Effects of Empty Sites on Cooperation in the Prisoner’s Dilemma Game Based on Social Diversity

We study the effects of empty sites in the prisoner’s dilemma game based on social diversity by introducing some empty sites into a square lattice. The results reveal that the empty sites dramatically enhance the cooperation level for a wide range of temptation to defection values if two types of players coexist. By calculating the chances of different type-combinations of the players located on the square lattice, we find that there is an intermediate region where five social ranks are induced to satisfy the certain rank distributions and the cooperation level is significantly enhanced. Moreover, simulation results also show that the moderate gaps among the social ranks can favor cooperation for a larger occupation density.

Related papers:

title: The genetical evolution of social behaviour. I.
abstract: A genetical mathematical model is described which allows for interactions between relatives on one another's fitness. Making use of Wright's Coefficient of Relationship as the measure of the proportion of replica genes in a relative, a quantity is found which incorporates the maximizing property of Darwinian fitness. This quantity is named “inclusive fitness”. Species following the model should tend to evolve behaviour such that each organism appears to be attempting to maximize its inclusive fitness. This implies a limited restraint on selfish competitive behaviour and possibility of limited self-sacrifices. Special cases of the model are used to show (a) that selection in the social situations newly covered tends to be slower than classical selection, (b) how in populations of rather non-dispersive organisms the model may apply to genes affecting dispersion, and (c) how it may apply approximately to competition between relatives, for example, within sibships. Some artificialities of the model are discussed.

title: The Evolution of Cooperation
abstract: Cooperation in organisms, whether bacteria or primates, has been a difficulty for evolutionary theory since Darwin. On the assumption that interactions between pairs of individuals occur on a probabilistic basis, a model is developed based on the concept of an evolutionarily stable strategy in the context of the Prisoner's Dilemma game. Deductions from the model, and the results of a computer tournament show how cooperation based on reciprocity can get started in an asocial world, can thrive while interacting with a wide range of other strategies, and can resist invasion once fully established. Potential applications include specific aspects of territoriality, mating, and disease.

title: Altruistic punishment in humans
abstract: Human cooperation is an evolutionary puzzle. Unlike other creatures, people frequently cooperate with genetically unrelated strangers, often in large groups, with people they will never meet again, and when reputation gains are small or absent. These patterns of cooperation cannot be explained by the nepotistic motives associated with the evolutionary theory of kin selection and the selfish motives associated with signalling theory or the theory of reciprocal altruism. Here we show experimentally that the altruistic punishment of defectors is a key motive for the explanation of cooperation. Altruistic punishment means that individuals punish, although the punishment is costly for them and yields no material gain. We show that cooperation flourishes if altruistic punishment is possible, and breaks down if it is ruled out. The evidence indicates that negative emotions towards defectors are the proximate mechanism behind altruistic punishment. These results suggest that future study of the evolution of human cooperation should include a strong focus on explaining altruistic punishment.

title: Chaos promotes cooperation in the spatial prisoner's dilemma game
abstract: We introduce chaotic variations, modelled by a spatially extended Lorenz system, to the payoffs of the spatial prisoner's dilemma game and study their effects on the evolution of cooperation. We show that chaotic variations of appropriate amplitude promote cooperation over a wide range of payoff parameters at which defection is the only strategy in a sterile environment. An appropriately pronounced chaotic environment can assure permanent domination of cooperation by full anonymity of players and without the aid of secondary strategies, thus designating chaotic payoff variations as a general and stand-alone mechanism for cooperation in the spatial prisoner's dilemma game.

title: THE SPATIAL DILEMMAS OF EVOLUTION
abstract: Evolutionary game theory can be extended to include spatial dimensions. The individual players are placed in a two-dimensional spatial array. In each round every individual “plays the game” with its immediate neighbours. After this, each site is occupied by its original owner or by one of the neighbours, depending on who scored the highest payoff. These rules specify a deterministic cellular automaton. We find that spatial effects can change the outcome of frequency dependent selection. Strategies may coexist that would not coexist in homogeneous populations. Spatial games have interesting mathematical properties. There are static or chaotically changing patterns. For symmetrical starting conditions we find “dynamical fractals” and “evolutionary kaleidoscopes.” There is a new world to be explored.

title: Evolutionary Dynamics of Biological Games
abstract: Darwinian dynamics based on mutation and selection form the core of mathematical models for adaptation and coevolution of biological populations. The evolutionary outcome is often not a fitness-maximizing equilibrium but can include oscillations and chaos. For studying frequency-dependent selection, game-theoretic arguments are more appropriate than optimization algorithms. Replicator and adaptive dynamics describe short- and long-term evolution in phenotype space and have found applications ranging from animal behavior and ecology to speciation, macroevolution, and human language. Evolutionary game theory is an essential component of a mathematical and computational approach to biology.

title: Scale-free networks provide a unifying framework for the emergence of cooperation.
abstract: We study the evolution of cooperation in the framework of evolutionary game theory, adopting the prisoner's dilemma and snowdrift game as metaphors of cooperation between unrelated individuals. In sharp contrast with previous results we find that, whenever individuals interact following networks of contacts generated via growth and preferential attachment, leading to strong correlations between individuals, cooperation becomes the dominating trait throughout the entire range of parameters of both games, as such providing a unifying framework for the emergence of cooperation. Such emergence is shown to be inhibited whenever the correlations between individuals are decreased or removed. These results are shown to apply from very large population sizes down to small communities with nearly 100 individuals.

title: Social games in a social network.
abstract: We study an evolutionary version of the Prisoner's Dilemma game, played by agents placed in a small-world network. Agents are able to change their strategy, imitating that of the most successful neighbor. We observe that different topologies, ranging from regular lattices to random graphs, produce a variety of emergent behaviors. This is a contribution towards the study of social phenomena and transitions governed by the topology of the community.

title: Evolutionary dynamics on graphs
abstract: Evolutionary dynamics have been traditionally studied in the context of homogeneous or spatially extended populations. Here we generalize population structure by arranging individuals on a graph. Each vertex represents an individual. The weighted edges denote reproductive rates which govern how often individuals place offspring into adjacent vertices. The homogeneous population, described by the Moran process, is the special case of a fully connected graph with evenly weighted edges. Spatial structures are described by graphs where vertices are connected with their nearest neighbours. We also explore evolution on random and scale-free networks. We determine the fixation probability of mutants, and characterize those graphs for which fixation behaviour is identical to that of a homogeneous population. Furthermore, some graphs act as suppressors and others as amplifiers of selection. It is even possible to find graphs that guarantee the fixation of any advantageous mutant. We also study frequency-dependent selection and show that the outcome of evolutionary games can depend entirely on the structure of the underlying graph. Evolutionary graph theory has many fascinating applications ranging from ecology to multi-cellular organization and economics.

title: Memory-based snowdrift game on networks.
abstract: We present a memory-based snowdrift game (MBSG) taking place on networks. We found that, when a lattice is taken to be the underlying structure, the transition of spatial patterns at some critical values of the payoff parameter is observable for both four- and eight-neighbor lattices. The transition points as well as the styles of spatial patterns can be explained by local stability analysis. In sharp contrast to previously reported results, cooperation is promoted by the spatial structure in the MBSG. Interestingly, we found that the frequency of cooperation of the MBSG on a scale-free network peaks at a specific value of the payoff parameter. This phenomenon indicates that properly encouraging selfish behaviors can optimally enhance the cooperation. The memory effects of individuals are discussed in detail and some non-monotonous phenomena are observed on both lattices and scale-free networks. Our work may shed some new light on the study of evolutionary games over networks.

title: Promotion of cooperation induced by appropriate payoff aspirations in a small-world networked game.
abstract: Based on learning theory, we adopt a stochastic learning updating rule to investigate the evolution of cooperation in the Prisoner's Dilemma game on Newman-Watts small-world networks with different payoff aspiration levels. Interestingly, simulation results show that the mechanism of intermediate aspiration promoting cooperation resembles a resonancelike behavior, and there exists a ping-pong vibration of cooperation for large payoff aspiration. To explain the nontrivial dependence of the cooperation level on the aspiration level, we investigate the fractions of links, provide analytical results of the cooperation level, and find that the simulation results are in close agreement with analytical ones. Our work may be helpful in understanding the cooperative behavior induced by the aspiration level in society.

title: Randomness enhances cooperation: a resonance-type phenomenon in evolutionary games.
abstract: We investigate the effect of randomness in both relationships and decisions on the evolution of cooperation. Simulation results show, in such randomness' presence, the system evolves more frequently to a cooperative state than in its absence. Specifically, there is an optimal amount of randomness, which can induce the highest level of cooperation. The mechanism of randomness promoting cooperation resembles a resonancelike fashion, which could be of particular interest in evolutionary game dynamics in economic, biological, and social systems.

title: Roles of mixing patterns in cooperation on a scale-free networked game.
abstract: We investigate how the degree-mixing pattern affects the emergence of cooperation in the networked prisoner's dilemma game. Our study shows that when a network becomes assortative mixing by degree, the large-degree vertices (hubs) tend to interconnect to each other closely, which destroys the sustainability among cooperators and promotes the invasion of defectors, whereas in disassortative networks, the isolation among hubs protects the cooperative hubs in holding onto their initial strategies to avoid extinction.

title: 2 2 D ec 2 00 6 Social Dilemmas and Cooperation in Complex Networks
abstract: In this paper we extend the investigation of cooperation in s ome classical evolutionary games on populations were the network of interactio ns among individuals is of the scale-free type. We show that the update rule, the pa yoff computation and, to some extent the timing of the operations, have a marke d influence on the transient dynamics and on the amount of cooperation that can be established at equilibrium. We also study the dynamical behavior of the pop ulations and their evolutionary stability.

title: The effect of the topology on the spatial ultimatum game
abstract: In this work we present an analysis of a spatially non homogeneous
ultimatum game. By considering different underlying topologies as
substrates on top of which the game takes place we obtain
nontrivial behaviors for the evolution of the strategies of the
players. We analyze separately the effect of the size of the
neighborhood and the spatial structure. Whereas this last effect
is the most significant one, we show that even for disordered
networks and provided the neighborhood of each site is small, the
results can be significantly different from those obtained in the
case of fully connected networks.

title: Robustness of cooperation in the evolutionary prisoner's dilemma on complex networks
abstract: Recent studies on the evolutionary dynamics of the prisoner's dilemma game in scale-free networks have demonstrated that the heterogeneity of the network interconnections enhances the evolutionary success of cooperation. In this paper we address the issue of how the characterization of the asymptotic states of the evolutionary dynamics depends on the initial concentration of cooperators. We find that the measure and the connectedness properties of the set of nodes where cooperation reaches fixation is largely independent of initial conditions, in contrast with the behaviour of both the set of nodes where defection is fixed, and the fluctuating nodes. We also check for the robustness of these results when varying the degree heterogeneity along a one-parametric family of networks interpolating between the class of Erdős–Renyi graphs and the Barabasi–Albert networks.

title: Diversity-optimized cooperation on complex networks.
abstract: We propose a strategy for achieving maximum cooperation in evolutionary games on complex networks. Each individual is assigned a weight that is proportional to the power of its degree, where the exponent alpha is an adjustable parameter that controls the level of diversity among individuals in the network. During the evolution, every individual chooses one of its neighbors as a reference with a probability proportional to the weight of the neighbor, and updates its strategy depending on their payoff difference. It is found that there exists an optimal value of alpha, for which the level of cooperation reaches maximum. This phenomenon indicates that, although high-degree individuals play a prominent role in maintaining the cooperation, too strong influences from the hubs may counterintuitively inhibit the diffusion of cooperation. Other pertinent quantities such as the payoff, the cooperator density as a function of the degree, and the payoff distribution are also investigated computationally and theoretically. Our results suggest that in order to achieve strong cooperation on a complex network, individuals should learn more frequently from neighbors with higher degrees, but only to a certain extent.

title: Diversity of rationality affects the evolution of cooperation.
abstract: By modifying the Fermi updating rule, we present the diversity of individual rationality to the evolutionary prisoner's dilemma game, and our results shows that this diversity heavily influences the evolution of cooperation. Cluster-forming mechanism of cooperators can either be highly enhanced or severely deteriorated by different distributions of rationality. Slight change in the rationality distribution may transfer the whole system from the global absorbing state of cooperators to that of defectors. Based on mean-field argument, quantitative analysis of the stability of cooperative clusters reveals the critical role played by agents with moderate degree values in the evolution of the whole system. The inspiration from our work may provide us a deeper comprehension toward some social phenomena.

title: Motion of influential players can support cooperation in Prisoner’s Dilemma
abstract: We study a spatial Prisoner’s dilemma game with two types (A and B) of players located on a square lattice. Players following either cooperator or defector strategies play Prisoner’s Dilemma games with their 24 nearest neighbors. The players are allowed to adopt one of their neighbor’s strategy with a probability dependent on the payoff difference and type of the given neighbor. Players A and B have different efficiency in the transfer of their own strategies; therefore the strategy adoption probability is reduced by a multiplicative factor (w < 1) from the players of type B. We report that the motion of the influential payers (type A) can improve remarkably the maintenance of cooperation even for their low densities.

title: Cooperation enhanced by inhomogeneous activity of teaching for evolutionary Prisoner's Dilemma games
abstract: Prisoner's Dilemma games with quenched inhomogeneities in the spatial dynamical rules are considered. The players following one of the two pure strategies (cooperation or defection) are distributed on a two-dimensional lattice. The rate of strategy adoption from randomly chosen neighbors is controlled by the payoff difference and a two-value pre-factor w characterizing the players whom the strategy learned from. The reduced teaching activity of players is distributed randomly with concentrations ν at the beginning and fixed further on. Numerical and analytical calculations are performed to study the concentration of cooperators as a function of w and ν for different noise levels and connectivity structures. Significant increase of cooperation is found within a wide range of parameters for this dynamics. The results highlight the importance of asymmetry characterizing the exchange of master-follower role during the strategy adoptions.

title: Evolutionary Prisoner's Dilemma Game Based on Division of Work
abstract: We propose a new two-type-player prisoner's dilemma game based on the division of work on a square lattice, in which a fraction of the population μ are assigned type A and the rest B. In a one-shot two-player game, we let both of their original payoffs be scaled by a same multiplicative factor α > 1, if two neighboring players are of different types; however we leave the payoffs unchanged if they are of the same type. Then we show that combined with the two-type setup, the square lattice can assist to induce different social ranks according to players' abilities to collect payoffs. Simulation results show that the density of cooperation is significantly promoted for a wide range of the temptation to defection parameters and that there are optimal values for both α and μ leading to the maximal cooperation level. We reach these results by analyzing the distribution of the players in the social ranks and we also show some typical snapshots of the system.

title: Evolutionary prisoner's dilemma game with dynamic preferential selection.
abstract: We study a modified prisoner's dilemma game taking place on two-dimensional disordered square lattices. The players are pure strategists and can either cooperate or defect with their immediate neighbors. In the generations each player updates its strategy by following one of the neighboring strategies with a probability dependent on the payoff difference. The neighbor selection obeys a dynamic preferential rule, i.e., the more frequently a neighbor's strategy was adopted by the focal player in the previous rounds, the larger probability it will be chosen to refer to in the subsequent rounds. It is found that cooperation is substantially promoted due to this simple selection mechanism. Corresponding analysis is provided by the investigation of the distribution of the players' impact weights, persistence, and correlation function.

title: Restricted connections among distinguished players support cooperation.
abstract: We study the evolution of cooperation within the spatial prisoner's dilemma game on a square lattice where a fraction of players mu can spread their strategy more easily than the rest due to a predetermined larger teaching capability. In addition, players characterized by the larger teaching capability are allowed to temporarily link with distant opponents of the same kind with probability p , thus introducing shortcut connections among the distinguished players. We show that these additional temporary connections are able to sustain cooperation throughout the whole range of the temptation to defect. Remarkably, we observe that, as the temptation to defect increases the optimal mu decreases, and moreover only minute values of p warrant the best promotion of cooperation. Our study thus indicates that influential individuals must be few and sparsely connected in order for cooperation to thrive in a defection-prone environment.

title: Effects of average degree on cooperation in networked evolutionary game
abstract: We study effects of average degree on cooperation in
the networked prisoner's dilemma game. Typical structures are
considered, including random networks, small-world networks and
scale-free networks. Simulation results show that the average
degree plays a universal role in cooperation occurring on all
these networks, that is the density of cooperators peaks at some
specific values of the average degree. Moreover, we investigated
the average payoff of players through numerical simulations
together with theoretical predictions and found that simulation
results agree with the predictions. Our work may be helpful in
understanding network effects on the evolutionary games.

title: Cooperation in spatial prisoner's dilemma with two types of players for increasing number of neighbors.
abstract: We study a spatial two-strategy (cooperation and defection) prisoner's dilemma game with two types ( A and B ) of players located on the sites of a square lattice. The evolution of strategy distribution is governed by iterated strategy adoption from a randomly selected neighbor with a probability depending on the payoff difference and also on the type of the neighbor. The strategy adoption probability is reduced by a prefactor (w<1) from the players of type B . We consider the competition between two opposite effects when increasing the number of neighbors ( k=4 , 8, and 24). Within a range of the portion of influential players (type A ) the inhomogeneous activity in strategy transfer yields a relevant increase (dependent on w ) in the density of cooperators. The noise dependence of this phenomenon is also discussed by evaluating phase diagrams.

title: Disordered environments in spatial games.
abstract: The Prisoner's dilemma is the main game theoretical framework in which the onset and maintainance of cooperation in biological populations is studied. In the spatial version of the model, we study the robustness of cooperation in heterogeneous ecosystems in spatial evolutionary games by considering site diluted lattices. The main result is that, due to disorder, the fraction of cooperators in the population is enhanced. Moreover, the system presents a dynamical transition at rho*, separating a region with spatial chaos from one with localized, stable groups of cooperators.

title: If players are sparse social dilemmas are too: Importance of percolation for evolution of cooperation
abstract: Spatial reciprocity is a well known tour de force of cooperation promotion. A thorough understanding of the effects of different population densities is therefore crucial. Here we study the evolution of cooperation in social dilemmas on different interaction graphs with a certain fraction of vacant nodes. We find that sparsity may favor the resolution of social dilemmas, especially if the population density is close to the percolation threshold of the underlying graph. Regardless of the type of the governing social dilemma as well as particularities of the interaction graph, we show that under pairwise imitation the percolation threshold is a universal indicator of how dense the occupancy ought to be for cooperation to be optimally promoted. We also demonstrate that myopic updating, due to the lack of efficient spread of information via imitation, renders the reported mechanism dysfunctional, which in turn further strengthens its foundations.

title: Percolation threshold determines the optimal population density for public cooperation
abstract: While worldwide census data provide statistical evidence that firmly link the population density with several indicators of social welfare, the precise mechanisms underlying these observations are largely unknown. Here we study the impact of population density on the evolution of public cooperation in structured populations and find that the optimal density is uniquely related to the percolation threshold of the host graph irrespective of its topological details. We explain our observations by showing that spatial reciprocity peaks in the vicinity of the percolation threshold, when the emergence of a giant cooperative cluster is hindered neither by vacancy nor by invading defectors, thus discovering an intuitive yet universal law that links the population density with social prosperity.

title: Population viscosity and the evolution of altruism.
abstract: The term population viscosity refers to limited dispersal, which increases the genetic relatedness of neighbors. This effect both supports the evolution of altruism by focusing the altruists' gifts on relatives of the altruist, and also limits the extent to which altruism may emerge by exposing clusters of altruists to stiffer local competition. Previous analyses have emphasized the way in which these two effects can cancel, limiting the viability of altruism. These papers were based on models in which total population density was held fixed. We present here a class of models in which population density is permitted to fluctuate, so that patches of altruists are supported at a higher density than patches of non-altruists. Under these conditions, population viscosity can support the selection of both weak and strong altruism.

title: Spatial games and the maintenance of cooperation.
abstract: The Prisoner's Dilemma (PD) is a widely employed metaphor for problems associated with the evolution of cooperative behavior. We have recently proposed an alternative approach to the PD, by exploring "spatial games" in which players--who are either pure cooperators, C, or pure defectors, D--interact with neighbors in some spatial array; in each generation, players add up the scores from all encounters, and in the next generation a given cell is retained by its previous owner or taken over by a neighbor, depending on who has the largest score. Over a wide range of the relevant parameters, we find that C and D persist together indefinitely (without any of the complex strategies that remember past encounters, and anticipate future ones, which characterize essentially all previous work on the iterated PD). Our earlier work, however, dealt with symmetric spatial lattices in two dimensions, deterministic winning and discrete time. We show here that the essential results remain valid in more realistic situations where the spatial distributions of cells are random in two or three dimensions, and where winning is partly probabilistic (rather than being determined by the largest local total). The essential results also remain valid (pace Huberman and Glance [Huberman, B. A. & Glance, N. S. (1993) ) when interactions occur in continuous rather than discrete time.

title: Evolutionary dynamics of group interactions on structured populations: a review
abstract: Interactions among living organisms, from bacteria colonies to human societies, are inherently more complex than interactions among particles and non-living matter. Group interactions are a particularly important and widespread class, representative of which is the public goods game. In addition, methods of statistical physics have proved valuable for studying pattern formation, equilibrium selection and self-organization in evolutionary games. Here, we review recent advances in the study of evolutionary dynamics of group interactions on top of structured populations, including lattices, complex networks and coevolutionary models. We also compare these results with those obtained on well-mixed populations. The review particularly highlights that the study of the dynamics of group interactions, like several other important equilibrium and non-equilibrium dynamical processes in biological, economical and social sciences, benefits from the synergy between statistical physics, network science and evolutionary game theory.

title: Collective behavior and evolutionary games - An introduction
abstract: This article is an introduction to a special issue in Chaos, Solitons & Fractals with the goal of attracting submissions that identify unifying principles that describe the essential aspects of collective behavior, and which thus allow for a better interpretation and foster the understanding of the complexity arising in such systems.

title: Coevolutionary Games - A Mini Review
abstract: Prevalence of cooperation within groups of selfish individuals is puzzling in that it contradicts with the basic premise of natural selection. Favoring players with higher fitness, the latter is key for understanding the challenges faced by cooperators when competing with defectors. Evolutionary game theory provides a competent theoretical framework for addressing the subtleties of cooperation in such situations, which are known as social dilemmas. Recent advances point towards the fact that the evolution of strategies alone may be insufficient to fully exploit the benefits offered by cooperative behavior. Indeed, while spatial structure and heterogeneity, for example, have been recognized as potent promoters of cooperation, coevolutionary rules can extend the potentials of such entities further, and even more importantly, lead to the understanding of their emergence. The introduction of coevolutionary rules to evolutionary games implies, that besides the evolution of strategies, another property may simultaneously be subject to evolution as well. Coevolutionary rules may affect the interaction network, the reproduction capability of players, their reputation, mobility or age. Here we review recent works on evolutionary games incorporating coevolutionary rules, as well as give a didactic description of potential pitfalls and misconceptions associated with the subject. In addition, we briefly outline directions for future research that we feel are promising, thereby particularly focusing on dynamical effects of coevolutionary rules on the evolution of cooperation, which are still widely open to research and thus hold promise of exciting new discoveries.

title: Social diversity and promotion of cooperation in the spatial prisoner's dilemma game.
abstract: 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 determine 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 payoffs 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 extrinsic 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 strategy 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 cooperation 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.