15. Suppose the assumptions of the median voter model hold and there is a population of 11 people with ideal points: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}. Note that the median is ? = 7. This question asks you to prove several propositions. In all cases, your proof should take the form of showing that at least one party would deviate from the proposed equilibrium.
a) Prove that it cannot be an equilibrium for both parties to adopt a platform ?1 = ?2 = 3.
b) Prove that it cannot be an equilibrium for parties to adopt ?1 = 3 and ?2 = 10.
c) Prove that it cannot be an equilibrium for parties to adopt ?1 = 2 and ?2 = 4.
d) Prove that it cannot be an equilibrium for only party to adopt the median position:
16. Consider a population of voters with idea points {1, 3, 5, 7, 9.1, 10, 11, 12, 13}:
a) Find the median voter outcome
b) Suppose a nonstrategic third party enters at ? = 1. Prove that the outcome in part a) is no longer an equilibrium. Can you find a new equilibrium?
c) Suppose a nonstrategic third party enters at ? = 7. Prove that the outcome in part a) is no longer an equilibrium. Can you find a new equilibrium?
