Choice under Uncertainty
Uncertainty means that the probability of a certain event is not known. To choose an option, different methods are available:
- Maximin rule
- Maximax rule
- Range rule
- Hurwicz rule
- Minimax-Regret rule
- Laplace Rule
Maximin Rule
This rule is the most pessimistic one. It assumes that the worst case happens. And it tries to maximize the pay-off in this situation. In short, we choose the option that grants us the highest (maximal) minimal pay-off.
Suppose we have the following table with three alternatives A and their pay-offs in the situations S 1 till 4
| S1 | S2 | S3 | S4 | Min | |
| A1 | 20 | 15 | 20 | 3 | 3 |
| A2 | 5 | 6 | 7 | 4 | 4 |
| A3 | 22 | 3 | 3 | -2 | -2 |
The minimal pay-off for A1 is 3, for A2 it is 4 and for A3 it is -2. Now we choose the alternative that yields us the highest of these pay-offs. This is A2.
Maximax Rule
The Maximax rule is the most optimistic one. It assumes that the best case happens. And it tries to maximize the pay-off in this situation. In short, we choose the option that grants us the highest (maximal) maximal pay-off.
| S1 | S2 | S3 | S4 | Max | |
| A1 | 20 | 15 | 20 | 3 | 20 |
| A2 | 5 | 6 | 7 | 4 | 7 |
| A3 | 22 | 3 | 3 | -2 | 22 |
The alternative with the highest pay-off is A3.
Range Rule
In the range rule, we assign ranks for each alternative in each Situation. Then, we sum up the ranks and choose the alternative with the best rank.
Suppose we have the following payoff table with two alternatives and 100 situations.
| S1 | S2 | S3 | … | S100 | |
| A1 | 0 | 3 | 3 | 3 | 3 |
| A2 | 1 | 1 | 1 | 1 | 1 |
The Maximin rule would tell us to choose A2. But this sounds unintuitive because there are 99 situations that would bring a better result.
Now, we assign a rank. The lower the rank, the better. In this example, our lowest rank is 0. (But it is also possible to use 1 as the lowest rank.)
| S1 | S2 | S3 | … | S100 | Sum | |
| A1 | 1 | 0 | 0 | 0 | 0 | 1 |
| A2 | 0 | 1 | 1 | 1 | 1 | 99 |
Then, we choose the alternative with the lowest total rank. In our case this is A1.
Hurwicz Rule
The Hurwicz rule introduces an optimism parameter a. The rank for each alternative is calculated by:
H = (1-a)*Min + a*Max.
Min: Minimal pay-off of an alternative
Max: Maximal pay-off of an alternative
Suppose we have the following table and choose an a = 0.4.
| S1 | S2 | S3 | S4 | Min | Max | H | |
| A1 | 20 | 15 | 20 | 3 | 3 | 20 | 3*(1-0.4)+20*0.4 = 9.8 |
| A2 | 5 | 6 | 7 | 4 | 4 | 7 | 5.2 |
| A3 | 22 | 3 | 3 | -2 | -2 | 22 | 7.6 |
Alternative A1 has the highest H. That’s why we would choose it.
Minimax Regret Rule
With this decision rule, we try to minimize the regret of a choice. Regret is the difference between the maximal possible pay-off in a situation and the realized pay-off.
R(ak,sj) = max U(sj) – U(ak,sj)
- Find for every situation sj the option with the maximal pay-off.
- Calculate for every option in situation sj the maximal loss.
- Find for each option the maximal loss from in all situations.
- Choose the option with the smallest loss R.
| S1 | S2 | S3 | S4 | |
| A1 | 20 | 15 | 20 | 3 |
| A2 | 5 | 6 | 7 | 4 |
| A3 | 22 | 3 | 3 | -2 |
| Max | 22 | 15 | 20 | 4 |
| S1 | S2 | S3 | S4 | Rmax() | |
| A1 | 22-20 = 2 | 15-15 =0 | 0 | 1 | 2 |
| A2 | 22-5 = 17 | 15-9 =6 | 13 | 0 | 17 |
| A3 | 22-22 =0 | 15-12 =3 | 17 | 6 | 17 |
A1 is the option with the smallest maximal regret.
LaPlace Rule
Now, we assume that every situation occurs with the same likelihood. Then we choose the option with the highest expected pay-off.
| S1 | S2 | S3 | S4 | Expected pay-off | |
| A1 | 20 | 15 | 20 | 3 | (20+15+20+3)/4=14.5 |
| A2 | 5 | 6 | 7 | 4 | 5.5 |
| A3 | 22 | 3 | 3 | -2 | 6.5 |
Option A1 has the highest expected value.
