Consider the following model of presidential elections. There are three candidates. Each candidate can be either good or bad. The type of a candidate is drawn at random in the beginning of the game. The probability of drawing a good type is . The type is a private information of a candidate.

There are two periods in this model. In the first period, the first two candidates participate in the election (the third candidate remains idle) and the candidate that wins becomes a president in the first period. The president's type is revealed to the public.

In the second period, the incumbent participates in the election against the third candidate, and the person that wins becomes a president in the second period. Each candidate is risk-neutral and values the office at v (this value is per term).

In the first period, the candidates can finance their political campaigns using their own funds. The amount e spent on the campaign is publicly observable. You can assume that the candidates have quasilinear utility (and no budget constraints): each candidate maximizesE[k]v e, where E[k] is the expected number of periods the candidate occupies the office.

Both candidates have to make their campaign finance decisions simultaneously and independently of each other. (Hint: the monies spent on the political campaign do not affect the quality of the candidates but they may affect the voters' perception of the candidates' quality)

The voters are all identical. They prefer a good candidate over a candidate of unknown quality and they also prefer a candidate of unknown quality over a bad one. You can assume that if the voters are indifferent, they elect either candidate with probability 1/2.

1. (a) Find an equilibrium in which candidates do not spend any money on their campaign.
2. (b) Find an equilibrium in which some candidates spend money on their campaign.
3. (c) A parliament introduces a law that imposes an upper limit on the amount of money that a candidate can spend on his campaign. How does such a law affect the equilibrium payoffs of the candidates and the welfare of the voters?