Chapter 7Choice under Uncertainty(高级微观经济学-上海财经大学,沈凌)

Chapter 7 Choice under Uncertainty

7.1 Expected utility function

z Simple gambles

Let {}n a a A ,...,1= be the set of outcomes. Then G , the set of simple gambles (on A), is given by ()⎭⎬⎫

⎩⎨⎧=≥∑i i i n n p p a p a p 1,0|,..,11o o

Compound gambles

z Axioms of choice under uncertainty

1. Completeness

2. Transitivity

1a ≿ 2a ≿ … ≿ n a

3. Continuity. For any gamble g in G, there is some probability, ]1,0[∈α, such

that )

)1(,(~1n a a g o o αα−. In words, continuity means that small changes in probabilities do not change the nature of the ordering between two gambles.

4. Monotonicity. For all probabilities ]1,0[,∈βα,))1(,(1n a a o o αα−≿

))1(,(1n a a o o ββ− iff βα≥. Monotonicity implies n a a f 1.

5. Substitution. If ),...,(11k k g p g p g o o =, and ),...,(11k k h p h p h o o = are in G ,

and if i g h i

i ∀~, then

g h ~. 6. Reduction to simple gambles. The decision maker cares about only the

effective probability. Hence, it can not model the behavior of vacationers in Las Vegas!

7. Independence axiom. For any three gambles 321,,g g g and )1,0(∈α, we

have 1g ≿ 2g iff 31)1(g g αα−+ ≿ 32)1(g g αα−+.

Question: does the preference under certainty satisfy this axiom? Why?

z V on Neumann-Morgenstern Utility

Utility functions possessing the expected utility property is VNM utility functions.

Expected utility property:

The utility function :u G Æ R has the expected utility property if, for every g ∈G , ∑==n

i i i a u p g u 1)()(, where ),...,(11n n a p a p o o is the simple gamble induced by g.

Theorem 7.1 Existence of a VNM function on G

A preference over gambles in G satisfying axioms above can be represented by at least one utility function which has the expected utility property.

Proof: 1. construct a utility function: )(g u is the number satisfying )))(1(,)((~1n a g u a g u g o o −. By axiom 3, there exists such a number, by axiom 4, this number is unique. Hence, we can define a real-valued function in this way.

2. u represents ≿. Because of axiom 2, we have g ≿ g’ iff )))(1(,)((1n a g u a g u o o −≿)))'(1(,)'((1n a g u a g u o o −. And axiom 4 tells us that )))(1(,)((1n a g u a g u o o −≿)))'(1(,)'((1n a g u a g u o o − iff )'()(g u g u ≥.

3. we must show ∑==n

i i i a u p g u 1)()(, where g is a simple gamble. By axiom

6, we can easily extend our result to any gambles.

By definition, i n i i i q a a u a a u a ≡−)))(1(,)((~1o o . Hence, we can know by