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
• Rank 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.

Rank Rule

In the rank 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 (A₁ and A₂) 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 to each alternative in each situation. The highest payout receives the lowest rank, the lowest payout receives the highest rank. Ranks are assigned by numbers. The lower the rank, the better. In this example, our lowest rank is 0. (But it is also possible to use 1 as the starting value for the lowest rank.)

	S1	S2	S3	…	S100	Sum
A1	1	0	0	0	0	1
A2	0	1	1	1	1	99

Then, we sum up the ranks of an alternative across all situations S. The alternative with the lowest sum is our choice. In our case this is A₁.

The downside of the rank rule is that while keeping the order of the preferences, we loose the distance between them.

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(a_k,s_j) = max U(s_j) – U(ak,s_j)

1. Find for every situation s_j the option with the maximal pay-off.
2. Calculate for every option in situation s_j the maximal loss.
3. Find for each option the maximal loss from in all situations.
4. 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.