Collaboration drives phase transitions towards cooperation in prisoner's dilemma

Abstract

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.

Figures (4)

• Figure 1: Emergence of cooperation, in a PD game, in a ring-structured population through phase transition. Subfigure (a): Schematic of the ring-structured population: white and black nodes denote defectors and cooperators, respectively, and solid black lines represent nearest neighbour interactions. Subfigure (b): Average order parameter, $\bar{x}^*$, as a function of the control parameters $b$ and $p$, shown using the colour bar. Critical lines where phase transitions occur are indicated by red dashed lines, with illustrative population configuration states shown in different parameter regions. Subfigure (c): Bar plots at $p=1$ display the fractions of nearest-neighbour pairs (DD, CD/DC, CC) for: (i) $3b-2c>0$ (only CC and CD/DC present), (ii) $3b-2c=0$ (all pair types present), and (iii) $3b-2c<0$ (only DD and CD/DC present). Numerical details are given in Appendix \ref{['numerics']}.
• Figure 2: All feasible microscopic transitions: This schematic presents all possible microscopic transitions arising from better-response and best-response updates. In the figure, the arrowheads indicate the direction of change, and above each arrow we specify the parameter range for which the corresponding transition occurs; in case of no specification, the transition is unconditional. Above the selected individuals (chosen to play the game either by forming a coalition or independently), we indicate the payoffs they obtain in the given configuration, where the actions are written inside the circles.
• Figure 3: Prediction of the average fraction of cooperators and its inflow rate using mean-field and pairwise ansatzs. This figure shows the average fraction of cooperators $\bar{x}^*$ [subfigure (a)], and the inflow rate $J_C$ [subfigure (b)] as functions of the coalition probability $p$ in an active phase with $p<1$, $b = \frac{5}{6}$ and $c = 1$ (green colour) and the phase at $p=1$ with $b = \frac{4}{6}$ and $c = 1$ (pink colour). Results generated from the numerical simulations are depicted by filled square. All dashed lines denote the mean-field prediction, which deviates significantly from the numerical simulation results. In contrast, the pairwise approximation ansatz, represented by solid line, closely matches the numerical simulation results across the full range of $p$, remaining within one standard deviation of the simulated time averages. Numerical details are provided in Appendix \ref{['numerics']}.
• Figure 4: One feasible transition in an irregular network: This figure illustrates the transition where two defectors [subfigure (a)] become two cooperators [subfigure (b)] in a coalition of size $m$, and vice versa, in an irregular network driven by better-response dynamics. Each individual in the coalition is labelled by an index $i \in \{1,\cdots, m\}$. Node $i$ has degree $l_i$ and therefore interacts with its $l_i$ nearest neighbours. Here $l = \max\{l_i,\, l_{i+1}\}$.