Comparing the structure of probabilistic 4- and 8-state finite transducer representations for Prisoner´s Dilemma

The fingerprint is a mathematical tool that turns a game-playing strategy into a representation-independent functional summary of its behaviour; several studies have used this to enable massive computational analyses of entire spaces of strategies. This study extends further by directly comparing two large (on the order of 10⁹⁶) state spaces, grids over the probabilistic 4- and 8-state finite transducers, as representations for playing iterated Prisoner´s Dilemma. We take uniformly random samples of size 65,536 from each, fingerprint each strategy in both samples, and compute all pairwise distances within each sample. Hierarchical clustering reduces each distance matrix to size 16,384 for embedding into Euclidean space with multidimensional scaling. Results indicate that several important dimensions are strongly preserved between the representations, and we can quantify them. We additionally find an important theoretical construct, two strategies that are not identical in behaviour, but with the same fingerprint: Random and periodic (CDDC).