Campaigning:
Two politicians, an incumbent (player 1) and a potential rival (player 2), are running for the local mayoralty. The incumbent has either a broad base of support (B) or a small base of support (S), each occurring with probability 1/2. The incumbent knows his level of support but the potential rival does not.
The incumbent first chooses how much soft money to spend on campaign financing: a low quantity (L) or a high quantity (H), a decision that is observed by the potential rival. The rival can then decide to run (R) or not to run (N). If the incumbent chooses a level of campaign financing L then given the support base and the reaction of the potential rival, the payoffs are given by the following payoff matrix (these are payoffs that represent the expectations from winning, campaigning, and so on; this is not a matrix game):
The cost in payoffs that an incumbent incurs for choosing H instead of L is 2 if he has a broad base of support and 4 if he has a small base of support (that is, these are costs that are deducted from the payoffs in the payoff matrix that are conditional on the type). A rival who runs against an incumbent with a broad base of support who chose H will obtain a payoff of -10, while a rival who runs against an incumbent with a small base of support who chose H will obtain a payoff of -4. If the rival chooses not to run then he obtains 0, as in the payoff matrix.
a. Draw the extensive form of this game and identify the proper subgames. Draw the matrix that represents the normal form of the extensive form.
b. If the rival could commit in advance to a certain pure strategy that he would follow regardless of the incumbent's choice of financing, anticipating that the incumbent would then choose his best response, what would that strategy be? What would be the incumbent's best response to this strategy? Is the pair of strategies you found a Bayesian Nash equilibrium?
c. Can the pair of strategies you found in (b) be part of a perfect Bayesian equilibrium?
d. Are there other pairs of strategies that can be part of a perfect Bayesian equilibrium?