Mackerels in the Moonlight: A Duopoly Model of Political Agency

I study political competition between two candidates who could differ in their ability, popularity, and ethics. In elections, each candidate proposes a flat (income) tax rate and a public good level. A high(er)-ability candidate can produce the public good using less funds. Collected taxes that are not used in public goods production are stolen by the elected politician. The voting decision is probabilistic; it depends on a candidate's fiscal policy and his popularity. I prove that the pure strategy Nash Equilibrium exists and that there are at most two separate equilibria. I also provide a fully solved example.