Chapter 7 Choice under Uncertainty.pdf

Chapter 7 Choice under Uncertainty
Expected utility function
z Simple gambles
Let A = {}a1,...,an be the set of es. Then G, the set of simple gambles (on
⎧⎫
A), is given by ⎨()p1 o a1,.., pn o an | pi ≥ 0,∑ pi = 1⎬
⎩ i ⎭
Compound gambles

z Axioms of choice under uncertainty
1. Completeness
2. Transitivity
a1 ≿ a2 ≿…≿ an
3. Continuity. For any gamble g in G, there is some probability, α∈[0,1], such
that g ~ (α o a1,(1−α) o an ) .
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 α,β∈[0,1] ,)(α o a1,(1−α) o an
(β o a1,(1−β) o an ) iff α≥β.
Monotonicity implies a1 f an .
1 k 1 k
5. Substitution. If g = ( p1 o g ,..., pk o g ) , and h = ( p1 o h ,..., pk o h ) are in G,
and if hi ~,g i ∀i then h ~.g
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 g1, g2 , g3 and α∈(0,1) , we
1
have g1 ≿ g2 iff αg1 + (1−α)g3 ≿αg2 + (1−α)g3 .
Question: does the preference under certainty satisfy this axiom? Why?

z Von 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
u(g) = ∑ piu(ai ) , where ( p1 o a1,..., pn o an ) is the simple gamble induced by g.
i=1

Theorem 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: u(g ) is the number satisfying
g ~ (u(g) o a1,(1− u(g))o an ) . By axiom 3, there exists such a number, by axiom 4,
this number is unique. H