A sensitivity analysis of matching coin game strategies

Consider a matching game where two players (labeled Player #1 and Player #2) each have a coin and for each trial, shows either a head or a tail. When either a head-tail or a tail-head is shown, Player #1 wins a specified amount; otherwise, Player #2 wins a specified amount. There exists a Nash equilibrium that is stable with respect to an individual player's deviation from it, hence, it is not advantageous for any of the players to use a strategy different from the optimal strategy, since the value of the player's expected winnings would not be increased. This note describes strategies for this game and shows the limitations of each player in securing guaranteed expected winnings. A generalization of this game to three people is discussed and analyzed. The primary contribution is a sensitivity analysis of the expected winnings with respect to small perturbations in the winnings parameters for this generalized game.