# One speaks decision

under risk of a decision under risk in the context of the decision theory if the decision maker knows the probabilities for occurring the possible environmental conditions. These probabilities can both objectively admit to be (Lotto, Roulette) or on subjective estimations (e.g. due to past data) are based.

## general

decision under risk is after the usual linguistic usage a Unterfall of decision under uncertainty. While one speaks with knowledge of probabilities of entrance of the environmental conditions of risk, a decision is present by uncertainty , if one knows the possible environmental conditions, however no probabilities of entrance to indicate can.

With decisions under risk a result matrix is present, which represents the decision problem: 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. The probabilities [itex] w_j [/itex] the different environmental conditions are well-known, whereby applies: [itex] 0 \ le w_j \ le 1< /math> and [itex] \ sum_ {j} w_j = 1< /math>.

• 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
the Entscheider counts on a probability of [itex] w_1 [/itex] with the fact that the share quotation rises, with a probability of [itex] w_2 [/itex] he counts on the sinking of the share quotation and on a probability of [itex] w_3 [/itex] the course remains unchanged.

## decision rules

with decisions under risk can find the following decision rules application:

### the Bayes rule

the Bayes rule is called also μ-rule. Here the Entscheider orients itself only after the expectancy values.

[itex] \ max_i: \ varphi {} _ {ai} = E (e_i) = \ mu {} _e = \ sum_j w {} _j \ e_ {ij} /math <cdot>

There only the expectancy value of the respective alternative [itex] a_i [/itex] , is risk neutral the Entscheider is evaluated, it is for example indifferent regarding the participation in a Lotterie by Münzwurf, in which it wins with 50% probability € 1 and loses with 50% probability € 1. In the above example is indifferent if applies: [itex] e_ {11}< /math> *< math> w_1< /math> + [itex] e_ {12}< /math> *< math> w_2< /math> + [itex] e_ {13}< /math> *< math> w_3< /math> = 100 (there independently of the probabilities [itex] w_j< /math> a safe “disbursement”), here thus: 120*< math> w_1< /math> + 80*< math> w_2< /math> + 100*< math> w_3< /math>. Indifferenz e.g. became. are present during uniform distribution, if thus applies: [itex] w_1< /math> = [itex] w_2< /math> = [itex] w_3< /math> = [itex] \ frac {1} {3}< /math>.

### it shows problems with

expectancy values the example Peter citizens of the paradox that the consideration of expectancy values does not correspond to the decision behavior of humans in the reality:

• (Ideal) a coin (i.e. Head and number appear in each case with a probability of 50%) are thrown.
• The player receives as disbursement:
• 4 €if only with the second throw head appears
• [itex] 2^n< /math> €, if only with the nth throw head appears
• the price to the Lotterie should a fair price be, i.e. correspond to the expectancy value.

A Entscheider, which decides only after the expectancy value, would be now thus ready to thus pay for the participation in this Lotterie this fair price, the expectancy value (it would be then exactly indifferent between the participation and the nonparticipation):

The expectation value determines itself as follows:

• The probability that with the first throw head appears is math <{>1 \ over2} /math<,> the disbursement is exact 2.
• The probability that with the second throw head appears is math <{>1 \ over4} /math<,> the disbursement is exact 4.
• ...
• The probability that with the nth throw head appears is math <\> frac {1} {2^n} /math<,> the disbursement is exact [itex] 2^n< /math>.

Thus E (X) is = [itex] {1 \ over2} * 2 + {1 \ over4} * 4 +… + \ frac {1} {2^n} * 2^n< /math> + ... = 1 + 1 +… + 1 + ... thus infinitely.

In the reality however nobody is ready to pay infinitely much money for the participation in the Lotterie.

### the μ-σ-rule

in the μ-σ-rule finds the risk attitude of the Entscheiders by the fact to consideration that also the standard deviation is considered. With risk-neutral Entscheidern it corresponds to the Bayes rule, with risikoaversen (risk-shy) Entscheidern sinks the attractiveness of an alternative [itex] a_i< /math> with increasing standard deviation. With risk-joyful Entscheidern attractiveness rises however.

[itex] \ max_i: \ varphi_ {ai} = \ Phi (\ mu_i, \ sigma_i) [/itex]

A possible form of the μ-σ-rule is for example:

[itex] \ Phi (\ mu_i, \ sigma_i) = \ mu_i - \ alpha \ cdot \ sigma_i [/itex]

For α < applies for 0: The Entscheider is risk joyful, an alternative with a higher σ an alternative with same expectancy value μ however lower σ is preferred. For α > applies for 0: The Entscheider is risikoavers, an alternative with lower σ an alternative with same expectancy value, but higher σ is preferred. For α = 0 the rule does not correspond to the Bayes rule, the Entscheider is risk neutral, the standard deviation σ has influence on the evaluation of the alternatives.

As a condition for application to the μ-σ-rule generally normaldistributed future net yields are considered or a square use function.

### the Bernoulli principle

with the application of the Bernoulli principle the results must [itex] e_ {ij} [/itex] only with the help of a risk use function into use values to be converted. The individual risk use function [itex] u (e_ {ij}) [/itex] thereby the risk attitude of the Entscheiders reflects again. A risk use function, which is a concave function, stands thereby for a risikoaversen Entscheider, while a convex function illustrates a risk-joyful Entscheider. It is however possible thereby that the risk use function exhibits both concave and convex ranges. This illustrates for example the empirically observable fact that humans play both Lotto (risk joy), and insurance lock (risk aversion).

Thereby the expectancy value of the risk use function is maximized.

[itex] \ max_i: \ varphi_ {ai} = \ sum_j w_j \ u (e_ {ij}) cdot [/itex]