From the next exercise of the book Game theory: An introduction of Steven Tadelis:

9.5 Campaign Spending: Two political candidates are destined to play the fol- lowing two-stage game. Assume throughout that there is no discounting 9.? Exercises ' 139 [ii = I}. First, they compete in the primaries of their respective parties. Each candidate i can spend sl- 33 D resources on ads that reach out to voters, which in turn increases the probability that candidate 1' wins the race. Given a pair of spending choices (3], s2}, the probability that candidate :3 wins is given by '71:}? If neither spends any resources then each wins with probability %. Each candidate values winning at a payoff of 16 .2 [I1 and the cost of spending sf is equal to 3;. After each player observes the resources spent by the other and a winner in the primaries is selected, they can choose how to interact. Each can choose to be pleasant [P for player 1 and p for player 2) or nasty [N and 11, respectively}. At this stage both players prefer that they be nice to each other rather than nasty, but if one player is nasty then the other prefers tobe nasty too. The payoffs from this stage are given by the following matrix (where w :5 {1}: Player 2 p n P mm Player] N mm a. Find the unique Nash equilibrium of the rst-stage game and the two pure-strategy Nash equilibria of the second-stage game. b. What are the Pareto-optimal outcomes of each stage-game\"? c. For which value of to can the players support the path of Pareto-optimal outcomes as a subgsrne-perfect equilibrium? d. Assume that to = i. What is the \"best" symmetric subgame-perfect equilibrium that the players can support? e. What happens to the best symmetric sub game-perfect equilibrium that the players can support as u: changes? In what way is this related to the role played by a discount factor