How an inferior player with a superior strategy can gain

QUESTION:

How an inferior player with a superior strategy can gain an advantage. Borb b about to play a two-game che:s:s match with an opponent, and wants to find the strategy that maximizes his winning chances. Each game ends with either a win by one of the players, or a draw. If the score is tied at the end of the two games, the match goes into sudden-death mode, and the players continue to play until the first time one of them wins a game (and the match). Boris has two playing styles. timid and bold, and he can choose one of the two at will in each game. no matter what style he chose in previous games. With timid play. he draws with probability Pd > 0, and he loses with probability 1 - Pd . With bold play. he wins with probability pw , and he loses with probability 1 - pw ' Boris will always play bold during sudden death, but may switch style between games 1 and 2. (a) Find the probability that Boris wins the match for each of the following strategies: (i) Play bold in both games 1 and 2. (ii) Play timid in both games 1 and 2. (iii) Play timid whenever he is ahead in the score. and play bold otherwise. (b) Assume that pw < 1 /2, so Boris is the worse player, regardless of the playing style he adopts. Show that with the strategy in (iii) above. and depending on the values of pw and Pd. Boris may have a better than a 50-50 chance to win the match. How do you explain this advantage?