Exercise 22.28. * This exercise involves generalizing the idea of single-crossing property used in Theorem 22.5 to multi-dimensional policy spaces. The appropriate notion of preferences of individuals turns out to be “intermediate preferences”. Let P ⊂ RI , where I is an integer, and policies p belong to P. We say that voters have intermediate preferences, if their indirect utility function U(p; αi) can be written as U(p; αi) = J1(p) + K(αi)J2 (p), here K(αi) is monotonic (monotonically increasing or monotonically decreasing) in αi, and the functions J1(p) and J2 (p) are common to all voters. Suppose that A2 holds and voters have intermediate preferences. We define the bliss point (vector) of individual i as in the text, as p (αi) ∈ P that maximizes individual i’s utility. Prove that when preferences are intermediate a Condorcet winner always exists and coincides with bliss point of the voter with the median value of αi, that is, pm = p (αm).