# Coevolution of Cooperation, Response to Adverse Social Ties and Network Structure

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## Abstract

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**PACS**87.23.Kg; 89.75.Fb

## 1. Introduction

## 2. Active Linking

- Coordination or Bistability: $T<R$ and $S<P$ leads to what is called coordination or SH games [65], in which it is always beneficial to follow the strategy of the opponent, turning both C’s and D’s advantageous when rare.

**Figure 1.**

**Game transformation through AL.**(a) 2-D parameter space, defined by T and S, characterizing four games: HG, SH, SG and PD ($R=4$ and $P=3$). The critical game parameters ${T}_{1}^{*}$ and ${S}_{1}^{*}$ separate the different regions. Greed comes into play when $T>{T}_{1}^{*}$, fear when $S<{S}_{1}^{*}$. (b) AL changes the nature of the game being played. Even though individuals engage in the same game, greed (fear) will come into play only when $T>{T}_{2}^{*}$ ($S<{S}_{2}^{*}$). ${T}_{2}^{*}$ and ${S}_{2}^{*}$ were calculated using $c=0.16$, ${\gamma}_{CC}=0.16$, ${\gamma}_{CD}=0.8$, ${\gamma}_{DD}=0.32$.

#### 2.1. Network Evolution

**Figure 2.**

**Frequency-dependent network evolution.**Blue circles indicate C’s, red circles indicate D’s. CC-links are depicted as cyan lines, CD-links as red lines and DD-links as gray lines. Each panel shows a snapshot of a network in the steady state of the network dynamics associated with given configuration of C’s and D’s. The parameters governing the network dynamics are $c=0.25,{\gamma}_{CC}=0.5,{\gamma}_{CD}=0.25$ and ${\gamma}_{DD}=0.5$. The total population size is $N=30$. Figure reproduced from [53].

#### 2.2. Strategy Evolution

#### 2.3. Separation of Time Scales

#### 2.4. Comparable Time Scales

**Figure 3.**

**AL changes the nature of the game, even outside the time scale separation limit.**We start from $50\%$ cooperating individuals. For small W, C’s never reach fixation. But already for $W=0.1$, fixation of C’s is almost certain. Thus, moderate AL is sufficient to turn cooperation into the dominant strategy here. Results are averages over 100 realizations. ($N=100$, $\beta =0.05$, $c=0.16$, ${\gamma}_{CC}=0.16$, ${\gamma}_{CD}=0.80$ and ${\gamma}_{DD}=0.32$). Figure reproduced from [52].

## 3. Diversity and Cooperation in the PD

#### 3.1. A Minimal Model

#### 3.2. Two Linking Strategies

**Figure 4.**

**Transition probabilities and stationary distributions for a population with**$M=2$

**different linking strategies.**In the limit of rare mutations, the dynamics reduces to transitions between homogeneous states of the population [80,81]. The arrows indicate those transitions for which the fixation probability is greater than neutral fixation, ${\rho}_{N}=\frac{1}{N}$. The explicit values were obtained analytically with the pairwise comparison rule (see Equation 5). Adaptive network dynamics allows C’s, in the form of SC or FC, to remain in the population for $7.2\%$ of the time. D’s dominate because of the flow from FD to SD, either directly or by using the alternative route via SC. (N = 100, β = 0.01, T = 2.1, R = 2, S = 0.9, P = 1, c = 0.16, δ = 0.3). Figure reproduced from [62].

#### 3.3. Arbitrary Number of Linking Strategies

**Figure 5.**

**Cooperation and behavioral diversity.**(a) The population spends more time in a cooperative state (of any type) when the number of possible types (M) increases, irrespective of the temptation to defect T in the PD. (N=100, β=0.1, c=0.16, δ=0.3$,R$=2, P=1, S=$3-T$) (b) While the majority of cooperator types is equally represented, only the slower types of defectors manage to survive. The defector population exhibits behavioral differences, which inhibits the dominance of slowest defector type, providing an escape hatch for cooperation to thrive (M=50, T=2.1, R=2, P=1, S=0.9). Figure reproduced from [62].

## 4. Selection Pressure and Behavioral Diversity

**Figure 6.**

**Transition probabilities and stationary distributions for the PD game with**$M=3$

**(a) and**$M=10$

**(b) types of individuals.**The population spends already $59.8\%$ of the time in a cooperative state when $M=10$. Numbering C’s and D’s according to their type, C0 (D0) being the slowest cooperator (defector), we see that increasing M splits the “outflow” of fast defectors among a wide range of different possibilities. As a result, cooperation emerges, since only few types of defectors are evolutionarily stable, whereas the vast majority of cooperative types work as “flow sinks”. ($N=100$, $T=2.1$, $R=2$, $S=0.9$, $P=1$, $c=0.16$, $\beta =0.01$, $\delta =0.3$). Figure reproduced from [62].

**Figure 7.**

**Role of the range of behavioral spectrum (**δ

**) and selection pressure (**β

**) in the evolution of cooperation.**Each contour region represents the fraction of time the population spends in a cooperative state (of any type), for given values of δ and β. The number of types in the population was changed so as to keep the interval in δ between consecutive types constant (0.01). Specifically, $\delta =0.3$ leads to 60 types in the population. ($N=100,T=2.1,R=2,P=1,S=0.9,c=0.16$).

**Figure 8.**

**Role of separation**d

**between consecutive rates in the evolutionary success of cooperators.**The plots shows the fraction of time the population spends in a cooperative state (of any type) for a given intensity of selection. Symbols represent the data. The lines are the result of a spline interpolation of the data. Each data set corresponds to a different distance d between consecutive rates (and types) in the population. Smaller values of d lead to more cooperation, an effect that becomes more pronounced for intermediate intensities of selection. In these regimes, small distances between types can effectively reduce the “flow” towards the slowest defector type, as discussed in Section 3. ($N=100,T=2.1,R=2,P=1,S=0.9,c=0.16,\delta =0.3$).

## 5. Diversity and Cooperation in the Snowdrift and Stag-hunt games

**Figure 9.**

**Transition probabilities and stationary distributions for the stag-hunt and snowdrift games.**(a) Two-dimensional parameter space defining all three social dilemmas. By fixing $R=2$ and $P=1$, while $0\le S\le 1$ and $1\le T\le 3$, we can represent the three most popular social dilemmas of cooperation: The PD ($S<1$ and $T>2$), the SG ($S>1$ and $T>2$) and the SH ($S<1$ and $T<2$). (b), (c) and (d) Transition probabilities and stationary distributions for the three dilemmas with two types of individuals. The dynamics is reduced to transitions between homogeneous states of the population. The arrows in black indicate those transitions for which the fixation probability ρ is greater than neutral fixation (${\rho}_{N}=\frac{1}{N}$). The explicit values are calculated using the pairwise comparison rule with parameters $N=100$, $\beta =0.01$, $c=0.16$ and $\delta =0.3$. The examples shown were obtained for $T=2.1$, $S=0.9$ (PD), $T=2.65$, $S=1.35$ (SG) and $T=1.65$, $S=0.35$ (SH).

**Figure 10.**

**Impact of behavioral differences on cooperation in the snowdrift (upper panels) and stag-hunt (lower panels) games.**The plots on the left depict the fraction of time the population spends in a cooperative state (of any type) as a function of the number of types M in the population. Each line corresponds to a different value of the temptation to defect T ($R=2$, $P=1$, $S=4-T$ for the SG and $S=2-T$ for the SH). Irrespective of T, cooperation increases with increasing number of types. The bar plots on the right ($T=2.65$ and $S=1.35$ for the SG and $T=1.65$, $S=0.35$ for the SH game), depict the fraction of time that each type of individual is present in a population with 50 different types. While all types of C’s survive, only the slower types of D’s remain in the population. However, the surviving D’s exhibit heterogeneity. As faster D’s manage to survive, part of the strategy flow is diverted into fast C’s ($N=100$, $\beta =0.01$, $M=50$, $c=0.16$ and $\delta =0.3$).

## 6. Conclusions

## Acknowledgements

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Van Segbroeck, S.; Santos, F.C.; Pacheco, J.M.; Lenaerts, T.
Coevolution of Cooperation, Response to Adverse Social Ties and Network Structure. *Games* **2010**, *1*, 317-337.
https://doi.org/10.3390/g1030317

**AMA Style**

Van Segbroeck S, Santos FC, Pacheco JM, Lenaerts T.
Coevolution of Cooperation, Response to Adverse Social Ties and Network Structure. *Games*. 2010; 1(3):317-337.
https://doi.org/10.3390/g1030317

**Chicago/Turabian Style**

Van Segbroeck, Sven, Francisco C. Santos, Jorge M. Pacheco, and Tom Lenaerts.
2010. "Coevolution of Cooperation, Response to Adverse Social Ties and Network Structure" *Games* 1, no. 3: 317-337.
https://doi.org/10.3390/g1030317