Papers International Journal of Bifurcation and Chaos, Vol. 12, No. 7 (2002) 1531���1548 c World Scientific Publishing Company EFFECTS OF SPACE IN 2 �� 2 GAMES CH. HAUERT* Institut f�� ur Mathematik, Universit�� at Wien, Strudlhofgasse 4, A-1090 Vienna, Austria Christoph.Hauert@univie.ac.at Received June 7, 2001 Revised August 17, 2001 A systematic analysis of the effects of spatial extension on the equilibrium frequency of cooper- ators and defectors in 2��2 games is presented and compared to well mixed populations where spatial extension can be neglected. We demonstrate that often spatial extension is indeed capa- ble of promoting cooperative behavior. This holds in particular for the prisoner���s dilemma for a small but important parameter range. For the hawk���dove game, spatial extension may lead to both, increases of the hawk- as well as the dove-strategy. The outcome subtly depends on the parameters as well as on the degree of stochasticity in the different update rules. For rectangu- lar lattices, the general conclusions are rather robust and hold for different neighborhood types i.e. for the von Neumann as well as the Moore neighborhood and, in addition, they appear to be almost independent of the update rule of the lattice. However, increasing stochasticity for the update rules of the players results in equilibrium frequencies more closely related to the mean field description. Keywords: Cooperation prisoner���s dilemma lattice games. 1. Introduction The essence of various ecological interactions among animals can be modeled by so-called 2 �� 2 games describing pairwise interactions between individu- als with two behavioral strategies to choose from. Depending on their joint behavior, each individual obtains a certain payoff. In biology, this payoff is usually related to the fitness of an individual, i.e. to its reproductive success. Consequentially, in a pop- ulation of interacting individuals, successful strate- gies achieving high payoffs will be prevalent. The most prominent representative of such 2��2 games is certainly the prisoner���s dilemma (PD) [Axelrod, 1984]. The PD has received widespread attention for explaining the emergence of coopera- tive and altruistic behavior among unrelated selfish individuals. In the PD, two players have to decide whether to cooperate (C) or defect (D). Mutual cooperation pays a reward R while mutual defec- tion results in a punishment P. If one player opts for D and the other for C, then the former obtains the temptation to defect T and the latter is left with the sucker���s payoff S. From the rank order- ing of the four payoff values T R P S fol- lows that a player is always better off defecting, regardless of the opponents decision. Consequen- tially, rational players will always end up with the punishment P instead of the higher reward for coop- eration R. Nevertheless, cooperative behavior can establish through basic discrimination mechanisms that enable individuals to target their altruistic acts towards certain other individuals only. Since Axelrod���s seminal work, numerous articles have been published on the subject suggesting differ- ent mechanisms to promote cooperative behavior. These mechanisms can be roughly divided into three *Current address: Department of Zoology, University of British Columbia, 6270 University Boulevard, Vancouver B.C., Canada V6T 1Z4. 1531

1532 Ch. Hauert categories: (a) repeated interactions of the same individuals, (b) identification through secondary traits such as reputation or (c) spatial extension. The latter will be the main topic of this article. In (a), individuals trigger their actions on the outcome of previous encounters (see e.g. [Nowak & Sigmund, 1993 Binmore & Samuelson, 1992 Lindgren, 1991 Milinski, 1987]). Memorizing the past, enables in- dividuals to cooperate with cooperative opponents only. In (b), individuals interact only once but carry an image score summarizing their past actions (see e.g. [Nowak & Sigmund, 1998 Wedekind & Milinski, 2000]). The opponents reputation, i.e. its image score, again allows to discriminate between cooperators with high and defectors with low scores. In spatially extended systems (c), individuals in- teract only with their local neighborhood (see e.g. [Nowak & May, 1992 Doebeli & Knowlton, 1998 Killingback & Doebeli, 1998 Hauert, 2001a Herz, 1994 Szab�� o et al., 2000]). Cooperators may sur- vive by forming clusters, thereby minimizing con- tacts with defecting players. Closely related to the PD is another 2 �� 2 game called chicken or hawk���dove game [Maynard Smith & Price, 1973] describing intra-species com- petition or, in the form of the snow-drift game [Sugden, 1986], explaining biproduct mutualism. Actually, it differs only in the payoff ranking with T R S P i.e. with the sucker���s payoff S being more favorable than the punishment P. In the general formulation, a 2 �� 2 game is de- termined by the payoff matrix R S T P ! , (1) where the rank ordering of the four payoff values R, S, T, P determines the characteristics of the game. Without loss of generality we may assume R P (if this does not hold, we simply interchange C and D) and normalize the payoff values such that R = 1, P = 0 holds. This leads to 12 different rank orderings corresponding to very different strategic situations (see e.g. [Rapoport et al., 1976 Binmore, 1992 Colman, 1995]). Each game corresponds to a region in the S, T-plane as shown in Fig. 1. If we now consider large populations of inter- acting individuals, further assumptions concerning the structure of the population are required. In the simplest case, each individual interacts with every other one with equal probability i.e. the population is well mixed and has no structure at all. For pop- ulations consisting of only two types of players ��� S T 2: chicken 3: leader 7: harmony 8: stag hunt 12: deadlock 11 10 9 6 5 1 1 0 0 1: prisoner���s dilemma 4: battle of the sexes Fig. 1. The rank orderings of the parameters R, S, T and P with R = 1, P = 0 divide the S, T-plane into 12 different regions. Each region determines the parameter range for a particular 2 �� 2 game. those who always cooperate and those who always defect ��� the equilibrium distribution of the two strategies can be calculated analytically in the so- called mean field approximation [Posch et al., 1999]. In the following section we briefly review this case and for the remaining sections, it provides a valu- able basis for comparisons and discussions of our results for spatially structured populations. The other extreme is represented by popula- tions with rigid spatial structures such as regular lattices where each individual is bound to a sin- gle lattice site and interacts pairwise with its lo- cal neighbors only. Even though this provides only a crude approximation to ecological scenarios in nature, it turned out to be a fruitful extension providing substantial and interesting new insights. Spatial extension is not only capable of promoting cooperative behavior but also produces very com- plex dynamics [Nowak & May, 1993 Killingback & Doebeli, 1998]. In this article, however, we concen- trate on the equilibrium frequencies of the strategies for different neighborhood types and different up- date rules on the player���s as well as the lattice level. For the lattice, we consider random or asynchronous and synchronized updates that model populations with overlapping and nonoverlapping generations, respectively. For the players, we consider several update rules with different degrees of stochasticity.

Effects of Space in 2 �� 2 Games 1533 2. Mean Field Games In the mean field description all spatial correlations are neglected. This corresponds to well mixed pop- ulations where individuals are randomly matched. In order to determine the equilibrium frequencies of both, the always cooperate and the always de- fect strategies, we consider a homogeneous popula- tion of residents, and determine the fate of a mu- tant strategy with frequency xm. In a well mixed population, the success of the mutant strategy de- pends only on xm as well as on the parameters S, T. The frequency of the resident is simply given by xr = 1 - xm. For cooperative mutants attempt- ing to invade a resident population of defectors we obtain the payoffs Pr and Pm for mutants and resi- dents, respectively: Pr = xmT Pm = xrS + xm . In biological context, the payoff denotes the fitness and hence the reproductive success of mutants and residents. Similarly, in terms of cultural evolution, an individual imitates the strategy of a randomly chosen ���model��� member of the population with a probability proportional to the difference between the model���s payoff and its own, provided the dif- ference is positive and with probability 0 otherwise [Weibull, 1995 Schlag, 1998]. In the continuous time limit both approaches lead to the replicator equation [Hofbauer & Sigmund, 1998]: ��i x = xi(Pi -P) with i ��� {r, m}. (2) The long term behavior of the population is deter- mined by the three fixed points of Eq. (2): ��1 f c = 0, ��2 f c = 1 and ��3 f c = S/(S + T - 1), where ��i f c denotes the equilibrium fractions of cooperators in the population as a function of S, T (see Fig. 2). By inspection, four different dynamical domains are -4 -2 0 2 4 S -4 -2 0 2 4 T (a) (b) (c) (d) Fig. 2. Fraction of cooperators �� f c as a function of S, T in the mean field description. Regions with low �� f c i.e. high fractions of defectors are colored red. Cooperative regions with �� f c close to 1 are blue. Intermediate values of �� f c are shown in yellow, green and light blue colors. The dashed lines divide the S, T-plane into four quadrants with different dynamical characteristics: (a) dominating defection, (b) coexistence, (c) bistability and (d) prevailing cooperation. In the case of bistability, the color codes indicate the size of the basin of attraction resulting in a cooperative state. In blue regions even a very small fraction of cooperators will spread and eventually dominate the population, while in reddish regions cooperators can spread only in populations that are already highly cooperative.