(Sequential truel) Each of persons A, B, and C has a gun contain- ing a single bullet. Each person, as long as she is alive, may shoot at any surviving person. First A can shoot, then B (if still alive), then C (if still alive). (As in the previous exercise, you may interpret the players as political candidates. In this exercise, each candidate has a budget sufficient to launch a negative campaign to discredit exactly one of its rivals.) Denote by pi the probability that player i hits her intended target; assume that 0 pi 1. Assume that each player wishes to maximize her probability of survival; among outcomes in which her survival probability is the same, she wants the danger posed by any other survivors to be as small as possible. (The last assumption is intended to capture the idea that there is some chance that further rounds of shooting may occur, though the possibility of such rounds is not incorporated explicitly into the game.) Model this situation as an extensive game with perfect information and chance moves. (Draw a diagram. Note that the subgames following histories in which A misses her intended target are the same.) Find the subgame perfect equilibria of the game. (Consider only cases in which pA, pB, and pC are all different.) Explain the logic behind A's equi- librium action. Show that "weakness is strength" for C: she is better off if pC pB than if pC > pB.

Now consider the variant in which each player, on her turn, has the additional option of shooting into the air. Find the subgame perfect equilibria of this game when pA pB. Explain the logic behind A's equilibrium action.