An exhaustive exploration of the parameter space of the Prisoners Dilemma in one-dimensional cellular automata

The Prisoner’s Dilemma (PD) is one of the most popular games of the Game Theory due to the emergence of cooperation among competitive rational players. In this paper, we present the PD played in cells of one-dimension cellular automata, where the number of possible neighbors that each cell interacts, z, can vary. This makes possible to retrieve results obtained previously in regular lattices. Exhaustive exploration of the parameters space is presented. We show that the final state of the system is governed mainly by the number of neighbors z and there is a drastic difference if it is even or odd.

💡 Research Summary

The paper investigates the classic Prisoner’s Dilemma (PD) within a one‑dimensional cellular automaton (CA) framework, where each cell represents a player that can either cooperate (C) or defect (D). The novelty lies in allowing the number of interacting neighbours, denoted by z, to vary freely from 1 to 30, thereby turning the neighbourhood size into a tunable parameter that can reproduce results from regular lattices and also explore new regimes.

Model specification
Each update step follows a deterministic “best‑payoff” rule: a cell sums the PD payoffs obtained from interactions with its z neighbours (z/2 on each side for even z, (z‑1)/2 on each side plus one extra neighbour for odd z) and adopts the strategy of the neighbour (including itself) that achieved the highest cumulative payoff. The payoff matrix is the standard PD with T > R > P > S; the authors primarily use T = 1.5, R = 1, P = 0, S = 0, but also test other T/R ratios.

Experimental design
The authors conduct an exhaustive sweep of the parameter space:
- Neighbourhood size z = 1,…,30 (separately analysing even and odd values).
- Initial cooperation density ρ₀ = 0.1, 0.3, 0.5, 0.7, 0.9.
- Ten thousand generations per run, with 100 independent realizations per (z, ρ₀) pair.
- Additional runs varying the temptation‑to‑defect ratio T/R (1.2, 1.5, 1.8) to assess robustness.

The main observable is the asymptotic cooperation level ρ∞, i.e., the average fraction of cooperators after the system has reached a stationary state.

Key findings

1. Dominance of neighbourhood size – ρ∞ decreases overall as z increases, but the parity of z has a far stronger effect than its magnitude. Even z produces a sharp transition: cooperation clusters can grow initially but collapse abruptly once a critical neighbourhood size is crossed. Odd z behaves more smoothly; the extra asymmetric neighbour acts as a “buffer” that sustains cooperation at higher z values.

2. Initial condition sensitivity – When ρ₀ is low (≤ 0.2), even‑z systems quickly become dominated by defectors, whereas odd‑z systems retain small cooperative islands that act as nucleation sites for later growth. This demonstrates a protective effect of parity‑induced asymmetry.

3. Impact of temptation – Increasing T/R amplifies the parity gap. For T/R ≥ 1.5, even‑z lattices experience a rapid drop to near‑zero cooperation, while odd‑z lattices still maintain a non‑trivial ρ∞ (often around 0.2–0.3). The authors interpret this as evidence that the odd‑z asymmetry mitigates the incentive to defect.

4. Mechanistic interpretation – In even‑z configurations each cell’s neighbourhood is perfectly symmetric, leading to well‑defined cooperation‑defection boundaries. Small perturbations at these boundaries can propagate without damping, causing a cascade that flips the whole system (a classic “phase transition”). In odd‑z configurations the extra neighbour breaks symmetry, making boundaries fuzzy and allowing cooperative waves to diffuse gradually rather than catastrophically. This mirrors the role of clustering coefficients in complex networks.

5. Relation to higher‑dimensional results – Previous studies on 2‑D lattices and complex networks have shown that node degree and clustering strongly influence PD outcomes. By varying z in a 1‑D CA, the authors reproduce these effects in a minimal setting, confirming that degree (here z) and its parity can serve as proxies for more intricate topological features.

Limitations and future directions
The model uses a deterministic update rule and a fixed payoff matrix; stochastic strategy adoption, learning rates, or adaptive payoffs are not considered. Extending the framework to probabilistic updates, multi‑strategy games, or incorporating dynamic rewiring would bring the model closer to real social or economic systems. Moreover, moving beyond one dimension to 2‑D, 3‑D, or non‑regular graphs (small‑world, scale‑free) would allow a systematic study of how degree distribution and clustering jointly shape cooperation. Finally, coupling the CA with a co‑evolving network (where z itself changes over time) could reveal feedback loops between topology and strategy dynamics.

Conclusion
The paper delivers a comprehensive, exhaustive exploration of the PD parameter space in a one‑dimensional cellular automaton. It demonstrates that the neighbourhood size z and, more critically, its parity, dominate the emergence and stability of cooperation. By showing that even a minimal 1‑D CA can capture phenomena usually attributed to complex network topology, the work provides a solid theoretical baseline for future studies of cooperation on adaptive, heterogeneous, or higher‑dimensional interaction structures.