Implications of contrarian and one-sided strategies for the fair-coin game

We derive some results on contrarian and one-sided strategies of Skeptic for the fair-coin game in the framework of the game-theoretic probability of Shafer and Vovk [G. Shafer and V. Vovk. Probability and Finance -- It's Only a Game!, Wiley, New York, 2001]. In particular, as regards the rate of convergence of the strong law of large numbers (SLLN), we prove that Skeptic can force that the convergence has to be slower than or equal to O(n-1/2). This is achieved by a very simple contrarian strategy of Skeptic. This type of result, bounding the rate of convergence from below, contrasts with more standard results of bounding the rate of SLLN from above by using momentum strategies. We also derive a corresponding one-sided result.