# Maximize expected score of a ranking

I am trying to figure out how to answer 1.1.c from here http://www.cc.gatech.edu/sites/default/files/images/200808_cse_qualifier.pdf. The problem setup is as follows

We are then asked to show

An earlier problem had us show that the maximizing ranking is in descending order of $y$ values. My intuition is that in expectation, if $y_i>y_j$, then in expectation, $p(y_i|x_i)>p(y_i|x_j)$, so that (with a big hand wave) in expectation $R(x_i)>R(x_j)$, but I'm having a hard time formalizing that or saying much about the other terms in the summation.

using the descending order of $R(x)$'s will give us the max based on the previous problem.