Evolution of cooperation in costly institutes

Abstract

We show that in a situation where individuals have a choice between a costly institute and a free institute to perform a collective action task, the existence of a participation cost promotes cooperation in the costly institute. Despite paying for a participation cost, costly cooperators, who join the costly institute and cooperate, can out-perform defectors, who predominantly join a free institute. This, not only promotes cooperation in the costly institute but also facilitates the evolution of cooperation in the free institute. A costly institute out-performs a free institute when the profitability of the collective action is low. On the other hand, a free institute performs better when the collective action's profitability is high. Furthermore, we show that in a structured population, when individuals have a choice between different institutes, a mutualistic relation between cooperators with different institute preferences emerges and helps the evolution of cooperation.

Figures (8)

• Figure 1: The density of different strategies in the $c_g-r$ plane. The densities of different strategies in the $c_g-r$ plane are color plotted. The phase diagram of the model is superimposed. For both small and large enhancement factors, $r$, the dynamics settle in a fixed point, denoted by FP. In between, the dynamics settle in a periodic orbit, denoted by PO. White lines show the boundary of the cyclic phase. For a small cost, the model is bistable for medium values of $r$. Green circles show the lower boundary of the bistable region, above which the cooperative orbit becomes stable. The red squares show the upper boundary of the bistable region above which the dynamics settle in the cooperative periodic orbit starting from all the initial conditions. The filled green circle shows the point where the transition between the two periodic orbits becomes a continuous transition. Parameter values: $g=5$, $nu=10^{-3}$, $\pi_0=2$. The replicator dynamic is solved for $8000$ time steps and time average are taken over the last $2000$ steps.
• Figure 2: Density of different strategies as a function of $r$ for three different values of cost. The replicator dynamics is solved for two different initial conditions, a cooperation favoring initial condition in which all the individuals are cooperators and prefer the costly institute (blue dots), and a uniform initial condition in which strategy and game preference of the individuals are assigned at random (red squares). The result of simulations in a population of size $N=10000$ starting from a random initial condition is shown by orange circles. The system shows two different cooperative phases. For small enhancement factors, cooperation in the costly institute, but not in the free institute, evolves. For larger enhancement factors, cooperation in both costly and free institutes evolves. While for a small cost, the transition between the two cooperative phases is discontinuous and shows bistability (a), for high cost, there is a cross-over between the two phases by increasing the enhancement factor (c). Parameter values: $g=5$, $nu=10^{-3}$, $\pi_0=2$. The replicator dynamic is solved for $9000$ time steps, and the time averages are taken over the last $2000$ time steps. The simulation is performed for $6000$ time steps, and the averages are taken for the last $3000$ time steps.
• Figure 3: Examples of the time evolution of the system. Examples of the time evolution of the system for the defective periodic orbit, (a) and (b), and the cooperative periodic orbit (c) and (d). The top panels show the replicator dynamics results, and the bottom panels show the result of a simulation in a population of size $N=40000$ individuals. While in the defective periodic orbit only in the costly institute, cooperation evolves, in the cooperative periodic orbit, cooperation in both institutes evolves. Parameter values: $g=5$, $nu=10^{-3}$, $\pi_0=2$, and $c_g=0.18$. In (a) and (b) $r=2.2$, and in (c) and (d) $r=2.35$.
• Figure 4: Evolution of cooperation. (a): Time average total density of cooperators, $\rho_C=\rho_C^1+\rho_C^2$ in the $r-c_g$ plane. (b): Time average difference between the probability that an individual in the costly institute is cooperator from the probability that an individual in the free institute is a cooperator, $\gamma=\rho_C^1/(\rho_C^1+\rho_D^1)-\rho_C^2/(\rho_C^2+\rho_D^2)$. Individuals are always more likely to be cooperator in the costly institute. (c) and (d): The time average total density of cooperators, $\rho_C=\rho_C^1+\rho_C^2$ in the $r_1-r_2$ plane for $c_g=0.1$ (c) and $c_g=0.4$ (d). Parameter values: $g=5$, $nu=10^{-3}$, and $\pi_0=2$. The replicator dynamics is solved for $8000$ time steps and time average are taken over the last $2000$ steps.
• Figure 5: The density of different strategies in the $r_1-r_2$ plane. The densities of different strategies in the $r_1-r_2$ plane, for two different costs, $c_g=0.1$ (top) and $c_g=0.4$ (bottom), are color plotted. The phase diagram of the model is superimposed. For a small cost ($c_g=0.1$, top), for both small and large enhancement factor values, $r$, the dynamics settle in a fixed point, denoted by FP. In between, the dynamics settle in a periodic orbit, denoted by PO. White lines show the boundary of the cyclic phases. For a small cost, the model is bistable for medium values of $r$. Green circles show the lower boundary of the bistable region, above which the cooperative periodic orbit becomes stable. The red squares show the upper boundary of the bistable region above which the dynamics settle in the cooperative periodic orbit starting from all the initial conditions. Blue triangles show the phase boundary, resulting from a uniform initial condition. For a large cost ($c_g=0.4$, bottom), the bistability is lost, and the periodic phase's domain increases. Parameter values: $g=5$, $nu=10^{-3}$, and $\pi_0=2$. The replicator dynamic is solved for $8000$ time steps and time average are taken over the last $2000$ steps.