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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.

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... Further on, the intuition that spatial effects have a major impact on evolutionary dynamics has led to the exploration of spatial games (Levin and Paine, 1974;Hassell et al., 1991;Nowak and May, 1992;Nowak and May, 1993;Lindgren and Nordahl, 1994;Szabó and Tke, 1998;Hauert, 2006;Perc and Szolnoki, 2008). These include deterministic and stochastic cellular automata (Nowak and May, 1992;Nowak and May, 1993;Hiebeler, 1997), exceedingly complex spatial patterns (Hassell et al., 1991), open-ended space-time chaos (Lindgren and Nordahl, 1994), and reaction-diffusion models with Turing patterns (Cressman and Vickers, 1997;Wakano et al., 2009). ...

... Further on, the intuition that spatial effects have a major impact on evolutionary dynamics has led to the exploration of spatial games (Levin and Paine, 1974;Hassell et al., 1991;Nowak and May, 1992;Nowak and May, 1993;Lindgren and Nordahl, 1994;Szabó and Tke, 1998;Hauert, 2006;Perc and Szolnoki, 2008). These include deterministic and stochastic cellular automata (Nowak and May, 1992;Nowak and May, 1993;Hiebeler, 1997), exceedingly complex spatial patterns (Hassell et al., 1991), open-ended space-time chaos (Lindgren and Nordahl, 1994), and reaction-diffusion models with Turing patterns (Cressman and Vickers, 1997;Wakano et al., 2009). It has been shown that space can influence game dynamics in a variety of ways, e.g., depending on the underlying structure (Lieberman et al., 2005), whether interaction is stochastic or deterministic (Nowak et al., 1994), and whether space is discrete or continuous (Durrett and Levin, 1994). ...

... Several extensive studies based on Prisoners Dilemma and Public Goods games have confirmed that population structure is beneficial for the persistence of cooperation, as it allows cooperators to form clusters that are less likely to be exploited by defectors (Hauert, 2006;Hauert and Szabo, 2003;Killingback et al., 2006;Nowak and May, 1992;Nowak et al., 2004). Additional features of our model include mutation and death by age in order to enable open-endedness (Lindgren and Nordahl, 1994), stochastic interaction to avoid the trivial patterns of deterministic cellular automata (Nowak and May, 1993), and memory to allow for strategies such as tit-for-tat (Axelrod and Axelrod, 1984). ...

For nearly three decades, spatial games have produced a wealth of insights to the study of behavior and its relation to population structure. However, as different rules and factors are added or altered, the dynamics of spatial models often become increasingly complicated to interpret. To tackle this problem, we introduce persistent homology as a rigorous framework that can be used to both define and compute higher-order features of data in a manner which is invariant to parameter choices, robust to noise, and independent of human observation. Our work demonstrates its relevance for spatial games by showing how topological features of simulation data that persist over different spatial scales reflect the stability of strategies in 2D lattice games. To do so, we analyze the persistent homology of scenarios from two games: a Prisoner's Dilemma and a SIRS epidemic model. The experimental results show how the method accurately detects features that correspond to real aspects of the game dynamics. Unlike other tools that study dynamics of spatial systems, persistent homology can tell us something meaningful about population structure while remaining neutral about the underlying structure itself. Regardless of game complexity, since strategies either succeed or fail to conform to shapes of a certain topology there is much potential for the method to provide novel insights for a wide variety of spatially extended systems in biology, social science, and physics.

... Further on, the intuition that spatial effects have a major impact on evolutionary dynamics has led to the exploration of spatial games (Levin and Paine, 1974;Hassell et al., 1991;Nowak and May, 1992;Nowak and May, 1993;Lindgren and Nordahl, 1994;Szabó and Tke, 1998;Hauert, 2006;Perc and Szolnoki, 2008). These include deterministic and stochastic cellular automata (Nowak and May, 1992;Nowak and May, 1993;Hiebeler, 1997), exceedingly complex spatial patterns (Hassell et al., 1991), open-ended space-time chaos (Lindgren and Nordahl, 1994), and reaction-diffusion models with Turing patterns (Cressman and Vickers, 1997;Wakano et al., 2009). ...

... Further on, the intuition that spatial effects have a major impact on evolutionary dynamics has led to the exploration of spatial games (Levin and Paine, 1974;Hassell et al., 1991;Nowak and May, 1992;Nowak and May, 1993;Lindgren and Nordahl, 1994;Szabó and Tke, 1998;Hauert, 2006;Perc and Szolnoki, 2008). These include deterministic and stochastic cellular automata (Nowak and May, 1992;Nowak and May, 1993;Hiebeler, 1997), exceedingly complex spatial patterns (Hassell et al., 1991), open-ended space-time chaos (Lindgren and Nordahl, 1994), and reaction-diffusion models with Turing patterns (Cressman and Vickers, 1997;Wakano et al., 2009). It has been shown that space can influence game dynamics in a variety of ways, e.g., depending on the underlying structure (Lieberman et al., 2005), whether interaction is stochastic or deterministic (Nowak et al., 1994), and whether space is discrete or continuous (Durrett and Levin, 1994). ...

... Several extensive studies based on Prisoners Dilemma and Public Goods games have confirmed that population structure is beneficial for the persistence of cooperation, as it allows cooperators to form clusters that are less likely to be exploited by defectors (Hauert, 2006;Hauert and Szabo, 2003;Killingback et al., 2006;Nowak and May, 1992;Nowak et al., 2004). Additional features of our model include mutation and death by age in order to enable open-endedness (Lindgren and Nordahl, 1994), stochastic interaction to avoid the trivial patterns of deterministic cellular automata (Nowak and May, 1993), and memory to allow for strategies such as tit-for-tat (Axelrod and Axelrod, 1984). ...

For nearly three decades, spatial games have produced a wealth of insights to the study of behavior and its relation to population structure. However, as different rules and factors are added or altered, the dynamics of spatial models often become increasingly complicated to interpret. To tackle this problem, we introduce persistent homology as a rigorous framework that can be used to both define and compute higher-order features of data in a manner which is invariant to parameter choices, robust to noise, and independent of human observation. Our work demonstrates its relevance for spatial games by showing how topological features of simulation data that persist over different spatial scales reflect the stability of strategies in 2D lattice games. To do so, we analyze the persistent homology of scenarios from two games: a Prisoner's Dilemma and a SIRS epidemic model. The experimental results show how the method accurately detects features that correspond to real aspects of the game dynamics. Unlike other tools that study dynamics of spatial systems, persistent homology can tell us something meaningful about population structure while remaining neutral about the underlying structure itself. Regardless of game complexity, since strategies either succeed or fail to conform to shapes of a certain topology there is much potential for the method to provide novel insights for a wide variety of spatially extended systems in biology, social science, and physics.

... In the Letter, we construct an evolutionary game model which features both kinds of effects: purely local interactions with nearest neighbors, and a selfconsistent, mean-field-type coupling to the order parameter. We start with the classic game of Nowak and May (NM) [14,26], where the order parameter can be chosen as the mean density of cooperators f c in the steadystate regime of evolution. The mean-field term is the coupling to the instantaneous density of cooperators, f c (t), which fluctuates due to the game dynamics and has a well-defined mean value in the steady-state regime. ...

... The rules of the Nowak-May spatial game [14,26]. -Consider a population of L 2 players arranged at the vertices of a regular L×L square lattice. ...

... discrete structure of the payoffs in the NM game (1)-(2) leads to a very specific dependence of the game dynamics on the payoff parameter b. Comparing the payoffs of C and D in the neighborhood of an agent, one finds for the NM game [18,26] that the dynamics of the game changes at the values of b=m/n with 1 m, n 8. Here m and n are integers and are related to the numbers of C in a local neighborhood. ...

We introduce a mean-field term to an evolutionary spatial game model. Namely, we consider the game of Nowak and May, based on the Prisoner's dilemma, and augment the game rules by a self-consistent mean-field term. This way, an agent operates based on local information from its neighbors and non-local information via the mean-field coupling. We simulate the model and construct the steady-state phase diagram, which shows significant new features due to the mean-field term. The main effects are observed for stationary steady states, which are parametrically close to chaotic states: the mean-field coupling drives such stationary states into spatial chaos. Mean-field games demonstrate continuous change between steady-states contrary to the original Nowak and May game discontinuous changes.

... CA is a lattice dynamics model proposed by von Neumann, which is discrete in time, space, and states of the systems, with local spatial interaction and temporal causality [38,39,41]. For a more complete discussion of CA, the reader is referred to Toffoli and Margolus [51] and Wolfram [54,55]. ...

Supply chain disruptions are inevitable and may bring about fatal influence to the system. In this article, we examine how up and downstream firms embedded in a supply chain network engage in decision-making under disruption risk. The decision evolution of the supply chain network, treated as a complex adaptive system (CAS), is investigated from both the aspects of time dynamic and spatial feature by employing the Evolutionary Game Theory (EGT) and Cellular Automata (CA) approach, respectively. Supply chain network agents (i.e., up and downstream firms) are supposed to have two optional responses, positive strategy or negative strategy when facing the disruption risk. By observing a series of simulation experiments on the developed EGT model and CA model, we offer some testable propositions regarding the behavioral nature of the agents in the supply chain network. Some managerial insights in managing supply chain disruption risk are presented as well. We find that the spatial feature of the supply chain network plays a vital role in the evolution process which can lead the system to a more satisfactory equilibrium, compared with the evolution process only considering the time dynamic.

... However, these abundant mechanisms raise a further question: what are the most-basic conditions in which cooperation can emerge? A longstanding literature proposes one such condition: spatial / network relations among organisms can promote the evolution of cooperation [37][38][39][40][41][42][43][44][45] . When the benefit-cost ratio of cooperation exceeds the average number of spatial neighbors, simply existing in that space can result in cooperation evolving [46] . ...

Researchers have identified numerous mechanisms that make cooperation in the prisoner's dilemma possible, yet recent research has proposed what ranks among the most basic of mechanisms: the presence of time. When organisms in spatial models can interact at multiple points in time within a generation, cooperation can evolve in a wider range of settings than in spatial models in which interaction occurs at a single moment. Here we further explore this mechanism via an analytic model that studies the effect of time on cooperation when no spatial dimension is present. The model shows that the mere presence of two or more points in time at which social interaction can occur creates an opportunity for mutant cooperators to invade a well-mixed population of defectors playing the one-shot prisoner's dilemma under the replicator dynamics. These invasions lead to a nonequilbrium cycling of strategies in which cooperation consistently reemerges at alternating time points.

... The relationship between the evolutionary process of individual organisms and the development of their environments remains one of the foremost questions in the field of ecology and evolution. It is well established that spatial interactions play an important role in these processes, as has been shown by spatial evolutionary game theory [143,144]. Although most theories in evolutionary ecology either do not consider space at all or assume that populations face a static underlying environmental heterogeneity, spatial or social structure can emerge spontaneously as a consequence of interactions between individuals [145]. ...

Liquid crystalline materials form a whole array of interesting textures and phases. While there are many cholesteric and smectic phases, the discovery of new nematic phases is rare since the space of possible configurations for achiral molecules is small. Here, we look at the novel class of modulated nematic structures -- the twist-bend, splay-bend and splay-twist. While twist-bend nematic phases have been extensively studied, the experimental observation of two dimensional, oscillating splay-bend phases is recent. We consider two theoretical models that have been used to explain the formation of twist-bend phases—flexoelectricity and bond orientational order—as mechanisms to induce splay-bend phases. Flexoelectricity is a viable mechanism, and splay and bend flexoelectric couplings can lead to splay-bend phases with different modulations. We show that while bond orientational order circumvents the need for higher order terms in the free energy, the important role of nematic symmetry and phase chirality rules it out as a basic mechanism. The Hopf fibration has inspired any number of geometric structures in physical systems, in particular, in chiral liquid crystalline materials. Because the Hopf fibration lives on the three sphere, some method of projection or distortion must be employed to realize textures in flat space. Here, we explore the geodesic preserving gnomonic projection of the Hopf fibration, and show that this could be the basis for a new modulated nematic texture with only splay and twist. We outline the structure and show that it is defined by the tangent vectors along the straight line rulings on a series of hyperboloids. The phase is defined by a lack of bend deformations in the texture, and is reminiscent of the splay-bend and twist-bend nematic phases. We show that domains of this phase may be stabilized through anchoring and saddle-splay. The second part of this thesis is about ecology and evolution in heterogenous environments. Organisms in nature have to be competent at multiple tasks in order to survive and a given phenotype cannot usually be optimal at all tasks at the same time. Recent studies employ the concept of Pareto optimality from economics and engineering to capture this inherent trade-off. If we associate each task with a different environmental niche, Pareto optimality is a useful framework to capture phenotypic plasticity. We compare Pareto optimal fronts to the well known ecological concept of fitness sets, and show how the shape of Pareto fronts in trait space can be connected to the determination of the optimal strategy in a heterogenous environment. We consider both temporal and spatial heterogeneity.

... Absent other mechanisms, defectors' higher fitness leads its portion of the population to grow, leaving defectors without cooperators to exploit, thus reducing the population's fitness and producing a worse outcome than if all members of the population had cooperated. Nowak and May [4][5][6] revealed that playing this game on a grid and updating strategies based on neighborhood comparisons yields clusters of cooperators that persist in the population alongside defectors. Subsequent research explored how modeling subtleties influenced the success of cooperation in this framework 25,26 and it extended the framework's reach into the study of heterogeneous networks 25,27,28 , including the development of universal rules to characterize the conditions in which cooperation evolves in any spatial structure 29 . ...

We study a spatial, one-shot prisoner's dilemma (PD) model in which selection operates on both an organism's behavioral strategy (cooperate or defect) and its choice of when to implement that strategy across a set of discrete time slots. Cooperators evolve to fixation regularly in the model when we add time slots to lattices and small-world networks, and their portion of the population grows, albeit slowly, when organisms interact in a scale-free network. This selection for cooperators occurs across a wide variety of time slots and it does so even when a crucial condition for the evolution of cooperation on graphs is violated--namely, when the ratio of benefits to costs in the PD does not exceed the number of spatially-adjacent organisms.

... The broad assumptions of Martin Nowak and colleagues' models differed from those of "typical" kin selection models initially [in the 1990s; reviewed in 2004 (13)] and converged to them only over time. Unlike typical kin selection models, the simulations of Nowak and May (46,47) do not allow for any chance effects in reproduction (i.e., genetic drift). The models are therefore fully deterministic, meaning that altruism can never spread when rare, since a single altruistic mutant will be immediately eliminated as it would be surrounded on all sides by nonaltruists receiving higher payoffs. ...

Significance
The canonical explanation for the evolution of altruism (“kin selection”)—which was mathematically derived in the 1960s by W. D. Hamilton—emphasizes the importance of genetic relatedness. Over the past three decades, numerous authors claim to have discovered alternative explanations. We systematically analyze the models substantiating these claims and reveal that in every model the interacting individuals are genetically related and that the authors have therefore unwittingly rediscovered Hamilton’s insight.

... In the fields of network evolutionary game theory, various strategy updating rules are proposed, such as best-takes-over updating rule [29,30] , so-called best imitation; proportional updating rule [31] , updating rule adopting a linear function of payoff difference; and proportional-comparison updating rule with Fermi function [32][33][34][35] , etc. Generally speaking, the animals in nature tend to observe and imitate more successful one's behavior, which is prime manifestation of social learning. Therefore, best strategy updating rules will be employed in the present model, where individuals will copy the strategy gaining highest payoff in their communities. ...

Individuals often move to distance themselves from defectors, or to seek better chances for higher payoffs, for example moving from rural to urban areas. Regardless of the reason , however, moving frequently also means alienation, which in turn means bearing costs for seeking new opportunities. With this motivation, we study a prisoner's dilemma game, where individuals with defectors in their communities either move or update their strategy. We find that the alienation from defectors reinforces larger and more compact cooperative clusters. However, the number of cooperative clusters depends on the viscosity of the interaction network, where network reciprocity still works well. And it is the fine-tuned interplay between the mobility to alienate from defectors and a still functioning network reciprocity that works best in promoting cooperation. Our results suggest that a limited mobility of minorities could spare public resources in social dilemma situations more effectively than reward and punishment.

... Dilemma strength determines the nature and the outcome of dyadic games irrespective of whether these games are enriched with any of the five social-viscosity mechanisms 7 15 , and extends to closely related common-goods exploitation games 16 . It is furthermore possible to conceive (i) two-player games with additional action choices, e.g., reward 17 and punishment 17,18 , (ii) multiplayer games with multiple actions 19,20 , and (iii) games embedded into a spatial structure [21][22][23][24][25] . Complexity ultimately escalates when social dilemmas are modelled after major societal concerns encompassing corruption 26,26 , vaccine hesitancy [27][28][29] , traffic congestion [30][31][32] , and countless others [33][34][35] . ...

What do corruption, resource overexploitation, climate inaction, vaccine hesitancy, traffic congestion, and even cancer metastasis have in common? All these socioeconomic and sociobiological phenomena are known as social dilemmas because they embody in one form or another a fundamental conflict between immediate self-interest and long-term collective interest. A shortcut to the resolution of social dilemmas has thus far been reserved solely for highly stylised cases reducible to dyadic games (e.g., the Prisoner’s Dilemma), whose nature and outcome coalesce in the concept of dilemma strength. We show that a social efficiency deficit, measuring an actor’s potential gain in utility or fitness by switching from an evolutionary equilibrium to a social optimum, generalises dilemma strength irrespective of the underlying social dilemma’s complexity. We progressively build from the simplicity of dyadic games for which the social efficiency deficit and dilemma strength are mathematical duals, to the complexity of carcinogenesis and a vaccination dilemma for which only the social efficiency deficit is numerically calculable. The results send a clear message to policymakers to enact measures that increase the social efficiency deficit until the strain between what is and what could be incentivises society to switch to a more desirable state.

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