The effect of migration on cooperation
An analysis of strategy update and neighbourhood assumptions
A thesis presented for the degree of
Master in Economics
Track: Behavioural Economics and Game Theory
Handed in by: Maj-Britt Sterba
12 July 2016
Statement of Originality
This document is written by Maj-Britt Sterba who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its
references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Abstract
This thesis investigates the outbreak and survival of cooperation in a spatial prisoner’s dilemma with migration. While recent research has anew fostered the debate on the impact of mobility on cooperation, the different assumptions used have not yet been explored systematically. The here presented study extends the comprehensive understanding by inspecting the system’s behaviour under an exponential update rule and different neighbourhood scales for game-play, updating and migration. The analysis is based on results from agent-based simu-lations. It is revealed that the outbreak and long term existence of cooperation is possible in a spatial prisoner’s dilemma with migration even if the update is not deterministic. Furthermore, the cooperation rates realized are increased by bringing the update on a bigger scale than the game-play and migration. Cooperation rates are enhanced for a broad range of parameter settings compared to the viscous case if the migration rule is not random.
Contents
1 Introduction 4
2 Literature review 8
3 Methodology 11
4 Model 13
4.1 Entities, State Variables and Scales . . . 13
4.2 Process overview and scheduling . . . 13
4.3 Design concepts . . . 14 4.4 Initialization . . . 15 4.5 Submodels . . . 15 4.5.1 Game play . . . 15 4.5.2 Update . . . 16 4.5.3 Migration . . . 16 4.5.4 Noise . . . 17 5 Results 17 5.1 Base Scenario . . . 18
5.2 Scenario Different betas . . . 27
5.3 Scenario Mixed neighbourhoods . . . 34
6 Conclusion and discussion 42
References 45
1
Introduction
This thesis investigates the effect of migration on the outbreak and survival of cooperation from an evolutionary game theoretical perspective. The basic study object in evolutionary game theory is a population of players inter-acting with each other. From these interactions players receive a payoff that determines how successful they will be in the next round. Through this mech-anism unsuccessful strategies are selected against and out-competed in the long run. One of the main questions evolutionary game theory tries to answer is the existence of cooperation in nature: while being of fundamental impor-tance for the world’s complex interactions and organizational structures it is not readily explained theoretically [31][36][2][7]. The game theoretical model preferably used to study the topic is the prisoner’s dilemma since it rep-resents the social dilemma in a parsimonious, yet essential, way [23][28][17]. Here, players can choose between cooperating and defecting. The payoff they get from the game interaction depends on their own choice and that of the other players: in the two-person version, if two cooperators encounter, they both receive the payoff R (reward), if two defectors encounter, they both receive the lower payoff, P (punishment). However, if a cooperator encoun-ters a defector, the former receives the lowest payoff, S (sucker’s payoff) and the latter the highest payoff, T (temptation). Since T > R > P > S and 2R > T + S, independent of the opponent’s choice, defecting is the dominant strategy. In a group of selfish individuals defection should hence overcome and erase cooperation [2].
One possible solution that raised a lot of interest is the introduction of a spatial structure as a cooperation supporting mechanism [29][22][9][4]. Here, individuals play the prisoners dilemma not in a well-mixed population, where they interact with each other randomly, but only with their nearest neigh-bours with the classic geometry for the interaction space being a regular lattice [30][18][34]. However, creating a structure where individuals repeat-edly interact with their neighbours is not per se helpful for cooperation. On
the one hand cooperation with the organisms around you offers mutual ben-efits, but on the other hand there is increased competition for local scarce resources that can outweigh the positive effects [35][41]. Nevertheless Nowak & May (1992) demonstrated that cooperation is possible for certain param-eter regions even with simple strategies such as ’always defect’ and ’always cooperate’. In this case, cooperators can group together with other coopera-tors in a way that the benefits of mutual cooperation within the cluster are greater than the losses against defectors at the boundaries. In nature, the way individuals interact is also systematically determined by how individuals move in space. In the model this can be represented by creating empty sites on the grid that players can migrate to. The overall density of the system than is the amount of players divided by the amount of sites. The extension of the spatial model to allow for movement poses new challenges to the evo-lution of cooperation, as already remarked decades ago by Hamilton (1964). With random migration the system gets closer to the scenario of random mixing which gives defectors the chance to flee from retaliation and exploit more cooperators. Additionally, the building up of clusters that are essential for preserving cooperation can be disturbed. Both factors help defection to take over so that it has generally been inferred that mobility hinders the evolution of cooperation [27][8]. Yet empty sites and mobile agents also re-duce the competition for local resources and give cooperators the chance to leave abusive environments and to join more cooperative clusters which both helps to sustain cooperation. In recent years there has been an increasing interest to specify migration mechanisms that emphasize the latter argument and reveal under which circumstances cooperation can be fostered by mobile agents [10][38][26][20][32][25].
There are several factors that determine the exact dynamics of a spatial prisoner’s dilemma with migration, most apparently the network structure, e.g. the above mentioned grid, that defines with whom an individual inter-acts, the migration rule that defines when and where an individual moves
to and the update rule, that defines how an individual chooses her strategy. Each of these components entails further structural choices. Since even sim-ple decision rules on a micro level can result in comsim-plex system dynamics on a macro level, it is hard to derive the outcomes of such a detailed model with purely mathematical tools. Due to that, the method used is to run com-puterized simulations of agents interacting with each other on the basis of individually prescribed behaviour, called agent-based modeling. Analyzing the simulation results is often the best way to gain an understanding of the causal inferences of the system [14][3]. There is always a risk, however, that the simulation results rely on assumptions that have not been judged impor-tant initially; a change in the assumptions can then lead to a significantly different result which would imply that the scope and causal inferences of the initial model have been misunderstood [11]. For spatial prisoner’s dilemma models with migration the effects of different assumptions have not yet been explored systematically which opens up the possibility of a misinterpretation of the effect of migration on cooperation levels [38][26][20]. For this rea-son, this thesis focuses on scrutinizing two model assumptions, namely the update rule and the neighbourhoods in which different actions take place. With respect to updating, many authors only consider deterministic updat-ing, where an individual copies the strategy of the player with the highest payoff in her neighbourhood, called best-takes-all rule. However, this rule does not take into account judgment errors, uncertainty and other unpre-dictable internal and external factors that possibly influence significantly how individuals interact [32]. With stochastic update rules on the other hand, these unpredictable factors are represented by introducing stochastic elements in the choice of the strategy. For static systems and systems with random movement there has already been confirmatory evidence provided that stochasticity in the update rule is a serious challenge for the outbreak and long term existence of cooperators [33][20]. The question remains in how far this discovery can be generalized. With respect to neighbourhoods, while
most authors only survey the case where all actions take place on the same local level [6][38][26], others allow players to move within a greater area than they play and update their strategy in [25][20]. What has not been consid-ered yet is the scenario where the game-play takes place on a smaller local level than the update and migration. This case seems especially interesting though since it brings the mutual benefits and competition aspects of the model on different levels. The thesis aims to extend the scope of the existing research by examining the effect of an update rule based on the exponential function as well as the effect of different neighbourhoods for the game play and the update and migration step on the outbreak of cooperation. The anal-ysis will consider three different migration rules: the first features random movement, the second movement based on local information and the third movement using non-local information. My research thereby brings several migration rules into a comparable format and sheds light on the question in how far migration fosters cooperation by examining the effect of the model assumptions.
The simulation results show that a stable coexistence of cooperation and defection is possible for a broad range of parameter settings in a spatial prisoner’s dilemma with migration. The cooperation rate realized is highly influenced by the local density - that is the share of neighbouring patches that are occupied - brought about under different migration rules. A high local density is positive to the extend that it allows a critical cluster of cooperators to come up and spread; it is negative if it prevents the possibilities for migra-tion given the migramigra-tion has a positive effect as is the case with non-random migration rules. Non-random migration rules also react less sensitively to the change from a deterministic to a stochastic update rule. The analysis provides first evidence that cooperation is fostered if the update and migra-tion take place on a bigger local level than the game-play. For random and adaptive migration this is solely due to the positive effect on the updating; the bigger migration area even has a negative effect. The thesis is structured
as follows: section 2 gives a brief literature review, section 3 explains the method used, section 4 describes the model, section 5 presents the results and section 6 draws a conclusion and discusses.
2
Literature review
Several papers have studied the effect of introducing migration into a spa-tial prisoner’s dilemma using agent-based modeling. The following review concentrates on models that use a regular grid as network structure, as will be the case in my model. Ichinose et al. (2013) investigate the promotion of cooperation if individuals move not only to neighboring but also to sites further away, the distance depending mostly on the number of defectors in their surrounding. They discover that this mechanism sustains cooperation in particular when there is a high temptation to defect. Cong et al. (2012) analyze a migration mechanism based on a relative reputation value that is adjusted over time and assumes agents to have the capability to memorize past states. Cheng et al. (2010) show that cooperation is enhanced by mi-gration if defectors have a higher probability of moving than cooperators. These papers share the deficit that their migration mechanisms are based on relatively complex formulas, while no attempt is made to justify them.
In contrast, the studies of Vainstein et al. (2007), Jiang et al. (2010) and Helbing & Yu (2009) are specially valuable since their migration and update rules are least based on certain unfunded assumptions about indi-vidual behaviour. Vainstein et al. (2007) elaborate the minimal conditions under which mobility leads to an increase in cooperation. In their model there are no explicit assortment rules or cognitively challenging strategies. Instead unconditional cooperators and defectors move with an adjustable probability in a random fashion. They use the best-takes-all update rule and all actions are performed within a von Neumann neighbourhood (see Fig. 1). They find that when the agents first play the game, afterwards migrate
Figure 1: Neighbourhoods in a regular lat-tice. Left: von Neumann, right: Moore.
and then update their strategy, co-operation is increased compared to the viscous case for a broad range of densities. A higher probability of moving is positive for the coopera-tion rate, which is surprising given that random movement is generally believed to bring the system closer
to the well-mixed state, where defection takes over [27][8]. Interchanging the migration and update step results in lower cooperation rates than for the viscous case and the effect of a higher probability of moving is negative. The authors do not study the effect of noise in strategy-updating and migration or the effect of a stochastic update rule which makes the robustness of the results questionable. Additionally, different from most other papers, play-ers update their strategy simultaneously instead of sequentially. This means that a player’s strategy update in round i cannot have an effect on the updat-ing decisions of other players in the same round which possibly qualitatively changes the results. Jiang et al. (2010) investigate the case where the migra-tion decision is not random but executed with a probability equal the number of defectors in her neighborhood divided by the number of spots that consti-tute the neighborhood. With regard to real ecosystems, this rule is intended to reflect that the number of hostile neighbours can serve as an indicator of the risk or danger of a location that will be consequently left behind. Again, the best-takes-all update rule is used and all actions are performed within a von Neumann neighbourhood. This local information based mech-anism promotes high cooperation levels for medium densities. The authors consider a noise in the strategy update but do not quantify it. In Helbing & Yu’s (2009) success-driven migration individuals calculate the expected payoff from occupying a neighboring empty site before deciding whether or not to migrate and only move if the latter is higher than the payoff of their
present location. Hence they not only leave an unfavorable location but also seek for a more favorable one. This migration mechanism tries to represent the phenomena that individuals relocate to improve their life conditions such as their living area or company. In their model the game play and strategy update take place in a von Neumann neighbourhood, the migration on the other hand in Moore neighbourhoods of different sizes. They find that pre-dominantly cooperative clusters are formed and maintained even with noise in the strategy-update and migration step. Again, the fraction of cooper-ators is highest for intermediate densities. Jiang et al. (2010) argue that the comparatively higher cooperation rates obtained with success-driven mi-gration for low densities are due to the non-local information that augments the players’ ability to find each other. They put forward that the game interactions necessary to assess a future neighbourhood are an unrealistic assumption about interactions in the real world. Considering the findings of Vainstein et al. (2007), Jiang et al. (2010) and Helbing & Yu (2009), a further highly influential factor seems to be the density of agents on the grid; with too high densities there is not enough room for assorting migration and with too low densities interaction between individuals is hindered. The problem that arises when comparing the preceding results is that the mod-els under consideration often vary in the simulation details, and it is hence hard to discern if the observed differences in the results are only due to the migration rule.
To sum up, several good attempts have been made to understand the effect of migration in a spatial prisoner’s dilemma, but until now it is hard to consolidate their findings due to the differences in the model settings. Furthermore, potentially important model assumptions have not been rigor-ously questioned yet. The contribution I aim to make with this thesis is to bring previous simulation models into a comparable format and to extend the comprehensive understanding of cooperation in spatial structures with mobile agents by systematically exploring the system’s behaviour under an
exponential update rule and different neighbourhood settings. Additionally I describe my model following the “Overview, Design concepts and Details protocol” proposed by Grimm et al. (2010) to ensure that the model is easily accessible and reproducible.
3
Methodology
I will investigate the evolution of cooperation under different combinations of migration mechanisms, densities, neighbourhood types and weights of the exponent in the update rule. With regard to migration mechanism I differ between random migration based on Vainstein et al. (2007), adaptive mi-gration based on Jiang et al. (2010) and success-driven mimi-gration based on Helbing & Yu (2009). These rules were chosen because the models they are embedded in are comparable in terms of their network structure and update dynamics but at the same time request different degrees of information for the migration mechanism. Furthermore, they are convincing because the individuals’ decision making is not based on parameters that are difficult to relate to real life observations. With respect to densities I consider all densities between 0.1 and 0.9 in steps of 0.1. This parameter is interesting because it affects how many other players one can actually interact with and how many possibilities there are for migration. Considering the update rule, it is enlightening to analyze what happens in the extremes. If the payoffs don’t matter for the the probability with which a strategy is chosen, this probability is determined only by how many players with this strategy are present in the neighbourhood. If only the payoffs matter, the strategy with the highest payoff among the neighbours is chosen for sure. With respect to neighbourhoods, I vary between all actions being undertaken in a von Neumann neighbourhood and the game-play being undertaken in a von Neu-mann, the update and migration in a Moore neighbourhood. While in a von Neumann neighbourhood the central player does not share any neighbours
with her four neighbours, in a Moore neighbourhood such mutual neighbours are possible, which could lead to different dynamics of the system [21]. Fur-thermore, as mentioned above, this constellation brings the benefits from the game interaction and the competition for resources on two different local levels. The hypothesizes that will guide the analysis of the system are the following:
Hypothesis 1: The results are qualitatively different for different densities since high densities decrease the possibilities for migration and low densi-ties the possibilidensi-ties for interaction.
Hypothesis 2: The system is more resistant to non deterministic updat-ing for migration rules that require more information, hence the least for random and the most for success-driven migration.
Hypothesis 3: The cooperation rate realized with the game-play and the update and migration step being on different local levels is higher than with all actions being on the same local level for all migration rules.
I will investigate the research question using an agent-based model pro-grammed with NetLogo [39]. The model is set up in a flexible way that allows to help understand the single as well as the interaction effects. The main pa-rameter of interest is the fraction of cooperators ρc= Nc/(N ∗ ρ) where Ncis
the number of cooperators and N the number of all agents. Since the model has stochastic effects, this value is obtained by averaging over five indepen-dent realizations. The use of an agent-based model is appropriate because heterogeneity among individuals, local interactions and adaptive behaviour are judged key for explaining the system’s behaviour on the macro level so that the outcomes are hard to derive with purely mathematical tools [14][3].
4
Model
To ensure that the model is easily accessible and reproducible I will describe it in the style of the “Overview, Design concepts and Details protocol” proposed by Grimm et al. (2010).
4.1
Entities, State Variables and Scales
The model has two entities: individuals and square patches. The patches form a 2-dimensional lattice of size LxL = 99x99 with periodic boundary conditions to prevent border effects. Different from the original setup by Nowak & May (1992), only a certain fraction of patches is occupied. A patch can be occupied by one individual only. The population density, ρ, of the system is defined by the number of individuals divided by the number of all sites. It does not change over a simulation run. Every individual is either an unconditional cooperator (C) or an unconditional defector (D) and has a score emerging from her interactions with the other individuals. Each individual has a neighborhood which defining with whom she interacts and where she can move. The neighbourhood is either von Neumann (4 nearest neighbours) or Moore (8 nearest neighbours). The assumption that individuals are either unconditional cooperators or unconditional defectors is not based on beliefs how real agents behave. It is rather a modeling technique to check if the maintenance of cooperation can be achieved even with this oversimplified assumption and hence can be hypothesized to hold true for a broader range of strategies.
4.2
Process overview and scheduling
The model contains three main actions: migration, game-play and updating. Firstly, all agents make a move according to their migration rule if there is an empty space within their neighbourhood. Secondly, agents are selected in a random sequential order to play and update their strategy (asynchronous
update). A selected agent plays the prisoner’s dilemma with all her neigh-bours which in turn play the game with all their neighneigh-bours. She afterwards updates her strategy. It should be noted that with this scheduling the up-date decision of a player in round i can influence the upup-date decision of other players the same round.
4.3
Design concepts
The main parameter of interest is the fraction of cooperators, which emerges from the update decisions made by the individuals. The update decision is determined by the scores of the individual herself and that of her neighbours, where the latter are determined by the migration mechanism. The individu-als adapt to their score by deciding whether or not to change their strategy. Depending on the migration rule, they adapt to characteristics of their neigh-bourhood by deciding whether to migrate or to stay. There is no learning present in the model. There is, however, the implicit prediction that a change of strategy to the strategy most successful in the neighbourhood will lead to a higher payoff in the next round. There is direct interaction when agents play the prisoners dilemma and update their strategy. Indirect interaction occurs when the density of the neighbourhood influences the possibilities of an individual to move to another place on the grid. In all scenarios, individu-als know if they themselves and their neighbors are cooperators or defectors, the payoff table, their own accumulated score and that of their neighbours as well as the location of themselves and their neighbours on the grid; this infor-mation is known without any error. Stochastic functions are used to initialize the location of the individuals, so that heterogeneous environments are cre-ated. Furthermore, the order in which individuals are chosen to execute each action is randomly shuffled between the individuals. There is stochasticity in the update decision as individuals choose their strategies with probabilities that depend on payoffs, according to an exponential function that is defined
below. For random and adaptive migration, stochasticity determines if an individual changes location or not and, if yes, to which of the empty sites. Lastly, stochastic functions are used to introduce noise.
4.4
Initialization
The number of individuals initialized is determined by the desired population density. There is the choice between starting with a state where everyone except one player in the center is a defector (called ”All-D” hereafter) and a state where everyone except one player in the center is a cooperator (”All-C” hereafter). Furthermore the migration rule, the neighbourhoods, the density and the weight of the exponent are adjustable.
4.5
Submodels
4.5.1 Game play
The individuals play the 2-person prisoner’s dilemma with all individuals within their neighbourhood. In case there are no neighbours, there is no game-play. In every game, two individuals simultaneously follow their strat-egy. In case two cooperators encounter they both receive the payoff R. If two defectors encounter, they both receive the lower payoff P. However, if a cooperator encounters a defector, the former receives the lowest payoff S and the latter the highest payoff T. As a matrix the payoffs look as follows:
C D C R S D T P
To create a prisoner’s dilemma the parameters are set so that T > R > P > S and 2R > T + S. Hence, independent of the opponents choice, defecting is the dominant strategy. If not stated otherwise, the parameters are set to T = 1.3, R = 1, P = 0.1 and S = 0. This certainly leaves a broad spectrum
of possible payoff combinations uncovered but presents a necessary limitation in order to concentrate on my main research interest within the given space.
4.5.2 Update
The individual updates her strategy according to the following rule:
Pr(D) = PnD i=1exiβ Pn j=1exj β and Pr(C) = PnC i=1exiβ Pn j=1exj β
where nD contains all defectors in the neighbourhood including the central
patch, nC contains all cooperators in the neighbourhood including the central
patch and n contains all neighbours including the central patch and β is the weight of the exponent. The term exiβ will be referred to as exponential
payoff in the following analysis. It should be noticed that a player only updates her strategy - and hence possibly mutates - if she has at least one neighbour and otherwise sticks to her old strategy. After one generation, all individuals with neighbours have been updated once.
4.5.3 Migration
• Random migration (RM): In the random migration scenario an indi-vidual migrates to a random empty site within her migration neigh-bourhood with Pr(migration) = 0.5.
• Adaptive migration (AM): In the adaptive migration scenario an indi-vidual migrates to a random empty site within her migration neigh-borhood with Pr(migration) = nD/npatches, where npatches equals the
number of patches that constitute her neighbourhood and nD the
• Success-driven migration (SM): In the success-driven migration sce-nario an individual inspects all empty sites within her migration neigh-bourhood with regard to the payoff she would accumulate there in the next round. She afterwards moves to the empty site with the highest potential payoff. In case two sites yield the same potential payoff, one is picked randomly.
In all scenarios the individual does not move if there is no empty site within her migration neighborhood.
4.5.4 Noise
Two sources of noise are introduced; a strategy noise α with which the in-dividual does not stick to the update rule, but sets her strategy randomly, and a migration noise γ with which the individual does not follow the chosen migration rule but makes a random move to an empty site within her neigh-bourhood. The noise depicts the fact that in nature all sorts of disturbances, such as mistakes in copying a strategy or a conscious trial-and-error behav-ior, hinder the correct execution of an action [20]. If not stated otherwise, α = γ = 0.01.
5
Results
The following chapter describes and explains the results. Firstly, a base sce-nario is considered in which players perform all actions within a von Neumann neighbourhood. The value for β is set to 130, which is the maximum value NetLogo can process in this context. Secondly, the results with different val-ues for β are presented, and lastly, the results with mixed neighbourhoods.
In all three scenarios, all selectable migration rules and densities are anal-ysed. If not stated otherwise, the output values are obtained for ρ = 0.1 − 0.4 by averaging over the last 1000 generations after 20.000 generations and for ρ = 0.5 − 0.9 by averaging over the last 100 generations after 5000 genera-tions. At these points the system has reliably reached the stable state.
5.1
Base Scenario
In the base scenario, the highest payoff among the neighbours has an expo-nential payoff that high, that the chance of choosing her strategy are virtually 1, which leads to the same results than deterministic updating. To gain an understanding of the system dynamics, it is helpful to first look at extreme scenarios. That is why, before the different migration rules are considered, in the following paragraph a preliminary analysis of the simpler scenario ρ = 1 is given. Additionally, the two initializations All-D and All-C are considered to verify that the results are independent of the initial state.
Full grid:
With ρ = 1 no movement is possible. Starting from an All-D scenario, coop-eration develops as follows: by chance a situation arises where a cooperator has two cooperating neighbours (called critical cluster hereafter). This situa-tion occurs if either three defectors mutate in the same round to cooperasitua-tion (called C-mutant hereafter), or if one (or two) defectors mutate in round i and two (or one) players mutate in round i + 1 before it is the turn of the players that mutated in round i to update their strategy (see Fig. 2).
The cooperator with two cooperating neighbours has a payoff of 2 whereas her defecting neighbours only have a payoff of 1.6. Consequently, all cooperators
Figure 2: The development and spreading of a critical cluster (upper figure). Snapshots of the typical dynamics in a full grid-scenario after t = 180, t = 250 and t = 500 generations (lower figure).
stay cooperators and the defecting neighbours of the central cooperator are converted to cooperation. This C-cluster would be stable in a world without noise since the central cooperator has a higher payoff than the surrounding defectors but the outer cooperators have a smaller payoff than the surround-ing defectors. With noise, the cluster can spread: for instance, by mutation a defector with two cooperating neighbours turns into a cooperator herself. As she has one cooperating neighbour more than her defecting neighbours, with the payoffs as they are, some of these defectors are converted to cooperators. Who and how many depends on the order in which they are updated. The spreading continues along straight boundaries where cooperators have more cooperating neighbours than their defecting neighbours. In the stable state cooperation has spread over the whole grid, with the cooperating clusters being separated by lines of defectors (see Fig. 2). These defectors have a lower payoff than their cooperating neighbors, who receive support from the
C-cluster, which leads to a stable coexistence of the two strategies. The co-operation rate is about 50 %. Starting from an All-C scenario, defection can spread until the same state is reached where there are so many defectors that a new defector is unlikely to find a position where she can earn more payoff than her cooperating neighbours.
Random migration:
With random migration no cooperation can develop for ρ = 0.1−0.7 (see Fig. 4). Since random movement destroys any sort of cluster, the local density is very low with enough room for movement. The low local density in turn leads to a low probability of three mutations arising next to each other. If, nevertheless, such a critical cluster comes up, it either falls apart before the players can benefit from it or survives for some time but cannot spread. For ρ = 0.8 −0.9 cooperators can stably coexist with defectors and reach an aver-age share of around 0.3 and 0.5 respectively. This is because the higher global density increases the local density and makes the occurrence of a critical clus-ter more likely. Moreover, its form becomes more stable because of the limited movement so that cooperators can profit from giving each other payoff. For ρ = 0.8 cooperation can spread where cooperators in a C-cluster move next to other cooperators or do not move at all. Cooperators that leave the C-cluster are converted to defection. The equal spread of empty spaces allows defectors to move into a C-cluster. As a result, the C-clusters transform constantly and small clusters often disappear over time. At the beginning cooperation can spread more than it is destroyed because cooperators have a higher chance to be next to cooperators than defectors so that their payoff is higher. With the rise of the cooperation rate, defectors get more cooperating neighbours and can better withstand the overtake of cooperation. In the stable state the point is reached where players face a mixture of defectors, cooperators and empty spots that on a global level equally supports the conversion from
Figure 3: Snapshots of the typical dynamics with random migration for ρ = 0.8 after t = 1300, t = 4400 and t = 5000 generations.
cooperation to defection as the other way around. For ρ = 0.9 the movement is even more limited and the stable state resembles that of the full grid.
Figure 4: Average cooperation rates starting with All-D. With success-driven migration for ρ = 0.1 − 0.2 the results are obtained by averaging over 1000 generations after 300.000 generations. Only then the stable state is reached.
Different from the other mi-gration rules, the restricted movement has no negative effect as the random move-ment does not additionally foster assortment; for the same densities, higher co-operation rates are realized without migration which im-plies that the impact of mi-gration is even negative (see table 1 in the appendix).
This is because in the static case, structures of C-clusters can survive that would not be stable if a defector was to fill out one of the empty spots in or around the cluster, as was already observed by Vainstein & Arenzon (2001). Starting with an All-C scenario, the system ends up in the same stable state than with All-D. The final outcome is hence independent of the initial state. For ρ = 0.1 − 0.7 a defector mutant in most cases has a higher payoff than
her neighbours since the local density is not high enough to give cooperators enough support from a cooperator within the cluster to resist the overtake. This underlines that in the All-D scenario, it is as much the incapability to build up cooperative clusters as the incapability of such an unlikely cluster to resist an overtake, that make it impossible to build up cooperation for low and medium densities. The constant movement also prohibits defectors from getting stuck in a neighbourhood with low payoffs so that they can carry on to exploit cooperators. For higher densities defection can spread until the point is reached where they face a mixture of defectors, cooperators and empty spots that allows parts of the cooperators to withstand defection taking over.
Adaptive migration:
With adaptive migration no cooperation can develop for ρ = 0.1 − 0.4. With everyone initially being a defector players try to have no neighbours at all. This leads to the local density being as low as possible. Consequently, there are very few interactions that could give rise to mutation and the chance that three neighbouring players mutate is infinitesimal small. For ρ = 0.5 − 0.8 cooperators can stably coexist with defectors and their fraction rises with the density. The higher density makes the occurrence of a critical cluster possible. If the cluster survives is determined by the probability with which the cooperators migrate away from each other. For medium densities, the probability of movement is small because there is a reasonable likelihood that the mutated neighbours make out a great share of all neighbours. For higher densities, the lack of space limits the movement. Since cooperators with adaptive migration do not attract other players, the amount of players at the boundaries of a C-cluster - and hence how much cooperation can spread - is determined only by the global density. With higher densities it is harder for the C-cluster to stay outside the reach of defectors. The latter are thus mostly
Figure 5: Snapshots of the typical dynamics with adaptive migration for ρ = 0.8 after t = 70, t = 150 and t = 1000 generations.
converted. Only few defectors survive, either by having no neighbours or by profiting from a situation where their neighbouring cooperators can’t move but don’t have a higher payoff. However, defectors cannot stably coexist with cooperators on a local level if there is enough room for migration: the neighbouring cooperators eventually move away and leave the defector with a low payoff so that she is likely to be converted back to cooperation, as was already pointed out by Jiang et al. (2010). Like that the building of streets of defectors at the boundaries of cooperating clusters is prevented. For ρ = 0.9 the cooperation rate decreases again because, with such few empty spaces, the possibilities to flee from a defecting neighbour are restricted. Structures as in the full grid scenario can develop. In contrast, Jiang et al. (2010) observe that cooperation cannot break out for high densities. The lack of information about their simulation details unfortunately prevents a further investigation of this phenomenon. Starting from an All-C scenario yields the same stable state. Since cooperators don’t move away from each other, the likelihood that at some point a cooperator will have a D-mutant as neighbour is very high. Since defectors do not rely on a cluster of players with the same strategy to have a high payoff, they convert their neigbouring cooperators even for low densities if the latter updates after her. Only in the next round the cooperator could flee. Naturally, the timing of the update
and the structure of the neighbourhood allow for different results of the interaction, but over time all cooperators will have been converted for low densities. For higher densities more cooperators have sufficient cooperating neighbours to withstand the overtake of defection. Compared to random migration, cooperators are also challenged less by defecting neighbours for medium densities as the latter don’t move if they are at a spot with no neighbours.
Success-driven migration:
With success-driven migration cooperators can coexist with defectors for all densities on high levels. Whoever has other players within her migration neighbourhood moves next to them because having any neighbour is better than having no neighbour. Clusters of defectors are formed so that the local density is high. This makes it more likely than for the other migration rules that a critical cluster appears. Cooperators in a cluster do not move because staying in each others neighbourhood is their payoff maximizing position; if they move, then only in an attempt to have more cooperating neighbours. Since the local density is high, for all densities, cooperation within the cluster spreads similar to the full grid case, with ”streets” of defectors splitting the clusters of cooperators. As having a defector as neighbour, for a cooperator with respect to payoff is as good as having no neighbour, the cooperators stay in this position even if their would still be room for migration. This coexistence is hence stable. The structure and size of the clusters never-theless change over time due to random moves and mutations that change the most profitable locations and thereby induce a movement. Lastly, D-mutants in a C-cluster cause some of their neighbours to become defectors as well. However, defection cannot take over because, due to the high density of the C-cluster, most cooperators neighbouring the defectors have neigh-bours that earn higher payoff than the defector herself. As with adaptive
Figure 6: Snapshots of the typical dynamics with success-driven migration for ρ = 0.3 after t = 880, t = 3000 and t = 20.000 generations (upper figure) and for ρ = 0.8 after t = 160, t = 300 and t = 1000 generations (lower figure).
migration, there is a decrease in the cooperation rate for ρ = 0.9. The de-crease is smaller, though, because the environment a D-mutant faces changes less than in the adaptive case. Additionally, even if there is space, it is not used by cooperators to flee defectors. I hypothesized that the reason why success-driven migration enables cooperation to stably coexist with defection for all densities lies in the fact that it always leads to a high local density and supports cooperators in coming and staying together. To test this hy-pothesis, I programmed a migration rule where players only explore their neighbourhood if they have no cooperating neighbour. They either move to a neighbouring empty site where they expect to have at least one cooperating neighbour or - if there are no cooperators at all - to the position where they expect to have the most neighbours. The resulting cooperation rates do not
differ from the success-driven case (see table 2). This lends support to the hypothesis and suggests that the precise payoffs players can earn are of little importance, given the fact that players with the adapted rule are likely not to end up at their payoff maximizing position. Starting with All-C does not change the stable state of the system in the long run. D-mutants migrate to the spot where they expect the most cooperating neighbours. They after-wards stay in stable coexistence, or they are converted back to cooperation if their neighbouring cooperators are more successful. When the stable state is reached, there exist so many defectors that a new defector is unlikely to find a position where she can earn more payoff than her cooperating neighbours.
From the above results it can be concluded that the most important factors determining if a migration rule leads to stable coexistence of cooperation and defection are firstly, the possibility of the emergence of a critical cluster, which in turn needs a sufficiently high local density, secondly, the sticking together of the critical cluster, thirdly, the possibilities of the cooperative cluster to spread and lastly, the reaction of cooperators in the cluster to defectors. Viewing the migration rules with respect to these factors, it can be said that with random migration there is an occasional emergence of a critical cluster from medium densities onward but the chance for it to stay together and spread is only existent for high densities. There is no migra-tion based defense against defectors. With adaptive migramigra-tion and All-D, the local density is as low as possible and hence the emergence of a criti-cal cluster only possible from medium densities onward. The chances that cooperators stay together are high, and the possibilities to spread depend on the overall density since the rule does not automatically lead to a high local density. If there is room for migration, cooperators have a good defense against defectors. With success-driven migration, there is a high likelihood of the emergence of a critical cluster for all densities because the local density is always high. The likelihood that cooperators stay together is high as well
because that is their payoff maximizing position. There is a high likelihood of spreading because of the high local density and the attraction of cooper-ators for other players. However, this rule offers no direct defense against D-mutants. It is confirmed that success-driven migration leads to higher co-operation rates than adaptive migration, as already detected by Jiang et al. (2010). The preceding analysis demonstrated that this difference can be re-produced without using the unrealistic assumption of game interactions with all potential neighbours.
5.2
Scenario Different betas
For β = 0 the outcome of the individual exponential payoff is always e0 = 1 so
that the probability with which a strategy is chosen is only determined by how many players with this strategy are present in the neighbourhood. For values of β between 0 and infinity, the lower the beta, the lower the probability that the strategy is picked that was played by the player with the highest payoff in the neighbourhood. Hence some defectors that in the deterministic case would become cooperators stay defectors, and some defectors that were meant to stay defectors become cooperators. The same logic of course applies to cooperators. An exemplary analysis of the case β = 3 will be done. All other parameters are kept as in the base scenario, and only the All-D initialization is considered.
Random migration:
For random migration the cooperation rate realized is around 50 % for all densities and hence higher than in the base scenario (see Fig. 7). Whereas with β = 130 (referred to as high β hereafter) the probability for a single cooperator to stay a cooperator is virtually going to 0, with β = 0 it is
at least 0.2, which would be the result of her being the only cooperator in a neighbourhood of four defectors. The exact case to case probability
Figure 7: Average cooperation rates with random migration for different values of β.
Figure 8: Conversion rates with random mi-gration for ρ = 0.9, β = 3 and t = 0 − 2000. Upper figure: Share of C (D) turning back to C (D) after having been converted to D (C). Lower figure: Share of C (D) staying C (D) after having been converted to C (D)
depends on the composition of the neighbourhood. A defector neigh-bouring the mutant, via the same mechanism, has a probability of at least 0.2 to switch to cooperation. Since the migration rule does not promote assortment, the chance that at some local point there will be more cooperators around is equal to the chance that there will be more defectors around. Where by chance more defectors were turned into co-operators, the chance of coopera-tion spreading further goes up. This leads to regions with almost only co-operators and regions with almost only defectors. In the long run, both strategies have an equal share. In-termediate values of β, such as β = 3, have a negative effect on the co-operation rate because the necessary structure of the cooperative clusters is destroyed without enough play-ers being converted to cooperation
to compensate the negative effect. Additionally, a defector that foregoes the strategic update to become a cooperator is more likely to stay a defector than is the cooperator to stay a cooperator (see Fig. 8). That is because there are less compositions of the neighbourhood in which a defector is
con-Figure 9: Snapshots of the typical dynamics with random migration for ρ = 0.3 after t = 50, t = 150 and t = 500 generations (upper figure) and for ρ = 0.8 after t = 10, t = 50 and t = 250 generations (lower figure).
verted to cooperation than compositions in which a cooperator is converted to defection. A defector that should have stayed a defector but becomes a cooperator is also less likely to stay a cooperator in the following rounds than a cooperator that became a defector. High noise levels, on the other hand, have a positive effect on the cooperation rate, especially for low and medium densities where a high rate of chance conversion is the only way for cooperative clusters to survive. For high densities a very low noise level is as good as a very high noise level since cooperative clusters can also be build up via the deterministic update rule (see base scenario).
Adaptive migration:
With β = 0 the cooperation rate realized is higher than in the base scenario
Figure 10: Average cooperation rates with adaptive migration for different values of β.
(see Fig. 10). Where for ρ = 0.1 − 0.4 with a high β no cooperation can develop, with β = 0 coopera-tion rates of up to almost 50 % can be reached. As in the base scenario, the lower the density, the lower the chance that two players interact and mutate. Different from the base
sce-nario, a single cooperator can now potentially survive and convert her neigh-bours, which explains the higher cooperation rate. The cooperation rate for low densities does not go up as high as with random migration since a great share of defectors are left with no neighbours. Cooperators are hence more likely to mutate which introduces a relative advantage for defection. For higher densities the difference between the two scenarios decreases because both allow cooperation to develop. Where for ρ = 0.5 − 0.6 in the base scenario the necessary clustering of cooperators is not always high enough to withstand defectors taking over, with β = 0 also less clustered coopera-tors have a positive probability to survive and spread their strategy. For a D-mutant the effect of the payoffs being unimportant is negative: different from a cooperator, she does not need a cluster to survive and loses this ad-vantage for β = 0. For ρ = 0.7 − 0.8 the proportion of empty sites is optimal for both spreading and fleeing so that the advantage of a D-mutant over a C-mutant is decreased and closer to the case of β = 0, where both mutants have equal chances to survive and spread. The advantage is regained in the base scenario for ρ = 0.9, where cooperators are less effective in preventing defectors to take over. This is reflected in a lower cooperation rate compared to β = 0.
Figure 11: Snapshots of the typical dynamics with adaptive migration for ρ = 0.3 after t = 50, t = 100 and t = 1000 generations (upper figure) and for ρ = 0.8 after t = 10, t = 50 and t = 150 generations (lower figure).
Figure 12: Conversion rates with adaptive migration for ρ = 0.5 (left), ρ = 0.9 (right), β = 3 and t = 0 − 2000.
One can clearly see that adaptive migration in itself leads to a cluster-ing of cooperators that is beneficial for their spreading. For lower den-sities the better chances to spread are outweighed by the constant chal-lenge of D-mutants, for higher densi-ties it is the other way around. With β = 3 there is a negative effect only for higher densities. Because it is harder to flee from defectors and leave them with a low payoff, defectors that come up by chance can spread. Co-operators that come up by chance, on the other hand, have no room to leave
the unfavourable neighbourhood: with ρ = 0.5 there is no big difference be-tween the share of cooperators that turned to defection but is going back to their original strategy the following round and the corresponding share of defectors (see Fig. 12). There is also no clear difference between the share of cooperators that would have converted to defection in the deterministic case, but stayed cooperators and managed to stick to cooperation in the fol-lowing round, and the corresponding share of defectors. For ρ = 0.9 a clear difference for the two strategies can be seen, with defectors more often going back to defection after a conversion to cooperation and more often staying a defector after a conversion to defection.
Success-driven migration:
With success-driven migration the cooperation rate obtained with β = 0 is higher than in the base scenario for all densities (see Fig. 13). Already in the base scenario, cooperation could come up and coexist with defection for lower densities because of the high local density. It is hence less important that with β = 0 cooperation can come up even without clustering. Since the migration mechanism is not directly influenced by the different value of β, cooperators still mutually seek to be next to each other whereas defec-tors seek cooperadefec-tors but are not sought in return. Hence cooperadefec-tors are more abundant in their neighbourhood and have a higher probability to con-vert their defecting neighbours than the other way around. Where in the base scenario defectors could stably exist at the boundaries of C-clusters, they are now constantly challenged by a positive probability to convert to cooperation. The same holds true for the cooperators at the boundaries though, and since their probabilities of conversion are the same, it does not explain the higher cooperation rate. What does change with β = 0 is that, as explained above, a D-mutant surrounded by cooperators has a signifi-cantly lower probability to stay a defector and to convert her neighbours.
Figure 13: Average cooperation rates with success-driven migration for different values of β.
Figure 14: Conversion rates with success-driven migration for ρ = 0.5 (left), ρ = 0.9 (right), β = 3 and t = 0 − 2000.
As a result, the system is bet-ter secured against the spreading of streets of defectors. The higher noise level induced with β = 3 does not alter the results for any den-sity. For ρ = 0.5 the difference between the share of cooperators that goes back to cooperation after having been turned to defection by chance and the share of defectors that goes back to defection is low. The same holds true for the share of players that stay with the opposed strategy even though following the deterministic rule they would have stuck to their old strategy (see Fig. 14). For ρ = 0.9 there is a differ-ence in the conversion rates but it
is smaller than for the other rules. The cooperation rate presumably does not decrease for the highest density, however, because already in the base scenario cooperation could spread over the whole grid and the maximum amount of defectors that could stably coexist with cooperation had already been reached.
Summing up, with β = 0 higher cooperation rates are realized for all den-sities and migration rules. How big the difference is depends on how well critical clusters could come up and spread already in the base scenario. For intermediate values of β it is important how likely the system dynamics are to establish the same structures than in the deterministic case. For random migration even a quite low noise level can make the system behave differ-ently than in the deterministic case, for adaptive migration a negative effect
Figure 15: Snapshots of the typical dynamics with success-driven migration for ρ = 0.2 after t = 10, t = 60 and t = 2500 generations (upper figure) and for ρ = 0.8 after t = 10, t = 60 and t = 250 generations (lower figure).
is only present for high densities and for success-driven migration there is no deviation of the final results from the base scenario at all.
5.3
Scenario Mixed neighbourhoods
In this scenario players play the prisoner’s dilemma with their von Neumann neighbours but migrate and update within the Moore neighbourhood. All other parameters are kept as in the base scenario.
Random migration:
Compared to the base scenario cooperation can coexist with defection from a
Figure 16: Average cooperation rates with random migration for the mixed and the base scenario.
lower density onward (see Fig. 16). Even though the chances that a crit-ical cluster comes up are the same in both scenarios, the chances of it to survive differ. For ρ = 0.7 in the base scenario clusters of cooperators do not convert enough cooperators to reach a size that allows them not to fall apart in the long term. In the mixed case, once a critical
clus-ter does not fall apart in the first migration step, its spreading is more likely: with the reach of the most successful cooperator in the cluster being broader, also defectors in the corners of the Moore neighbourhood can be converted where at this stage these defectors in turn cannot convert the cooperator. The broader spreading of cooperation increases the chance that after the next migration step enough cooperators are in each others von Neumann neigh-bourhood to survive and spread further. On the other hand, a D-mutant in a cooperating neighbourhood can convert more players to defection than in the base scenario, an advantage not shared by the C-mutant. The effect seems to be not big enough, however, to outweigh the increased regular spreading of cooperation so that cooperation in the long term reaches a higher level. The cooperation rate goes up further for ρ = 0.8 but decreases for ρ = 0.9. This is presumably because in the mixed scenario with ρ = 0.9 it is more likely that a situation arises where a cooperator in the center of a cooperating von Neumann neighbourhood has defectors within her Moore neighbourhood that have the same payoff than her (4 ∗ R = 4 and 3 ∗ T + P = 4). The defector that would have stayed a defector in the base scenario can hence be
Figure 17: Snapshots of the typical dynamics with random migration for ρ = 0.7 after t = 480, t = 700 and t = 1000 generations.
converted to cooperation, but, at the same time, the cooperator that before for sure stayed a cooperator can be converted to defection. The negative impact of a converted cooperator is greater than the positive one of a con-verted defector since the conversion of the cooperator is more likely to lead to further conversions of cooperators. In the base scenario the situation of equal payoffs is excluded, and no drop in the cooperation rate can be ob-served. It is also worth noticing that if only the migration step is executed in the Moore neighbourhood, no cooperation can develop for ρ = 0.1 − 0.8 and only to a very low extent (0.2) for ρ = 0.9 (see table 3). If on the other hand only the update is executed in the Moore neighbourhood, the cooper-ation rate realized is higher than in the mixed scenario (see table 4). The highest cooperation rates are reached when the update happens in the Moore neighbourhood and there is no migration at all (see table 1). This further stresses the point that the update happening on a higher level is positive for the development of cooperation while random movement has a negative effect that increases with the migration area.
Adaptive migration:
With the migration taking place in the Moore neighbourhood, players also move proportionally to the amount of players in the Moore neighbourhood. In contrast to the base scenario cooperation can stably coexist with de-fection from lower densities onward (see Fig. 18). For ρ = 0.1 − 0.4
Figure 18: Average cooperation rates with adaptive migration for the mixed and the base scenario.
with the update happening in a Moore neighbourhood, there are more mutations than in the base sce-nario because it is harder for players not to be within each others reach. For ρ = 0.1 − 0.2 there are always more cooperators around than can be explained purely by mutation. That is because one effect of the up-date happening in a different
neigh-bourhood than the game-play is, that two players with different strategies that are within each others Moore neighbourhood but have no neighbours in their von Neumann neighbourhood have the same score, namely 0. If there are no other players within their Moore neighbourhood, this leads to a fifty-fifty chance for either strategy to be selected in the updating (see Fig. 19). With game-play and update happening on the same local level, two neighbouring players with different strategies would not have the same score so that this situation is excluded. As a result, in the mixed case groups of cooperators come about that are out of each others von Neumann but within each others Moore neighbourhood. These clusters are not stable, however: they fall apart either through a further mutation or a defector coming into their von Neumann neighbourhood; since the cooperators are not interacting with each other for the game play, they cannot withstand the overtake of the defector.
Figure 19: Typical snap-shot of two players with a payoff of 0.
The same dynamics can be observed with random migration but to a smaller extent since the chances of having no one in your Moore neighbourhood are smaller. For higher densities the cooperation rate first rises but then decreases for ρ = 0.8 − 0.9. With a Moore update neighbourhood already one mutation and hence a lower den-sity is enough to convert other players, as explained above. If the mutant moves in between the two players she converted, a row of three cooperators is formed. As in the base scenario, with medium densities, the probability of
Figure 20: Snapshots of the typical dynamics with adaptive migration for ρ = 0.3 after t = 350, t = 1100 and t = 3000 generations (upper figure) and ρ = 0.8 after t = 50, t = 120 and t = 200 generations (lower figure).
these cooperators to move is reasonable since there are not many defectors in the Moore neighbourhood; with higher densities, there are limited possibili-ties to move so the C-cluster stays together. Cooperation can spread better and reach a higher level because it is harder for the clustered cooperators to
stay away from defectors, which leads to more neighbours to pass the strat-egy on for lower densities. The spreading of cooperation is further enhanced as there is still enough room to flee from defectors to prevent them spread-ing on their turn. With the density gettspread-ing higher, defectors are likely to have enough cooperating neighbours on their own to withstand a conversion or spread their strategy so that the long term cooperation rate goes down. The difference to the base scenario presumably shrinks with higher densities because the effect of the higher reach of cooperators loses importance. The fact that, different from the random case, the cooperation rate does not fall below the level of the base scenario is possibly explained by the fact that with adaptive migration less defectors survive and can potentially end up in a position where they can convert a focal cooperator. With only the mi-gration happening in the Moore neighbourhood, the obtained cooperation rate is lower than in the base and mixed scenario (see table 3). This in the first instance is surprising since the effect of adaptive migration in the base scenario is positive. However, the bigger migration area does not increase a cooperator’s chances not to end up next to a defector after the migration step while it could increase her chances to move away from her cooperating neighbours. The highest cooperation rates are obtained with only the update happening in the Moore neighbourhood (see table 4). One can hence con-clude that adaptive migration has a positive effect, but this effect decreases as the migration area grows bigger.
Success-driven migration:
The cooperation rate realized is higher than in the base scenario for all den-sities (see Fig. 21). In the base scenario streets of defectors could stably exist between cooperative clusters because their score (2.8) was higher than the one of their direct neighbours (2) but lower than the one of the focal cooperator (4) (see Fig. 22). This is not possible anymore with the update
Figure 21: Average cooperation with success-driven migration for the mixed in comparison to the base scenario.
happening in the Moore neighbour-hood as the defector also takes the focal cooperator into consideration for her update decision and is thus converted. Different from the other two migration rules, there is no drop in the cooperation rate for the high-est densities. As mentioned in the base scenario, this could be due to the fact that the local density is
al-ways high so that players face similar environments for different densities. With only the migration taking place in a Moore neighbourhood, the results are still lower than in the mixed case but different to the other two migration rules they are slightly higher than in the base scenario (see table 3 and 4).
Figure 22: Frozen border with success-driven migration.
The positive effect of the migration is hence increased by the greater migration area.This can be explained by the fact that with success-driven migration individuals not only chose to leave but also chose where to go. A greater migra-tion area rather enhances the possibilities of cooperators to find each other than that it makes them move away from each other, as is the case with the other two rules. With only the update happening in the Moore neighbourhood results similar to the mixed scenario are obtained. This suggests that given that the update is in the Moore neighbourhood, a greater migra-tion area does not have a positive effect on its own. This is because if your Moore neighbourhood is constituted mainly of cooperators, which is fostered by an update on this level, so is your von Neumann neighbourhood and there is no need to inspect a greater area in order to find the most profitable spot to be.
Figure 23: Snapshots of the typical dynamics with success-driven migration for ρ = 0.3 after t = 20, t = 8000 and t = 20.000 generations (upper figure) and for ρ = 0.8 after t = 225, t = 2700 and t = 350 generations (lower figure).
Summing up, it can be stated that game-play and update happening on different local level is beneficial for the cooperation rate because it can spread already for lower densities and reach higher levels due to the wider reach of successful players. The effect of a bigger migration area is only positive for success-driven migration but becomes redundant if the update also takes place in the Moore neighbourhood. For random and adaptive migration the effect of a bigger migration area is negative.
6
Conclusion and discussion
The impact of migration on the outbreak of cooperation is a disputed topic in academic research. The preceding analysis shows that an outbreak is possible in a spatial prisoner’s dilemma with migration even if the update is not de-terministic and that the cooperation rates realized are increased by bringing the update on a bigger scale than the game-play and migration. Coopera-tion rates are enhanced for a broad range of parameter settings compared to the viscous case if the migration rule is not random. The crucial factors for cooperation to stably exist in the long term are the possibility of the emergence of a critical cluster of cooperators, the sticking together of the cluster and its possibilities to spread and lastly the reaction of cooperators to defectors. As predicted, the density highly influences to which extend these factors are present. However, it is primarily the local density that is important. This local density can vary substantially for the same low and medium global density under different migration rules: with random migra-tion the local density equals the global density, with adaptive migramigra-tion the local density depends on the strategies present in the neighbourhood and with success-driven migration the local density is always high. It thus can-not be claimed that low densities in general lead to cooperation rates that are different from those at high densities. For high global densities, the local density is inevitably high which decreases the possibilities for migration. For random migration the limited migration is positive as the movement is in fact detrimental to cooperation. For adaptive migration the limited migra-tion has a negative effect since cooperators can flee less likely from defectors. For success-driven migration the effect is merely slightly negative as the local density is always high. It is noticeable that the constantly high cooperation rates brought about by success-driven migration are not due to the extensive fictitious game-plays. They are rather due to the fact that players seek to have neighbours and cooperators come and stay together which seem to be
less demanding assumptions on real interactions. The hypothesis that co-operation rates obtained with success-driven migration are least influenced by a change in the value of β and cooperation rates obtained with random migration most, is shown to be true. This is because random migration offers neither mechanisms to prevent defectors that come up by chance to take over nor to incorporate cooperators into a cooperative cluster. In a noisy environ-ment migration hence only presents a benefit if it features a mechanism that leads to the necessary clustering of cooperators. It is revealed that coopera-tion is addicoopera-tionally fostered if the game-play takes place in a von Neumann, the update and migration step on the other hand in a Moore neighbourhood. Yet, for random and adaptive migration this is solely thanks to the positive effect of the updating. The bigger migration area even has a negative effect. On the base of this evidence it seems fair to propose that more attention should be directed at the role of the level of updating.
Since the scope of this thesis is too narrow to examine all initial assump-tions, it is possible that the results don’t hold under different specifications. The most important parameters that were hold constant and could represent a modeling artefact are presented in the following discussion and should be subject to further research. The model is limited to the case of a regular square lattice, though there are other networks possible such as small-world networks [1] or scale-free networks [12]. The results emerging from these models could be qualitatively different, and some enlightening patterns are possibly not brought to light by regular models [21]. Furthermore, only asyn-chronous strategy update was considered. It has been pointed out, however, that for individuals embedded in some sort of social setting a synchronous update would be an unrealistic premise since there is no universal instance that induces individuals to revise their behaviour at the same point of time. The opinions regarding the robustness towards changes in synchronization are diverging [13][21]. Lastly, a smaller mutation rate could lead to different results. Yet, most likely, merely the time until the stable state would have
been prolonged as is observed in single test simulations and confirmed by Ichinose et al. (2013) and Helbing & Yu (2010). Given these limitations the results should not be generalized without confirmatory research but never-theless provide additional evidence that migration serves as a key explanation for the existence of cooperation. This is even more true if one believes that most real world agents don’t act randomly but base their movement on a heuristic that proves advantageous for them.
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