1. Choice under the following Assumptions
List of alternatives is well‑defined,
one decision maker,
no time dimension,
no involvement of other decision makers,
clear objective of maximizing reward.
1. Static problem with certainty:
Every alternative has a certain outcome.
No conceptual problem of defining the best alternative.
The list may be too big to allow for a direct sorting.

2. 2. Static problem under uncertainty
You are an impresario who considers signing Madonna for a show in the national stadium. Depending of the weather you profits, in 1000s, are:
Sign
Do not sign
500
sunny
-200
rainy
cloudy
Insurance against a bad weather costs 50,000.

3. Outcomes depend on states of nature
Outcomes depend on states of nature whose realizations will be known only later.
A - Set of alternatives;
S – set of states of nature (disjoint and exhaustive);
r: AxS  R - set of outcomes (not necessarily money);
and, possibly, p(s), sS - a probability function.
PAYOFF TABLE

4. It is a process in two stages:
1. making a decision and then
2. the (true) state of nature is revealed and an outcome is attained.
It is static in the sense that the decision is taken only once.
Dominance: Alternative a dominates alternative b if
rsa ³ rsb for all states of nature s (and rsa > rsb for at least one s).
Sign & Insure
Sign
Do not sign
450
500
sunny
-50
-200
rainy
cloudy

5. Choice rules under uncertainty or risk
Decision making under -
uncertainty - probabilities of states of nature are not considered
risk - probabilities of states of nature are considered.
1. Laplace. Select the alternative which maximizes the simple average reward. Principle of insufficient reason. i.e. if we cannot tell which outcome is more likely, we ought to assign equal probabilities.

6. 2. Maximax (of reward). Select the alternative which maximizes its maximum possible reward - maxa{maxs[rsa]}.
An optimistic criterion.

7. 3. Maximin (of reward). Select the alternative which maximizes its minimum possible reward - maxa{mins[rsa]}.
A pessimistic criterion.
4. ‑ criterion. 0££1. Select the alternative which maximizes some weighted average of its maximum and minimum possible rewards - maxa{maxs[rsa]+(1)mins[rsa]}.

8. 5. Minimum (maximum) Regret
5. Minimum (maximum) Regret The regret value of alternative a in state s is Gsa=maxa{rsa}rsa.
Select the alternative which minimizes its maximum possible regret - mina{maxs[Gsa]}.
rewards
regret values

9. These are descriptive rules! but each of them makes some sense.
However, they violate reasonable norms.
Laplace: refining the states of nature may change the selection.
‑criterion: Adding a constant to each reward in a certain state of nature may change the order between the alternatives.

10. Regret: Adding an alternative may change the order between existing alternatives.
nonexists
Sign & Insure
Sign
Do not sign
700
450
250
500
200
sunny
regret values
-50
-200
rainy
cloudy
Max regret