The evolution of cooperation among unrelated individuals is studied in a lattice-structured habitat, where individuals interact locally only with their neighbors. The initial population includes Tit-for-Tat (abbreviated as TFT, indicating a cooperative strategy) and All Defect (AD, a selfish strategy) distributed randomly over the lattice points. Each individual plays the iterated Prisoner’s Dilemma game with its nearest neighbors, and its total pay-off determines its instantaneous mortality. After the death of an individual, the site is replaced immediately by a copy of a randomly chosen neighbor. Mathematical analyses based on mean-field approximation, pair approximation, and computer simulation are applied. Models on one and two-dimensional regular square lattices are examined and compared with the complete mixing model. Results are: (1) In the one-dimensional model, TFT players come to form tight clusters. As the probability of iteration w increases, TFTs become more likely to spread. The condition for TFT to increase is predicted accurately by pair approximation but not by mean-field approximation. (2) If w is sufficiently large, TFT can invade and spread in an AD population, which is impossible in the complete mixing model where AD is always ESS. This is also confirmed by the invasion probability analysis. (3) The two-dimensional lattice model behaves somewhat in between the one-dimensional model and the complete mixing model. (4) The spatial structure modifies the condition for the evolution of cooperation in two different ways: it facilitates the evolution of cooperation due to spontaneously formed positive correlation between neighbors, but it also inhibits cooperation because of the advantage of being spiteful by killing neighbors and then replacing them. � 1997 Academic Press Limited

Abstract not found

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 four sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fifth section surveys the topological complications implied by non-mean-field-type social network structures in general. The next 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.

Understanding the mechanisms that can lead to the evolution of cooperation through natural selection is a core problem in biology. Among the various attempts at constructing a theory of cooperation, game theory has played a central role. Here, we review models of cooperation that are based on two simple games: the Prisoner’s Dilemma, and the Snowdrift game. Both games are two-person games with two strategies, to cooperate and to defect, and both games are social dilemmas. In social dilemmas, cooperation is prone to exploitation by defectors, and the average payoff in populations at evolutionary equilibrium is lower than it would be in populations consisting of only cooperators. The difference between the games is that cooperation is not maintained in the Prisoner’s Dilemma, but persists in the Snowdrift game at an intermediate frequency. As a consequence, insights gained from studying extensions of the two games differ substantially. We review the most salient results obtained from extensions such as iteration, spatial structure, continuously variable cooperative investments, and multi-person interactions. Bridging the gap between theoretical and empirical research is one of the main challenges for future studies of cooperation, and we conclude by pointing out a number of promising natural systems in which the theory can be tested experimentally.

The term population viscosity means limited dispersal, which increases the genetic relatedness of neighbors. This 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 overall population density was 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.

In this paper I describe how the spatial dynamics of autocatalytic interactions among entities in a virtual world led to a type of co-operation that typically would be studied from a game theoretical perspective. The entities were very simple and completely identical at the start of the simulation. They just aggregated and performed aggressive interactions in which winning was self-reinforcing. Patterns of reciprocation emerged at the level of the group, particularly in loose assemblages. These patterns appeared not to be due to global spatial structures as I have suggested before (Hemelrijk, 1996ab), but arose from local series of Titfor -Tat like interactions. These involved pairs of individuals that 'collaborated` by taking turns in chasing away a third entity. Runs with varying parameter values showed that particularly those entities that were designed to be more aggressive, were more prone to co-operate. The processes responsible for the Tit-for-Tat like patterns are outlined. It ...

We consider an individual-based spatial model of a generic host-pathogen system and explore the differences between such models and mean-field systems. We find a range of new dynamical and evolutionary phenomena. In particular, (i) in this system selective pressure is substantially reduced compared to the corresponding mean-field models and artificial suppression of the pathogen population leads to faster evolution and reduces evolutionary stability; (ii) unlike the mean-field models, there exists a critical transmissibility c above which the pathogen dies out and (iii) the system displays self-evolved criticality. If the transmissibility is allowed to mutate, it evolves to the critical value c . Thus the system evolves so as to put itself at the boundary of where it can exist. Observations of the individual-based spatial model motivate an explanation for these phenomena in terms of the dynamics of host patches involving their connections and disconnections. We therefore c...

We have performed calculations on reaction-diffusion equations with an aim to study two-dimensional spatial patterns. The systems explicitly studied are three different catalytic networks: A 4-component network displaying chaotic dynamics, a 5-component hypercycle network and a simple 1-component system. We have obtained cluster states for all these networks, and in all cases the clusters have the ability to divide. This contradicts recent conclusions that only systems with chaotic dynamics may give cluster states: On the contrary, we think that any network architecture may display cluster formation and cluster division. Our conclusion is in agreement with experimental results reported for an inorganic system corresponding to the simple 1-component system studied in this paper. In a partial differential equations model, the clusters do not provide resistance to parasites, which are assumed to arise by mutations: Parasites may spread from one cluster to another, and eventually kill all ...

We analyze the role of social structure in maintaining cooperation within a population of adaptive agents for whom cooperative behavior may be costly in the short run. We use the example of a collection of agents playing pairwise Prisoner's Dilemma. We call sustained cooperative behavior in such circumstances a 'cooperative regime'. We show that social structure, by channeling which agents interact with which others, can sustain cooperative regimes against forces that frequently dissolve them. We show in detail the process through which structured interaction in a population creates a “shadow of the adaptive future ” allowing even a small set of cooperative strategies to grow into a cooperative regime, a coherent, self-sustaining entity that is something more than the sum of the pairwise interactions among its members.

We study evolutionary games on graphs. Each player is represented by a vertex of the graph. The edges denote who meets whom. A player can use any one of n strategies. Players obtain a payoff from interaction with all their immediate neighbors. We consider three different update rules, called ‘birth–death’, ‘death–birth ’ and ‘imitation’. A fourth update rule, ‘pairwise comparison’, is shown to be equivalent to birth–death updating in our model. We use pair approximation to describe the evolutionary game dynamics on regular graphs of degree k. In the limit of weak selection, we can derive a differential equation which describes how the average frequency of each strategy on the graph changes over time. Remarkably, this equation is a replicator equation with a transformed payoff matrix. Therefore, moving a game from a well-mixed population (the complete graph) onto a regular graph simply results in a transformation of the payoff matrix. The new payoff matrix is the sum of the original payoff matrix plus another matrix, which describes the local competition of strategies. We discuss the application of our theory to four particular examples, the Prisoner’s Dilemma, the Snow-Drift game, a coordination game and the Rock–Scissors–Paper game.