Decision under uncertainty

In decision theory, decision-making under uncertainty refers to decision- making situations in which the alternatives , possible environmental conditions and the results of choosing a certain alternative and the occurrence of a certain environmental condition are known, but in which the probability of occurrence of the environmental conditions is unknown. These are sometimes called decisions in the event of objective uncertainty .

General

Decision under uncertainty in the decision theory , a sub-category of decisions under uncertainty . Decisions under uncertainty differ from decisions under risk in that in the latter case the probabilities of the occurrence of certain environmental conditions are assumed to be known or at least can be assigned by an estimate.

The decision-making situation for decisions under uncertainty can be represented by a result matrix . The decision maker has the choice between various alternatives which, depending on the possible environmental conditions, produce n different results . However, the decision maker does not know beforehand the probability with which the environmental conditions and thus the results will occur. ${\displaystyle a_{i}}$ ${\displaystyle s_{j}}$ ${\displaystyle e_{ij}}$

Results matrix

	${\displaystyle s_{1}}$	${\displaystyle \dots }$	${\displaystyle s_{j}}$	${\displaystyle \dots }$	${\displaystyle s_{n}}$
${\displaystyle a_{1}}$	${\displaystyle e_{11}}$		${\displaystyle e_{1j}}$		${\displaystyle e_{1n}}$
${\displaystyle \vdots }$
${\displaystyle a_{i}}$	${\displaystyle e_{i1}}$		${\displaystyle e_{ij}}$		${\displaystyle e_{in}}$
${\displaystyle \vdots }$
${\displaystyle a_{m}}$	${\displaystyle e_{m1}}$		${\displaystyle e_{mj}}$		${\displaystyle e_{mn}}$

The distinction between uncertainty in uncertainty and risk has linguistically not yet fully established in the literature. In some cases, there is only a division into uncertainty (probabilities unknown) and risk (probabilities known).

Decision rules

The following decision rules are to be explained in more detail using an exemplary decision situation.

Example: € 100 should be invested for one year. You can choose between a share ( ) or the savings stocking that does not generate any interest ( ). The possible environmental conditions are: The share price rises ( ), it falls ( ) or it stays the same ( ). ${\displaystyle a_{1}}$ ${\displaystyle a_{2}}$ ${\displaystyle s_{1}}$ ${\displaystyle s_{2}}$ ${\displaystyle s_{3}}$

The result matrix looks like this, for example:

	${\displaystyle s_{1}}$	${\displaystyle s_{2}}$	${\displaystyle s_{3}}$
${\displaystyle a_{1}}$	120	80	100
${\displaystyle a_{2}}$	100	100	100

Decisions under uncertainty can be made rationally according to different rules:

Minimax rule

→ Main article : Minimax rule

The minimax rule or maximin rule ( also Wald rule after Abraham Wald ) is very pessimistic . Here, the most unfavorable event is considered, which can occur if a certain alternative action is selected in the various environmental conditions. The alternatives are only compared on the basis of the worst result in each case (which can occur in different environmental conditions); all other possible results of an alternative are not considered. ${\displaystyle a_{i}}$

${\displaystyle \max _{i}:\varphi _{a_{i}}=\min _{j}e_{ij}}$

	${\displaystyle s_{1}}$	${\displaystyle s_{2}}$	${\displaystyle s_{3}}$	${\displaystyle \min _{j}e_{ij}}$
${\displaystyle a_{1}}$	120	80	100	80
${\displaystyle a_{2}}$	100	100	100	100

${\displaystyle \max _{i}(\varphi _{a_{i}})=\max(80,100)=100}$

In the present example, the decision maker chooses the savings stocking (alternative 2, ), as this guarantees a payout of 100 € regardless of the environmental conditions, while with alternative 1, in the worst case (price drops, environmental status ), only 80 € are to be booked at the end of the year . The maximum is then selected from these line minima. The name of the decision rule is derived from this procedure. ${\displaystyle a_{2}}$ ${\displaystyle s_{2}}$

A concrete application of the MaxiMin rule can be found in John Rawls in A Theory of Justice . Many chess programs use a corresponding Minimax algorithm when choosing a move.

An extension of the Maximin rule is the Leximin rule from Amartya Sen , according to which, in the event that two alternatives show the worst condition in each case, the one in which the second worst case has the highest value, etc. is avoided. that an overall worse version can be chosen just because it corresponds to the Maximin principle.

Maximax rule

The Maximax rule is a very optimistic decision rule . Here, each alternative is only assessed on the basis of the result that can occur in the environmental condition that is most favorable for this alternative . The decision maker therefore chooses the alternative course of action with the maximum line maximum.

${\displaystyle \max _{i}:\varphi _{a_{i}}=\max _{j}e_{ij}}$

	${\displaystyle s_{1}}$	${\displaystyle s_{2}}$	${\displaystyle s_{3}}$	${\displaystyle \max _{j}e_{ij}}$
${\displaystyle a_{1}}$	120	80	100	120
${\displaystyle a_{2}}$	100	100	100	100

${\displaystyle \max _{i}(\varphi _{a_{i}})=\max(120,100)=120}$

In the present example, the decision maker consequently chooses the alternative . ${\displaystyle a_{1}}$

If, instead of maximizing, the aim is to minimize a target value, then we speak of the minimin principle.

Criticism of the Maximin and Maximax rules

Both of these rules do not take into account all possible results of an alternative course of action, but only select the best (Maximax) or the worst (Maximin) result of an alternative. This can lead to undesirable results, as the following examples show.

	${\displaystyle s_{1}}$	${\displaystyle s_{2}}$	${\displaystyle s_{3}}$	${\displaystyle s_{...}}$	${\displaystyle s_{99}}$	${\displaystyle s_{100}}$	${\displaystyle \max _{j}e_{ij}}$
${\displaystyle a_{1}}$	0	0	0	0	0	120	120
${\displaystyle a_{2}}$	119	119	119	119	119	119	119

According to the Maximax rule, the alternative would be chosen here, since only the result in the most favorable environmental condition is considered, which is greater than 119. The payout of zero for alternative , which occurs in all other environmental conditions, would not be taken into account. ${\displaystyle a_{1}}$ ${\displaystyle s_{100}}$ ${\displaystyle e_{1;100}=120}$ ${\displaystyle a_{1}}$

	${\displaystyle s_{1}}$	${\displaystyle s_{2}}$	${\displaystyle s_{3}}$	${\displaystyle s_{...}}$	${\displaystyle s_{99}}$	${\displaystyle s_{100}}$	${\displaystyle \min _{j}e_{ij}}$
${\displaystyle a_{1}}$	120	120	120	120	120	99	99
${\displaystyle a_{2}}$	100	100	100	100	100	100	100

According to the minimax rule, the alternative would be selected here , since only the result occurring in the most unfavorable environmental condition is considered, i.e. for the alternative the result = 99 and for alternative 100. The payout of 120 for alternative , which occurs in all other environmental conditions, would not be taken into account . ${\displaystyle a_{2}}$ ${\displaystyle a_{1}}$ ${\displaystyle e_{1;100}}$ ${\displaystyle a_{2}}$ ${\displaystyle a_{1}}$

Hurwicz rule

The Hurwicz rule , named after Leonid Hurwicz , even optimism / pessimism rule called, allows trade-offs between pessimistic and optimistic decision rules because the decision-makers while his personal and subjective adjustment by the so-called optimism parameters (with can bring) expressed. ${\displaystyle \lambda }$ ${\displaystyle 0\leq \lambda \leq 1}$

The respective line maxima are thus with (which is between 0 and 1) and the respective line minima with ( ) - d. H. the amount resulting in a total of 1 - multiplied. ${\displaystyle \lambda }$ ${\displaystyle 1-\lambda }$ ${\displaystyle \lambda }$

The larger the value , the more optimistic the basic setting , with = 1 the Maximax rule is applied, with = 0 the Maximin rule is applied. ${\displaystyle \lambda }$ ${\displaystyle \lambda }$ ${\displaystyle \lambda }$

${\displaystyle \max _{i}:\varphi _{a_{i}}=\lambda \cdot \max _{j}e_{ij}+(1-\lambda )\min _{j}e_{ij}}$

In this example, the decision maker chooses the share for > 0.5 and the savings stocking for <0.5. ${\displaystyle \lambda }$ ${\displaystyle \lambda }$

The Hurwicz rule also does not consider all possible outcomes, but rather evaluates the alternatives on the basis of a weighted average of their best and worst possible results. Another problem with her is that the choice of the optimism parameter can vary greatly depending on the mood.

Example:

when one would therefore be for the alternative decide. ${\displaystyle \lambda =0{,}3}$ ${\displaystyle a_{2}}$

	${\displaystyle s_{1}}$	${\displaystyle s_{2}}$	${\displaystyle s_{3}}$	Hurwicz rule
${\displaystyle a_{1}}$	120	80	120	${\displaystyle (120\cdot 0{,}3+80\cdot 0{,}7)=92}$
${\displaystyle a_{2}}$	100	100	100	${\displaystyle (100\cdot 0{,}3+100\cdot 0{,}7)=100}$

Laplace's rule

The Laplace rule : on one assumes that the probability for the occurrence of possible outcomes for all choices are equal ( principle of indifference ). The option which then promises the best result is chosen; H. the alternative is chosen whose expected value is maximum:

${\displaystyle \max _{i}:\varphi _{ai}={\frac {1}{n}}\sum _{j}e_{ij}}$

Laplace's rule is based on the following assumption: Since no probabilities of occurrence with regard to environmental conditions are known, there is no reason to assume that one environmental condition is more probable than another, so one must assume that the probability of occurrence is equally distributed. The Laplace rule takes into account all environmental conditions when evaluating the alternatives. In this example, the decision maker is indifferent between the share and the savings stocking.

Laplace's rule is a special case of Bayes' rule .

Savage-Niehans rule

The Savage-Niehans rule (also minimax-regret rule or rule of least regret): With this rule, the assessment of the alternative courses of action is not based on the immediate benefit of the results, but on their damage values ​​or loss of opportunity compared to the maximum possible gain. One chooses the alternative that minimizes the potential damage.

In the example: assuming four possible states of the environment ( , , and ), and three available alternatives ( , and ) ${\displaystyle Z_{1}}$ ${\displaystyle Z_{2}}$ ${\displaystyle Z_{3}}$ ${\displaystyle Z_{4}}$ ${\displaystyle A_{1}}$ ${\displaystyle A_{2}}$ ${\displaystyle A_{3}}$

	${\displaystyle Z_{1}}$	${\displaystyle Z_{2}}$	${\displaystyle Z_{3}}$	${\displaystyle Z_{4}}$
${\displaystyle A_{1}}$	2180	1640	1750	480
${\displaystyle A_{2}}$	1840	2560	690	810
${\displaystyle A_{3}}$	720	1970	2320	860

In order to determine the optimal alternative according to the Savage-Niehans rule, the maximum result value for all alternatives must be determined in each state and this must be subtracted from all other result values. ${\displaystyle Z_{j}}$

Example:

• Consideration of the state . ${\displaystyle Z_{2}}$
• Determination of the maximum result value ${\displaystyle \max(e_{i2})=2560}$
• Subtract the on all ${\displaystyle \max(e_{i2})}$ ${\displaystyle e_{i2}}$
• ${\displaystyle e_{22}-e_{12}=2560-1640=920}$
• ${\displaystyle e_{22}-e_{22}=2560-2560=0}$
• ${\displaystyle e_{22}-e_{32}=2560-1970=590}$

This process must be carried out for each state. The highest values ​​of the three alternatives (rows) are then compared with each other. The lowest value here represents the lowest loss of opportunity and is therefore the cheapest alternative.

Overall, the invoice looks like this:

	${\displaystyle Z_{1}}$	${\displaystyle Z_{2}}$	${\displaystyle Z_{3}}$	${\displaystyle Z_{4}}$	${\displaystyle {\text{max. Nachteil}}}$
${\displaystyle A_{1}}$	2180 - 2180 = 0	2560-1640 = 920	2320-1750 = 570	860-480 = 380	920
${\displaystyle A_{2}}$	2180-1840 = 340	2560-2560 = 0	2320 - 690 = 1630	860 - 810 = 50	1630
${\displaystyle A_{3}}$	2180 - 720 = 1460	2560-1970 = 590	2320 - 2320 = 0	860 - 860 = 0	1460

We note that the minimum value of the maximum penalty (max penalty) is 920. The loss of opportunity in alternative is the lowest and therefore the alternative to be chosen. ${\displaystyle A_{1}}$

Krelle rule

Another decision rule was proposed by Wilhelm Krelle . It is based on that all with an action associated utility values , , ..., with a relevance to the decision uncertainty preference function are transformed and are then added. ${\displaystyle a_{i}}$ ${\displaystyle u_{i1}}$ ${\displaystyle u_{i2}}$ ${\displaystyle u_{in}}$ ${\displaystyle \omega }$

${\displaystyle \Phi (a_{i})=\sum _{j=1}^{n}\omega (u_{ij})}$

The best alternative is now the one with the greatest quality measure.

Affordable loss to Sarasvathy

The individually achievable loss or effort (and not the expected return) determine which opportunities are taken and which steps are actually taken in a project. This is a decision heuristic which, according to start-up research, is preferred by very experienced entrepreneurs under uncertainty (see Effectuation - Theory of Entrepreneurial Expertise).

Experience criterion from Hodges and Lehmann

This rule forms a compromise between the Maximin rule and the Bayes rule for an a priori variable . In addition, the confidence parameter is introduced, which indicates to what extent the decision maker trusts the a priori probability . ${\displaystyle \pi }$ ${\displaystyle \lambda }$

See also

• Decision under security
• Decision under uncertainty
• Decision under risk
• Expected utility theory

literature

• W. v. Zwehl: Decision rules . In: Concise Dictionary of Business Administration , Volume 1. 5th Edition. Schäffer-Poeschel, 1993
• G. Bamberg, AG Coenenberg : Business decision-making . 14th edition. Verlag Vahlen, 2008, ISBN 978-3-8006-3506-1

Individual evidence

1. ↑ Decision-making rules - definition in the Gabler Wirtschaftslexikon
2. ↑ Amartya Sen: Equality of What ?, in: S. Murrin (ed.): The Tanner Lectures on Human Values, Cambridge University Press 1980, 196–220, also in: Amartya Sen: Choice, Welfare and Measurement, Oxford 1982
3. ↑ Klaus Birker: B2B manual General Management: Companies market-oriented tax . Ed .: Werner Pepels. 2nd Edition. Symposion Publishing GmbH, Düsseldorf 2008, ISBN 978-3-939707-06-6 , p. 52 ( Google Books ).