# Decision theory#Choice under uncertainty

As a decision under uncertainty decision-making situations are referred to in the decision theory, in which, although the alternatives, potential environmental conditions and the results of choosing a particular alternative and the occurrence of a particular environmental condition are known in which the probabilities of the environmental conditions are unknown but. These are sometimes also referred to decisions from an objective uncertainty.

## General

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

The decision situation when making decisions under uncertainty can be represented by a matrix result. The decision maker has the choice between different alternatives that have the potential, depending on environmental conditions different results may occur. However, the decision maker does not know before, how likely arrive environmental conditions and thus the results.

The distinction between uncertainty in uncertainty and risk has language not yet completely established in the literature. So in some cases only a dichotomy in uncertainty ( unknown probabilities ) and risk ( known probabilities ) is made.

## Decision rules

The following decision rules are to be explained in greater detail an exemplary decision-making situation.

Example: 100 € to be invested for one year. Choices are: one share ( ) or the piggy bank, which does not generate any interest (). The possible environmental states are: The share price rises (), it decreases () or remain constant ().

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

### Minimax rule

The minimax rule or maximin rule (after Abraham Wald and Wald- rule), is very pessimistic. Here, the most unfavorable event is considered, which can i enter the different environmental conditions in choosing a particular alternative course of action. The alternatives are only based on this worst result in each case (which can occur both at different environmental conditions ) compared to all other possible results of an alternative are not considered.

In this example, the decision maker chooses the piggy bank (Alternative 2), since this regardless of the environmental conditions of a payout of 100 guaranteed, while Alternative 1 in the worst case ( sinks course, environmental state s2) at the end of the year only 80 to book. For these Zeilenminima one chooses then the maximum. For this procedure, the name of the decision rule is derived.

A concrete application of MaxiMin rule is found in John Rawls in A Theory of Justice. Many chess programs use a corresponding minimax algorithm in the choice of move.

An extension of the maximin rule is the Leximin rule of Amartya Sen, which for the case that two alternatives have the worst in each state, that is to be selected, in which the second-worst case has the highest value, etc. This addition prevents an overall poorer version can be chosen only because it corresponds to the maximin principle.

### Maximax rule

The Maximax is usually a very optimistic decision rule. Here, each alternative is only based on the result, which may occur when more favorable to this alternative environmental status assessed. Thus, the decision maker chooses the alternative course of action with the maximum line maximum.

In this example, the decision maker chooses the share ( Alternative 1 ), da = 120 = 100 is greater than

### Criticism of minimax and Maximax rule

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

After Maximax rule, the alternative would be chosen, since only the result in the best state of the environment is therefore considered = 120, which is greater than 119 here. The entering in all other environmental conditions payment of zero at alternative would not be considered.

After the minimax rule, the alternative would be chosen because only the incoming respectively in the worst environmental condition result is considered, ie for the alternative, the result = 99 and at Alternative 100 The entering in all other environmental conditions payment of 120 at alternative here would not be considered.

### Hurwicz Rule

The Hurwicz rule, named after Leonid Hurwicz, also known as optimism / pessimism rule that allows tradeoffs between pessimistic and optimistic decision rules because of the decision makers here can bring his personal and subjective adjustment by the so-called optimism parameter ( with 0 ≤ ≤ 1) expressed.

The respective maxima lines are thus with (which lies between 0 and 1 ) and the respective Zeilenminima with (1 -) - ie, in the sum with a value of 1 resulting amount - multiplied.

The larger, more optimistic is the default setting, at = 1 is the application of Maximax rule, at = 0, the application of the maximin rule before.

In this example, the decision maker selects for > 0.5, the stock and for < 0.5 the piggy bank.

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

Example:

At = 0.3 would therefore opt for the alternative.

### Laplace rule

Laplace's rule: It is believed that the chances for the occurrence of the possible results are the same for all choices. The option, then promising the best result is selected, that is, it is chosen that alternative whose expected value is maximal:

Laplace's rule is based on the following assumption: Since no probabilities with respect to the environmental conditions are known, there is no reason to assume that an environmental condition is more probable than another, therefore, have to assume equal distribution of probabilities. This takes into account the Laplace rule all environmental conditions in the evaluation of alternatives. In this example, the decision maker is indifferent between the stock and the piggy bank.

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

### Savage Niehans rule

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

In the example: Adoption four possible environmental conditions (Z1, Z2, Z3 and Z4), and three available alternatives (A1, A2 and A3)

To determine the optimal alternative according to the Savage Niehans rule, the maximum result value must be determined based on all alternatives and that of all other result values are subtracted in each state ().

Example:

- Consideration of the state.
- Determination of the maximum resulting value
- Subtraction of all

This operation must be performed for each state. Is the highest value in each of the three alternatives (lines ) are then compared. The lowest value in this connection in this case represents the least opportunity loss and is therefore the cheapest alternative.

In the overall analysis, the calculation looks like this:

We note that the minimum value of the maximum handicap ( max disadvantage ) is 920. The opportunity losses in Alternative 1 are the lowest and thereby determine the choice of alternative.

### Krelle rule

Another decision rule was proposed by Wilhelm Krelle. It is based on that all associated with an action -worths, , ..., be transformed with a relevant for the decision-maker preference function and uncertainty are then added.

The best alternative is now the one with the largest figure of merit.

### Affordable loss after Sarasvathy

The individually affordable loss or application (and not the expected income ) determine what opportunities are perceived and what steps are being effectively put into a project. It is a decision heuristic, which is preferably used according to start-up research by highly experienced entrepreneurs under uncertainty (see effectuation - Theory entrepreneurial expertise ).

### Experience criterion of Hodges and Lehmann

This rule is a compromise between the maximin rule and the Bayes rule π to an a priori size. In addition, the confidence parameter λ is introduced which indicates the extent to which the decision-makers of the a priori probability familiar.