The structure of a 3-state finite transducer representation for Prisoner's Dilemma

Abstract

Abstract-To facilitate systematic and automated analysis of game playing strategies, the fingerprint, a mathematical technique that generates a functional summary independent of representation, was developed. This study attempts to push the boundaries of full state space investigation, looking at 3-state finite transducers as a representation for playing Prisoner's Dilemma. There are a staggering 23,000 unique strategies in this space, which severely limits the choice of analysis methods. These strategies are fingerprinted and pairwise distances computed, then hierarchical clustering reduces them to a manageable size for further experiments with multidimensional scaling and the mutational connectivity network. Results indicate there are no obvious cutoff scales of structure; mutational distance is not correlated with fingerprint distance; and a level of similarity with past results on smaller state spaces.

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

“…These are summed in a binary divide-and conquer fashion to decrease roundoff error. From results in previous studies [8], the integration error is at most on the order of 10-10, where distances are generally on the order of 0.01-l. For all intents and purposes, the integral is basically exact.…”

“…The three top dimensions that we have found in all previous studies [7], [11], [8] are, up to linear combinations, the cooperativity of the strategy in general, the responsiveness of the strategy to the opponent, and the probability of initial cooperation. RAND and per(CDDC) cannot be separated using any of the above measures.…”

“…Since several studies [9], [10] have demonstrated that the choice of representation used in an evolutionary simulation can have a drastic effect on the results, we 978-1-4799-3547-5/1411.00 ©2014 IEEE are interested in investigating the structure of entire rep resentations. Some of the small spaces have been studied in [7], [8], [11], [12] and we would like to extend the analysis; however, due to the exponential number of possible strategies, enumeration quickly becomes impossible for any reasonably large space.…”

“…For special properties, we will use the family of geometric distributions: JL ( n; 0: ) = (1 o:)o: n -l , 0: E [0,1) , at 0: = 0.8 in continuity with prior work [7], [8].…”

“…From [8], a metric has been de fined on the space of fingerprints, which allows mathematical quantification of the distance between particular strategies.…”