When designated decision under uncertainty 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 information

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 a_i, which entail different results in dependence of the possible environmental conditions s_j e_ {ij}. 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 (a_1) or the saving trunk, which does not bear interest (a_2). The possible environmental conditions are: The share quotation rises (s_1), it sinks (s_2) or it remains directly (s_3).

The result matrix looks then for example as follows:

{| border= " 0 "

! width= " 5% " |! width= " 10% " align= " right " |s_1! width= " 10% " align= " right " |s_2! width= " 10% " align= " right " |s_3 | -! a_1 | align= " right " | 120 | align= " right " | 80 | align= " right " | 100 | -! a_2 | align= " right " | 100 | align= " right " | 100 | align= " right " | 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.

\ max_ i:\varphi _ {ai} = \ min_j e_ {ij}

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

\ max_ i:\varphi _ {ai} = \ max_j e_ {ij}

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

{| border= " 0 "

! width= " 5% " |! width= " 10% " align= " right " |s_1! width= " 10% " align= " right " |s_2! width= " 10% " align= " right " |s_3! width= " 10% " align= " right " |s_ {"…} ! width= " 10% " align= " right " |s_ {99}! width= " 10% " align= " right " |s_ {100} | -! a_1 | align= " right " | 0 | align= " right " | 0 | align= " right " | 0 | align= " right " | 0 | align= " right " | 0 | align= " right " | 120 | -! a_2 | align= " right " | 119 | align= " right " | 119 | align= " right " | 119 | align= " right " | 119 | align= " right " | 119 | align= " right " | 119 |} 

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

{| border= " 0 "

! width= " 5% " |! width= " 10% " align= " right " |s_1! width= " 10% " align= " right " |s_2! width= " 10% " align= " right " |s_3! width= " 10% " align= " right " |s_ {"…} ! width= " 10% " align= " right " |s_ {99}! width= " 10% " align= " right " |s_ {100} | -! a_1 | align= " right " | 120 | align= " right " | 120 | align= " right " | 120 | align= " right " | 120 | align= " right " | 120 | align= " right " | 99 | -! a_2 | align= " right " | 100 | align= " right " | 100 | align= " right " | 100 | align= " right " | 100 | align= " right " | 100 | align= " right " | 100 |} 

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

Hurwicz rule

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

\ max_ i:\varphi _ {ai} = \ lambda \ cdot \ max_j e_ {ij} + (1 \ lambda) \ min_j e_ {ij}

In the available example the Entscheider for \ lambda > 0.5 and for \ lambda < 0.5 the saving trunk selects the share.

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.

\ max_ i:\varphi _ {ai} = \ frac {1} {n} \ sum_j e_ {ij}

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 Alternativen.Im available example is indifferent the Entscheider 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 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:

{| border= " 0 "

! width= " 5% " |! width= " 10% " align= " right " |s_1! width= " 10% " align= " right " |s_2! width= " 10% " align= " right " |s_3 | -! a_1 | align= " right " | 0 | align= " right " | 20 | align= " right " | 0 | -! a_2 | align= " right " | 20 | align= " right " | 0 | align= " right " | 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 linked use values with an action a_i u_ {i1}, u_ {i2},"… , u_ {into} with an uncertainty preference function relevant for the decision maker \ omega to be transformed and afterwards added.

\ Phi (a_i) = \ sum^n_ {j=1} \ omega (u_ {ij})

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., Poeschel, 1993
• Bamberg, G., Coenenberg A., economical decision teachings, 11. Edition, publishing house Vahlen, 2002

Articles in category "Decision under uncertainty"

We found here 10 articles.

Related Websites

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• Decision and Reasoning under Uncertainty
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• Decisions under Uncertainty - Cambridge University Press
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