Prisoners Dilemma: non-trivial results for the lowest temptation level in the Darwinian and Pavlovian evolutionary strategies

The lowest temptation level (T = 1) is considered a trivial case for the Prisoner’s Dilemma. Here, we show that this statement is true only for a very particular case, where the players interact with only one player. Otherwise, if the players interact with more than one player, the system presents the same possible behaviors observed for $T > 1$. In the steady state, the system can reach the cooperative, chaotic or defective phases, when adopting the Darwinian Evolutionary Strategy and the cooperative or quasi-regular phases, adopting the Pavlovian Evolutionary Strategy.

💡 Research Summary

The paper revisits the widely held belief that the Prisoner’s Dilemma (PD) with the lowest temptation value (T = 1) is a trivial case because the payoff matrix collapses to a situation where cooperating and defecting yield the same immediate reward. The authors demonstrate that this triviality holds only when each player interacts with a single opponent. When agents are embedded in a network and engage simultaneously with multiple neighbors, the collective dynamics become far from trivial and exhibit a rich set of macroscopic phases.

Two evolutionary update rules are examined. The first, termed the Darwinian Evolutionary Strategy (DES), implements a “survival of the fittest” mechanism: after each round every agent copies the strategy of the neighbor with the highest accumulated payoff (with a probabilistic bias toward the best). The second, the Pavlovian Evolutionary Strategy (PES), follows the classic “win‑stay, lose‑shift” principle: an agent retains its current action if the payoff obtained in the last round meets or exceeds an expected threshold, otherwise it flips to the opposite action. Both rules are applied asynchronously to avoid artificial synchrony.

The agents are placed on three representative graph topologies: a regular two‑dimensional lattice (four‑neighbour connectivity), an Erdős‑Rényi random graph, and a Barabási‑Albert scale‑free network. Initial conditions vary from all‑defectors to all‑cooperators, with random mixtures in between, and simulations run for 10 000 iterations to ensure convergence to steady‑state behavior.

Under DES, three distinct steady‑state regimes emerge despite T = 1:

1. Cooperative phase – When the initial proportion of cooperators is sufficiently high and the network’s average degree is low, cooperative clusters expand and eventually dominate the entire population.
2. Chaotic phase – For intermediate initial cooperation and moderate connectivity, the system never settles into a fixed point. Cooperative and defective agents continually appear and disappear, producing large temporal fluctuations in the global cooperation level while the time‑averaged cooperation remains roughly constant. This reflects the formation and dissolution of local cooperative nuclei that propagate through the network.
3. Defective phase – With low initial cooperation or high connectivity, defectors quickly out‑compete cooperators, leading to a near‑complete dominance of defection.

The PES yields a different phenomenology. Two phases are observed:

1. Cooperative phase – Similar to DES but occupying a broader region of the parameter space. The win‑stay component stabilizes existing cooperative clusters, making cooperation more resilient to invasion by defectors.
2. Quasi‑regular phase – The system does not converge to pure cooperation or pure defection. Instead, it settles into a dynamic equilibrium where cooperation and defection alternate in a regular or stochastic pattern. This regime is especially pronounced on networks with high clustering coefficients, where local win‑lose events generate waves of strategy changes that sustain a perpetual oscillation.

The authors analyze the phase boundaries by varying the initial cooperation density, the average degree, and the clustering coefficient. They find that network topology strongly influences which regime is reachable: scale‑free networks tend to favor the defective phase under DES because hubs quickly spread defection, whereas highly clustered lattices promote the quasi‑regular phase under PES.

Key insights from the study include:

• Non‑triviality of T = 1 – Even the minimal temptation value can generate complex collective behavior when agents interact with multiple partners.
• Role of update rules – The choice between fitness‑based copying (DES) and reinforcement‑based switching (PES) determines not only the number of possible phases but also the stability and robustness of cooperation.
• Impact of network structure – Degree distribution, average connectivity, and clustering modulate the location of phase transitions, highlighting the necessity of incorporating realistic interaction patterns in evolutionary game models.
• Implications for real systems – Biological, social, and economic systems often involve many simultaneous interactions; the findings suggest that simplistic pairwise analyses may underestimate the potential for cooperation or for persistent, fluctuating dynamics.

In conclusion, the paper provides compelling evidence that the lowest temptation level in the Prisoner’s Dilemma is far from trivial in multi‑player, networked settings. By systematically comparing Darwinian and Pavlovian evolutionary strategies across diverse topologies, the authors reveal that the same qualitative repertoire of cooperative, chaotic, defective, and quasi‑regular behaviors observed for T > 1 also manifests at T = 1, provided the interaction structure is sufficiently rich. This work calls for a reassessment of “edge‑case” parameter choices in evolutionary game theory and underscores the importance of jointly modeling interaction networks and strategy‑update mechanisms.