We present a collaboration ring model -- a network of players playing the prisoner's dilemma game and collaborating among the nearest neighbours by forming coalitions. The microscopic stochastic updating of the players' strategies are driven by their innate nature of seeking selfish gains and shared intentionality. Cooperation emerges in such a structured population through non-equilibrium phase transitions driven by propensity of the players to collaborate and by the benefit that a cooperator generates. The robust results are qualitatively independent of number of neighbours and collaborators.
Following Axelrod's famous prisoner's dilemma (PD) tournament [1], more than four decades of research on the evolution of cooperation has converged to the socalled "five rules of cooperation": kin selection [2], direct reciprocity [3], indirect reciprocity [4], network reciprocity [5], and group selection [6]. Nevertheless, when humans are specifically in focus, it cannot be ignored that they are cognitively able; consequently, a complete explanation of cooperation must naturally also include shared intentionality [7] as an ingredient which manifests as mutualistic collaborative behaviour. It is interesting to note that direct reciprocity may arise from the phenomenon of collaborative strategy choice as it involves strategy rule conditional on opponents' strategy.
The collaborative actions of multiple selfish individuals lead to formation of coalitions to obtain individual benefits. Such collaborative behaviour can be observed across all forms of life, including the ones with limited cognitive ability, such as bacteria [8], trees [9], and animals [10]; e.g., in mobbing behaviour of birds during the breeding season, when a predator approaches a nest, birds nesting nearby-often genetically unrelated-form temporary coalitions to collectively harass and drive away the predator [11][12][13][14]. Notably, humans-cognitively superior to any other organism-are far more collaborative than any other living organisms, including their closest evolutionary relatives, the great apes [15,16]. This observation has further inspired the shared intentionality hypothesis [7], which posits that the cooperative nature and societal structure of human beings have been crucial for the evolution of sophisticated cognitive abilities [17][18][19].
A recent study [20] numerically demonstrates that collaboration among individuals playing a PD game in sparsely connected populations sustain a stable fraction of cooperators. In this paper, we intend to put this observation into a firm footing by introducing a mathematical model that not only provides an analytically tractable template of the phenomenon but also provides insights about microscopic reasoning behind the emergence of cooperation via this route.
Of course, the cooperation level in any population is result of a collective behaviour resulting from the interactions between its players. A change in the collective global behaviour of the population due to changing character of local interactions is readily reminiscent of the phenomenon of phases transition. The statistical physical concept of phase transition has been researched extensively in the context of games [21][22][23][24][25][26][27]. The model introduced here presents a novel methodology of investigating transitions in population’s collaboration induced cooperation level using the combined framework of nonequilibrium statistical physics and non-linear dynamicswhile the former is used in framing and simulating the inherent stochastic Markov process, the latter is used in analysing the deterministic equations extracted therefrom by adopting pairwise approximation (cf. Bethe approximation [28][29][30][31][32]).
Specifically, consider a simple yet non-trivial structured population modelled by a ring network [33] [see Fig. 1(a)]: Each of N nodes represent individuals, and the edges represent connections between them. Thus, each individual is directly connected to exactly two neighbours (they are the nearest neighbours) and plays Prisoner’s Dilemma (PD) with both of them. The mathematical form of the PD game matrix is given in Fig. 1 Let the number of cooperators (action C) be N C and so rest of the N D = N -N C individuals are defectors (action D). The updating rule is as follows: At any time instant, τ , an individual is selected randomly, say i, who chooses one action, either C or D, while playing a PD game with her two neighbours, i + 1 and i -1. Player i updates her action in two ways:
With probability p (propensity of collaboration), she forms a collaboration with one of her two nearest neighbours, chosen randomly (either i -1 or i + 1). In the collaboration thus established, they jointly choose a (coalitional) better response [34][35][36] such that each player in the coalition receives a payoff at least as high as her payoff in the current configuration. If there is no better response, they stick with their current actions. If multiple better responses exist, they choose one of them randomly with equal probability.
Otherwise with probability 1 -p, she independently adopts a best response [37]-an action that maximizes her payoff in a play against her opponents whose actions are held fixed. If multiple best responses exist, she chooses one of them randomly with equal probability.
We want to understand how the fraction of cooperators, x = N C N = P(C) (the probability that a randomly chosen node is a cooperator), varies with the two control parameters b and p.
To this end, we first perform simulate the aforemen
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