in an influential paper , robert axelrod addressed the question  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ if people tend to become more alike in their beliefs , attitudes , and behavior when they interact , why do not all differences eventually disappear ? ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  to investigate this problem , axelrod introduced an agent - based model to explore mechanisms of competition between the tendency towards globalization and the persistence of cultural diversity .culture in this model is defined as a set of individual attributes subject to social influence .the state of an agent is described by a set of cultural features , each with possible values or traits .the parameter gives a measure of initial culture diversity for a random distribution of trait values .the local interaction between neighboring agents follows two basic social principles .the probability of interaction is determined by a principle of homophily : it is proportional to the cultural overlap between the agents , that is to the amount of cultural similarities ( number of features ) that they share .the result of the interaction is determined by a principle of social influence , so that similarity is enhanced when interaction occurs . in a typical dynamical evolution ,the system freezes in a multicultural state with coexisting spatial domains of different cultures , illustrating how a simple mechanism of local convergence can lead to global polarization .later systematic studies of axelrod s model have identified a globalization - polarization transition depending on the value of for a fixed  .there is a threshold value such that for globalization ( cultural homogeneity ) occurs , while for a persistent multicultural state ( polarization or cultural diversity ) is reached .several extensions of this model have been investigated , some of them already suggested in axelrod s paper .for example , it has been shown that frozen polarized configurations are not robust against cultural drift modelled as random perturbations .other extensions include the consideration of quantitative instead of qualitative values for the cultural traits , the extension of the model to continuous values of the cultural traits and the inclusion of heterophobic interactions , the simulation of technology assimilation , the consideration of specific historical contexts , or the effect of a fixed external cultural influence .a number of works have addressed the issue of the role of the social network of interactions .the existence of long range links of interaction in a social network with a small world structure naturally promotes cultural globalization .however , a study of the increase in the range of interaction suggests that an increase in communication promotes the emergence of a global culture , but it can also function to maintain areas of cultural uniqueness .a related question is the co - evolution of the social network of interaction and the dynamics of cultural changes , so that possible social interaction are not fixed from the outset . within this general context of the studies of the effects of different forms of social interactions ,shibanai et al. considered the process of information feedback into the social system .such feedback mechanism is one of the functions of mass media . from a general perspective, this question can be addressed as the competition of local agent interactions with mechanisms of global coupling .shibanai et al . concluded that  _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ mass media , contrary to lay beliefs of their strong uniforming power , would rather contribute to creating differences in the long run ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  this result is certainly surprising since mass media are believed to be powerful instruments to influence people s attitudes and opinions to homogenize society. conclusions in reference are based on the analysis for a single value of the parameter and few values of the intensity of the influence of mass media . in this paperwe consider the full range of values of when addressing the general question of the effects of different forms of information feedback on cultural dynamics in the framework of axelrod s model , including the study of local feedback .we consider information feedback mechanisms as different types of mass media influences acting on a social system .this extension of axelrod s model was referred to as public education and broadcasting " .our aim is to explore mechanisms , and their efficiency , by which mass media modifies processes of cultural dynamics based on local agent interactions .we investigate two main mechanisms that differ on how the information feedback affects the agents : ( i ) direct mass media influence , and ( ii ) indirect mass media influence . in the first case , mass media acts as an additional element in the network of interactions of each agent . as applications of direct mass media influence we study models of global and local mass media and compare their effects on the system . in the second feedback mechanism , mass media acts as a moderator or filter of the influence of the existing network of interactions of each agent .the assumption of this second model is that mass media have indirect influential power which reinforces the effect of personal networks .in particular , we focus on a model of indirect influence of global mass media.  we deal with states of the agents and mass media influences described by vectors whose components can take discrete values .we consider mass media influences that originate endogenously but can act either directly or indirectly on the system . in the first case ,the agent - agent interaction and the interaction of the agents with the mass media are two independent processes and both are based on the same homophily and social influence principles of axelrod s model .this scheme constitutes a model for a social system interacting with global or local mass media that represent endogenous cultural influences or plurality information feedback at different levels . in the case of indirect action ,agent - agent interactions is not independent of the state of global mass media , the interaction being reinforced when the states of the agent and mass media message are connected .our results show that the conclusions of shibanai et al . are not generally valid , but that different effects are obtained for different values of the parameters of the system .we find that cultural diversity builds - up by increasing the probability of interaction of the agents with the mass media , in agreement with shibanai et al .this occurs independently of the form of action ( either direct or indirect ) of the mass media message .however , a new finding that follows from our simulations is that cultural diversity is only promoted by strong enough mass media messages , while weak mass media messages , in combination with agent - agent interaction , are efficient in producing cultural homogenization .moreover , this homogenizing effect in the way to globalization turns out to be much more efficiently implemented by direct local mass media messages , as for example local or regional tv ( narrowcast ) than by direct global mass media messages as worldwide tv channels ( broadcast ) , or by an indirect influence of global mass media as in a filtering process of local interactions .the model of direct mass media influence , including the description of the local and global mass media , is presented in section ii .this section shows the effects of the direct action of these two types of mass media .we describe different effects occurring in the culturally homogeneous state of the system ( ) as well as in the multicultural state ( ) .section iii describes the model of indirect global mass media influence and the effects of this mechanism in the collective states of the agents .section iv contains the conclusions of our work , including a comparison of the effects of the different models of information feedback that we have considered .we consider a system of agents at the nodes of a two - dimensional regular lattice .the state of agent is defined by a vector of components . in axelrod s model ,the components of correspond to the cultural features ( language , religion , etc . ) describing the -dimensional culture of agent .each component can take any of the values in the set ( called cultural traits in axelrod s model ) . as an initial condition ,each agent is randomly and independently assigned one of the possible state vectors with uniform probability .we introduce a vector representing the mass media message , with components .we distinguish two types of direct mass media influences , schematically shown in fig . 1.    \(i ) the _ global mass media _ is spatially uniform and may vary in time .each is assigned the most abundant value exhibited by the -th component of the state vectors of al the agents , which we denote by .if the maximally abundant value is not unique , one of the possibilities is chosen at random with equal probability . in this type of influencethe same global information is feedback to each agent at any given time .the components of may change as the system evolves . in the context of cultural models , this represents a global mass media influence shared identically by all the agents and containing the most predominant trait in each cultural feature present in a society ( a global cultural trend " ) .\(ii ) the _ local mass media _ , is spatially non - uniform and non - constant on time . herethe component is assigned the most frequent value of the -th component of the state vectors of the agents belonging to the von neumann neighborhood of agent . if there are two or more maximally abundant values of component one of these is chosen at random with equal probability .this type of influence can be interpreted as a local mass media conveying the local cultural trend " of its neighborhood to each agent in a social system .systems subject to either local or global mass media describe social systems with endogenous cultural influences .cultural influences generated endogenously represent a plurality information feedback , which is one of the functions of mass media , but this can occur at a global ( broadcast `` ) or at a local ( narrowcast '' ) level .the strength of the coupling to the mass media is controlled by a parameter $ ] that measures the probability of interaction of the agents with the mass media message .we shall assume that is uniform , i.e. , the mass media influence reaches all the agents with the same probability .the parameter can be interpreted as the probability that the mass media message attracts the attention of the agents in the social system .the parameter takes into account factors of the mass media influence that can be varied , such as its amplitude , frequency , attractiveness , etc . atany given time , we assume that any agent can either interact with the mass media message or with other agents in the system .each agent in the network possesses a probability of interacting with the message and a probability of interacting with its neighbors .formally , we treat the mass media acting on each agent as an additional neighbor of with whom an interaction is possible . the mass media message is represented as an additional agent such that in the following definition of the dynamics .the dynamical evolution proceeds iterating the following steps : + ( 1 ) select at random an agent on the lattice ( called active agent ) .\(2 ) select the source of interaction . with probability set as an interaction with the mass media vector . otherwise , choose agent at random among the four nearest neighbors of on the network .\(3 ) calculate the cultural overlap ( number of shared features ) .if , sites and interact with probability . in case of interaction, choose randomly such that and set . herewe use the definition of the kroneckr s delta function . + ( 4 ) update the mass media vector if required .resume at ( 1 ) .+ the role of mass media can be seen as an additional neighbor of each agent with whom an interaction can take place .fifth " neighbor competes with the other four neighbors to influence an active agent . is the probability of capturing the attention of any agent to interact with either the global or the local mass media message .the effect of that interaction on each agent depends on the specific cultural overlap between the agent and the message .the total probability to interact with the mass media is , where , and is the overlap between agent and the message . on the other hand ,the active agent has a probability to interact with a randomly chosen neighbor .therefore , the parameter weights the influence of the mass media with respect to the influence of the neighbors on an agent .this dynamics is based on the same principles of axelrod s model , namely homophily or similarity and social influence .but now the agents interact either between them or with m. the competition of these interactions in measured by .  to characterize the collective final state reached by the dynamics, we calculate the average fraction of cultural domains .a cultural domain is a set of contiguous sites with identical cultural traits . is the number of cultural domains formed in the final state of the dynamics for a given realization of initial conditions .a culturally homogeneous state is characterized by values .when the system settles into a culturally diverse state we have .our numerical results in this work are based on averages over realizations for systems of size , and .figure 2 shows the average fraction of cultural domains as a function of time under the direct action of global mass media , for two values of the number of traits with , and for different values of the probability .fig . 2 ( left ) : for small values of the system reaches a culturally homogeneous state in absence of mass media influence ( ) and also for small values of . when the probability increases , we see that global mass media is able to induce cultural diversity in the system .this result agrees with the results obtained by shibanai et al . about the ability of global mass media to induce cultural diversity in the system ( ). however we observe in fig . 2 ( right ) that , for larger values of , where the system would be in a culturally polarized state with no mass media influence ( ) , small values of the probability can reduce the number of cultural groups .effects similar to those shown in fig .2 are observed when a local mass media message interacts with the system .we therefore conclude that mass media can increase or decrease cultural diversity depending on the value of the parameter .   in a system subject to a global mass media message for different values of the probability , with fixed .time is measured in number of events per site .system size . left : ; ( crosses ) ; ( squares ) ; ( diamonds ) ; ( circles ) .right : ; ( crosses ) ; ( squares ) ; ( circles ) ; ( diamonds).,title="fig : " ]  in a system subject to a global mass media message for different values of the probability , with fixed .time is measured in number of events per site .system size . left : ; ( crosses ) ; ( squares ) ; ( diamonds ) ; ( circles ) .right : ; ( crosses ) ; ( squares ) ; ( circles ) ; ( diamonds).,title="fig : " ]  a global picture of direct mass media influences is provided by figure 3 , which shows the asymptotic value of for long time as a function of , with , for different values of the probability . in absence of any mass media influences ( )there is a threshold value of the number traits , such that for dynamical evolution always leads to one of the possible homogeneous states . on the other hand , for values of , the behavior of the system changes and it settles into a multicultural state when . when the probability is increased , the threshold value of decreases .there is a value below which the system still reaches in its evolution a homogeneous cultural state ( ) under the action of any of these mass media messages .   as a function of , for different values of the probability and for different types of mass media influences . ( diamonds ) ; ( solid squares , direct global mass media ) ; ( empty squares , direct global mass media ) ; ( solid circles , direct local mass media ) ; ( empty circles , direct local mass media ) . ]a color code can be assigned to each cultural state for visualization of the spatial patters during simulations , as shown in figure 10 in the appendix .figure 4 shows the spatial configurations of the final states of the system subject to a global mass media message , when . in the absence of mass media influence ,i.e. , the system settles into any of the possible homogeneous cultural states . when the probability is increased , the system is driven towards a homogeneous cultural state that depends on the evolution of the global mass media from the initial conditions .thus , for small values of , global mass media contributes to maintain a globalized state in the system , as one may expect .however , there is a value of the strength above which there is no convergence to a homogeneous state , but a state of cultural diversity emerges in the system .a similar behavior is observed when the system is under the influence of local mass media .these results describe for a transition at a threshold value of from a culturally homogeneous state to a state of cultural diversity characterized by an increasing number of cultural domains as is increased .thus , we find the counterintuitive result that for , above some threshold value of the probability of interaction , mass media induces cultural diversity in a situation in which the system would be culturally homogeneous under the effect alone of local interactions among the agents .   for a direct global mass media influence , for , , and .top left : ; top right : ; bottom left : ; bottom right : .,scaledwidth=50.0% ]  figure 5 shows as a function of .the threshold value for each type of media decreases with increasing for . in each case , the threshold curve versus in fig .4 separates the region of cultural diversity from the region where homogeneous , monocultural states occur on the space of parameters .figure 5 can also be understood as a threshold curve of versus .for each value of there is a critical value , such that for , a homogeneous state in reached by the system while for multiculturality sets in . for values of above this curve, the interaction with the mass media dominates over the local interactions among the individual agents in the system .consequently , agents whose states exhibit a greater overlap with the state of the mass media have more probability to converge to that state .this process contributes to the differentiation of states between neighboring agents and to the formation of multiple cultural domains in the system for large enough values of the probability .   vs. for corresponding to the global and mass media .each line separates the region of cultural diversity ( above the line , in grey ) from the region of a global culture ( below the line ) for direct global ( circles ) and local ( triangles ) mass media influences.,scaledwidth=50.0% ]  when there are no mass media influences ( ) , the system always freezes into culturally polarized states for .figure 3 shows that the effect of mass media for depends on the magnitude of .for the two types of mass media messages that we consider , small values of produce a drop in to values below the reference line corresponding to its value when .thus , the limit does not recover the behavior of the model with only local agent - agent interactions .the fact that for small values of the interaction with mass media promotes cultural homogeneity is related to the non - stable nature of the inhomogeneous states in axelrod s model .when the probability of interaction is very small , the action of mass media can be seen as a sufficient perturbation that allows the system to escape from polarized states with frozen dynamics .the role of mass media in this situation is similar to that of cultural drift .the drop in the asymptotic values of for small values of from the reference value ( ) that is observed for the local mass media in fig .3 is much more pronounced than the corresponding drops for global mass media .this can be understood in terms of a greater efficiency of a local mass media influence as a nonuniform perturbation that allows the system to escape from a frozen inhomogeneous configuration .increasing the value of results , in both types of mass media , in an enhancement of cultural polarization in the system , but the local mass media always keeps the amount of cultural diversity , as measured by , below the value obtained for .information feedback at the regional level is more efficient in promoting cultural homogenization .figure 6 shows configurations reached under the influence of the direct local and global mass media in the multicultural region . a smaller number of cultural domains ( smaller ) that for are observed for global mass media field . but this number in much smaller under the action of local mass media field .in addition , the time to reach the final state is much longer for a local than for a global field ( see movie 1 ) .local mass media does not lead to an early frozen state , leaving room for agent - agent interactions that result in a culturally homogenized state .   for different mass media influences in the multicultural region , for , , and .left : no mass media ; center : global mass media with ; right : local mass media with .,scaledwidth=60.0% ]  we have focused on the effects of information feedback when both the probability of interaction with mass media b and the number of options q are varied , since they give rise to different qualitative behavior .variations of other parameters , such as the number of neighbours or the system size , have little effect .in particular , the system size may become relevant only in the limit of very small values of b for , when the role of mass media is similar to that of noise added to the system .in this section we analyze a model of global information feedback where the global mass media acts as a moderator or filter of the local influence of neighbors , as proposed by shibanai et al . . in the original axelrod s model one feature with different traits for two neighboring agentsis chosen , and the trait of the active agent is changed to that of the neighbor .this is modified in the model of indirect global mass media influence analyzed here , taking into account the agreement of the chosen trait of the neighbor and that of the global mass media or _ the plurality _ of the population .if the trait of the neighbor is concordant with the dominant one , that is , the same as that of the global mass media message , the feature of the active agent will be changed to that of the neighbor .but if the feature of the neighbor is different from that of the global mass media message , then , with probability the active agent will not change .thus , this model assumes that agents are more likely to adopt a trait from those neighbors that are concordant with the plurality .we use the definition of a uniform global mass media as in section ii , .the dynamical evolution of the filter model can be described in terms of the following iterative steps : + ( 1 ) select at random an agent on the lattice ( active agent ) .\(2 ) select at random one agent among the four neighbors of .\(3 ) calculate the overlap .if , sites and interact with probability . in case of interaction, choose randomly such that . if , then set ; otherwise with probability the state of agent does not change and with probability set   \(4 ) update the global mass media vector if required .resume at ( 1 ) .+ figure 7 shows a diagram of the filter model .the parameter describes the intensity of the filtering effect of the global mass media on the local interactions .the case corresponds to the original axelrod s model , while implies that cultural interaction only causes a change if the chosen trait of the neighbor was equal to that of the global mass media .the overall probability of interaction between an active agent and a chosen neighbor is if the chosen trait of is different from that corresponding to , and if the chosen trait is equal to that corresponding to .figure 8 shows the average fraction of cultural domains as a function of time in the global mass media filter model , for two values of with , and for different values of the filtering probability . in fig .8 ( left ) , when the system reaches a homogeneous state for and also for small values of .however , when the probability increases , the filtering influence of the global mass media can induce cultural diversity .our results for support the results obtained by shibanai et al . about the ability of the filtering process to induce cultural diversity in the same fashion as the model with direct global mass media influence .but comparison with fig .2 ( left ) shows that direct interaction with global mass media is more efficient in promoting cultural diversity than the filtering mechanism of agreement with the global plurality .the analysis of reference was restricted to a single value of .we have also explored values of , where the system would be in a heterogeneous cultural state in absence of any filtering ( ) . for these values of find ( fig .8 , right ) that the filtering mechanism has no appreciable effects for small , in contrast with the case of direct global mass media influence where for small values of the probability of interaction with the media message , the number of cultural groups is reduced as a consequence of this interaction .   in the filter model for different values of the probability , with fixed .time is measured in number of events per site .system size . left : ; ( crosses ) ; ( squares ) ; ( diamonds ) ; ( circles ) .right : ; ( crosses ) ; ( squares ) ; ( circles ) ; ( diamonds).,scaledwidth=75.0% ]  a systematic analysis of the filtering effect for different values of is summarized in figure 9 which shows the asymptotic value for long times of the average fraction of cultural domains as a function of , with , for different values of the filtering probability .when no filtering acts on the system ( ) the behavior is that of the original axelrod s model and also coincides with the direct mass media models for .the effects of the filtering process in the culturally homogeneous region , i.e. , for parameter values , is similar to that of a direct influence of endogenous mass media . when the probability is increased , the threshold value of decreases .there is a value below which the system still reaches a homogeneous cultural state under the influence of the filter .an increase in for parameters leads to cultural diversity .thus , both mechanisms of feedback information , either direct or indirect , promote multiculturality in the region of parameters where globalization prevails in the absence of any feedback .the similar behavior found for the three types of mass media influences considered here suggests that the phenomenon of mass media - induced diversity should be robust in this region of parameters , regardless of the type of feedback mechanism at work .however , in the region of parameters where multiculturality occurs for or , the behavior of the filter model differs from those of the direct mass media influence .the filtering mechanism has little effect for values of the probability . as there is a small decrease in the number of cultural groups formed in the system .but at a discontinuity appears : the fraction of cultural groups jumps from a value close to the one for to a value close to corresponding to maximum multiculturality ( number of cultural groups equal to the number of agents in the system ) .the case corresponds to an extreme restriction on the dynamics , when no adoption of cultural features from neighbors is allowed unless the state of the neighbor coincides with the one of the global mass media .since we are considering random initial conditions , when is large enough , the probability that the features of any agent coincide with those of the global mass media message is quite small , making the convergence to globalization , i.e. , a common state with the media , very unlikely . as a consequence, the random multicultural state subsists in the system and manifests itself as a maximum value of .the small probability of interaction with the global mass media for large values of when is also reflected in the very long convergence time needed to reach the final multicultural state as compared with the convergence time for .we note that the case in the model of direct global mass media influence is less restrictive than the condition in the filter model .although local agent - agent interactions produce negligible effects in both cases , in the model of direct influence , an agent can still interact with the global mass media when there is some cultural overlap between the agent and the mass media message .   as a function of , for different values of the probability for the filter model . ( circles ) ; ( squares ) ; ( triangles down ) ; ( diamonds ) ; ( triangles up ) ; ( stars ) ; ( plus signs),scaledwidth=60.0% ]we have studied the effects of different forms of information feedback on a social system in the framework of axelrod s model for cultural dissemination , including a model of local mass media influence .two basic mechanisms of information feedback have been interpreted as models of direct or indirect mass media influence on the system . for the direct information feedback mechanism we have considered the cases of global mass media and local mass media influences .the dynamics of direct interaction with the mass media is based on the similarity or overlap between vector states . although their origin is different , at the local level the direct global and local mass media influences act in the same manner , as a fifth " effective neighbor whose specific source is not essential .consequently they produce similar effects in the system .first , we find the nontrivial result that direct mass media can enhance cultural diversity , in agreement with the results of : for parameter values for which the system reaches a global culture due to the local interaction among the agents , there is a threshold value of the probability of interaction with mass media. for values of above this threshold cultural diversity emerges .this happens because there is a competition between the consequences of the similarity rule applied to the agent - agent interactions , and applied to the agent - mass media interaction .this competition leads to the formation of cultural domains and to polarized states .secondly , we find another surprising effect at variance with the conclusions of : for parameter values for which the dynamics based on the local interaction among the agents produces a frozen multicultural configuration , very weak directly interacting mass media produces cultural homogeneity .this range of parameters was not explored in ref . , and we find here that mass media interaction for produces the opposite effect of the one described before .that is , it favors cultural homogeneization.the limit is discontinuous and the homogenizing effect for occurs because the interaction with the mass media acts as a perturbation on the non stable multicultural configurations with frozen dynamics that appear for . as a general result for any value of , we find that for small values of , the interaction with mass media promotes cultural homogenization : for this interaction preserves homogeneity , while for it causes a drop in the degree of cultural diversity in the system as measured by the average fraction of cultural domain .we have calculated , for the two types of direct mass media influences considered , the corresponding boundary in the space of parameters that separates globalization and polarization states .the spatially nonuniform local mass media has a greater ordering effect than the uniform global mass media in the regime .the range of values of for which globalization occurs for is also larger for the nonuniform local mass media .the effect of the indirect global mass media influence as a filtering process in the culturally homogeneous region , i.e. , for , is similar to that caused by a direct influence of mass media . for small values of the filtering probability the system reaches a culturally homogeneous state . for values of greater than a threshold valuethe system converges to a state of cultural diversity .thus , both mechanisms of feedback information , either direct or indirect , promote multiculturality in a region of parameters where it would not be present in the absence of any feedback . in the region of parameters where multiculturality occurs for either or ,the filtering mechanism has , for values of the probability , a very weak effect in comparison to the one caused by a direct mass media influences : there is only a small decrease in the number of cultural groups formed .however , when the extreme restriction is imposed , the number of cultural groups jumps discontinuously to a value corresponding to maximum multiculturality .the similarity of the behaviors of the different types of mass media influences considered here suggests that the phenomenon of mass media - induced diversity should be robust , regardless of the type of feedback mechanism . in spite of the differences mentioned between direct and indirect information feedback processes as well as between uniform and nonuniform mass media influences ,it is remarkable that the collective behavior of the agents displays analogous phenomenology for the three types of mass media considered .  generally speaking , our analysis unveils the delicate compromise between direct agent - agent interactions and feedback processes .mass media reflects local or global cultural trends created by local agent - agent interaction , but mass media information is processed by agent interactions , while the agent - mass media interaction is conditioned by the overlap of the cultural features of the agent and the mass media message .we have analyzed the effect of different forms of mass media for the full range of the parameter that measure an initial cultural diversity .our results indicate qualitatively different effects when globalization or polarization would prevail when no mass media feedback is taken in account . in summary , we find that , when the probability of interacting with the mass media is sufficiently large , mass media actually contribute to cultural diversity in a social system , independently of the nature of the media . butdirect mass media influences are found to be efficient in promoting cultural homogeneity in conditions of weak broadcast of a message , so that local interactions among individuals can be still effective in constructing some cultural overlap with the mass media message .strong media messages do not lead to cultural homogenization because agent - agent interaction becomes inefficient .these results identify the power of being subtle in mass media massages .in addition , direct local mass media appear to be more effective in promoting uniformity in comparison to direct global broadcasts , which identifies the importance of local media ( feedback at regional levels ) in the cultural globalization path .color code for visualization of patterns .a different color can be assigned to each cultural state .cultural domains correspond to sets of elements that share the same color .figure 10 illustrates the case for and . in general for , weights the basic colours ( r , g , b ) : .   and there are different states that can be assigned different color as shown.,scaledwidth=60.0% ]  [ cols= " < , < " , ]j. c. g - a . , v. m. e. and m. sm acknowledge financial support from mec ( spain ) through project conoce2 ( fis2004 - 00953 ) .m. g. c. acknowledges support from fonacit ( venezuela ) under grant f-2002000426 .k k. k. acknowledges support from dfg bioinformatics initiative biz-6/1 - 2 and from deutscher akademischer austausch dienst ( daad ) .99 r. axelrod , j. of conflict resolution * 41 * , 203 ( 1997 ) . c. castellano , m. marsili , and a. vespignani , phys .. lett . * 85 * , 3536 ( 2000 ) .k. klemm , v. m. eguluz , r. toral , and m. san miguel , phys .e * 67 * , 045101(r ) ( 2003 ) .m. san miguel , v. m. egu ' iluz , r. toral , and k. klemm , computing in science & engineering * 7 * , 67 ( 2005 ) .k. klemm , v. eguluz , r. toral , and m. san miguel , physica a , * 327 * , 1 ( 2003 ). k. klemm , v. eguluz , r. toral , and m. san miguel , j. econ .control , * 29 * , 321 ( 2005 ) . a. flache and m. macy , los alamos arxiv : physics/0604201 ( 2006 ) .macy , j. kitts , a. flace and s. bernard , in dynamic social network modeling and analysis . eds .r. breiger et al . , national academies press ( washington , 2003 ) .l. leydesdorff ( 2001 ) , journal of artificial societies and social simulation * 4 * , no .3 , http://www.soc.surrey.ac.uk/jasss/4/3/5.html . r. bhavnani ( 2003 ) , journal of artificial societies and social simulation * 6 * , no .4 , http://jasss.soc.surrey.ac.uk/6/4/1.html .j. c. gonzlez - avella , m. g. cosenza , and k. tucci , phys .e * 72 * , 065102(r ) ( 2005 ) .k. klemm , v. m. eguluz , r. toral , and m. san miguel , phys .e * 67 * , 026120 ( 2003 ) .j. greig , j. of conflict resolution * 46 * , 225 ( 2002 ) .d. centolla , j. c. gonzlez - avella , v. m. eguluz and m. san miguel , arxiv : physics/0609213 .y. shibanai , s. yasuno , and i. ishiguro , j. conflict resolution * 45 * , 80 ( 2001 ) .j. c. gonzlez - avella , v. m. eguluz , m. g. cosenza , k. klemm , j.l .herrera and m. san miguel , phys .e * 73 * , 046119 ( 2006 ) .

Abstract: we study the effects of different forms of information feedback associated with mass media on an agent - agent based model of the dynamics of cultural dissemination . in addition to some processes previously considered , we also examine a model of local mass media influence in cultural dynamics . two mechanisms of information feedback are investigated : ( i ) direct mass media influence , where local or global mass media act as an additional element in the network of interactions of each agent , and ( ii ) indirect mass media influence , where global media acts as a filter of the influence of the existing network of interactions of each agent . our results generalize previous findings showing that cultural diversity builds - up by increasing the strength of the mass media influence . we find that this occurs independently of the mechanisms of action ( direct or indirect ) of the mass media message . however , through an analysis of the full range of parameters measuring cultural diversity , we establish that the enhancement of cultural diversity produced by interaction with mass media only occurs for strong enough mass media messages . in comparison with previous studies a main different result is that weak mass media messages , in combination with agent - agent interaction , are efficient in producing cultural homogeneity . moreover , the homogenizing effect of weak mass media messages are more efficient for direct local mass media messages than for global mass media messages or indirect global mass media influences .


one test recently proposed in the literature to discriminate between different homogeneous and inhomogeneous models is the redhshift - drift .the redshift - drift is the change in the redshift of a given source observed at different proper times by a comoving observer in an expanding universe [ see fig . 1 ] .   of a source , initially at a redshift on the past light cone of an observer at , as measured by the same observer at after an elapsed time of the observer s proper time.,width=302 ]  sandage and then mcvittie had first calculated its expression in the friedmannian framework . in friedmann - lematre - robertson - walker ( flrw ) models , when the universe expansion decelerates , the redshift decreases with time , hence negative redshift - drift and when the expansion accelerates , like in the model , sources with redshifts exhibit a positive redshift - drift .the redshift - drift has been recently calculated for inhomogeneous spherically symmetric ltb models .we have generalized this calculation to the model proposed in and derived the equation for the redshift - drift in the axially symmetric qss model .we derive the following equation for the redshift - drift in the axially symmetric qss model : we solve eqn .( [ eq19 ] ) numerically to get the redshift - drift ( ) .  ) as a function of the redshift for the axially symmetric qss swiss - cheese model , the model , the cgbh ltb void model ( courtesy ) , the alnes et al .ltb void model ( courtesy : ) and the yoo ltb hump model ( courtesy : ) ]  in fig .[ drift_compfig ] , we display the redshift - drift for the szekeres model , the model , and for three ltb models : alnes et al.s void model and the constrained gbh ( cgbh ) void model , both studied in , and yoo s hump model , studied in .  in all three ltb models the observer is located at the center of the void . the drift for all three ltb as well as for the qss model is negative and its magnitude increases monotonically with the redshift .the magnitude of the drift in the qss model is higher by a factor of about two , at a given redshift , than that in the ltb models ( at those redshifts where the ltb curves show a decline with increasing ) .the drift in the model is positive up to , while it is negative in qss model . at redshift ,its magnitude is much higher than that in the model by a factor of ,

Abstract: exact inhomogeneous solutions of einstein s equations have been used in the literature to build models reproducing the cosmological data without dark energy . however , owing to the degrees of freedom pertaining to these models , it is necessary to get rid of the degeneracy often exhibited by the problem of distinguishing between them and accelerating universe models . we give an overview of redshift drift in inhomogeneous cosmologies , and explain how it serves to this purpose . one class of models which fits the data is the szekeres swiss - cheese class where non - spherically symmetric voids exhibit a typical size of about 400 mpc . we present our calculation of the redshift drift in this model , and compare it with the results obtained by other authors for alternate scenarios .


evolutionary game theory has been used extensively to study the evolution of cooperation in social dilemmas .a social dilemma is typically modeled as a game with two strategies , cooperate ( ) and defect ( ) , whose payoffs for pairwise interactions are defined by a matrix of the form     . for a focal player using a strategy on the left - hand side of this matrix against an opponent using a strategy on the top of the matrix , the first ( resp .second ) coordinate of the corresponding entry of this matrix is the payoff to the focal player ( resp .opponent ) .that is , a cooperator receives when facing another cooperator and when facing a defector ; a defector receives when facing a cooperator and when facing another defector . since the same argument applies to the opponent ,the game defined by ( [ generic2by2 ] ) is _symmetric_. if defection pays more than cooperation when the opponent is a cooperator ( ) , but the payoff for mutual cooperation is greater than the payoff for mutual defection ( ) , then a social dilemma arises from this game due to the conflict of interest between the individual and the group ( or pair ) .the nature of this social dilemma depends on the ordering of , , , and .biologically , the most important rankings are given by the prisoner s dilemma ( ) and the snowdrift game ( ) .since matrix ( [ generic2by2 ] ) defines a symmetric game , any two players using the same strategy are indistinguishable for the purpose of calculating payoffs . in nature , however , asymmetry frequently arises in interspecies interactions such as parasitic or symbiotic relationships .interactions between subpopulations , such as in dawkins battle of the sexes game , also give rise to asymmetry that can not be modeled by the symmetric matrix ( [ generic2by2 ] ) .even intraspecies interactions are essentially always asymmetric : ( i ) phenotypic variations such as size , strength , speed , wealth , or intellectual capabilities ; ( ii ) differences in access to and availability of environmental resources ; or ( iii ) each individual s history of past interactions , all affect the interacting individuals differently and result in asymmetric payoffs .the winner - loser effect , for example , is a well - studied example of effects of previous encounters on future interactions and has been reported across taxa , including even mollusks .asymmetry may also result from the assignment of social roles , such as the roles of parent " and offspring " : cooperation may be tied to individual energy or strength , for example , which is , in turn , determined by a player s role . in the realm of continuous strategies , adaptive dynamics has been used to study asymmetric competition , which applies to the resource consumption of plants , for instance . in social dilemmas containing many cooperators , accumulated benefits may be synergistically enhanced ( or discounted ) in a way that depends on who or where the players are , thereby making larger group interactions asymmetric . to model such interactions using evolutionary game theory, the payoff matrix must reflect the asymmetry .  in the _ donation game _, a cooperator pays a cost , , to deliver a benefit , , to the opponent , while a defector pays no cost and provides no benefit . in terms of matrix ( [ generic2by2 ] ), this game satisfies , , , and .provided and are positive , mutual defection is the only nash equilibrium . if , then this game defines a prisoner s dilemma .perhaps the simplest way to modify this game to account for possible sources of asymmetry is to allow for each pair of players to have a distinct payoff matrix ; that is , the payoff matrix for player against player in the donation game is    for some , , , and .if player cooperates , then this player donates to his or her opponent and incurs a cost of for doing so . as before ,defectors provide no benefit and pay no cost .the index could refer to a baseline trait of the player , the player s location , his or her history of past interactions , motivation , or any other non - strategy characteristic that distinguishes one player from another .games based on matrices of the form ( [ donationmatrix ] ) , with payoffs for _ both _ players in each entry of the matrix , are sometimes called _ bimatrix _ games .although bimatrix games have appeared in the context of evolutionary dynamics , most of the focus on these games has been in the setting of classical game theory and economics ( see * ? ? ?* ) where matrix game " generally means bimatrix game . "bimatrix games may be used to model classical asymmetric interactions such as those arising from sexual asymmetry in the battle of the sexes game .the asymmetric , four - strategy hawk - dove game of consisting of the strategies hawk , dove , bourgeois , and anti - bourgeois may also be framed as a ( ) bimatrix game ( see * ? ? ?symmetric matrix games , such as ( [ generic2by2 ] ) , are special cases of bimatrix games .we explore here the ways in which bimatrix games can be incorporated into evolutionary dynamics and used to model natural asymmetries in biological populations .we treat two particular forms of asymmetry : _ ecological _ and _ genotypic_. ecological asymmetry is derived from the locations of the players , whereas genotypic asymmetry is based on the players themselves . with ecological asymmetry , is the payoff matrix for a player at _ location _ against a player at _ location _ .since the payoffs depend on the locations of the players , this form of asymmetry requires a structured population .ecological asymmetry is a natural consideration in evolutionary dynamics since it ties strategy success to the environment . in the donation game , for instance, cooperators might be donating goods or services , but the costs and benefits may depend on the environmental conditions , i.e. the location of the donor .  on the other hand, players might instead differ in ability or strength , and strong " cooperators might contribute greater benefits ( or incur lower costs ) than weak " cooperators .this variation results in genotypic asymmetry , where each player has a baseline genotype ( strength ) and a strategy ( or ) .this form of asymmetry turns out to be subtler than it seems at first glance , however , since genotypes are generally represented by strategies in evolutionary game theory .in particular , it might seem that the genotype and strategy of a player could be combined into a single composite strategy and that the symmetric game based on these composite strategies could replace the original asymmetric game . as it happens , whether genotypic asymmetry can be resolved by a symmetric game depends on the details of the evolutionary process .classically , evolutionary games were studied in infinite populations via replicator dynamics , and more recently these games have been considered in finite populations . because every biological population is finite , we focus on finite populations ( which , for technical reasons , we assume to be large ) . since ecological asymmetry requires distinguishing different locations within the population , we assume that the population is structured and that a network defines the structure .network - structured populations have received a considerable amount of attention in evolutionary game theory and provide a natural setting in which to study social dilemmas .compared to well - mixed populations , in which each player interacts with every other player , networks can restrict the interactions that occur within the population by specifying which players are `` neighbors , '' i.e. share a link .we represent the links among the players in the population using an adjacency matrix , , which is defined by letting if there is a link from vertex to vertex and otherwise ( and satisfies for each and ) .  in an evolutionary game ,the _ state _ of a population of players is defined by specifying the strategy of each player .each player interacts with all of his or her neighbors .the total payoff to a player is multiplied by a selection intensity , , and then converted into _ fitness _ ( see methods ) .once each player is assigned a fitness , an update rule is used to determine the state of the population at the next time step .for example , with a birth - death update rule , a player is chosen from the population for reproduction with probability proportional to relative fitness .a neighbor of the reproducing player is then randomly chosen for death , and the offspring , who inherits the strategy of the parent , fills the vacancy .this process is a modification of the moran process , adapted to allow for ( i ) frequency - dependent fitnesses and ( ii ) population structures that are not necessarily well - mixed .the order of birth and death could also be reversed to get a death - birth update rule . in this rule, death occurs at random and the neighbors of the deceased compete to reproduce in order to fill the vacancy .these two rules result in the update of a single strategy in each time step , but one could consider other rules , such as wright - fisher updating , in which _ all _ of the strategies are revised in each generation .the rules mentioned to this point define strategy updates via reproduction and inheritance ; as such , we refer to them as _ genetic _ update rules .another popular class of update rules is based on revisions to the existing players strategy choices .we refer to rules falling into this class as _ cultural _ update rules .examples include imitation updating , in which a player is selected at random to evaluate his or her strategy and then probabilistically compares this strategy to those of his or her neighbors . a more localized version of this update rule is known as pairwise comparison updating , in which a player chooses a random neighbor for comparison rather than looking at the entire neighborhood . under _ best response dynamics _, an individual adopts the strategy that performs best given the current strategies of his or her neighbors . in each of these cultural processes ,the strategy of a player can change , but the underlying genotype is always the same , which suggests that baseline genotype and strategy need to be treated separately .genotypic asymmetry needs to be handled more carefully if the update rule is genetic since the nature of genotype transmission affects the dynamics of the process .in contrast to cultural processes , the genotype _ and _ strategy of a player at a given location may both change if the update rule is genetic : genotype may be inherited but not imitated .we will see that this property results in cultural and genetic processes behaving completely differently in the presence of genotypic asymmetry ._ phenotype _ may have both genetic and environmental components , and after treating the genetic ( genotypic ) and environmental components separately , these two forms of asymmetry may be combined in order to get a model in which the asymmetry is derived from varying baseline phenotypes .thus , with a theory of both ecological asymmetry and genotypic asymmetry based on inherited genotypes , one can account for more complicated forms of asymmetry appearing in biological populations .here we develop a framework for _ ecologically _ asymmetric games in which the payoffs depend on the locations of the players as well as their strategies .we assume that all of the players have the same set of strategies ( or actions " ) available to them , .the payoff matrix for a player at vertex against a player at vertex is    that is , a player at vertex using strategy against an opponent at vertex using strategy realizes a payoff of , whereas his opponent receives . since depends on and , these payoff matrices capture the asymmetry of the game .in the simpler setting of symmetric games , the pair approximation method has been used successfully to describe the dynamics of evolutionary processes on networks . for each , this method approximates the frequency of strategy , which we denote by , using the frequencies of strategy pairs in the population .pair approximation is expected to be accurate on large _ random regular _ networks , so we assume that the network is regular ( of degree ) and that is sufficiently large .( for , the network is just a cycle , which we do not treat here . )we also take , meaning that selection is _ weak _ , which results in a separation of timescales : the local configurations equilibrate quickly , while the global strategy frequencies change much more slowly .this separation allows us to get an explicit expression for the expected change ,  ] for each ( where is the frequency of strategy and  ] , is independent of for each . in particular , the dynamics of the process do not depend on the particular network configuration .  as an example ,the asymmetric donation game is spatially additive _ and _ possesses the equal gains from switching property , which greatly simplifies the analysis of its dynamics :  [ ex : donationecological ] the asymmetric donation game with payoff matrices defined by eq .( [ donationmatrix ] ) is spatially additive and satisfies    where and .therefore , the dynamics of the asymmetric game are the same as those of its symmetric counterpart with benefit , , and cost , , regardless of network configuration or resource distribution . under death - birth ( resp .imitation ) updating , this result implies that cooperation is expected to increase if and only if ( resp . ) , where is the degree of the ( regular ) network . fig .[ fig : ecological](a ) compares the predicted result obtained from to simulation data for imitation updating when benefit and cost values are distributed according to gaussian random variables .  in order to illustratewhen corollary [ cor : spatiallyadditive ] fails , we turn to cooperation in the snowdrift game . in this game , two drivers find themselves on either side of a snowdrift . if both cooperate in clearing the snowdrift , they share the cost , , equally , and both receive the benefit of being able to pass , . if one player cooperates and the other defects , both players receive but the cooperator pays the full cost , .if both players defect , each receives no benefit and pays no cost . in order to incorporate ecological asymmetry, we assume that the benefits are all the same since they are derived from being able to pass in the absence of a snowdrift . on the other hand ,the cost a player pays to clear the snowdrift may depend on his or her location : the snowdrift may appear on an incline , for example , in which case one player shovels with the gradient and the other player against it .moreover , when two cooperators meet , they might clear unequal shares of the snowdrift . thus , the payoff matrix for a player at location against a player at location should be of the form    where and  .intuitively , when two cooperators face one other , they each begin to clear the snowdrift and stop once they meet ; the quantity indicates the fraction of the snowdrift a cooperator at location clears before meeting the cooperator at location .a natural choice for is    which is the unique value that gives for each and , ensuring that the game is _ fair _ ,i.e. that the cooperator with the higher cost clears a smaller portion of the snowdrift than the one with the lower cost .averaging the payoff to one cooperator against another over all possible locations gives    which is the upper - left entry of .in contrast , the remaining three entries of do not depend on .therefore , provided there are at least two locations with distinct cost values , the dynamics of an evolutionary process depend on the particular network configuration ( theorem [ thm : maintheorem ] ) .this network dependence is illustrated in fig .[ fig : networkdependence ] .suppose now that we set to model ecological asymmetry in the snowdrift game ; that is , if two cooperators meet , they each clear exactly half of the snowdrift .if there are two cost values in the population , and , with , then a player who incurs a cost of finds it beneficial to cooperate against a defector , but a player who incurs a cost of would rather defect in this situation .thus , based on the social dilemma implied by the ranking of the payoffs , a player who incurs a cost of for cooperating is always playing a snowdrift game while a player who incurs a cost of is always playing a prisoner s dilemma .it follows that ecological asymmetry can account for multiple social dilemmas being played within a single population , even if the players all use the same set of strategies ( and ) .the payoff matrices of this particular game are spatially additive , so , by corollary [ cor : spatiallyadditive ] , the dynamics do not depend on the network configuration .if is the fraction of vertices with cost value then is the average cost of cooperation for a particular location and the dynamics are the same as those of the symmetric snowdrift game in which the cost of clearing a snowdrift is ( see fig . [fig : ecological](b ) ) . fig .[ fig : ecologicalstrongerselection ] demonstrates that this result does not extend to stronger selection strengths , so theorem [ thm : maintheorem ] is unique to weak selection .  based on theorem [ thm : maintheorem ] and the relative rank of payoffs , the social dilemma defined by the asymmetric game ( [ eq : adaptedsnowdrift ] ) ( for general ) is a prisoner s dilemma if and a snowdrift game if when selection is weakthat is , microscopically , there is a mixture of prisoner s dilemmas and snowdrift games , but , macroscopically , the process behaves like just one of these social dilemmas . consequently , although the dynamics of this evolutionary process may depend on the network configuration , the type of social dilemma implied by this game does not .    , as a function of the frequency of cooperators , , for a spatially non - additive snowdrift game , eq .( [ eq : adaptedsnowdrift ] ) , with selection intensity .the blue and green data are obtained using pairwise comparison updating and differ only in the configuration of the underlying network , which in both cases is a random regular graph of size and degree .every vertex has a benefit value of , and the cost values are split equally , with half of the vertices having and the remaining half having .the average payoff for mutual cooperation , eq .( [ eq : cnetwork ] ) , is ( blue ) and ( green ) , which suggests that the former arrangement is more attractive for cooperation . the analytical predictions ( solid lines ) are obtained from eq .( [ expectationpc ] ) in methods ( and are linear in for ) .[ fig : networkdependence],scaledwidth=50.0% ]     another form of asymmetry is based on the genotypes of the players rather than their locations .each player in the population has one of possible genotypes , and these genotypes are enumerated by the set . for an -strategy game ,the payoff matrix for a player whose genotype is against a player whose genotype is is    we explore genotypic asymmetry for cultural and genetic processes separately :   if genotypic asymmetry is incorporated into a cultural process , then the genotypes of the players never change ; only the strategies of the players are updated . in a structured population , it follows that each player s genotype may be associated with his or her location , and this association is an invariant of the process .thus , if denotes the genotype of the player at location , then we may apply theorem [ thm : maintheorem ] to the matrices defined by for each and . in this sense ,genotypic asymmetry may be reduced " to ecological asymmetry in evolutionary games with cultural update rules .note that , unlike ecological asymmetry , genotypic asymmetry does not _ require _ a structured population .however , one can always think of a population as structured ( even in the well - mixed case ) , and doing so allows one to make sense of the locations " of the players and to apply theorem [ thm : maintheorem ] to cultural processes with genotypic asymmetry .  [ex : culturaldonation ] in the donation game , a cooperator of genotype donates at a cost of .defectors contribute no benefit and pay no cost , irrespective of genotype .consider imitation updating on a large , regular network of degree , and let denote the genotype of the player at location ( henceforth player " ) .suppose that player is a cooperator , player is a defector , and that player imitates player and becomes a cooperator . despite this strategy change , the genotype of player is still , and the payoff matrix for player against player is still . on the other hand , consider the same process but with the genotypic asymmetry replaced by ecological asymmetry ( and with as the payoff matrix for the player at location against the player at location ) .since the genotype of a player at a given location never changes in an imitation process , the process with ecological asymmetry is well - defined ; that is , is independent of the dynamics of the process for each and . therefore , we may instead study the evolution of cooperation in the process with ecological asymmetry , and we already know from example [ ex : donationecological ] that , in the limit of weak selection , the frequency of cooperators in this donation game is expected to increase if and only if .in contrast , for genetic update rules , the asymmetry present due to differing genotypes can be removed completely if the genotypes of offspring are determined by genetic inheritance :   genetic update rules are defined by the ability of players to propagate their offspring to other locations in the population by means of births and deaths . in other words , there is a reproductive step in which genetic information is passed from parent(s ) to child .both the death - birth and birth - death processes have genetic update rules , but reproduction need not be clonal for the update rule to be genetic .if the genotypes of offspring are determined by genetic inheritance , then the strategy _ and _ genotype at each location are updated simultaneously : if the offspring of a player whose genotype is and whose strategy is replaces a player whose genotype is and whose strategy is , then is updated to and is updated to synchronously .therefore , rather than treating genotypes and strategies separately , we may consider them together in the form of pairs , , linking genotype and strategy .these pairs may be thought of as composite strategies of a larger evolutionary game whose payoff matrix , , is defined by    for genotypes , and , and strategies , and .the map    resolves a collection of asymmetric payoff matrices with a single symmetric payoff matrix , , of size .this argument holds for any population structure , so evolutionary processes with genotypic asymmetry that are based on genetic update rules can be studied in any setting in which there is a theory of symmetric games . for example, we may use the results from pair approximation on large , regular networks to study the donation game with genotypic asymmetry and genetic updating :  as in example [ ex : culturaldonation ] , a cooperator of genotype in the donation game donates at a cost of .defectors contribute no benefit and pay no cost , irrespective of genotype . for the death - birth and birth - death update rules ,defectors may be modeled as cooperators whose benefit and costs are both . in the larger symmetric game defined by ( [ strategyblowup ] ) , it follows that there are distinct composite strategies : , , , , and . for death - birth updating on a large , regular network of degree , cooperators of genotype are expected to increase if and only if    where , for each , denotes the frequency of cooperators of genotype ( i.e. the frequency of strategy in the larger symmetric game ) .the terms and are the average population benefit and cost values , respectively . therefore , the condition for the expected increase in cooperators of a particular genotype depends on the average level of cooperation within the population .( [ eq : genotypicdb ] ) may be thought of as an analogue of the ` ' rule of with replaced by the benefit premium , " , and replaced by the cost premium , " .  in the birth - death process, on the other hand , cooperators of genotype are expected to increase if and only if    interestingly , this condition is independent of the benefit values and says that cooperators of genotype increase in abundance if they incur , on average , smaller costs for cooperating than the other cooperators .( [ eq : genotypicdb ] ) and ( [ eq : genotypicbd ] ) are obtained by noticing that the expected change in the frequency of cooperators of genotype ,  ] denote the expected change in in one step of the process .a pair denotes a player on vertex using strategy .given pairs and , we denote by the expected payoff to a player at vertex playing strategy given that they have as a neighbor an individual playing strategy at vertex .if is a parameter representing the intensity of selection , then payoff , , is converted to fitness , , via    when defined in this way , fitness is always positive .the main theorem we prove is the following :  in the limit of weak selection , the dynamics of the ecologically asymmetric death - birth , birth - death , imitation , and pairwise comparison processes on a large , regular network may be approximated by the dynamics of a _ symmetric _ game with the same update rule and payoff matrix , i.e.    where for each and .theorem [ thm : maintheorem ] is established for each of these four update rules separately :   if an individual is playing strategy at node , at , and if , then    suppose that an individual is selected for death . the probability that replaces this focal individual is proportional to . for each , let be an enumeration of the indices with ( say , in increasing order ) and let be the strategy used by the player at vertex . if is chosen for death , then the probability that it is replaced by is    the taylor expansion of this term for small is    this expansion will be used frequently in the displays that follow .let be the kronecker delta ( defined to be if and otherwise ) .the probability of choosing the player on vertex for death is .the chance that this player is using strategy is .suppose that is a -tuple of strategies .if the focal player at vertex uses strategy , then the probability that the player on vertex uses strategy for each is .thus ,   & = \frac{1}{n}\sum_{i=1}^{n}\sum_{h\neq r}p_{h}\sum_{s_{i_{1}},\dots , s_{i_{k}}=1}^{n}q_{s_{i_{1}}\vert h}\cdots q_{s_{i_{k}}\vert h}\sum_{\ell = 1}^{k}\delta_{s_{i_{\ell}},r}\left(\frac{f_{\beta}\left(\pi_{\left(a_{r},i_{\ell}\right)}\left(a_{h},i\right)\right)}{\sum_{j=1}^{k}f_{\beta}\left(\pi_{\left(a_{s_{i_{j}}},i_{j}\right)}\left(a_{h},i\right)\right)}\right)\left(\frac{1}{n}\right ) \nonumber \\ & \quad + \frac{1}{n}\sum_{i=1}^{n}p_{r}\sum_{s_{i_{1}},\dots , s_{i_{k}}=1}^{n}q_{s_{i_{1}}\vert r}\cdots q_{s_{i_{k}}\vert r}\sum_{h\neq r}\sum_{\ell = 1}^{k}\delta_{s_{i_{\ell}},h}\left(\frac{f_{\beta}\left(\pi_{\left(a_{h},i_{\ell}\right)}\left(a_{r},i\right)\right)}{\sum_{j=1}^{k}f_{\beta}\left(\pi_{\left(a_{s_{i_{j}}},i_{j}\right)}\left(a_{r},i\right)\right)}\right)\left(-\frac{1}{n}\right)\end{aligned}\ ] ]  for each strategy , .the taylor expansion to first - order yields   & \approx \beta\left(\frac{\left(k-1\right)p_{r}}{k^{2}n^{2}}\right)\big(\left(\textrm{a}\right ) - \left(\textrm{b}\right ) -\left(\textrm{c}\right ) + \left(\textrm{d}\right)\big ) + o\left(\beta^{2}\right ) , \end{aligned}\ ] ]  where     if , then   & = \frac{1}{n}\sum_{i=1}^{n}\sum_{h\neq r , s}p_{h}\sum_{s_{i_{1}},\dots , s_{i_{k}}=1}^{n}q_{s_{i_{1}}\vert h}\cdots q_{s_{i_{k}}\vert h } \nonumber \\ & \quad\quad \times\sum_{\ell = 1}^{k}\delta_{s_{i_{\ell}},r}\left(\frac{f_{\beta}\left(\pi_{\left(a_{r},i_{\ell}\right)}\left(a_{h},i\right)\right)}{\sum_{j=1}^{k}f_{\beta}\left(\pi_{\left(a_{s_{i_{j}}},i_{j}\right)}\left(a_{h},i\right)\right)}\right)\left(\frac{2\sum_{\alpha = 1}^{k}\delta_{s_{i_{\alpha}},s}}{kn}\right ) \nonumber \\ & \quad + \frac{1}{n}\sum_{i=1}^{n}\sum_{h\neq r , s}p_{h}\sum_{s_{i_{1}},\dots , s_{i_{k}}=1}^{n}q_{s_{i_{1}}\vert h}\cdots q_{s_{i_{k}}\vert h } \nonumber \\ & \quad\quad \times\sum_{\ell = 1}^{k}\delta_{s_{i_{\ell}},s}\left(\frac{f_{\beta}\left(\pi_{\left(a_{s},i_{\ell}\right)}\left(a_{h},i\right)\right)}{\sum_{j=1}^{k}f_{\beta}\left(\pi_{\left(a_{s_{i_{j}}},i_{j}\right)}\left(a_{h},i\right)\right)}\right)\left(\frac{2\sum_{\alpha = 1}^{k}\delta_{s_{i_{\alpha}},r}}{kn}\right ) \nonumber \\ & \quad + \frac{1}{n}\sum_{i=1}^{n}p_{r}\sum_{s_{i_{1}},\dots , s_{i_{k}}=1}^{n}q_{s_{i_{1}}\vert r}\cdots q_{s_{i_{k}}\vert r } \nonumber \\ & \quad\quad \times\sum_{\ell = 1}^{k}\delta_{s_{i_{\ell}},s}\left(\frac{f_{\beta}\left(\pi_{\left(a_{s},i_{\ell}\right)}\left(a_{r},i\right)\right)}{\sum_{j=1}^{k}f_{\beta}\left(\pi_{\left(a_{s_{i_{j}}},i_{j}\right)}\left(a_{r},i\right)\right)}\right)\left(\frac{2\sum_{\alpha = 1}^{k}\left(\delta_{s_{i_{\alpha}},r}-\delta_{s_{i_{\alpha}},s}\right)}{kn}\right ) \nonumber \\ & \quad + \frac{1}{n}\sum_{i=1}^{n}p_{r}\sum_{s_{i_{1}},\dots , s_{i_{k}}=1}^{n}q_{s_{i_{1}}\vert r}\cdots q_{s_{i_{k}}\vert r } \nonumber \\ & \quad\quad \times\sum_{h\neq r , s}\sum_{\ell = 1}^{k}\delta_{s_{i_{\ell}},h}\left(\frac{f_{\beta}\left(\pi_{\left(a_{h},i_{\ell}\right)}\left(a_{r},i\right)\right)}{\sum_{j=1}^{k}f_{\beta}\left(\pi_{\left(a_{s_{i_{j}}},i_{j}\right)}\left(a_{r},i\right)\right)}\right)\left(-\frac{2\sum_{\alpha = 1}^{k}\delta_{s_{i_{\alpha}},s}}{kn}\right ) \nonumber \\ & \quad + \frac{1}{n}\sum_{i=1}^{n}p_{s}\sum_{s_{i_{1}},\dots , s_{i_{k}}=1}^{n}q_{s_{i_{1}}\vert s}\cdots q_{s_{i_{k}}\vert s } \nonumber \\ & \quad\quad \times\sum_{\ell = 1}^{k}\delta_{s_{i_{\ell}},r}\left(\frac{f_{\beta}\left(\pi_{\left(a_{r},i_{\ell}\right)}\left(a_{s},i\right)\right)}{\sum_{j=1}^{k}f_{\beta}\left(\pi_{\left(a_{s_{i_{j}}},i_{j}\right)}\left(a_{s},i\right)\right)}\right)\left(\frac{2\sum_{\alpha = 1}^{k}\left(\delta_{s_{i_{\alpha}},s}-\delta_{s_{i_{\alpha}},r}\right)}{kn}\right ) \nonumber \\ & \quad + \frac{1}{n}\sum_{i=1}^{n}p_{s}\sum_{s_{i_{1}},\dots , s_{i_{k}}=1}^{n}q_{s_{i_{1}}\vert s}\cdots q_{s_{i_{k}}\vert s } \nonumber \\ & \quad\quad \times\sum_{h\neq r , s}\sum_{\ell = 1}^{k}\delta_{s_{i_{\ell}},h}\left(\frac{f_{\beta}\left(\pi_{\left(a_{h},i_{\ell}\right)}\left(a_{s},i\right)\right)}{\sum_{j=1}^{k}f_{\beta}\left(\pi_{\left(a_{s_{i_{j}}},i_{j}\right)}\left(a_{s},i\right)\right)}\right)\left(-\frac{2\sum_{\alpha = 1}^{k}\delta_{s_{i_{\alpha}},r}}{kn}\right ) .\end{aligned}\ ] ]  on the other hand ,   & = \frac{1}{n}\sum_{i=1}^{n}\sum_{h\neq r}p_{h}\sum_{s_{i_{1}},\dots , s_{i_{k}}=1}^{n}q_{s_{i_{1}}\vert h}\cdots q_{s_{i_{k}}\vert h } \nonumber \\ & \quad\quad \times\sum_{\ell = 1}^{k}\delta_{s_{i_{\ell}},r}\left(\frac{f_{\beta}\left(\pi_{\left(a_{r},i_{\ell}\right)}\left(a_{h},i\right)\right)}{\sum_{j=1}^{k}f_{\beta}\left(\pi_{\left(a_{s_{i_{j}}},i_{j}\right)}\left(a_{h},i\right)\right)}\right)\left(\frac{2\sum_{\alpha = 1}^{k}\delta_{s_{i_{\alpha}},r}}{kn}\right ) \nonumber \\ & \quad + \frac{1}{n}\sum_{i=1}^{n}p_{r}\sum_{s_{i_{1}},\dots , s_{i_{k}}=1}^{n}q_{s_{i_{1}}\vert r}\cdots q_{s_{i_{k}}\vert r } \nonumber \\ & \quad\quad \times\sum_{h\neq r}\sum_{\ell = 1}^{k}\delta_{s_{i_{\ell}},h}\left(\frac{f_{\beta}\left(\pi_{\left(a_{h},i_{\ell}\right)}\left(a_{r},i\right)\right)}{\sum_{j=1}^{k}f_{\beta}\left(\pi_{\left(a_{s_{i_{j}}},i_{j}\right)}\left(a_{r},i\right)\right)}\right)\left(-\frac{2\sum_{\alpha = 1}^{k}\delta_{s_{i_{\alpha}},r}}{kn}\right ) .\end{aligned}\ ] ]  the zeroth - order taylor expansion yields   & \approx \frac{4p_{r}}{kn}\left(- kq_{s\vert r } + \left(k-1\right)\sum_{h=1}^{n}q_{s\vert h}q_{h\vert r}\right ) + o\left(\beta\right)\end{aligned}\ ] ]  if , and   & \approx \frac{2p_{r}}{kn}\left(1-kq_{r\vert r } + \left(k-1\right)\sum_{h=1}^{k}q_{r\vert h}q_{h\vert r}\right ) + o\left(\beta\right).\end{aligned}\ ] ]  therefore , = o\left(\beta\right) ] ( by eqs .( [ dbpairdifferent ] ) and ( [ dbpairsame ] ) ) for each and , which results in a separation of timescales between the strategy frequencies and the pair frequencies .in particular , the pair frequencies will reach their equilibrium much more quickly than the strategy frequencies will , so we can examine the expression for  ] and  ] ) , we have     that this equation implies that    assuming the system has reached this local equilibrium , we then have   & \approx \beta\left(\frac{\left(k-1\right)p_{r}}{k^{2}n^{2}}\right ) \bigg(\left(k+1\right)\sum_{i , j=1}^{n}w_{ij}\sum_{s=1}^{n}a_{rs}^{ij}q_{s\vert r } \nonumber \\ & \qquad\qquad\qquad\qquad\qquad -k\sum_{i , j=1}^{n}w_{ij}\sum_{s , t=1}^{n}a_{st}^{ij}q_{t\vert s}q_{s\vert r}-\sum_{i , j=1}^{n}w_{ij}\sum_{s ,t=1}^{n}a_{st}^{ij}q_{s\vert t}q_{t\vert r}\bigg ) + o\left(\beta^{2}\right ) \nonumber \\ & = \beta\left(\frac{\left(k-1\right)p_{r}}{kn}\right)\bigg(\left(k+1\right)\sum_{s=1}^{n}\overline{a}_{rs}q_{s\vert r}-k\sum_{s , t=1}^{n}\overline{a}_{st}q_{t\vert s}q_{s\vert r}-\sum_{s , t=1}^{n}\overline{a}_{st}q_{s\vert t}q_{t\vert r}\bigg ) + o\left(\beta^{2}\right ) \nonumber \\ & = \beta\left(\frac{\left(k-2\right ) p_{r}}{k\left(k-1\right ) n}\right ) \bigg(-\left(k-2\right)\left(k+1\right)\sum_{s , t=1}^{n}\overline{a}_{st}p_{s}p_{t}+\left(k^{2}-k-1\right)\sum_{s=1}^{n}\overline{a}_{rs}p_{s } \nonumber \\ & \qquad\qquad\qquad\qquad\qquad -\sum_{s=1}^{n}\overline{a}_{sr}p_{s}-\left(k+1\right)\sum_{s=1}^{n}\overline{a}_{ss}p_{s}+\left(k+1\right ) \overline{a}_{rr}\bigg ) + o\left(\beta^{2}\right)\end{aligned}\ ] ]  as long as is small .therefore , if we choose an appropriate time scale and set  }{\delta t } ; \\ \overline{b}_{rs } & = \frac{\overline{a}_{rr}+\overline{a}_{rs}-\overline{a}_{sr}-\overline{a}_{ss}}{k-2 } ; \\ \phi & = \sum_{s , t=1}^{n}p_{s}p_{t}\overline{a}_{st } , \end{aligned}\ ] ]  then , recovering the replicator equation of .it follows that the dynamics depend on , proving theorem [ thm : maintheorem ] for death - birth updating .in the birth - death process , an individual is selected for reproduction with probability proportional to relative fitness .the offspring of the selected player then replaces a random neighbor . rather than trying to approximate the total fitness of the population, we will simply denote this value by .since this value is positive , it does not influence the sign of the expectation values and as such we will largely ignore it .we have   & = \frac{1}{f^{\textrm{pop}}}np_{r}\left(\frac{1}{n}\right)\sum_{i=1}^{n}\sum_{s_{i_{1}},\dots , s_{i_{k}}=1}^{n}q_{s_{i_{1}}\vert r}\cdots q_{s_{i_{k}}\vert r}f_{\beta}\left(\sum_{\ell = 1}^{k}a_{rs_{i_{\ell}}}^{ii_{\ell}}\right)\sum_{h\neq r}\left(\frac{\sum_{j=1}^{k}\delta_{s_{i_{j}},h}}{k}\right)\left(\frac{1}{n}\right ) \nonumber \\ & \quad + \frac{1}{f^{\textrm{pop}}}\sum_{h\neq r}np_{h}\left(\frac{1}{n}\right)\sum_{i=1}^{n}\sum_{s_{i_{1}},\dots , s_{i_{k}}=1}^{n}q_{s_{i_{1}}\vert h}\cdots q_{s_{i_{k}}\vert h}f_{\beta}\left(\sum_{\ell = 1}^{k}a_{hs_{i_{\ell}}}^{ii_{\ell}}\right)\left(\frac{\sum_{j=1}^{k}\delta_{s_{i_{j}},r}}{k}\right)\left(-\frac{1}{n}\right ) .\end{aligned}\ ] ]  the local equilibrium conditions for birth - death updating turn out to be the same as those for death - birth updating ( eq .( [ localequilibrium ] ) ) .these local equilibrium conditions do not take into account selection as long as is close to , so they are essentially based on a neutral process in which at most one strategy is update at each time step .therefore , it is perhaps not surprising that these conditions are the same for different processes based on one strategy update in each time step .  in the following expressions , by we mean that is proportional to with _ positive _ constant of proportionality . letting and using the local equilibrium conditions ( as well as the same separation - of - timescales argument we used in [ subsubsec : weakdynamics ] ) , we find that   & \propto \beta p_{r}\left(k\sum_{i , j=1}^{n}w_{ij}\sum_{s=1}^{n}a_{rs}^{ij}q_{s\vert r}-\left(k-1\right)\sum_{i , j=1}^{n}w_{ij}\sum_{s , t=1}^{n}a_{st}^{ij}q_{t\vert s}q_{s\vert r}-\sum_{i , j=1}^{n}w_{ij}\sum_{s=1}^{n}a_{sr}^{ij}q_{s\vert r}\right ) + o\left(\beta^{2}\right ) \nonumber \\ & \propto \beta p_{r}\left(k\sum_{s=1}^{n}\overline{a}_{rs}q_{s\vert r}-\left(k-1\right)\sum_{s , t=1}^{n}\overline{a}_{st}q_{t\vert s}q_{s\vert r}-\sum_{s=1}^{n}\overline{a}_{sr}q_{s\vert r}\right ) + o\left(\beta^{2}\right ) \nonumber \\ & \propto \beta p_{r}\bigg ( -\left(k-2\right)\sum_{s , t=1}^{n}\overline{a}_{st}p_{s}p_{t } + \left(k-1\right)\sum_{s=1}^{n}\overline{a}_{rs}p_{s } - \sum_{s=1}^{n}\overline{a}_{sr}p_{s } - \sum_{s=1}^{n}\overline{a}_{ss}p_{s}+ \overline{a}_{rr } \bigg ) + o\left(\beta^{2}\right ) .\end{aligned}\ ] ]  just as we saw with the death - birth process , after choosing an appropriate time scale and letting    we have , proving theorem [ thm : maintheorem ] for birth - death updating .   in the imitation process , an individual is selected uniformly at random from the population to evaluate his strategy .the chosen player then compares his fitness with the fitness of each neighbor and either adopts a new strategy or retains his or her current strategy ( with probability proportional to relative fitness ) .suppose that an individual at vertex , playing , is selected to evaluate his or her strategy . if , then the probability that he or she adopts strategy is    and the probability that his strategy remains unchanged is    we let be the same as it was for death - birth updating . for small ,     for ,   & = \frac{1}{n}\sum_{i=1}^{n}\sum_{h\neq r}p_{h}\sum_{s_{i_{1}},\dots , s_{i_{k}}=1}^{n}q_{s_{i_{1}}\vert h}\cdots q_{s_{i_{k}}\vert h } \nonumber\\ & \quad\quad \times\sum_{\ell = 1}^{k}\delta_{s_{i_{\ell}},r}\left(\frac{f_{\beta}\left(\pi_{\left(a_{r},i_{\ell}\right)}\left(a_{h},i\right)\right)}{\sum_{j=1}^{k}f_{\beta}\left(\pi_{\left(a_{s_{i_{j}}},i_{j}\right)}\left(a_{h},i\right)\right ) + f_{\beta}\left(\sum_{j=1}^{k}a_{hs_{i_{j}}}^{ii_{j}}\right)}\right)\left(\frac{1}{n}\right ) \nonumber \\ & \quad + \frac{1}{n}\sum_{i=1}^{n}p_{r}\sum_{s_{i_{1}},\dots , s_{i_{k}}=1}^{n}q_{s_{i_{1}}\vert r}\cdots q_{s_{i_{k}}\vert r } \nonumber \\ & \quad\quad \times\sum_{h\neq r}\sum_{\ell = 1}^{k}\delta_{s_{i_{\ell}},h}\left(\frac{f_{\beta}\left(\pi_{\left(a_{h},i_{\ell}\right)}\left(a_{r},i\right)\right)}{\sum_{j=1}^{k}f_{\beta}\left(\pi_{\left(a_{s_{i_{j}}},i_{j}\right)}\left(a_{r},i\right)\right ) + f_{\beta}\left(\sum_{j=1}^{k}a_{rs_{i_{j}}}^{ii_{j}}\right)}\right)\left(-\frac{1}{n}\right ) .\end{aligned}\ ] ]  the local equilibrium conditions are exactly the same as they were for the death - birth process . assuming that the system has reached this local equilibrium , the separation - of - timescales argument we used in [ subsubsec : weakdynamics ] gives   & \approx \beta\left(\frac{p_{r}}{\left(k+1\right)^{2}n^{2}}\right ) \bigg(\left(k^{2}+2k-1\right)\sum_{i , j=1}^{n}w_{ij}\sum_{s=1}^{n}a_{rs}^{ij}q_{s\vert r } \nonumber \\ & \qquad\qquad\qquad\qquad\qquad -\left(k^{2}+k-2\right)\sum_{i , j=1}^{n}w_{ij}\sum_{s , t=1}^{n}a_{st}^{ij}q_{t\vert s}q_{s\vert r } \nonumber \\ & \qquad\qquad\qquad\qquad\qquad -\left(k-1\right)\sum_{i , j=1}^{n}w_{ij}\sum_{s , t=1}^{n}a_{ts}^{ij}q_{t\vert s}q_{s\vert r } -2\sum_{i , j=1}^{n}w_{ij}\sum_{s=1}^{n}a_{sr}^{ij}q_{s\vert r}\bigg ) + o\left(\beta^{2}\right ) \nonumber \\ & = \beta\left(\frac{kp_{r}}{\left(k+1\right)^{2}n}\right ) \bigg(\left(k^{2}+2k-1\right)\sum_{s=1}^{n}\overline{a}_{rs}q_{s\vert r } -\left(k^{2}+k-2\right)\sum_{s , t=1}^{n}\overline{a}_{st}q_{t\vert s}q_{s\vert r } \nonumber \\ & \qquad\qquad\qquad\qquad\qquad -\left(k-1\right)\sum_{s , t=1}^{n}\overline{a}_{ts}q_{t\vert s}q_{s\vert r } -2\sum_{s=1}^{n}\overline{a}_{sr}q_{s\vert r}\bigg ) + o\left(\beta^{2}\right ) \nonumber \\ & = \beta\left(\frac{k\left(k-2\right ) p_{r}}{\left(k-1\right)\left(k+1\right)^{2}n}\right ) \bigg ( -\left(k-2\right)\left(k+3\right ) \sum_{s , t=1}^{n}\overline{a}_{st}p_{s}p_{t}+\left(k^{2}+k-3\right)\sum_{s=1}^{n}\overline{a}_{rs}p_{s } \nonumber \\ & \qquad\qquad\qquad\qquad\qquad -3\sum_{s=1}^{n}\overline{a}_{sr}p_{s } -\left(k+3\right)\sum_{s=1}^{n}\overline{a}_{ss}p_{s } + \left(k+3\right)\overline{a}_{rr}\bigg ) + o\left(\beta^{2}\right ) .\end{aligned}\ ] ]  with and , we have    which establishes theorem [ thm : maintheorem ] for imitation updating .   in the pairwise comparison process ,a focal individual is selected uniformly at random from the population .a model individual is then chosen uniformly at random from the neighbors of the focal individual .if and denote the payoffs to the focal and model individuals , respectively , then the focal player will adopt the strategy of the model player with probability    where is a real parameter representing the intensity of selection .in addition to the expected payoff ( defined in the same way as for death - birth updating ) , we let    if has as a neighborhood . with this notation in place , we have   & = \frac{1}{n}\sum_{i=1}^{n}\sum_{h\neq r}p_{h}\sum_{s_{i_{1}},\dots , s_{i_{k}}=1}^{n}q_{s_{i_{1}}\vert h}\cdots q_{s_{i_{k}}\vert h } \nonumber \\ & \quad\quad \times\sum_{\ell = 1}^{k}\left(\frac{1}{k}\right)\delta_{s_{i_{\ell}},r}\left(\frac{f_{\beta}\left(\pi_{\left(a_{r},i_{\ell}\right)}\left(a_{h},i\right)\right)}{f_{\beta}\left(\pi_{\left(a_{r},i_{\ell}\right)}\left(a_{h},i\right)\right)+f_{\beta}\left(\pi_{\left(a_{h},i\right)}\right)}\right)\left(\frac{1}{n}\right ) \nonumber \\ & \quad + \frac{1}{n}\sum_{i=1}^{n}p_{r}\sum_{s_{i_{1}},\dots , s_{i_{k}}=1}^{n}q_{s_{i_{1}}\vert r}\cdots q_{s_{i_{k}}\vert r } \nonumber \\ & \quad\quad \times\sum_{h\neq r}\sum_{\ell = 1}^{k}\left(\frac{1}{k}\right)\delta_{s_{i_{\ell}},h}\left(\frac{f_{\beta}\left(\pi_{\left(a_{h},i_{\ell}\right)}\left(a_{r},i\right)\right)}{f_{\beta}\left(\pi_{\left(a_{h},i_{\ell}\right)}\left(a_{r},i\right)\right)+f_{\beta}\left(\pi_{\left(a_{r},i\right)}\right)}\right)\left(-\frac{1}{n}\right ) .\end{aligned}\ ] ]  as , we have    consequently , in the limit of weak selection ,   & \approx \beta\frac{p_{r}}{2kn^{2}}\left(k\sum_{i , j=1}^{n}w_{ij}\sum_{s=1}^{n}a_{rs}^{ij}q_{s\vert r } -\left(k-1\right)\sum_{i , j=1}^{n}w_{ij}\sum_{s , t=1}^{n}a_{st}^{ij}q_{t\vert s}q_{s\vert r } -\sum_{i ,j=1}^{n}w_{ij}\sum_{s=1}^{n}a_{sr}^{ij}q_{s\vert r}\right ) + o\left(\beta^{2}\right ) \nonumber \\ & = \beta\frac{p_{r}}{2n}\left(k\sum_{s=1}^{n}\overline{a}_{rs}q_{s\vert r } -\left(k-1\right)\sum_{s , t=1}^{n}\overline{a}_{st}q_{t\vert s}q_{s\vert r } -\sum_{s=1}^{n}\overline{a}_{sr}q_{s\vert r}\right ) + o\left(\beta^{2}\right ) \nonumber \\ & = \beta\left(\frac{\left(k-2\right ) p_{r}}{2\left(k-1\right ) n}\right ) \bigg(-\left(k-2\right)\sum_{s , t=1}^{n}\overline{a}_{st}p_{s}p_{t}+\left(k-1\right)\sum_{s=1}^{n}\overline{a}_{rs}p_{s } \nonumber \\ & \qquad\qquad\qquad\qquad\qquad -\sum_{s=1}^{n}\overline{a}_{sr}p_{s}-\sum_{s=1}^{n}\overline{a}_{ss}p_{s}+\overline{a}_{rr}\bigg ) + o\left(\beta^{2}\right ) .\end{aligned}\ ] ]  the local equilibrium conditions are exactly the same as they were for the other processes , but in this case they are not needed to arrive at this last expression for  ] , is independent of for each . in particular , the dynamics of the process do not depend on the particular network configuration .if for each , then    which is independent of . the corollary then follows directly from theorem [ thm : maintheorem ] .   in each simulation , a random -regular network ( with ) of vertices is generated .the selection intensity is for figs .[ fig : ecological ] and [ fig : networkdependence ] , for fig .[ fig : ecologicalstrongerselection](a ) , and for fig .[ fig : ecologicalstrongerselection](b ) .the figures are generated based on data collected from a number of cycles : in each cycle , the network is given an initial configuration of cooperators by first choosing a density , , uniformly at random from the interval  ] by a factor of .  small deviations from the expected results are seen in each of the figures , and these deviations are due to the effects of finite selection parameter ( ) and the finiteness of the set of possible values of ( is a multiple of ) . as an example of how these properties can give rise to small deviations ,consider the donation game under imitation updating in fig .[ fig : ecological](a ) .( [ expectationim ] ) predicts that $ ] is always positive , yet we observe in fig .[ fig : ecological](a ) that this change becomes negative as .if and , then the only defector in the population has a higher payoff than all of the other cooperators .let denote the fitness of the player at location .thus , with just a single defector ( at location ) in a population of cooperators , we have for each , with equality if and only if . the expected change in the frequency of cooperators in the next time step is   & = \left(\frac{1}{n}\right)\left(\frac{1}{n}\right)\left(1-\frac{f_{\beta}^{\left(i\right)}}{f_{\beta}^{\left(i\right)}+\sum_{\left\{j\ : \w_{ij}=1\right\}}f_{\beta}^{\left(j\right)}}\right ) \nonumber \\ & \quad -\left(\frac{1}{n}\right)\sum_{\left\{j\ : \ w_{ij}=1\right\}}\left(\frac{1}{n}\right)\left(\frac{f_{\beta}^{\left(i\right)}}{f_{\beta}^{\left(j\right)}+\sum_{\left\{l\ : \w_{jl}=1\right\}}f_{\beta}^{\left(l\right)}}\right ) .\end{aligned}\ ] ]  the first ( resp .second ) summation runs over all of the neighbors of ( resp . ) . for each ,  both with equality if and only if .therefore , we see that   & \leqslant \left(\frac{1}{n}\right)\left(\frac{1}{n}\right)\left(1-\frac{1}{k+1}\right ) - \left(\frac{1}{n}\right)\left(\frac{k}{n}\right)\left(\frac{1}{k+1}\right ) = 0\end{aligned}\ ] ]  with equality if and only if . the same argument explains the negative average changes as .since can only take on finitely many values for a given population size , similar arguments explain the small discrepancies between the actual and expected results for intermediate values of ( see fig .[ fig : ecological ] ) .a. m. thanks farhan abedin and gyrgy szab for helpful discussions .a. m. and c. h. acknowledge financial support from the natural sciences and engineering research council of canada ( nserc ) and c. h. from the foundational questions in evolutionary biology fund ( fqeb ) , grant rfp-12 - 10 .         c. hauert , f. michor , m. a. nowak , and m. doebeli .synergy and discounting of cooperation in social dilemmas ._ journal of theoretical biology _ , 2390 ( 2):0 195202 , mar 2006 .doi : 10.1016/j.jtbi.2005.08.040 .w. g. wright and a. l. shanks .previous experience determines territorial behavior in an archaeogastropod limpet . _ journal of experimental marine biology and ecology _ , 1660 ( 2):0 217229 , mar 1993 .doi : 10.1016/0022 - 0981(93)90220-i .a. l. shanks .previous agonistic experience determines both foraging behavior and territoriality in the limpet lottia gigantea ( sowerby ) . _behavioral ecology _ , 130 ( 4):0 467471 , jul 2002 .doi : 10.1093/beheco/13.4.467 .m. doebeli and i. ispolatov .symmetric competition as a general model for single - species adaptive dynamics ._ journal of mathematical biology _ , 670 ( 2):0 169184 , may 2012 .doi : 10.1007/s00285 - 012 - 0547 - 4 .m. bergman , m. olofsson , and c. wiklund .contest outcome in a territorial butterfly : the role of motivation ._ proceedings of the royal society b : biological sciences _ , 2770 ( 1696):0 30273033 , may 2010 .doi : 10.1098/rspb.2010.0646 .a. e. magurran and m. a. nowak .another battle of the sexes : the consequences of sexual asymmetry in mating costs and predation risk in the guppy , poecilia reticulata ._ proceedings of the royal society b : biological sciences _ , 2460 ( 1315):0 3138 , oct 1991 .doi : 10.1098/rspb.1991.0121 .m. mesterton - gibbons .ecotypic variation in the asymmetric hawk - dove game : when is bourgeois an evolutionarily stable strategy ?_ evolutionary ecology _ , 60 ( 3):0 198222 , may 1992 .doi : 10.1007/bf02214162 .     c. taylor , d. fudenberg , a. sasaki , and m. a. nowak .evolutionary game dynamics in finite populations ._ bulletin of mathematical biology _, 660 ( 6):0 16211644 , nov 2004 .doi : 10.1016/j.bulm.2004.03.004 .a. traulsen , j. m. pacheco , and m. a. nowak .pairwise comparison and selection temperature in evolutionary game dynamics ._ journal of theoretical biology _ , 2460 ( 3):0 522529 , jun 2007 .doi : 10.1016/j.jtbi.2007.01.002 .h. matsuda , n. ogita , a. sasaki , and k. sato .statistical mechanics of population : the lattice lotka - volterra model ._ progress of theoretical physics _, 880 ( 6):0 10351049 , dec 1992 .doi : 10.1143/ptp/88.6.1035 .j. vukov , g. szab , and a. szolnoki .cooperation in the noisy case : prisoner s dilemma game on two types of regular random graphs ._ physical review e _ , 730 ( 6 ) , jun 2006 .doi : 10.1103/physreve.73.067103 .o. p. hauser , a. traulsen , and m. a. nowak .heterogeneity in background fitness acts as a suppressor of selection ._ journal of theoretical biology _, 343:0 178185 , feb 2014 .doi : 10.1016/j.jtbi.2013.10.013 .a. szolnoki and g. szab .cooperation enhanced by inhomogeneous activity of teaching for evolutionary prisoner s dilemma games . _ europhysics letters _ , 770 ( 3 ) , jan 2007 .doi : 10.1209/0295 - 5075/77/30004 .h. ohtsuki , m. a. nowak , and j. m. pacheco .breaking the symmetry between interaction and replacement in evolutionary dynamics on graphs . _ physical review letters _ , 980 ( 10 ) ,doi : 10.1103/physrevlett.98.108106 .h. ohtsuki , j. m. pacheco , and m. a. nowak .evolutionary graph theory : breaking the symmetry between interaction and replacement ._ journal of theoretical biology _ , 2460 ( 4):0 681694 , jun 2007 .doi : 10.1016/j.jtbi.2007.01.024 .j. m. pacheco , f. l. pinheiro , and f. c. santos .population structure induces a symmetry breaking favoring the emergence of cooperation ._ plos computational biology _ , 50 ( 12):0 e1000596 , dec 2009 .doi : 10.1371/journal.pcbi.1000596 .w. maciejewski , f. fu , and c. hauert .evolutionary game dynamics in populations with heterogenous structures . _plos computational biology _ , 100 ( 4):0 e1003567 , apr 2014 .doi : 10.1371/journal.pcbi.1003567 .

Abstract: evolutionary game theory is a powerful framework for studying evolution in populations of interacting individuals . a common assumption in evolutionary game theory is that interactions are symmetric , which means that the players are distinguished by only their strategies . in nature , however , the microscopic interactions between players are nearly always asymmetric due to environmental effects , differing baseline characteristics , and other possible sources of heterogeneity . to model these phenomena , we introduce into evolutionary game theory two broad classes of asymmetric interactions : ecological and genotypic . ecological asymmetry results from variation in the environments of the players , while genotypic asymmetry is a consequence of the players having differing baseline genotypes . we develop a theory of these forms of asymmetry for games in structured populations and use the classical social dilemmas , the prisoner s dilemma and the snowdrift game , for illustrations . interestingly , asymmetric games reveal essential differences between models of genetic evolution based on reproduction and models of cultural evolution based on imitation that are not apparent in symmetric games .