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(Voting by alternating veto, exercise 163.2 in Osborne (2004)) Two people select a policy that affects them both by alternately vetoing policies...

163.2. (Voting by alternating veto, exercise 163.2 in Osborne (2004)) 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 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.
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 the previous exercise. 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?

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