# Decision under uncertainty

as decision under uncertainty is designated in the decision theory a decision situation, with which the alternatives, which admits possible environmental conditions and the results with choice of a certain alternative and entrance of a certain environmental condition are, are unknown in which however the probabilities of entrance of the environmental conditions. Sometimes these are called also decisions with objective uncertainty.

## general

decision under uncertainty is in the decision theory a Unterfall of the decision under uncertainty. Decisions under uncertainty differ from decisions under risk by the fact that with the latter probabilities for occurring certain environmental conditions are expected to be familiar.

The decision situation with decisions under uncertainty can be represented by a result matrix. The Entscheider has the choice between different alternatives [itex] a_i [/itex], those in dependence of the possible environmental conditions [itex] s_j [/itex] different results [itex] e_ {ij} [/itex] have as a consequence. However the Entscheider does not know before, with which probability the environmental conditions and thus the results arrive.

## decision rules

### exemplary decision situation

example: 100 € is to be put on for one year. Are available: a share ([itex] a_1 [/itex]) or the saving trunk, which does not bear interest ([itex] a_2 [/itex]). The possible environmental conditions are: The share quotation rises ([itex] s_1 [/itex]), it sinks ([itex] s_2 [/itex]) or it remains directly ([itex] s_3 [/itex]).

The result matrix looks then for example as follows:
[itex] s_1< /math> [itex] s_2< /math> [itex] s_3< /math>
[itex] a_1< /math> 120 80 100
[itex] a_2< /math> 100 ,100 ,100

decisions under uncertainty can rationally according to different rules please to become:

### Maximin rule

the Maximin rule, which is called after Abraham forest also forest rule, is very pessimistic, it here only the most unfavorable in each case event is regarded, which can occur with choice of a certain alternative i in the possible environmental conditions. Only on the basis this worst in each case result (that in each case with different environmental conditions to occur can) compared, all other possible results of an alternative will not become regarded the alternatives.

In the available example the Entscheider selects the saving trunk (alternative 2), there [itex] e_ {12} [/itex] = 80 smaller than [itex] e_ {22} [/itex] = 100.

### Maximax rule

the MaxiMax rule is very optimistically, here each alternative only on the basis the result, which can occur with in each case for this alternative of most favorable environmental condition, is judged.

In the available example the Entscheider selects the share (alternative 1), there [itex] e_ {11} [/itex] = 120 more largely than [itex] e_ {21} [/itex] = 100.

### criticism at Maximin and Maximax rule

both available rules consider not all possible results of an alternative course of action, but pick out themselves only in each case the best (Maximax) or the worst (Maximin) result of an alternative. This can lead to unwanted results, as the following examples show.

[itex] s_1< /math> [itex] s_2< /math> [itex] s_3< /math> [itex] s_ {…}[/itex] [itex] s_ {99}< /math> [itex] s_ {100}< /math>
[itex] a_1< /math> 0 0 0 0 0 120
[itex] a_2< /math> 119 ,119 ,119 ,119 ,119 ,119

according to the Maximax rule here the alternative became [itex] a_1< /math> selected, there only the result in the most favorable environmental condition [itex] s_ {100}< /math> thus [itex] e_ {1; 100}< /math> = 120 one regards, which than 119 is larger. Into all other environmental conditions occurring disbursement of zero with alternative [itex] a_1< /math> one did not consider.

[itex] s_1< /math> [itex] s_2< /math> [itex] s_3< /math> [itex] s_ {…}[/itex] [itex] s_ {99}< /math> [itex] s_ {100}< /math>
[itex] a_1< /math> 120 ,120 ,120 ,120 ,120 99
[itex] a_2< /math> 100 ,100 ,100 ,100 ,100 ,100

according to the Maximin rule here the alternative became [itex] a_2< /math> selected, since only the result occurring in each case in the most unfavorable environmental condition is regarded, thus for the alternative [itex] a_1< /math> the result [itex] e_ {1; 100}< /math> = 99 and with alternative [itex] a_2< /math> 100. Into all other environmental conditions occurring disbursement of 120 with alternative [itex] a_1< /math> one did not consider.

### Hurwicz rule

the Hurwicz rule permits compromises between pessimistic and optimistic decision rules, because the decision maker thereby its personal and subjective attitude by the so-called optimism parameter [itex] \ lambda< /math> (with 0< =< math> \ lambda< /math>< =1) to express can.

In the available example the Entscheider for math <\> lambda /math< selects> > 0,5 the share and for [itex] \ lambda< /math> < 0,5 the saving trunk.

Also the Hurwicz rule regards not all possible results, but evaluates the alternatives on the basis a gewichteteten average value of their optimum and their bad-possible result. Problematic it is with it further that the choice of the optimism parameter can vary strongly tendency-dependently.

### Laplace rule

the Laplace rule: all possible event entrances receive the same probability. The alternative then the best result promises, is selected.

The Laplace rule is based on the following acceptance: Since concerning the environmental conditions it does not admit probabilities of entrance is there is no reason to assume that an environmental condition is more probable than another, therefore must one from uniform distribution of the probabilities of entrance proceed. Thus the Laplace rule considers all environmental conditions during the evaluation of the alternatives. In the available example the Entscheider is indifferent between the share and the saving trunk.

### Savage Niehans rule

the Savage Niehans rule: the evaluation of the alternatives are based thereby not on the direct basis of the results, but due to appropriate regret values. One selects that alternative, which minimizes the potential regret (rule of the smallest regret), also Minimax Regret rule mentioned.

In the example: If environmental condition 1 occurs (share rises), then one with choice of the saving trunk 20 would have lost (Opportunitätskosten). If environmental condition 2 occurs (share sinks), then one with choice of the share 20 would have lost. With environmental condition 3 it is no matter, which I would have selected. The result matrix looks then as follows:

[itex] s_1< /math> [itex] s_2< /math> [itex] s_3< /math>
[itex] a_1< /math> 0 20 0
[itex] a_2< /math> 20 0 0

for the selection of the best alternative one must line by line the largest value look for (maximum regret) and then the alternative (line) select, which exhibits the smallest value (minimize maximum regret). The Savage Niehans rule is not suitable for decision making.

### Krelle rule

a further decision rule was suggested by William Krelle. It is based on the fact that all with an action [itex] a_i< /math> linked use values [itex] u_ {i1}< /math>, [itex] u_ {i2}< /math> ,... ,< math> u_ {in}< /math> with an uncertainty preference function relevant for the decision maker [itex] \ omega< /math> are transformed and to be afterwards added.

[itex] \ Phi (a_i) = \ sum^n_ {j=1} \ omega (u_ {ij})< /math>

The best alternative is now that one with the largest quality measure.

## literature

• v. Zwehl, W., decision rules, in: Hand dictionary of the marketing and management, volume 1, 5. Aufl., Schäffer Poeschel, 1993
• Bamberg, G., Coenenberg A., economical decision teachings, 11. Edition, publishing house Vahlen, 2002