Peter, Andrew, and James are playing the following game in which the winner is awarded M dollars. Each of the three players receives a coupon and is to decide whether or not to bet on it. If a player chooses to bet, he or she loses the coupon with probability 1/2 and wins an additional coupon with probability 1/2 (thus resulting in two coupons in total). The success of each player in the bet is independent of the results of the bets of the other players. The winner of the prize is the player with the greatest number of coupons.
If there is more than one such player, the winner is selected from among them in a lottery where each has an equal chance of winning. The goal of each player is to maximize the probability of winning the award.
(a) Describe this game as a game in strategic form and find all its Nash equilibria.
(b) Now assume that the wins and losses of the players are perfectly correlated: a single coin flip determines whether all the players who decided to bid either all win an additional coupon or all lose their coupons. Describe this new situation as a game in strategic form and find all its Nash equilibria.