1. Osborne 163.2

2. Osborne 173.3. (Hint: use backward induction)

3. Find all the Subgame perfect equilibria (SPEs) in the following game (including all the mixed ones).

1. Osborne 163.2 2. Osborne 173.3. (Hint: use backward induction) 3. Find all the Subgame perfect equilibria (SPEs) in the following game (including all the mixed ones). C H 210to fight in the event of entry, then the analysis would be different. Such a com- mitment would induce the challenger to stay out, an outcome that the incumbent prefers. In the absence of the possibility of the incumbent's making a commitment, we might think of its announcing at the start of the game that it intends to fight; but such a threat is not credible, because after the challenger enters the incumbent's only incentive is to acquiesce. EXERCISE 163.1 (Nash equilibria of extensive games) Find the Nash equilibria of the games in Exercise 156.2a and Figure 160.1. (When constructing the strategic form of each game, be sure to include all the strategies of each player.) EXERCISE 163.2 (Voting by alternating veto) Two people select a policy that affects them both by alternately vetoing policies until only one remains. First person 1 vetoes a policy. If more than one policy remains, person 2 then vetoes a policy. If more than one policy still remains, person 1 then vetoes another policy. The process continues until a single policy remains unvetoed. Suppose there are three possible policies, X, Y, and Z, person 1 prefers X to Y to Z, and person 2 prefers Z to Y to X. Model this situation as an extensive game and find its Nash equilibria.Figure 173.1 One of the games for Exercise 173.2. . EXERCISE 173.3 (Voting by alternating veto) Find the subgame perfect equilibria of the game in Exercise 163.2. Does the game have any Nash equilibrium that is not a subgame perfect equilibrium? Is any outcome generated by a Nash equilibrium not generated by any subgame perfect equilibrium? Consider variants of the game in which player 2's preferences may differ from those specified in Exercise 163.2. Are there any preferences for which the outcome in a subgame perfect equilibrium of the game in which player 1 moves first differs from the outcome in a subgame perfect equilibrium of the game in which player 2 moves first? . EXERCISE 173.4 (Buming a bridge) Army 1, of country 1, must decide whether to attack army 2, of country 2, which is occupying an island between the two coun- tries. In the event of an attack, army 2 may fight, or retreat over a bridge to its mainland. Each army prefers to occupy the island than not to occupy it; a fight is the worst outcome for both armies. Model this situation as an extensive game with