Chapter 11.2, Problem 37E

Golf In a simplified variation of the Ryder Cup golf competition between the United States and Europe, each team captain must choose between a very good golfer (V) for the first game of a one-to-one match and a good golfer (G) for the second, or vice versa. Suppose that p_vv represents the probability that a very good golfer from one team will beat a very good golfer from another, with analogous meanings for p_GG, p_VG and p_GV. Assume that for either side to win the tournament, they must win both games, so the payoff for choosing a strategy is the probability of winning both games. Assume that these probabilities are independent. Source: Interfaces.

(a) Explain why the payoff matrix (from the point of view of the United States) is given by

(b) We will assume that p_VVp_GG ≠ p_VGp_GV. (Why would the problem be trivial if this assumption were not true?) Explain why this implies dial there is not a saddle point.

(c) Under the assumption of part (b), find the optimal strategy for each team. What is the value of the game?

(d) Explain why, because of symmetry, the optimal strategy found in part (c) is obvious, without doing any calculations.