Download Homework 4

ECON 3810/CS 5846/ECON 6676: Problem Set 4
Handed out: 4/22/14; due: 5/6/14
1. Exercises on null sets:
(a) Show that ∅ is null.
(b) Show that if B ⊆ A and A is null, then B is null.
(c) Show that if A and B are disjoint and null, then A ∪ B is null.
2. GRAD: Let S denote a set of states, O a set of outcomes, L the set of
all Savage acts from S to O, and a preference order on L. Suppose
that has an expected utility representation with payoff function u
and probability distribution p. Show that if A is not a null event, then
f A g iff the conditional expected utility of f given A exceeds that of
g given A.
3. Show exactly where the standard choices made fail the Independence
Postulate in (a) the Allais Paradox and (b) Ellsberg’s paradox. (Note
that different versions of the Independence Postulate are involved; the
first is one is for von Neumann-Morgenstern, the second is for Savage.)
4. Let A4w be the following weakening of A4:
A4w . If p and q are probabilities on prizes and s and s0 are non-null,
then p{s} f q{s} f implies p{s0 } f q{s0 } f .
The key difference between A4 and A4w is that in A4w , the conclusion
has “”, not .
Show that MMEU satisfies A2 and A4w . More precisely, given a set
P of probability measure and a utility function u, consider the preference order P,u induced by MMEU: f P,u g iff inf Pr∈P EPr (u ◦ f ) >
inf Pr∈P EPr (u ◦ g) Show that P,u satisfies A2 and A4w . For extra
credit, show that by means of a coutnerexample that MMEU does not
in general satisfy A4 (i.e., A4w with replaced by ).
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5. Prove that, for a probability measure µ and a random variable mapping
a finite state space S to the real numbers, the definition of expectation
given by Choquet coincides with the standard definition; that is, if the
values of the function f are x1 < x2 < ... < xn and µ is a probability
on S, then
X
f (s)µ(s) = x1 +(x2 −x1 )µ(f > x1 )+· · ·+(xn −xn −1)(µ(f > xn−1 ).
s∈S
(As usual, f > xi is the set {s ∈ S : f (s) > xi }.)
6. GRAD: This exercise shows that A1 and A4 imply A40 . So assume
that is an order on acts (horse lotteries) for which A1 and A4 hold.
If p is a lottery over prizes, let p denote the constant function on states
that always gives p. Recall that fX g is the act that agrees with f on
X and with g on X c .
(a) Show that if p{s} f q {s} f and s is non-null, then p q. (Hint:
Start by showing that if p{s} f q {s} f and s and s0 are non-null,
then p{s0 } f 0 q {s0 } f 0 for an arbitrary act f 0 . The case that f = f 0
is just A4. Proceed by induction on |D(f, f 0 )|, where D(f, f 0 ) is
the number of states where f and f 0 differ. Then show that for
all sets S 0 such that S 0 is non-null, pS 0 f 0 q S 0 f 0 . This can be
done easily by induction on S 0 . Finally, observe that pS f = p and
q S f = q.)
(b) Prove A40 : show that if f (s) g(s) for all s ∈ S, then f g.
(Hint: Show by induction on |S 0 | that if f (s) g(s) for all s ∈ S,
then fS 0 g g.)
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