Abstract
We study the viability of conditional cooperation in a dynamically evolving social network. The network possesses the small world property, with high clustering coefficient but low characteristic path length. The interaction among linked individuals takes the form of a multiperson prisoners’ dilemma, and actions can be conditioned on the past behavior of one’s neighbors. Individuals adjust their strategies based on performance within their neighborhood, and both strategies and the network itself are subject to random perturbation. We find that the long-run frequency of cooperation is higher under the following conditions: (i) the interaction radius is neither too small nor too large, (ii) clustering is high and characteristic path length low, (iii) the mutation rate of strategies is small, and (iv) the rate of adjustment in strategies is neither too fast nor too slow.
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Notes
Recently, Wu et al. (2006) consider a model of cooperation on a disordered square lattice, in which some links on the lattice are randomly rewired.
There are some structural differences between a random network and the limiting case of the α-model. However, when k ≫ 1, the limit very closely approximates a random graph. See Watts (1999, pp. 51–52) for details.
In Fig. 3, \(L\left( \Gamma \right) \) and \(C\left( \Gamma \right) \) are normalized by dividing them by \(L\left( \Gamma \right) \) for α = 0 and \(C\left( \Gamma \right) \) for α = 0, respectively.
The fact that the characteristic path length and clustering coefficient cannot be tuned independently is a disadvantage of the α-model. However, varying one of these while holding the other constant would cause the average degree k to change, and this would prevent us from identifying the independent effects of changes in the average degree on the incidence of cooperation.
The notion of hardness allows for a multiperson generalization of the Tit-for-Tat strategy. In a two-person repeated interaction, Tit-for-Tat cooperates in the first period and then simply mimics the last action of the opponent. In a multiperson setting, opponent actions are multidimensional. Hardness allows us to aggregate these actions in a natural way, by identifying the threshold proportion of cooperation in one’s neighborhood that is sufficient to induce a player to cooperate.
This unconditional response is not only realistic but also technically important when we introduce mutation in hardness. For example, without the unconditional response, mutations in hardness in a homogenous neighborhood cannot lead to changes in actions.
The manner in which hardness evolves ensures that \(h_{i}\left( t\right) \in \left[ 0,1\right] \) for all t.
An important determinant of real world social networks is the endogenous formation and breakage of links (Jackson and Wolinsky 1996). One might expect that individuals would seek to form links with those with a higher propensity to cooperate, and break links with defectors. These effects, which we neglect here, would make it easier for cooperative clusters to form and spread. We show that despite the unbiased formation and breakage of links, the incidence of cooperation can be significant.
Note that mutations in the propensity to cooperate are sufficient to ensure that there are no absorbing states, so random rewiring of the network is not necessary for this purpose. We allow for changes in network structure because it occurs in practice, and is likely to affect the extent to which cooperation can persist in the long run.
The simulation was run for 100,000 periods and first 30,000 observations deleted. There are 300 periods between adjacent data points in the figure, to smooth out extreme short run volatility.
Eshel et al. prove that the level of cooperation must lie between 60% and 100%. It is interesting that we find cooperation to lie roughly at the midpoint of this range.
In a regular network with degree k, a large enough cluster of cooperators survives in the steady state if the benefit-cost ratio β exceeds than 4k/(3(k/2) − 1); see Jun and Sethi (2007). Accordingly, we choose a value higher than this threshold.
For each of the following four cases, we change only one parameter of the model, while fixing all other parameters at the values of Section 4.2.
Although not shown, the frequency is almost zero for values of k ≥ 34.
The matrix is based on parameter values n = 1,000, k = 16, α = 7, δ = 0.85, ε = 0.005, p = 10 − 10, β = 4.783, ω = 2, T = 100,000, and t = 30,000.
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Acknowledgements
The first author acknowledges financial support from Kyung Hee University (KHU-20050406). We thank Beom Jun Kim for help with simulations and for comments on an earlier draft, and an anonymous referee for a number of helpful suggestions.
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Jun, T., Sethi, R. Reciprocity in evolving social networks. J Evol Econ 19, 379–396 (2009). https://doi.org/10.1007/s00191-008-0117-5
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DOI: https://doi.org/10.1007/s00191-008-0117-5