A condition of cooperation. Games on network

Natural selection is often regarded as a result of severe competition. Defect seems beneficial for a single individual in many cases.However, cooperation is observed in many levels of biological systems ranging from single cells to animals, including human society. We have yet known that in unstructured populations, evolution favors defectors over cooperators. On the other hand, there have been much interest on evolutionary games^1,2^ on structured population and on graphs^3-16^. Structures of biological systems and societies of animals can be taken as networks. They discover that network structures determine results of the games. Together with the recent interest of complex networks^17,18^, many researchers investigate real network structures. Recently even economists study firms' transactions structure^19^. Seminal work^11^ derives the condition of favoring cooperation for evolutionary games on networks, that is, benefit divided by cost, _b/c_, exceeds average degree, (_k_). Although this condition has been believed so far^20^, we find the condition is _b/c_ (_k~nm~_) instead. _k~nm~_ is the mean nearest neighbor degree. Our condition enables us to compare how network structure enhances cooperation across different kinds of networks. Regular network favors most, scale free network least. On ideal scale free networks, cooperation is unfeasible. We could say that (_k_) is the degree of itself, while _k~nm~_ is that of others. One of the most interesting points in network theory is that results depend not only on itself but also on others. In evolutionary games on network, we find the same characteristic

Evolutionary dynamics on any population structure

Evolution occurs in populations of reproducing individuals. The structure of a biological population affects which traits evolve. Understanding evolutionary game dynamics in structured populations is difficult. Precise results have been absent for a long time, but have recently emerged for special structures where all individuals have the same number of neighbors. But the problem of determining which trait is favored by selection in the natural case where the number of neighbors can vary, has remained open. For arbitrary selection intensity, the problem is in a computational complexity class which suggests there is no efficient algorithm. Whether there exists a simple solution for weak selection was unanswered. Here we provide, surprisingly, a general formula for weak selection that applies to any graph or social network. Our method uses coalescent theory and relies on calculating the meeting times of random walks. We can now evaluate large numbers of diverse and heterogeneous population structures for their propensity to favor cooperation. We can also study how small changes in population structure---graph surgery---affect evolutionary outcomes. We find that cooperation flourishes most in societies that are based on strong pairwise ties.Comment: 68 pages, 10 figure

Upstream reciprocity in heterogeneous networks

Many mechanisms for the emergence and maintenance of altruistic behavior in social dilemma situations have been proposed. Indirect reciprocity is one such mechanism, where other-regarding actions of a player are eventually rewarded by other players with whom the original player has not interacted. The upstream reciprocity (also called generalized indirect reciprocity) is a type of indirect reciprocity and represents the concept that those helped by somebody will help other unspecified players. In spite of the evidence for the enhancement of helping behavior by upstream reciprocity in rats and humans, theoretical support for this mechanism is not strong. In the present study, we numerically investigate upstream reciprocity in heterogeneous contact networks, in which the players generally have different number of neighbors. We show that heterogeneous networks considerably enhance cooperation in a game of upstream reciprocity. In heterogeneous networks, the most generous strategy, by which a player helps a neighbor on being helped and in addition initiates helping behavior, first occupies hubs in a network and then disseminates to other players. The scenario to achieve enhanced altruism resembles that seen in the case of the Prisoner's Dilemma game in heterogeneous networks.Comment: 10 figures, Journal of Theoretical Biology, in press (2010

Evolutionary consequences of behavioral diversity

Iterated games provide a framework to describe social interactions among groups of individuals. Recent work stimulated by the discovery of "zero-determinant" strategies has rapidly expanded our ability to analyze such interactions. This body of work has primarily focused on games in which players face a simple binary choice, to "cooperate" or "defect". Real individuals, however, often exhibit behavioral diversity, varying their input to a social interaction both qualitatively and quantitatively. Here we explore how access to a greater diversity of behavioral choices impacts the evolution of social dynamics in finite populations. We show that, in public goods games, some two-choice strategies can nonetheless resist invasion by all possible multi-choice invaders, even while engaging in relatively little punishment. We also show that access to greater behavioral choice results in more "rugged " fitness landscapes, with populations able to stabilize cooperation at multiple levels of investment, such that choice facilitates cooperation when returns on investments are low, but hinders cooperation when returns on investments are high. Finally, we analyze iterated rock-paper-scissors games, whose non-transitive payoff structure means unilateral control is difficult and zero-determinant strategies do not exist in general. Despite this, we find that a large portion of multi-choice strategies can invade and resist invasion by strategies that lack behavioral diversity -- so that even well-mixed populations will tend to evolve behavioral diversity.Comment: 26 pages, 4 figure

Graph Transformations and Game Theory: A Generative Mechanism for Network Formation

Many systems can be described in terms of networks with characteristic structural properties. To better understand the formation and the dynamics of complex networks one can develop generative models. We propose here a generative model (named dynamic spatial game) that combines graph transformations and game theory. The idea is that a complex network is obtained by a sequence of node-based transformations determined by the interactions of nodes present in the network. We model the node-based transformations by using graph grammars and the interactions between the nodes by using game theory. We illustrate dynamic spatial games on a couple of examples: the role of cooperation in tissue formation and tumor development and the emergence of patterns during the formation of ecological networks

Social Network Reciprocity as a Phase Transition in Evolutionary Cooperation

In Evolutionary Dynamics the understanding of cooperative phenomena in natural and social systems has been the subject of intense research during decades. We focus attention here on the so-called "Lattice Reciprocity" mechanisms that enhance evolutionary survival of the cooperative phenotype in the Prisoner's Dilemma game when the population of darwinian replicators interact through a fixed network of social contacts. Exact results on a "Dipole Model" are presented, along with a mean-field analysis as well as results from extensive numerical Monte Carlo simulations. The theoretical framework used is that of standard Statistical Mechanics of macroscopic systems, but with no energy considerations. We illustrate the power of this perspective on social modeling, by consistently interpreting the onset of lattice reciprocity as a thermodynamical phase transition that, moreover, cannot be captured by a purely mean-field approach.Comment: 10 pages. APS styl

Participation costs dismiss the advantage of heterogeneous networks in evolution of cooperation

Real social interactions occur on networks in which each individual is connected to some, but not all, of others. In social dilemma games with a fixed population size, heterogeneity in the number of contacts per player is known to promote evolution of cooperation. Under a common assumption of positively biased payoff structure, well-connected players earn much by playing frequently, and cooperation once adopted by well-connected players is unbeatable and spreads to others. However, maintaining a social contact can be costly, which would prevent local payoffs from being positively biased. In replicator-type evolutionary dynamics, it is shown that even a relatively small participation cost extinguishes the merit of heterogeneous networks in terms of cooperation. In this situation, more connected players earn less so that they are no longer spreaders of cooperation. Instead, those with fewer contacts win and guide the evolution. The participation cost, or the baseline payoff, is irrelevant in homogeneous populations but is essential for evolutionary games on heterogeneous networks.Comment: 4 figures + 3 supplementary figure