political game theory
2. Senate Race Game Revisited
Earlier we solved the Senate Race Game assuming that the incumbent, Staton, first decided whether or not to advertise and that the potential challenger, Reenock, then decided whether to enter or stay out. What happens, though, if we reverse the order in which the choices are made? In other words, what happens if Reenock has to decide whether to enter or stay out before Staton decides whether to advertise or not? The game tree for this scenario is shown in Figure 3.14. Assume that the two players have the same preference orderings as before; that is, Reenockâ€™s preference ordering is
â€¢ (enter; donâ€™t advertise) > (stay out; advertise) > (stay out; donâ€™t advertise) > (enter; advertise).
Statonâ€™s preference ordering is
â€¢ (stay out; donâ€™t advertise) > (stay out; advertise) > (enter; donâ€™t advertise) > (enter; advertise).
(a) Put the payoâ†µs for each player associated with the four possible outcomes into the game tree in Figure 3.14. Use the numbers 4, 3, 2, and 1 to indicate the preference ordering for each player as we did with the original Senate Race Game.
(b) Solve the game by backward induction. Write down the expected outcome of the game, the payoâ†µs that each player receives, and the subgame perfect equilibrium.3
(c) Based on what you have found, does it matter which player gets to move first? If it does matter, explain why it matters. If it does not matter, explain why it does not matter.
4. Legislative Pay Raise Game
Imagine a strategic situation in which three legislators vote sequentially on whether they should receive a pay raise. Letâ€™s assume that decisions are made by majority rule. This means that if at least two legislators vote yes, then each legislator will receive a pay raise. Although all the legislators would like to receive a pay raise, they know that they will pay a cost with their constituents if they are seen to vote for the raise. From the perspective of each legislator, four possible outcomes can occur. The most preferred outcome for all three legislators is that they get the pay raise even though they personally vote no. The worst possible outcome is that they do not get the pay raise and they voted yes. Of the remaining two outcomes, let us assume that the legislators prefer the outcome in which they get the pay raise when they voted yes to the outcome in which they do not get the pay raise when they voted no. As a result, the preference ordering for each legislator is
â€¢ Get raise, vote no > Get raise, vote yes > No raise, vote no > No raise, vote yes.
(a) Imagine that you are one of the legislators. Would you prefer to vote first,second, or third? Explain your answer.
(b) The game tree for the Legislative Pay Raise Game is shown in Figure 3.16. Using the preference orderings shown above, we have written in appropriate payoffs for some, but not all, of the outcomes. As before, we have used the numbers 4, 3, 2, and 1 to indicate the preference ordering of the players. Legislator 1â€™s payoff is shown first, Legislator 2â€™s payoffs is shown second, and Legislator 3â€™s payoff is shown third. Fill in the missing payoffs to complete the game.
(c) Solve the game by backward induction. Be careful to make sure that you are comparing the payoffs of the correct legislator. What is the expected outcome of the game? What are the payoffs that each player receives? What is the subgame perfect equilibrium?
(d) Imagine that you are one of the legislators again. Now that you have solved the game, would you prefer to vote first, second, or third? Explain your answer. Did your answer change from before?
(e) Does the Legislative Pay Raise Game have any implications for whether you would want to be in a position to set the rules or the agenda in business meetings or other ocial settings? Explain your answer.