Optimizing Wealth by a Game within Cellular Automata

TL;DR

This work tackles the global-to-local problem of evolving 2D cellular automata to maximize a global fitness defined as the sum of local PD payoffs. It combines a genetic algorithm to discover optimal master patterns, a gliding-window extraction of 3×3 templates, and a probabilistic, asynchronous CA rule that uses these templates to scale to larger grids. The main findings show that, for even grid sizes, optimal patterns achieve a wealth $W$ approaching $43/36=1.19444$ with a defector density of $1/4$, while odd sizes yield mixtures of points and dominoes with a single singularity and the same asymptotic wealth. The results illustrate a practical method to design CA rules that realize globally optimized structures from local interactions and offer a template-driven path to scalable pattern formation.

Abstract

The objective is to find a Cellular Automata (CA) rule that can evolve 2D patterns that are optimal with respect to a global fitness function. The global fitness is defined as the sum of local computed utilities. A utility or value function computes a score depending on the states in the local neighborhood. First the method is explained that was followed to find such a CA rule. Then this method is applied to find a rule that maximizes social wealth. Here wealth is defined as the sum of the payoffs that all players (agents, cells) receive in a prisoner's dilemma game, and then shared equally among them. The problem is solved in four steps: (0) Defining the utility function, (1) Finding optimal master patterns with a Genetic Algorithm, (2) Extracting templates (local neighborhood configurations), (3) Inserting the templates in a general CA rule. The constructed CA rule finds optimal and near-optimal patterns for even and odd grid sizes. Optimal patterns of odd size contain exactly one singularity, a 2 x 2 block of cooperators.

Figures (18)

• Figure 1: The expected wealth vs the rate of cooperation, for different parameter settings $(T,R,P,S)$. The parameters shown for case (a) are used in our problem.
• Figure 2: Algorithm that generates optimal patterns with respect to a global fitness function. The fitness function used here is the Wealth of the community, taking into account the local payoffs of all agents.
• Figure 3: 6 x 6 Optimal pattern evolved by a Genetic Algorithm. The total payoffs are shown below the patterns. Fitness is 387, the maximum.
• Figure 4: 6 x 6 Near-optimal patterns evolved by a GA. Fitness is 386 for (1a)--(2a), and 385 for (2b)--(2c).
• Figure 5: (a) 5 x 5 optimal pattern evolved by a Genetic Algorithm. The total payoffs are shown below it. Fitness is 265 and wealth is 1.17778, the maximum. Inside the red marked area there is a block of 4 zeroes. (b) The pattern (a) (inside the white marked boundaries) is 4 times replicated, twice horizontal and twice vertical. This "quad" representation emphasizes the inherent structures.