evolutionary Snowdrift Game

The effects of networking on the extent of cooperation emerging in a competitive setting are studied. The evolutionary snowdrift game, which represents a realistic alternative to the well-known Prisoner's Dilemma, is studied in the Watts-Strogatz network that spans the regular, small-world, and random networks through random re-wiring. Over a wide range of payoffs, a re-wired network is found to suppress cooperation when compared with a well-mixed or fully connected system. Two extinction payoffs, that characterize the emergence of a homogeneous steady state, are identified. It is found that, unlike in the Prisoner's Dilemma, the standard deviation of the degree distribution is the dominant network property that governs the extinction payoffs.Comment: Changed conten