  We study capital process behavior in the fair-coin game and biased-coin games
in the framework of the game-theoretic probability of Shafer and Vovk (2001).
We show that if Skeptic uses a Bayesian strategy with a beta prior, the capital
process is lucidly expressed in terms of the past average of Reality's moves.
  From this it is proved that the Skeptic's Bayesian strategy weakly forces the
strong law of large numbers (SLLN) with the convergence rate of O(\sqrt{\log
n/n})$ and if Reality violates SLLN then the exponential growth rate of the
capital process is very accurately described in terms of the Kullback
divergence between the average of Reality's moves when she violates SLLN and
the average when she observes SLLN. We also investigate optimality properties
associated with Bayesian strategy.
