The Dynamics of Cooperation in Society. The ECHO-EGN Model

11 Understanding the dynamics of cooperative behavior of individuals in 12 complex societies represents a fundamental research question which puz- 13 zles scientists working in heterogeneous ﬁelds. Many studies have been 14 developed using the unitary agent assumption, which embeds the idea 15 that when making decisions, individuals share the same socio-cultural 16 parameters. In this paper, we propose the ECHO-EGN model, based 17 on Evolutionary Game Theory, which relaxes this strong assumption by 18 considering the heterogeneity of three fundamental socio-cultural aspects 19 ruling the behavior of groups of people: the propensity to be more co- 20 operative with members of the same group (Endogamy), the propensity 21 to cooperate with the public domain (Civicness) and the propensity to 22 prefer connections with members of the same group (Homophily). The 23 ECHO-EGN model is shown to have high performance in describing real 24 world behavior of interacting individuals living in complex environments. 25 Extensive numerical experiments allowing the comparison of real data and 26 model simulations conﬁrmed that the introduction of the above mecha- 27 nisms enhances the realism in the modelling of cooperation dynamics. 28 Additionally, theoretical ﬁndings allow us to conclude that Endogamy 29 may limit signiﬁcantly the emergence of cooperation. 30


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The modelling of the evolution of cooperation in social networks is a consoli-32 dated stream of research at the boundary of mathematics and social sciences. 33 The main focus of this line of work is the understanding of how cooperation can 34 develop in a population of rational agents based on selfish motivations. Evolu- 35 tionary Game Theory assumed the role of main framework (e.g. [1]), with many 36 studies adopting the Prisoner Dilemma as analytical tool to model the emer-37 gence of mutually beneficial interactions among rational decision makers [2][3][4][5][6][7].  It has to be noted that in recent years more complex models of population 48 have been introduced, characterized by spatial structure like lattices [8]. Also 49 more complex organized structures have been taken into account, assuming that 50 agent interactions take place according to the topology of a network of inter-51 connections (e.g. [6,7,9]). However, these innovations concern the second aspect 52 of the unitary agent assumption only -the well-mixed population design -and 53 only partially. Indeed, they imply a variable distribution of connections among 54 agents, yet as a product of the network's spatial structure, rather than as a 55 function of a priori inherently variable agents' preferences.

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The idealized, input-centered, normative approach implied in the unitary 58 agent assumption limits the realism of research and therefore the chance to 59 model natural social settings for the sake of understanding the current and 60 future actual evolution of historically concrete human groups, which are inher-61 ently plural (e.g. [10][11][12]). The latter aim requires an output-centered approach, 62 namely an approach which is mainly interested in understanding the potential 63 evolution of given input states, and to draw from it the identification of struc-64 tural and individual conditions improving the overall level of cooperation.
to increase the member's Endogamy, namely to make them more inclined to 124 cooperate with in-group members than with out-group members. In contrast, 125 universalist values make adherents cooperate with in-group and out-group sim-  that v and w are not connected. We will refer to the number of neighbors of a 156 generic player v as its degree, then continuously over time. The games played are assumed to be Prisoner's dilemmas, where the payoff earned by player v against w is described by the matrix: where R v,w is the reward for mutual cooperation, T v,w is the temptation to 165 defect when the opponent cooperates, S v,w is the sucker's payoff earned by a 166 cooperative player when the opponent is a free rider, and P v,w is the punish-167 ment for mutual defection. A Prisoner's dilemma game is characterized by the In this work, we assume that R v,w = 1, Thus, taken as a whole, the level of cooperation of a generic player v is 177 denoted by x v ∈ [0, 1], and its dynamics is ruled by the following equation: whereẋ v denotes the time derivative of x v , i.e.ẋ v = dx v /dt and The population V is assumed to be subdivided into M groups, namely The size 197 of group G g is N g . Hence, the share of population belonging to group G g is player v with affine and non-affine players, respectively, i.e.
In the following subsections, the ECHO-EGN model specifications of Ho-207 mophily, Endogamy, and Civicness will be introduced. As stated above, Homophily is the propensity of an individual to prefer connec-210 tions with members of the same group. In order to account for this property, 211 a rewiring process has been carried out to modify the initial network of con-212 nections, according to a given probability, specific for each group, and denoted 213 by the Homophily factor h g ∈ [0, 1]. All details on the algorithm used for the 214 rewiring phase are given in Appendix A. als. Specifically, given v ∈ G g and w ∈ V with a v,w = 1, we define: is the payoff matrix used with non affine players, and is the payoff matrix used with affine players, where T A g = (1−e g )T N and S A g =

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(1 − e g )S N . Coherently, the payoff matrix of the self-game is constraining the agent's selfish attitude.

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Moreover, we assume that all members of a given group share the same self-229 regulation parameter, which depends on the Civicness value c g of the group, 230 according to the following formula: where ρ N = 1−T N S N and k ′ g is the effective average degree of group g.   Using these assumptions, equation (1) for player v ∈ G g can be rewritten as 249 follows: where is the affine equivalent player of v, and is the non affine equivalent player of v. We also have that the equivalent player 251 of v is: Notice that equation (3) Table 1). The distribution is given in terms of the size of the segments 292 of population, each of them defined by individuals characterized by one of five 293 symbolic universes described below.   They support people in accomplishing their projects.    .

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Accordingly, this index lends itself to be interpreted as a proxy of the propensity 342 to limit the relational contact to the in-group.        Table 4: Validation data: trust in people Y P , trust in institution Y I and the corresponding standard deviations σ P and σ I for each region considered.
Group IPC α g G 1 10.00 G 2 9.77 G 3 10.28 G 4 9.48 G 5 9.75 The average cooperation X V and its standard deviation σ X V obtained for each  Table 5) of each natural group/symbolic universe over the whole R = 22 422 sample, and the corresponding level generated by the simulation models.      540 then the steady state x ALLC is asymptotically stable.
The Theorem essentially states that Endogamy restrains cooperation. In-
then the steady state x ALLD is asymptotically stable. The proofs of Theorems 1 and 2 are reported in Appendix C.

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It is interesting to investigate the relationship between values of parameters β v 564 and the thresholds found in Theorems 1 and 2.

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To this end, in Table 6 the self-regulation parameters β v , averaged over 566 the regions and the simulation trials, are reported in column 2 for each group. (2) β G1 16.1 5.5 yes 5.1 no β G2 9.6 6.2 yes 5.4 no β G3 15.6 6.5 yes 5.6 no β G4 5.3 7.8 no 6.0 yes β G5 10.6 41.4 no 9.8 no Table 6: Relationship between average self-regulation parameters β v and the thresholds of Theorems 1 and 2. Col. 2: the value β v averaged over a given group, over the R = 22 regions and over the 100 trials. Col. 3: the value of threshold η v averaged over a given group, over the R = 22 regions and over the 100 trials. Col. 4: Theorem 1 satisfaction (in average). Col. 5: the value the value ζ v averaged over a given group, over the R = 22 regions and over the 100 trials. η v averaged over a given group, over the R = 22 regions and over the 100 trials. Col. 6: Theorem 2 satisfaction (on average).
defection, related to Theorem 2. Notice that in this case, "yes" means that the 572 value of column 2 is lower than the value in column 5, and "no" the opposite, 573 as shown in column 6. Only group G 4 satisfies the requirements of Theorem 2.

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This means that for this group it is not only more difficult to cooperate, but it 575 is also more easier to defect.   affine one, then the probability of changing an affine link with a non affine one 683 is 1 − h g . It can therefore be seen that, due to the stochasticity of this process, Due to the stochasticity of the network, the effective average degree k ′ g of a 686 group g is in general slightly different than the average k g . To distinguish the 687 two cases, we introduce the effective parameters k ′ g and k ′ as k According to the assumption discussed in Section 3.1, for which the topology of the connection network is random with a scale-free distribution, we notice that the network obtained by the above procedure is still scale-free. Indeed: where δ g is the share of group g in the considered population and k = M g=1 δ g k g 2 690 is the average degree of the resulting complete network.

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Since in this paper we assume that k g = 4, then k = 4 M g=1 δ g = 4.

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In this appendix the proofs of the theoretical results concerning the stability of the steady states x ALLC and x ALLD of the ECHO-EGN equation (3), referred to in Section 5, are reported. To this aim, the equations (3) are linearized locally near x ALLC and x ALLD by evaluating the entries of the Jacobian matrix It is useful to rewrite equation (3) as follows: In particular, given a player v ∈ G g , the diagonal entries of J(x) are: On the other hand, the off-diagonal entries of J(x) are: From the theory of nonlinear dynamic systems, the stability of a steady state 706 x * depends on the sign of the real part of the eigenvalues of the corresponding 707 Jacobian matrix J(x * ) [34]. For x ALLC and x ALLD , since x v ∈ {0, 1}, then the 708 off-diagonal entries reported in (A.6) are identically null. Therefore, the Jaco-709 bian matrix has a diagonal structure, and hence its eigenvalues coincide with 710 the expressions reported in (A.5), i.e. λ v = j v,v (x * ) = ∂ẋv ∂xv (x * ). In the following 711 Theorems, we find the sufficient conditions to have negative λ v ∀v ∈ {1, . . . , N }.
then the steady state x ALLC is asymptotically stable.
Proof. Considering a player v ∈ G g , from equation (A.5) we get that: .
Since T A g > 1 and β v > η v , then λ v < 0. This holds for any v. Hence, all 715 eigenvalues are negative, and x ALLC is asymptotically stable.

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In accordance to Theorem 3 in [32], we know that, considering a population which uses only the payoff matrix B N , then x ALLC is asymptotically stable if Comparing the thresholds obtained in Theorem 1 of this paper and in Theorem 3 of [32], we notice that: Indeed: T N > T A g . This corresponds to Remark 1. Theorem 2. If β v < ζ v ∀v ∈ V, where then the steady state x ALLD is asymptotically stable.

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Proof. Considering a player v ∈ G g , from equation (A.5) we get that: .
Since S A < 0 and β v < ζ v , then λ v < 0. This holds for any v. Hence, all 726 eigenvalues are negative, and x ALLD is asymptotically stable.

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In accordance to Theorem 4 of [32], we know that, considering a population which uses only the payoff matrix B N , then x ALLD is asymptotically stable if Comparing the thresholds obtained in Theorem 2 of this paper and in The-728 orem 4 of [32], we notice that: of European societies' cultural milieu. PLoS ONE 13(1): e0189885.