Evolutionary games of multiplayer cooperation on graphs

AbstractThere has been much interest in studying evolutionary games in structured populations, often modelled as graphs. However, most analytical results so far have only been obtained for two-player or linear games, while the study of more complex multiplayer games has been usually tackled by computer simulations. Here we investigate evolutionary multiplayer games on graphs updated with a Moran death-Birth process. For cycles, we obtain an exact analytical condition for cooperation to be favored by natural selection, given in terms of the payoffs of the game and a set of structure coefficients. For regular graphs of degree three and larger, we estimate this condition using a combination of pair approximation and diffusion approximation. For a large class of cooperation games, our approximations suggest that graph-structured populations are stronger promoters of cooperation than populations lacking spatial structure. Computer simulations validate our analytical approximations for random regular graphs and cycles, but show systematic differences for graphs with many loops such as lattices. In particular, our simulation results show that these kinds of graphs can even lead to more stringent conditions for the evolution of cooperation than well-mixed populations. Overall, we provide evidence suggesting that the complexity arising from many-player interactions and spatial structure can be captured by pair approximation in the case of random graphs, but that it need to be handled with care for graphs with high clustering.Author SummaryCooperation can be defined as the act of providing fitness benefits to other individuals, often at a personal cost. When interactions occur mainly with neighbors, assortment of strategies can favor cooperation but local competition can undermine it. Previous research has shown that a single coefficient can capture this trade-off when cooperative interactions take place between two players. More complicated, but also more realistic models of cooperative interactions involving multiple players instead require several such coefficients, making it difficult to assess the effects of population structure. Here, we obtain analytical approximations for the coefficients of multiplayer games in graph-structured populations. Computer simulations show that, for particular instances of multiplayer games, these approximate coefficients predict the condition for cooperation to be promoted in random graphs well, but fail to do so in graphs with more structure, such as lattices. Our work extends and generalizes established results on the evolution of cooperation on graphs, but also highlights the importance of explicitly taking into account higher-order statistical associations in order to assess the evolutionary dynamics of cooperation in spatially structured populations.

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Ordering structured populations in multiplayer cooperation games

10.1101/021550 ◽

2015 ◽

Author(s):

Jorge Peña ◽

Bin Wu ◽

Arne Traulsen

Keyword(s):

Population Structure ◽

Spatial Structure ◽

Structured Populations ◽

Evolution Of Cooperation ◽

Multiplayer Games ◽

Spatial Games ◽

Population Structures ◽

Single Structure ◽

Containment Order ◽

Update Rules

AbstractSpatial structure greatly affects the evolution of cooperation. While in two-player games the condition for cooperation to evolve depends on a single structure coefficient, in multiplayer games the condition might depend on several structure coefficients, making it difficult to compare different population structures. We propose a solution to this issue by introducing two simple ways of ordering population structures: the containment order and the volume order. If population structure 𝒮1 is greater than population structure 𝒮2 in the containment or the volume order, then 𝒮1 can be considered a stronger promoter of cooperation. We provide conditions for establishing the containment order, give general results on the volume order, and illustrate our theory by comparing different models of spatial games and associated update rules. Our results hold for a large class of population structures and can be easily applied to specific cases once the structure coefficients have been calculated or estimated.

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Ordering structured populations in multiplayer cooperation games

Journal of The Royal Society Interface ◽

10.1098/rsif.2015.0881 ◽

2016 ◽

Vol 13 (114) ◽

pp. 20150881 ◽

Cited By ~ 15

Author(s):

Jorge Peña ◽

Bin Wu ◽

Arne Traulsen

Keyword(s):

Population Structure ◽

Spatial Structure ◽

Structured Populations ◽

Evolution Of Cooperation ◽

Multiplayer Games ◽

Spatial Games ◽

Population Structures ◽

Single Structure ◽

Containment Order ◽

Update Rules

Spatial structure greatly affects the evolution of cooperation. While in two-player games the condition for cooperation to evolve depends on a single structure coefficient, in multiplayer games the condition might depend on several structure coefficients, making it difficult to compare different population structures. We propose a solution to this issue by introducing two simple ways of ordering population structures: the containment order and the volume order. If population structure is greater than population structure in the containment or the volume order, then can be considered a stronger promoter of cooperation. We provide conditions for establishing the containment order, give general results on the volume order, and illustrate our theory by comparing different models of spatial games and associated update rules. Our results hold for a large class of population structures and can be easily applied to specific cases once the structure coefficients have been calculated or estimated.

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Eco-evolutionary spatial dynamics of non-linear social dilemmas

10.1101/660266 ◽

2019 ◽

Author(s):

Chaitanya S. Gokhale ◽

Hye Jin Park

Keyword(s):

Spatial Structure ◽

Public Goods ◽

Public Good ◽

Spatial Dynamics ◽

Social Dilemmas ◽

Evolutionary Games ◽

Evolution Of Cooperation ◽

Evolutionary Trajectory ◽

The Public ◽

Non Linear

AbstractSpatial dynamics can promote the evolution of cooperation. While dispersal processes have been studied in simple evolutionary games, real-world social dilemmas are much more complicated. The public good, in many cases, does not increase linearly as per the investment in it. When the investment is low, for example, every additional unit of the investment may help a lot to increase the public good, but the effect vanishes as the number of investments increase. Such non-linear behaviour is the norm rather than an exception in a variety of social as well as biological systems. We take into account the non-linearity in the payoffs of the public goods game as well as the natural demographic effects of population densities. Population density has also been shown to impact the evolution of co-operation. Coupling these non-linear games and population size effect together with an explicitly defined spatial structure brings us one step closer to the complexity of real eco-evolutionary spatial systems. We show how the non-linearity in payoffs, resulting in synergy or discounting of public goods can alter the effective rate of return on the cooperative investment. Synergy or discounting in public goods accumulation affects the resulting spatial structure, not just quantitatively but in some cases, drastically changing the outcomes. In cases where a linear payoff structure would lead to extinction, synergy can support the coexistence of cooperators and defectors. The combined eco-evolutionary trajectory can thus be qualitatively different in cases on non-linear social dilemmas.

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Evolution of Cooperation for Multiple Mutant Configurations on All Regular Graphs with N ≤ 14 Players

Games ◽

10.3390/g11010012 ◽

2020 ◽

Vol 11 (1) ◽

pp. 12

Author(s):

Hendrik Richter

Keyword(s):

Regular Graph ◽

Good Predictor ◽

Computational Approach ◽

Structured Populations ◽

Evolution Of Cooperation ◽

Regular Graphs ◽

Weak Selection ◽

Structure Coefficient ◽

Low Degree ◽

Emergence Of Cooperation

We study the emergence of cooperation in structured populations with any arrangement of cooperators and defectors on the evolutionary graph. In a computational approach using structure coefficients defined for configurations describing such arrangements of any number of mutants, we provide results for weak selection to favor cooperation over defection on any regular graph with N ≤ 14 vertices. Furthermore, the properties of graphs that particularly promote cooperation are analyzed. It is shown that the number of graph cycles of a certain length is a good predictor for the values of the structure coefficient, and thus a tendency to favor cooperation. Another property of particularly cooperation-promoting regular graphs with a low degree is that they are structured to have blocks with clusters of mutants that are connected by cut vertices and/or hinge vertices.

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Parent-preferred dispersal promotes cooperation in structured populations

Proceedings of The Royal Society B Biological Sciences ◽

10.1098/rspb.2018.1949 ◽

2019 ◽

Vol 286 (1895) ◽

pp. 20181949 ◽

Cited By ~ 9

Author(s):

Xiaojie Chen ◽

Åke Brännström ◽

Ulf Dieckmann

Keyword(s):

Pairwise Comparison ◽

Small World ◽

Structured Populations ◽

Evolution Of Cooperation ◽

Pair Approximation ◽

Scale Free ◽

Scale Free Networks ◽

Different Types ◽

Population Structures ◽

Update Rules

Dispersal is a key process for the emergence of social and biological behaviours. Yet, little attention has been paid to dispersal's effects on the evolution of cooperative behaviour in structured populations. To address this issue, we propose two new dispersal modes, parent-preferred and offspring-preferred dispersal, incorporate them into the birth–death update rule, and consider the resultant strategy evolution in the prisoner's dilemma on random-regular, small-world, and scale-free networks, respectively. We find that parent-preferred dispersal favours the evolution of cooperation in these different types of population structures, while offspring-preferred dispersal inhibits the evolution of cooperation in homogeneous populations. On scale-free networks when the strength of parent-preferred dispersal is weak, cooperation can be enhanced at intermediate strengths of offspring-preferred dispersal, and cooperators can coexist with defectors at high strengths of offspring-preferred dispersal. Moreover, our theoretical analysis based on the pair-approximation method corroborates the evolutionary outcomes on random-regular networks. We also incorporate the two new dispersal modes into three other update rules (death-birth, imitation, and pairwise comparison updating), and find that similar results about the effects of parent-preferred and offspring-preferred dispersal can again be observed in the aforementioned different types of population structures. Our work, thus, unveils robust effects of preferential dispersal modes on the evolution of cooperation in different interactive environments.

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Origin of diversity in spatial social dilemmas

10.1101/2021.03.21.436334 ◽

2021 ◽

Author(s):

Christoph Hauert ◽

Michael Doebeli

Keyword(s):

Spatial Structure ◽

Adaptive Dynamics ◽

Social Dilemmas ◽

Finite Size ◽

Structured Populations ◽

Pair Approximation ◽

Evolutionary Diversification ◽

Invasion Fitness ◽

Population Structures ◽

Mixed Populations

Cooperative investments in social dilemmas can spontaneously diversify into stably co-existing high and low contributors in well-mixed populations. Here we extend the analysis to emerging diversity in (spatially) structured populations. Using pair approximation we derive analytical expressions for the invasion fitness of rare mutants in structured populations, which then yields a spatial adaptive dynamics framework. This allows us to predict changes arising from population structures in terms of existence and location of singular strategies, as well as their convergence and evolutionary stability as compared to well-mixed populations. Based on spatial adaptive dynamics and extensive individual based simulations, we find that spatial structure has significant and varied impacts on evolutionary diversification in continuous social dilemmas. More specifically, spatial adaptive dynamics suggests that spontaneous diversification through evolutionary branching is suppressed, but simulations show that spatial dimensions offer new modes of diversification that are driven by an interplay of finite-size mutations and population structures. Even though spatial adaptive dynamics is unable to capture these new modes, they can still be under-stood based on an invasion analysis. In particular, population structures alter invasion fitness and can open up new regions in trait space where mutants can invade, but that may not be accessible to small mutational steps. Instead, stochastically appearing larger mutations or sequences of smaller mutations in a particular direction are required to bridge regions of unfavourable traits. The net effect is that spatial structure tends to promote diversification, especially when selection is strong.

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Spatial social dilemmas promote diversity

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.2105252118 ◽

2021 ◽

Vol 118 (42) ◽

pp. e2105252118

Author(s):

Christoph Hauert ◽

Michael Doebeli

Keyword(s):

Spatial Structure ◽

Adaptive Dynamics ◽

Social Dilemmas ◽

Finite Size ◽

Structured Populations ◽

Pair Approximation ◽

Evolutionary Diversification ◽

Invasion Fitness ◽

Population Structures ◽

Mixed Populations

Cooperative investments in social dilemmas can spontaneously diversify into stably coexisting high and low contributors in well-mixed populations. Here we extend the analysis to emerging diversity in (spatially) structured populations. Using pair approximation, we derive analytical expressions for the invasion fitness of rare mutants in structured populations, which then yields a spatial adaptive dynamics framework. This allows us to predict changes arising from population structures in terms of existence and location of singular strategies, as well as their convergence and evolutionary stability as compared to well-mixed populations. Based on spatial adaptive dynamics and extensive individual-based simulations, we find that spatial structure has significant and varied impacts on evolutionary diversification in continuous social dilemmas. More specifically, spatial adaptive dynamics suggests that spontaneous diversification through evolutionary branching is suppressed, but simulations show that spatial dimensions offer new modes of diversification that are driven by an interplay of finite-size mutations and population structures. Even though spatial adaptive dynamics is unable to capture these new modes, they can still be understood based on an invasion analysis. In particular, population structures alter invasion fitness and can open up new regions in trait space where mutants can invade, but that may not be accessible to small mutational steps. Instead, stochastically appearing larger mutations or sequences of smaller mutations in a particular direction are required to bridge regions of unfavorable traits. The net effect is that spatial structure tends to promote diversification, especially when selection is strong.

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Modes of migration and multilevel selection in evolutionary multiplayer games

10.1101/029470 ◽

2015 ◽

Author(s):

Yuriy Pichugin ◽

Chaitanya S. Gokhale ◽

Julián Garcia ◽

Arne Traulsen ◽

Paul B. Rainey

Keyword(s):

Structured Populations ◽

Multilevel Selection ◽

Evolution Of Cooperation ◽

Cost Ratio ◽

Fixation Probability ◽

Multiplayer Games ◽

Cooperative Strategy ◽

Individual Actions

The evolution of cooperation in group-structured populations has received much attention, but little is known about the effects of different modes of migration of individuals between groups. Here, we have incorporated four different modes of migration that differ in the degree of coordination among the individuals. For each mode of migration, we identify the set of multiplayer games in which the cooperative strategy has higher fixation probability than defection. The comparison shows that the set of games under which cooperation may evolve generally expands depending upon the degree of coordination among the migrating individuals. Weak altruism can evolve under all modes of individual migration, provided that the benefit to cost ratio is high enough. Strong altruism, however, evolves only if the mode of migration involves coordination of individual actions. Depending upon the migration frequency and degree of coordination among individuals, conditions that allow selection to work at the level of groups can be established.

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