Problem 8. Let (A,0,0, (=;: A)) be an assignment problem. Let v; : O* R represent >, for each A. A random allocation z' ez-ante Pareto dominates x for vy,...,v, if . ' v - T = E ; ,vi(0) > E z; ,0;(0) 0e0* oe0* for all i A, with at least one strict inequality. A random allocation is ez-ante efficient if it is not ex-ante Pareto dominated by any other random allocation. 1. Consider the example with random allocation 5/12 1/2 1/12 5/12 5/12 1/2 1/12 5/12 that we discussed in lecture. Exhibit explicit functions v;, representing >, in that example, = 1, 2, 3,4, for which x is ex-ante Pareto dominated. 2. Suppose that z solves the problem of maximizing Y ", \\v;z} for fixed weights A; > 0, over all random assignments :C'E Show that x is ex-ante efficient. 3. Show that if a random allocation is ex-ante efficient, then it is ordinally efficient