Chapter 5: Choices under Uncertainty

Question 1

State-Space Model Representation (general)

Answer 1

u(a)=\sum_{s\inS} u_s (a(s))

Question 2

State Space Model Constant Utility

Answer 2

u(a)=\sum_{s\inS} p(s)U(a(s))

Question 3

Savage’s Sure Thing Principal

Answer 3

Suppose a, a’, b, b’ are four acts and there is T\subseteq S in the state space, where a(s)=a’(s) and b(s)=b’(s) for all s in T. If a(s)=b(s) and a’(s)=b’(s) for all s in T^c.
Then a\succeq b iff a’\succeq b’.
Bringing the problem to another context doesn’t matter. If the action is identical outside the space (T^c), but within the space T there exists a transformation, then that transformation preserves the ordering in that subspace.

Question 4

vNM assumptions

Answer 4

1. complete and transitive preferences over outcomes X
2. Additive separability (mixing in a second lottery instead of the first preserves order)
3. Continuity (if pi \succ \rho, there is some a for all p s.t. ap + (1-a)pi \succ \rho

Question 5

vNM conclusions

Answer 5

iff pi\succeq \rho implies the additive utility representation is greater than or equal to for some U. Any affine transformation of U is also satisfactory.

Question 6

AA Setup

Answer 6

Define H=Pi^S, and h(s)\in Pi, based on the probability. define mixtures between h and h’ and \alpha h + (1-\alpha)h’ = \alpha h(s)+(1-\alpha)h’(s) for all s. This satisfies mixture space.

Question 7

AA Preference Form

Answer 7

\sum_{s\in S}\sum_{x\in supp(h(s))} U_s(x) * h(s)(x). Preferences are affine as well.