The tragedy of the commons in a multi-population complementarity game

We study a complementarity game with multiple populations whose members’ offered contributions are put together towards some common aim. When the sum of the players’ offers reaches or exceeds some threshold K, they each receive K minus their own offers. Else, they all receive nothing. Each player tries to offer as little as possible, hoping that the sum of the contributions still reaches K, however. The game is symmetric at the individual level, but has many equilibria that are more or less favorable to the members of certain populations. In particular, it is possible that the members of one or several populations do not contribute anything, a behavior called defecting, while the others still contribute enough to reach the threshold. Which of these equilibria then is attained is decided by the dynamics at the population level that in turn depends on the strategic options the players possess. We find that defecting occurs when more than 3 populations participate in the game, even when the strategy scheme employed is very simple, if certain conditions for the system parameters are satisfied. The results are obtained through systematic simulations.

💡 Research Summary

The paper investigates a complementarity game in which several distinct populations contribute resources toward a common target. Each individual chooses a non‑negative integer contribution c_i. If the total contribution C = ∑ c_i reaches or exceeds a predefined threshold K, every player receives a payoff equal to K minus his own contribution (K − c_i). If the sum falls short of K, all players receive zero. This payoff structure creates a tension: each player wishes to contribute as little as possible while still hoping that the collective sum will be sufficient to trigger the reward.

At the level of a single individual the game is perfectly symmetric, and the standard Nash analysis yields a continuum of equilibria. In particular, (i) a “uniform contribution” equilibrium in which every player offers K/N (N being the total number of individuals) and (ii) a family of “defector” equilibria in which one or more populations contribute nothing (c = 0) while the remaining populations increase their contributions enough to meet the threshold. Both types of equilibria are feasible; which one is realized depends on the evolutionary dynamics that operate at the population level.

The authors model each population as a homogeneous group of agents that share a single strategy (a common contribution level). The game is thus described by M populations, each of size N_j, and a vector of contributions (c_1,…,c_M). The total contribution is C = ∑ N_j c_j. Evolution proceeds in discrete generations using a replication‑selection process: after each round the payoff of each population is computed, and populations with higher payoffs reproduce proportionally more copies of their strategy. A small mutation probability μ allows an individual to switch to a randomly chosen contribution in the set {0,1,…,K}. The authors explore a range of parameters: K ∈