Fingerprinting codes provide a means for the digital content distributor to trace the origin of an unauthorized redistribution. The maximum achievable rate, or capacity, has recently been derived as the value of a two-person zero-sum game between the fingerprinting embedder and the pirates. Under the so-called Boneh-Shaw marking assumption, we prove rigorously that the asymptotic capacity is 1/(k22 ln 2), where k is the number of pirates. Furthermore, we confirm our earlier conjecture that Tardos' choice of the arcsine distribution asymptotically maximizes the mutual information payoff function while the interleaving attack minimizes it. Along with the asymptotic behavior, numerical solutions to the game for small k are also presented.