Problem 10: Let Z {21, 22, 5K} be a set of alternatives, so that (Z) represents

Question

Problem 10: Let Z {21, 22, 5K} be a set of alternatives, so that (Z) represents all lotteries (or probability distributions) over Z. Let u : Z R and V : (Z) R be such that V(p) piui (21) + +pKu(5K) and V represents a preferende> over (Z). . Prove that satisfies the independence axiom. Note: you cannot claim "that's what the theorem in the class-notes says". I want you to prove t directly ·Prove that if a E R satisfies a > 0 and b E R is any real number then U(p) = pu ) +.. + PKü(2K) also represents preferences, where u au b . Two friends are talking about this problem. The first says that if u(z) 2u(z2) then that means the person in question (i.e. the person with preferences ) likes alternative z1 twice as much as what he likes alternative z2. The second friend says this is anncorrect claim. One of them is right, the other is wrong: who is who? Justify your answer

Explanation / Answer

Patrick is right as all of these utility functions represent different prefrences but VNM axioms are satisfied by them.

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