Evolution of cooperation in costly institutes

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.

💡 Research Summary

The paper investigates how the availability of two alternative institutions—a costly institute that requires a participation fee and a free institute that does not—affects the evolution of cooperation in public‑goods games. Individuals belong to a population of size N and, at each time step, are randomly grouped into sets of size g. Within each group, each individual chooses one of the two institutions. In the costly institute (institution 1) a fixed entry cost c_g is paid; in the free institute (institution 2) there is no entry cost. After the choice, participants play a standard public‑goods game: cooperators invest a cost c, defectors invest nothing, the total contribution is multiplied by an enhancement factor r_i (i = 1, 2) and the resulting benefit is shared equally among all members of the chosen institution. In addition, every individual receives a baseline payoff π₀ from activities unrelated to the public‑goods game. Reproduction occurs proportionally to total payoff, preserving the population size. Offspring inherit both the strategy (C or D) and the institution preference (1 or 2), each subject to independent mutation with probability ν, which flips the trait to its opposite state.

In a well‑mixed population the authors derive replicator‑mutator equations for the four possible strategy‑preference types (C₁, D₁, C₂, D₂). By numerically solving these equations and by stochastic simulations, they map the long‑term dynamics in the (c_g, r) plane. Four distinct regimes emerge:

1. Fixed‑point regime (low r): The system converges to a stable fixed point where only defectors in the free institute (D₂) survive; cooperation is extinct.
2. Defective periodic orbit (intermediate r): As r crosses a first threshold, cooperators in the costly institute (C₁) appear and the system settles into a limit cycle in which cooperation fluctuates only within the costly institute while the free institute remains dominated by defectors. This orbit is termed the “defective periodic orbit.”
3. Cooperative periodic orbit (higher r): When r exceeds a second threshold, both institutions support cooperation, and the limit cycle now involves oscillations of all four types. This is called the “cooperative periodic orbit.”
4. Bistability (medium r, low c_g): For small participation costs, the two periodic orbits coexist; the final attractor depends on initial conditions, producing a bistable region. Increasing c_g shrinks the bistable region and eventually yields a single continuous transition (a critical point) between the two orbits.

The authors also explore the case of asymmetric enhancement factors (r₁ ≠ r₂). The qualitative picture remains the same: low r₁ or r₂ favors the fixed point, intermediate values generate periodic dynamics, and high values enable cooperation in both institutions. Larger participation costs expand the domain of the cooperative periodic orbit.

To assess spatial effects, the model is placed on a square lattice with von Neumann neighborhoods and periodic boundaries. Each individual participates in five overlapping groups (its own and those centered on its four neighbors). Evolution proceeds via an imitation‑birth‑death process: an individual copies the strategy‑preference pair of a randomly chosen neighbor with probability proportional to the neighbor’s payoff, followed by possible mutation. Simulations reveal a striking mutualistic relationship: cooperators that prefer the costly institute (C₁) and cooperators that prefer the free institute (C₂) tend to cluster near each other, protecting each other from exploitation by defectors. This spatial mutualism raises the overall level of cooperation in both institutions, even when the participation cost is relatively high. Moreover, the coexistence of the two institutions in structured populations reduces the parameter range where defectors dominate, compared with the well‑mixed case.

Overall, the study demonstrates that a participation cost can act as a selective barrier that keeps free‑riders out of the costly institute, thereby allowing cooperators to thrive there. When the collective‑action benefit (r) is low, the costly institute outperforms the free one; when r is high, the free institute becomes more advantageous. In structured populations, the presence of both institutions fosters a synergistic interaction between different types of cooperators, further promoting cooperation. These findings provide theoretical insight into the design of real‑world institutions such as subscription‑based clubs, paid scientific consortia, or free public services, suggesting that a modest entry fee can paradoxically enhance collective welfare.