EXERCISE $173\mathrm{.}3($Voting by alternating veto$\}$ Find the subgame perfect equilibria of the game in Exercise $163\mathrm{.}2.$ 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 $2\text{'}$s preferences may differ from those specified in Exercise $163\mathrm{.}2.$ 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 $163\mathrm{.}2($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.