Ponente
Descripción
Overfishing, tax evasion, and environmental pollution all share the same structure: individuals exploit a shared resource while costs fall on everyone. Under natural selection, selfish strategies are favored, yet cooperative behavior persists across scales, from microbial communities to international climate agreements. Explaining this persistence is a central challenge in evolutionary biology, economics, and social science (Hardin, G., Science 162, 1243–1248, 1968).
Many mechanisms promote cooperation: kin selection, network reciprocity, and reward (Nowak, M.A., Science 314, 1560–1563, 2006). Punishment has received the most attention as a direct suppressor of defection(Botta, R. et al., CLEI Electron. J. 17, 7, 2014). Peer punishment generates a second-order free-rider problem since cooperators who sanction are outcompeted by those who do not. Pool punishment distributes costs through a shared fund but incurs enforcement costs even when no defectors are present (Sigmund, K. et al., Dyn. Games Appl. 1, 149–171, 2011). Both leave the punisher population vulnerable to erosion.
We introduce a lottery-based punishment mechanism in the spatial optional public goods game (SOPGG) on a two-dimensional square lattice with periodic boundary conditions where each agent holds one of three strategies: Cooperator(C), Defector(D), and Loner(L). At each round, each agent plays within its Moore neighborhood of radius 1, forming groups of up to nine players. Cooperators contribute $c = 1$ to a common pool multiplied by $r > 1$ and redistributed equally among participating members. Defectors contribute nothing but share equally, and loners abstain and receive a fixed payoff $\sigma$.
\begin{equation}
\pi_C = \frac{r \cdot n_C}{n} - 1.
\end{equation}
\begin{equation}
\pi_D = \frac{r \cdot n_C}{n}.
\end{equation}
Rather than treating punishment as a choosable strategy, every cooperator is independently assigned the punisher role with probability $P_0$ via a Bernoulli trial, local to each neighborhood and independent across neighborhoods. A punisher imposes a fine $\beta$ on each neighboring defector at personal cost $\alpha$ per defector:
\begin{equation}
\pi_C^P = \frac{r \cdot n_C}{n} - 1 - \alpha \cdot n_D.
\end{equation}
Each defector has its payoff reduced by $\beta$ per punisher in its neighborhood:
\begin{equation}
\pi_D^P = \frac{r \cdot n_C}{n} - \beta \cdot n_{cp}
\end{equation}
Strategy updates follow proportional imitation: each agent adopts a neighbor's strategy with probability proportional to the positive payoff difference across all eight neighbors.
Simulations show that above a critical $P_0 \approx 0.197$ the system undergoes a sharp phase transition to full cooperation, with loners going extinct firstand defectors following. Below this threshold the three strategies cycle in a Red Queen coexistence. Between the two regimes there is a narrow band where defection dominates before cooperation takes over. Because strategies update by imitation, once loners disappear the game becomes compulsory and extinct strategies cannot return, so the transition is irreversible. Larger populations($N \geq 100 \times 100$) reach full cooperation consistently, while smaller grids collapse to defection, cooperation or loner at random. The loner payoff $\sigma$ has no significant effect within its valid range, and higher $r$ lowers thecritical $P_0$. Randomly distributing enforcement can stabilize cooperation in spatial populations without punishment being an independently selectable strategy.