Evolution and Game Theory - Abstracts

Satellite Workshop at ECCS 07
Dresden, 4th-5th October
Organizers:
Jens Christian Claussen (Kiel)
Christoph Hauert (Harvard)
György Szabó (Budapest)

Tag-based cooperation in finite populations
Arne Traulsen (Harvard)

In tag-based models for cooperation individuals recognize each other via arbitrary signals, so-called tags. Cooperators use a tag until they are discovered by defectors, who then destroy cooperation based on this tag by exploiting cooperators. If many tags are available, then cooperators can always establish new signals of recognition. This leads to a constant chase: Cooperators establish new signals for cooperation, while defectors constantly try to find out about these signals. Tag based cooperation is intimately related to the green beard effect. While conventional wisdom leads to the breakdown of cooperation based on such green beards, it can be shown that green beards can lead to cooperation if they are worn by red queens.
[1] Arne Traulsen and Martin A. Nowak, Chromodynamics of cooperation in finite populations, PLoS ONE. 2007;2:e270

Love and hatred in a world of feedback
Hanna Kokko (Helsinki)

Game theory has been used in evolutionary ecology since the 1970s, when George Price and John Maynard Smith asked the question why animals do not fight as hard as they can - instead aggression levels seem to be limited. Applications of game theory in evolutionary studies have evolved since then and have become much more ecologically realistic. Ecology (the study of the abundance and distribution of organisms) is important because there are numerous phenomena where the adaptive value of a behaviour depends not only on what others are doing - the familiar reason why one uses game theory - but also on how many of those 'others' exist in the vicinity - i.e. population density. On the other hand, behaviour itself can alter this number, which creates a two-way feedback loop from individuals to populations and vice versa. While this statement has an abstract feel to it, I will show how this feedback really can influence our understanding of important evolutionary processes. I will focus on two examples: territoriality and a modern view of hawk-dove games in this context, and the evolution of sex roles, i.e. why do females often perform the majority of parental care duties?

Co-evolution and the Red Queen
Stefan Bornholdt (Bremen)

Replicator Equations for Simple (Bio)chemical Replicons
Peter Stadler (Vienna and Leipzig)

Replicator equations have been studied for three decades as a generic dynamical system modelling replication processes. In models of "protocells" they arise naturally as description of the internal selection dynamics of the genetic component of such systems. The interesting aspect is that this internal dynamics remains well-separated from the external boundary conditions. Even the internal replicator dynamics is in many cases - as far as the structure of the equations is concerned - independent of mechanistic details of the chemcial reaction system.

Cooperation in Social Dilemmas: The Role of Punishment and Volunteering
Christoph Hauert (Harvard)

The emergence and maintenance of cooperative behavior that is beneficial to others but costly to the individual represents a major conundrum in evolutionary biology. Punishment represents an efficient mechanism to stabilize and maintain cooperation in social dilemmas and is ubiquitous in animal and human societies - ranging from toxin producing microorganisms to law enforcement institutions - but it remains unresolved how initially rare and costly punishment behavior can gain a foothold and spread through the population. In nature, animals and humans often carefully select their interaction partners or adjust their behavioral patterns in response to them. In the simplest case they simply refuse to participate in risky enterprises. Such voluntary participation in social dilemmas is an efficient mechanism to prevent deadlocks in states of mutual defection and thus represents a potent promoter of cooperation but fails to stabilize it. However, the combined efforts of punishment and volunteering may change the odds in favor of cooperation. Interestingly, even though the combined efforts fail in infinite populations they nevertheless provide an efficient mechanism to stabilize cooperation (and punishment) under the stochastic dynamics of finite populations with mutation and selection. Thus, the freedom to withdraw leads to prosocial coercion. This implements Hardin's principle: ``mutual coercion mutually [and voluntarily] agreed upon''.

Evolution of Robustness and Phenotypic Fluctuation
Kunihiko Kaneko (Tokyo)

How is phenotypic fluctuation related with evolution? Here fluctuations have two sources, one due to intrinsic, isogenic phenotypic fluctuations, arising from stochasticity through developmental process, the other due to genetic variation. From the analysis of evolutionary stability in population distribution, we propose a general inequality between the phenotype variance due to genetic variation and the intrinsic phenotype variance of clones, as well as proportionality between the two through evolution. Following this theoretical proposition, we derive a relationship between (isogenic) phenotype fluctuation andevolution speed, which is confirmed in bacteria evolution experiments. As an application of our theory, we discuss relationship between phenotype robustness to mutation and that to noise through development. Through simulations of a simple stochastic gene expression network that undergoes mutation and selection, we show how noise that cells encounter during growth and development shapes networks' robustness to stochasticity in development, which in turn shapes networks' robustness to mutation. The necessary condition for evolution of robustness as well as relationship between genetic and developmental robustness is derived quantitatively through the variance of phenotypic fluctuations. In a brief outlook, I also discuss how interaction among organisms and phenotype dynamics leads to robust sympatric speciation, as a result of Baldwin-type genetic fixation of interaction-induced phenotypic diversification.
[1] Life: An Introduction to Complex Systems Biology, Springer (2007).

Altrusim before the transition to extinction
Konstantin Klemm (Leipzig)

We study the evolution of altruism in spatially extended populations at the survival-extinction transition. At contrast with earlier spatial models, e.g. [Nowak and May, Nature (1992)], we consider variations in the population density by allowing lattice sites to be empty. As the selective pressure p, defined as the death rate of agents in the absence of cooperating neighbors, approaches the critical value pc from below, the dominance of defectors becomes unstable at a value pu < pc. Cooperators can invade and become dominant. This effect - cooperation before extinction - is observed whenever the total benefit is larger than the cost incurred by the altruistic act. The increase in the number of cooperators is larger than the decrease in the number of defectors as p rises from pu to pc. Thus the overall population density increases as a function of the pressure. These properties of the phase diagram are obtained by simulations on lattices and by using pair approximation.

Evolution as diffusion in type space
Daniel Lawson (London)

The analysis of low dimensional type spaces is of particular interest for the comparison of trait experiments, but is difficult to approach by considering lineages because lineages can converge. We consider a novel form of "moment-closure" based on minimum information assumptions that permits a solution for the trait distribution of a finite population. The mean position performs a random walk with speed independent of population size, and the fluctuations in the width of the distribution are always as large as the distribution itself. Hence there is a simple "steady state" solution that is subject to very large fluctuations, which can be used as a null model, or form part of a more complex models. The population can separate into distinct clusters, and we discuss to what degree this can be considered as speciation.

Cooperation in repeated games with unlimited stochastic payoffs
Anders Eriksson and Kristian Lindgren (Gothenburg)

In repeated interactions between individuals, we do not expect that exactly the same situation will occur from one time to another. Contrary to what is common in models of repeated games in the literature, most real situations may differ a lot and they are seldom completely symmetric. In previous papers, we have discussed the evolution and stability of cooperative behaviour for a repeated game with random observable payoffs, where payoff values are uniformly distributed between 0 and 1 [Eriksson and Lindgren 2002, 2006]. The players are thus faced with a more complex situation, compared to the Prisoner's Dilemma. Still, there are robust cooperating strategies that select an action that allows for maximizing the sum of the payoff of the two players in each round, regardless of the own payoff. In this presentation, we analyse a variation of the model in which payoff values are unlimited, e.g., exponentially distributed. Then, depending on the discount rate, a player may be faced with a temptation that is hard to reject. The main question we discuss is: Under what circumstances can cooperation still be expected in an evolutionary context?
[1] A. Eriksson and K. Lindgren (2002). Cooperation in an unpredictable environment, in proceedings of Artificial Life VIII (Sydney).
[2] A. Eriksson and K. Lindgren (2006). A simple model for cognitive processing in repeated games, in proceedings for European Conference for Complex Systems 2006 (Oxford).

Equilibrium transitions in finite populations of players
Jacek Miękisz (Warsaw)

We analyze the long-run behaviour of stochastic dynamics in well-mixed populations and in spatial games with local interactions. We review results concerning the effect of the number of players and the noise level on the stochastic stability of Nash equilibria.
To address the problem of equilibrium selection in spatial games with many players, we introduce a concept of ensemble stability. The standard stochastic stability describes a long-run behaviour of systems with a fixed number of players in the zero-noise limit. On the contrary, the ensemble stability is concerned with a fixed (but nevertheless low) noise level in the limit of the infinite number of players.
We present examples of games in which when the number of players increases or the noise level decreases, a population undergoes a transition between its equilibria.
In particular, it may happen that a risk-dominant and Pareto-efficient strategy, which is stochastically stable, in the long run is played with an arbitrarily small probability if the noise level is low and the number of players is big enough.
1. J. Miękisz, Stochastic stability in spatial games, J. Stat. Phys. 117: 99-110 (2004).
2. J. Miękisz, Statistical mechanics of spatial evolutionary games, J. Phys. A: Math. Gen. 37: 9891-9906 (2004).
3. J. Miękisz, Equilibrium selection in evolutionary games with random matching of players, J. Theor. Biol. 232: 47-53 (2005).
4. D. Kaminski, J. Miękisz, and M. Zaborowski, Stochastic stability in three-player games, Bull. Math. Biol. 67: 1195-1205 (2005).

Evolutionary dynamics on networks: from multiple networks to active linking dynamics
Jorge Pacheco (Lisboa)

We investigate the impact of networks - as means of defining who interacts with whom in a given population - on the evolutionary game dynamics associated with social dilemmas. We find that sparse graphs provide an efficient mechanism to promote cooperation. On static graphs, we show that it is best that those we interact with are also our role models. When links are allowed to follow individual's preferences, dynamical graphs provide compelling evidence for the role of strategic assortment in the evolution of cooperation. We discuss analytic treatments of both static and dynamic, single and multiple, graphs, and show that often one can map the dynamics on graphs to a mean-field dynamics for a game involving a rescaled payoff matrix. The re-scaling depends on the evolutionary dynamics employed.

Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games
Tobias Reichenbach (Munich)

Counterintuitive to a naive understanding of Darwinian evolution, where among two interacting species one is expected to be fitter than the other and therefore outcompetes it, a surprising biodiversity exists within the earth's ecosystems. Rock-paper-scissors games, where three strategies cyclically dominate each other, have emerged as a fruitful metaphor for the explanation of biodiversity. In this talk we discuss populations spatially coevolving with local cyclic dominance, and show that they are capable of preserving coexistence of all subpopulations, and in this way ensuring biodiversity. We find that the individuals' mobility competes with the locality of interactions (cyclic dominance) such that biodiversity gets lost above a certain mobility threshold. Below this critical value, all subpopulations coexist forming fascinating moving patterns composed of entangled spirals, which we describe analytically.
[1] Tobias Reichenbach, Mauro Mobila and Erwin Frey, Mobility promotes and jeopardizes biodiversity in rock-paper-scissors games, to appear in Nature (2007).

Evolution of cooperators on static and dynamic graphs
Istvan Scheuring (Budapest)

Population structure has been proposed as one of the mechanism promoting cooperation. Until recently, most studies assumed that the interaction network can be described by a regular graph. Recently it is shown for a number of other interaction topologies that selection favours cooperation (i.e. the fixation probability of a single cooperator is higher than the fixation probability of a neutral mutant) in the prisoner's dilemma game if the benefit (b) of the altruistic act divided by its cost (c) exceeds the average number of neighbours (k), that is, if b/c > k. This relation is approximately valid in populations of different structure, in which interaction topology is described variously by regular, random regular, random, or scale-free graphs. We studied the game dynamics numerically on scale-free graphs in a more detailed manner than the previous work did. We concluded that fixation probability of cooperators depends linearly on the number of neighbours of the site from where the invasion begins. According our measures fixation probability scales with k as exp(-ak)k-l, (where a and l are positive parameters) and it depends not simply on b/c, as the previous approximation suggested. Since animal and human societies can not be described by static graphs, we studied fixation probability of cooperators on dynamical graphs. For biological reasons we focused on scale-free graphs. We have studied the following three algorithms for the graph dynamics:
(a) Random choice: An edge between the focal site and a randomly chosen site is established. This randomly chosen site loses one of its randomly selected neighbour. This scenario implies that the focal individual searches randomly for a new interaction partner, and the chosen new partner abandons one of its connections at random.
(b) Preferential choice 1. : An edge between the focal site and a randomly chosen individual is established. This randomly chosen individual loose one of its defector neighbour with a given probability. If it has no defector neighbour then a randomly chosen cooperating neighbour is exchanged.
(c) Preferential choice 2. : The focal individual establish a link with one of its cooperating neighbour's neighbour with a given probability. First a cooperating neighbour is selected randomly (or a random individual if none of them cooperates), then one of its neighbour is selected randomly. The focal individual will establish a connection with this individual. The selected new interaction partner tries to loose a defecting neighbour as in the previous scenario.
On the basis of comprehensive numerical simulations we concluded that when the benefit of the cooperative interaction is low, preferential choices does not increase fixation probability to a great extent. On the other hand at higher benefits (b), the fixation probability increases significantly with increasing level of preference. Already at moderate level of preference the fixation probability will be higher than the fixation probability observed for the static graph if the average connectivity is above 6. On the other side random choice has always a detrimental effect on the fixation probability of cooperators.

Cooperation in diffusive spatial games
Mendeli H. Vainstein, Ana T. C. Silva, and Jeferson J. Arenzon (Porto Alegre)

Random diffusion is shown to be an important mechanism on fostering cooperative behavior among simple agents (memoryless, unconditional cooperators or defectors) living on a spatially structured environment. In particular, under the Prisoner's Dilemma framework, when allowing the agents to move with the simple ``always-move'' rule, we find that cooperative behavior is not only possible but may even be enhanced. In addition, for a broad range of densities, mobile cooperators can more easily invade a population of mobile defectors, when compared with the fully viscous, immobile case. Thus, such simple mobility pattern may have played a fundamental role both in the onset and development of cooperative behavior, paving the way to more complex, individual and group, motility rules.
[1] M.H. Vainstein, and J.J. Arenzon, Phys. Rev. E 64, 051905 (2001).
[2] M.H. Vainstein, A.T.C. Silva, and J.J. Arenzon, J. Theor. Biol. 244, 722--728 (2007).

Cooperation out of noise
Matjaž Perc (Maribor)

We study how the addition of noise to the payoffs affects the outcome of the spatial prisoner's dilemma game. We show that there exists an optimal intensity of noise for which cooperation thrives best, and perhaps even more interestingly, prevails for defection temptation values substantially exceeding the one marking the transition point to homogeneous defection by deterministic payoffs. By employing a pair-approximated version of the prisoner's dilemma game we outline certain similarities with the phenomenon of coherence resonance observed previously in nonlinear dynamical systems, where an optimal intensity of noise warrants the most ordered temporal or spatial dynamics of the system.

Hierarchical meanfield theory of evolutionary games on structured populations
Gergely J. Szollosi and Imre Derenyi (Budapest)

Evolutionary games played out in populations structured by spatial embedding or more general networks of interaction have been shown to have fundamentally different dynamics and outcomes compared to those taking place in well mixed ones. Recent experimental and theoretical work - published largely in interdisciplinary journals such as Nature and PNAS - has demonstrated that longstanding open problems in biology, sociology, and the economic sciences (ranging from the maintenance of diversity to the evolution of altruism and reciprocity) can only be understood if we look beyond well mixed populations and take into account the effects of spatial structure. The question of how one goes about choosing the relevant model to describe population structures, however, stands unanswered. Models where individuals are confined to the sites of some lattice or the nodes of some random graph have proved highly sensitive to seemingly minor changes in the implementation of the dynamics and the details of the underlying topology of interactions. In our paper we introduce a minimal model of population structure that is described by two distinct hierarchical levels of interaction. We derive the dynamics governing the evolution of such a system starting from fundamental individual level stochastic processes and find that the simple and straightforward hierarchical application of the mean-field approximation (the assumption of being well mixed) at both levels surprisingly unveils a new level of dynamical complexity. We believe that such minimal structure is more relevant in a wide range of natural systems, than more subtle setups with a delicate dependence on the details and symmetries of the model. We show that such minimal structure is sufficient for the emergence of effects previously only observed for explicit spatial embedding and demonstrate the potential of our model to identify robust effects of population structure on the dynamics and outcome of evolutionary games.
[1] Gergely J Szollosi, Imre Derenyi, http://arxiv.org/abs/0704.0357

The Tangled Nature Model
Simon Laird (London), Daniel Lawson (Edinburgh) and Henrik J. Jensen (London)

The Tangled Nature model is developed to focus on the effect of evolution and multiple interactions on ecological and evolutionary observables. The model is individual based and ecological structures, such as species, are emergent quantities. The dynamics consists of a simplistic mutation prone multiplication in which the probability of producing an odetermined by the occupancy in genotype space. The macroscopic long time dynamics is intermittent and exhibit a slow decrease in the macroscopic extinction rate. Ecological quantities such as the Species Abundance Distribution and the Species Area Relation compare qualitatively well with observations, as does the relation between interaction and diversity. The effect of correlations between parents and mutants has been studied as has the effect of a conserved resource.

Coevolutionary dynamics from finite to infinite populations
Arne Traulsen (Harvard), Jens Christian Claussen (Kiel), and Christoph Hauert (Harvard)

Coevolutionary dynamics arises in a wide range from biological to social dynamical systems. For infinite populations, a standard approach to analyze the dynamics are deterministic replicator equations, however lacking a systematic derivation. In finite populations modelling finite-size stochasticity by Gaussian noise is not in general warranted [1]. We show that for the evolutionary Moran process and a Local update process, the explicit limit of infinite populations leads to the adjusted or the standard replicator dynamics, respectively [2]. In addition, the first-order corrections in the population size are given by the finite-size update stochasticity and can be derived as a generalized diffusion term of a Fokker-Planck equation [2]. This framework can be readily transferred to other microscopic processes, as the local Fermi process [3] or the inclusion of mutations in the process [4]. We explicitely discuss the differences for the Prisoner's Dilemma, and Dawkin's Battle of the Sexes, where we show that the stochastic update fluctuations in the Moran process lead to a finite-size dependent drift reversal [2].
[1] J.C. Claussen and A. Traulsen, Phys. Rev. E 71, 025101(R) (2005)
[2] A. Traulsen, J.C. Claussen, C. Hauert, Phys. Rev. Lett, 95, 238701
[3] A.Traulsen, M.A.Nowak, J.M.Pacheco, Phys. Rev. E 74, 011909 (2006)
[4] A. Traulsen, J.C. Claussen, C. Hauert, Phys. Rev. E 74, 011901 (2006)

Evolutionary Prisoner's Dilemma game on the Newman-Watts networks
Jeromos Vukov, György Szabó and Attila Szolnoki (Budapest)

Maintenance of cooperation was studied on rarely linked chain-like structures by considering an evolutionary Prisoner's Dilemma game. In these systems the players are located on a one-dimensional chain and their payoff comes from games with the nearest and next-nearest neighbor interactions. The evolutionary rule (pairwise comparison) involves some noise characterized by a temperature and allows irrational strategy adoptions between the interacting players. Using Monte Carlo simulations and the extended versions of dynamical mean-field theory we determined the density of cooperators as a function of temperature and temptation to choose defection. The resultant (unique) phase diagram reveals that the cooperative behavior can be supported by choosing an optimal temperature (noise level) in the one-dimensional system. The disadvantageous consequences of the one-dimensional connectivity structure can be reduced gradually (especially for the low temperature limit) when adding extra (random) links to the connectivity structure according to the Newman-Watts construction.

Evolution under alliance-specific cyclical invasion rates
Matjaž Perc (Maribor), Attila Szolnoki (Budapest), and György Szabó (Budapest)

We study a six-species Lotka-Volterra type system on different two-dimensional lattices when each species has two superior and two inferior partners. The invasion rates from predator sites to a randomly chosen neighboring prey's site depend on the predator-prey pair, whereby cyclic symmetries within the two three-species defensive alliances are conserved. Monte Carlo simulations reveal an unexpected non-monotonous dependence of alliance survival on the difference of alliance-specific invasion rates. The invasion probabilities are supplemented by Gaussian noise and conditions are identified that warrant the largest impact of noise on the evolutionary process. Our findings are conceptually related to the coherence resonance phenomenon in dynamical systems via the mechanism of threshold duality that is not limited to predator-prey cyclical interactions, but may apply to models of evolutionary game theory as well, thus indicating its applicability in several different fields of research.

Asynchronous snowdrift game with synergistic effect as a model of cooperation
Ádám Kun, Gergely Boza, István Scheuring (Budapest)

The Snowdrift (or Chicken) game emerges as a new paradigm in the study of non-kin cooperation in animals. Many situations, e.g. cooperative hunting, group foraging, territorial defence, predator watching or parental care, can be adequately described as a Snowdrift game. In this paper, we investigate the asynchronous version of the game in which, contrary to the rather unrealistic assumption of simultaneous moves, one of the players acts first, and the other responds by knowing its decision. Players are assigned to be first or second movers randomly and with the same probability. We found that both a synergistic effect of cooperation (i.e. cooperative effort is better than the sum of the individual efforts) and population structure (low dispersal, spatial confinement or group formation) are crucial for mutual cooperation to emerge. Otherwise, only one of the players will carry the burden of cooperation.

Stable Limit Cycles and Complex Behaviour in Smoothed Best Response Dynamics
Marius I. Ochea (Amsterdam)

A large part of the research on evolutionary game dynamics focused on identifying conditions, both in the class of dynamics and underlying games, for uniqueness of and convergence towards point-attractors such as Nash Equilibrium and Evolutionary Stable Strategy (ESS). In the realm of non-convergence literature an important result is Zeeman (1980) conjecture that there are no generic Hopf bifurcations in the case of three strategies games: When n = 3 all Hopf bifurcations are degenerate under Replicator Dynamics''. Hofbauer (1981) proves that starting with the 3-simplex stable limit cycles are possible under Replicator Dynamics. The main goal of this paper is to show that, even for ``simple'' three-strategy games, periodic attractors do occur under an alternative, rationalistic way of modelling evolution in games, namely the Logit Dynamics.

Sampling dynamics: an alternative to payoff-monotone selection dynamics
Rainer Berkemer (Lyngby)

Osborne and Rubinstein introduced sampling equilibria which are based on the concept of ``procedural rationality''. Sethi extended their idea to a dynamic framework which leads to the so called sampling dynamics. Unlike e.g the replicator dynamics this selection dynamics turns out to be neither payoff-monotone nor payoff-positive which has interesting consequences. This can be demonstrated by application to the travelers dilemma, a deliberately constructed social dilemma. The game has just one symmetric Nash equilibrium which is Pareto inefficient. Especially when the travelers have many options this result is rather counter intuitive and indeed there is experimental evidence which indicates that deviation will be likely even though the Nash equilibrium is strict. One can use the fact that strict Nash equilbria must be also sampling equilibria to test for the ``plausibility'' of the standard game theory result. Both, analytical tools and agent based simulation are used to investigate the dynamic stability of sampling equilibria in a generalized travelers dilemma. Two parameters are of interest: the number of strategy options (m) available to each traveler and an experience parameter (k), which indicates the number of samples an agent would evaluate before fixing his decision. The special case (k=1) can be treated analytically. The stationary points of the dynamics must be sampling equilibria and one can calculate that for m>3 there will be an interior solution in addition to the pure Nash equilibrium. Furthermore one can prove that this interior solution is asymptotically stable under the sampling dynamics while the strict Nash equilibrium is unstable (for m>3). This could not happen with any payoff-positive selection dynamic. Even more interesting is the dynamical behavior for k>1. For sufficiently large experience parameters one can observe limit cycles by means of agent based simulation. On the other hand, if k grows too large these limit cycles will be destroyed and all trajectories approach the Nash equilibrium. For any number of options one can can analytically derive a threshold k(m) such that above k(m) the Jacobean of the dynamical system, evaluated for the Nash equilibrium, can only have eigenvalues with negative real parts. One might well argue that for biological systems payoff-monotonicity of selection dynamics should be better preserved. For social systems, on the other hand, the sampling dynamics offers an interesting alternative which may help to explain deviations from Nash behavior in experiments.
[1] Osborne, M. J. and Rubinstein, A.: Games with Procedurally Rational Players, in: American Economic Review, 88(4), p.834-847, 1998
[2] Sethi, R.: Stability of Equilibria in Games with Procedurally Rational Players, in: Games and Economic Behavior, 32(1), p.85-104, 2000

Combined problems of cooperation and coordination
Hans-Ulrich Stark (Zurich)

In game theory, much attention has been paid to symmetrical 2-players games with binary decisions of the players. Within this frame, questions of social cooperation and social dilemmas have mostly been attached to investigations of the Prisoner's Dilemma (PD) with T > R > P > S and 2R > T + S. In this context, the readiness of individuals to resist the temptation to defect is studied in various settings. These investigations aim at explaining the origin and stability of cooperation among selfish individuals. But what if the readiness to resist temptation is not enough to reach a desired outcome? Maybe there are more than one desired solutions and the individuals additionally have to coordinate their actions to realize one of them. In this work, I focus on game theoretical conflicts that exhibit a combination of cooperation and coordination problems in the same game. Examples are (i) the Turn-Taking Dilemma (Neill, 2003) and (ii) the Route Choice Game (Helbing et al., 2005; Stark et al., 2007). The first one, (i), is similar to the above described PD, but the second inequality is reversed to T + S > 2R. The Pareto-inefficient equilibrium, and, thereby, the cooperation dilemma remains the same, but the system optimal solution (maximal cumulative payoff) is shifted to the off diagonal of the bimatrix. When considering an iterated game, this leads to a non-trivial, temporal coordination problem as flipping between the upper right and the lower left solutions of the bimatrix would lead to the only Pareto-efficient solution of the supergame. The latter point also holds for the Route Choice Game with T > P > S > R and T + S > 2P, that represents the problem of efficient usage of networks with capacity-restricted links (traffic networks, data-communication networks). Of course, investigations regarding the performance of systems with this underlying conflict yield completely different results than those with a PD game underlying. However, currently there is very little work done in this direction. In this contribution, I will present my current research on this topic as well as empirical results of previous work.
[1] Helbing, D.; Schönhof, M.; Stark, H.-U.; Holyst, J. A. (2005). How individuals learn to take turns: Emergence of alternating cooperation in a congestion game and the prisoner's dilemma. Adv. Complex Syst. 8, 87-116.
[2] Neill, D. B. (2003). Cooperation and coordination in the turn-taking dilemma. In: TARK. pp. 231-244.
[3] Stark, H.-U.; Helbing, D.; Schönhof, M.; Holyst, J. A. (2007). Alternating cooperation strategies in a route choice game: Theory, experiments, and effects of a learning scenario. In: A. Innocenti; P. Sbriglia (eds.), Games, Rationality, and Behaviour, Palgrave, MacMillan.

Group size and cooperation in n-player Prisoner's Dilemma threshold game
Gergely Boza, Balazs Könnyű and Sz. Számadó (Budapest)

There are many cooperative phenomena in Nature, where more than two players (n) interact with each other at the same time. A variant of the commonly used Public Goods Game dealing with social dilemmas with the participation of more than two players, is the n-player Prisoner's Dilemma threshold game. In this game a cooperating individual pays the cost of cooperation (c) but receives nothing if the number of cooperating individual is below a certain threshold. In the same situation a defector pays no cost, and receives no benefit. When there are more cooperating individuals in a group than the threshold, then a cooperator's payoff is P = r - c, while a defector receives a higher payoff, P = r. The r is the reward of cooperation, and its value is independent of the number of cooperators. It was shown analytically (1) that high levels of cooperation can be achieved in this system, unlike in the original Public Goods Game with variable investments. However, with the increase of the cost of cooperation an ESS bifurcation occurs. At intermediate levels of the cost of cooperation a hysteresis point exists, after which the cooperative effort evolves to zero. Here we investigated the effect of group size and threshold value on the outcome of an n-player threshold game. How does the frequency of cooperative act change with different group size and/or different threshold for cooperative reward? In our evolutionary model each individual had a trait x (0-1), which measures their tendency to act cooperatively. A value of 0 means that the individual always defects, and 1 means that it always cooperates. The average x of the population indicates the levels of cooperation. Players interacted with a given number of partners, and this defined the size of the group. We varied the threshold level of cooperators needed in a group for receiving the benefit. Our results show that the position of the hysteresis point strongly depends on the threshold value and on group size. The bigger the group, the smaller the final cost of cooperation where the hysteresis point appears. The lowest cost of cooperation where hysteresis appears is at intermediate threshold values. Thus, it is the hardest to preserve cooperation in large groups with intermediate threshold values. We also investigated the effect of different update rules. We found that the location of the instable fix points changes rather than the location of the stable fix points with different update rules.
Acknowledgements: G. B. would like to thank Ádám Kun for his helpful comments.
[1] Bach, L. A., Helvik, T., Christiansen, F. B., 2006. The evolution of n-player cooperation - threshold games and ESS bifurcations, Journal of Theoretical Biology, 238: 426-434.

Prebiotic evolution: A chemically more realistic version of the metabolic model
Balazs Könnyű, T. Czárán

The problem of prebiotic information integration is still one of the most difficult issues of all recent theories on the early evolution of replicators. After the seminal work of Eigen and Schuster [1] the problem has become well known as Eigen's paradox which roughly states: long replicators cannot be reliably copied without the catalytic help of long replicators. That is, the information necessary for coding a useful enzyme cannot be transmitted to the next generation without a useful enzyme. This leaves us with the chicken-or-egg puzzle: how did the first useful replicator enzyme appear on prebiotic Earth? Eigen and Schuster offered the hypercycle model as a possible solution to the problem, but later the hypercycle was shown to be vulnerable due to the effects of parasitic replicators [2]. In an earlier study [3] we analyzed the dynamics of metabolically coupled replicators on a surface by a cellular automaton model implementing a discrete-event reaction-diffusion system. In that model, replicators contribute to metabolism synergistically, described by a multiplicative function. Despite this coupling, the corresponding ordinary differential equation system leads to competitive exclusion by the faster replicating species, which in turn results in extinction of the whole system. To the contrary, coexistence of replicators with different per capita fitnesses occurs in the spatial, discrete system, due to an advantage of rarity of less vigorously growing replicators. Synergism means obligatory complementation, and a rare species is more likely to be complemented by synergistic partners in a finite neighbourhood. Increasing neighbourhood size leads to system collapse because thus the limiting case of the homogeneous, well-stirred reactor arises. In contrast, increasing diffusion rate does not result in competitive exclusion, because a so-called trait-group model rather than a homogeneous system is approached. Harmful parasites are selected against in the model [3, 4]. The interaction of metabolism and the replicators was implemented in a rudimentary manner without any chemical detail in this early model. Our present work elaborates on this part of the system: simple but specific metabolic reaction topologies are assumed, and also the diffusion of the metabolites is explicitly modelled. This allows the separation and independent tuning of time scales for metabolic reactions and replication. We show that introducing these details into the metabolic system does not change the conclusion in the qualitative sense: a considerable part of the parameter space admits replicator coexistence.
[1] Eigen, M. and P. Schuster, The hypercycle. 1979, Springer-Verlag: Berlin.
[2] Maynard Smith, J., Hypercycles and the origin of life. Nature, 1979. 20: p. 445-446.
[3] Czárán, T. and E. Szathmáry, Coexistence of replicators in prebiotic evolution, in The Geometry of Ecological Interactions, U. Dieckmann, R. Law, and J.A.J. Metz, Editors. 2000, Cambridge University Press: Cambridge.
[4] T. Czárán, Balazs Könnyű, and E. Szathmáry, Prebiotic replicase evolution in a metabolic system. (in prep.)

How public are public goods games?
Marta D. Santos (Lisboa), Francisco C. Santos (Bruxelles) and Jorge M. Pacheco (Lisboa)

Throughout their life, humans often engage in public goods games (PGG) in situations ranging from family related issues to global warming. In all cases, the tragedy of the commons threatens the possibility of reaching the optimal solution associated with global cooperation, a scenario predicted by theory and demonstrated by many experiments. Up to now individuals have been treated as equivalent in all respects, in sharp contrast with real life situations, where diversity is overwhelming. Here we investigate the impact of social diversity in the evolution of cooperation modeled as a PGG. Each group of M individuals may participate in an M-player PGG. We show how the diversity of PGG associated with different group sizes promotes cooperation. The enhancement of cooperation is particularly strong when social ties follow a scale-free distribution. Global cooperation is found to rely mainly on the fact that individuals engage in the PGG of others.

Topology control with IPD network creation games
Jan Scholz (Frankfurt) and Martin Greiner (Munich)

Network creation games couple a two-players game with the evolution of network structure. A vertex player may increase its own payoff with a change of strategy or with a modification of its edge-defined neighbourhood. By referring to the Iterated Prisoners Dilemma (IPD) game we show that this evolutionary dynamics converges to network-Nash equilibria, where no vertex is able to improve its payoff. The resulting network structure exhibits a strong dependence on the parameter of the payoff matrix. Degree distributions and cluster coefficients are also strongly affected by the specific interactions chosen for the neighbourhood exploration. This allows to see network creation games as a promising artificial-social-systems approach for a distributive topology control of complex networked systems [1].
[1] Jan C. Scholz and Martin O.W. Greiner. Topology control with IPD network creation games. New Journal of Physics, 9(6):185, 2007.

Anomalous fluctuations in Dawkins Battle of the Sexes
Jonas Cremer, Tobias Reichenbach and Erwin Frey (Munich)

We investigate finite-size fluctuations in Dawkins Battle of the Sexes. This famous game describes mating behavior of males and females, where males can be either philanderer or faithful, females are fast or coy. The dynamics is cyclic with a deterministic drift towards coexistence of all four strategies. We show that finite-size fluctuations unavoidably lead to extinction of two strategies in the population. However, the typical time until extinction occurs strongly prolongs with increasing system size. In the meantime, a quasi-stationary probability distribution forms that is nongaussian in the vicinity of the coexistence state.

Evolution of norms in a multi-level selection model of conflict and cooperation
Francisco C. Santos (Brussels), Jorge M. Pacheco and Fabio A. C. C. Chalub

We develop a multi-level selection model in the framework of indirect reciprocity. Using two levels of selection, one at the individual level and another at the community level, we propose a competitive scenario among social norms, in which all individuals in each community undergo pairwise interactions, whereas all communities also engage in pairwise conflicts, modeled by different games. Norms evolve as a result of inter-community conflicts whereas evolution inside each community promotes the selection of the best strategies compatible with each ruling social norm. Different types of inter- community conflict and intensities of selection are considered. The model leads to the emergence of a social norm, which we call stern- judging, and which turns out to be one of the recently obtained leading eight social norms. The result is robust to changes in the type of conflict between communities. We also compared the individual performance of stern-judging with that associated with several other popular norms, showing that stern-judging outperforms the other norms. Stern-judging is characterized by an unambiguous response to each individual behavior, where prompt forgiving coexists with implacable punishment.

Influence of Altruistic Behaviour on Game Evolution
Dorota Marciniak (Warsaw)

We consider the Iterated Prisoner's Dilemma game with additional two factors, which describe altruism and negotiation strength of participants. We construct a model which simulates an iterated game between such players. We give a solution of the derived equation, which is a differential equation, and conclude with examples of solutions of our model.