Fingerprinting operators generate functional signatures of game players and are useful for their automated analysis independent of representation or encoding. The theory for a fingerprinting operator which returns the length-weighted probability of a given move pair occurring from playing the investigated agent against a general parametrized probabilistic finite-state transducer (PFT) is developed, applicable to arbitrary iterated games. Results for the distinguishing power of the 1-state opponent model, uniform approximability of fingerprints of arbitrary players, analyticity and Lipschitz continuity of fingerprints for logically possible players, and equicontinuity of the fingerprints of bounded-state probabilistic transducers are derived. Algorithms for the efficient computation of special instances are given; the shortcomings of a previous model, strictly generalized here from a simple projection of the new model, are explained in terms of regularity condition violations, and the extra power and functional niceness of the new fingerprints demonstrated. The 2-state deterministic finite-state transducers (DFTs) are fingerprinted and pairwise distances computed; using this the structure of DFTs in strategy space is elucidated.