Pavlovian Prisoner's Dilemma in one-dimensional cellular automata: analytical results, the quasi-regular phase, spatio-temporal patterns and parameter space exploration

TL;DR

This paper analyzes a one-dimensional cellular automaton model of the Prisoner's Dilemma using Pavlovian strategies, deriving analytical critical points, exploring spatio-temporal patterns, and identifying a novel quasi-regular phase that promotes cooperation.

Contribution

It provides analytical results for critical temptation values, characterizes emergent patterns, and introduces the quasi-regular phase in a one-dimensional PD model with Pavlovian updating.

Findings

Pavlovian strategy promotes cooperation and prevents defection.

Analytical critical temptation thresholds match numerical simulations.

Discovery of a quasi-regular phase with stable cooperation levels.

Abstract

The Prisoner's Dilemma (PD) game is used in several fields due to the emergence of cooperation among selfish players. Here, we have considered a one-dimensional lattice, where each cell represents a player, that can cooperate or defect. This one-dimensional geometry allows us to retrieve the results obtained for regular lattices and to keep track of the system spatio-temporal evolution. Players play PD with their neighbors and update their state using the Pavlovian Evolutionary Strategy. If the players receive a positive payoff greater than an aspiration level, they keep their states and switch them, otherwise. We obtain analitycally the critical temptation values, we present the cluster patterns that emerge from the players local interaction and we perform an exploration of paramater space. The numerical results are in accordance to the critical temptation analitycal results, it confirms that the Pavlovian strategy foment the cooperation among the players and avoid the defection. The system also presented a new phase in the steady state, the quasi-regular phase, where several players switch their states during round to round, but the proportion of cooperators does not alter significantly.