Three NYU students, call them 1, 2, and 3, are the only people bidding in the NYU charity auction for a painting. Student i has private value vi for the painting. These three values are independently uniformly distributed on the interval [0, 12]. Each student knows their own value but not the values of the other two students. They only know how the other values are distributed. Throughout this question, you can ignore ties in values or in bids. It will also help to know the following math fact: If xi and xj are each independently uniformly distributed on the interval [0, k] then the expectation of the maximum (i.e., the larger) of xi and xj is 2k 3 , and the expectation of the minimum (i.e., the smaller) of xi and xj is k 3