Decision under risk

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In the context of decision theory , one speaks of a decision under risk if the decision maker knows the probabilities for the occurrence of the possible environmental conditions . These probabilities can be known objectively ( lottery , roulette ) or based on subjective estimates (e.g. based on empirical values).

General

“Decision under risk” is, according to common usage, a sub-case of decision under uncertainty . While knowledge of the probability of occurrence of the environmental conditions is used to speak of risk , a decision is made under uncertainty if one knows the possible environmental conditions but cannot state the probability of occurrence .

In the case of decisions under risk, a so-called result matrix is available, which shows the decision problem: The decision maker has the choice between various alternatives , which, depending on the possible environmental conditions, lead to different results . The probabilities of occurrence of the various environmental conditions are known, where: and .

Result
matrix decision under risk

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 ( ).

The result matrix looks like this, for example:




120 80 100
100 100 100

The decision maker reckons with a probability of that the share price will rise, with a probability of he reckons with a decrease in the stock price and with a probability of that the price will remain unchanged.

Classic decision-making rules

The following decision rules are also known as classic decision rules.

Bayes' rule

With Bayes' rule (also called the μ rule , expected value rule or expected value principle ), the decision maker only orientates himself according to the expected values .

Since only the expected value of the respective alternative is assessed, the decision maker is risk-neutral ; he is, for example, indifferent to participating in a coin toss lottery, in which he wins € 1 with 50% probability and € 1 with 50% probability loses. In the above example, the then indifferent, if: (because regardless of the probabilities secure "payment"), in this case: . Indifference would e.g. As present in equal distribution, so if: .

If there is equal probability, there is a special case of Bayes' rule, Laplace's rule .

rating

The example of the Saint Petersburg paradox shows that taking expected values ​​into account does not always correspond to the decision-making behavior of people in reality. In the Saint Petersburg lottery, a fair coin is tossed (i.e. heads and tails each have a 50% probability). The player receives as a payout:

  • if head appears on the first throw
  • if head only appears on the second throw
  • if head only appears on the third throw
  • ...
  • if head only appears on the -th throw

The expected value corresponds here

According to Bayes' rule, a decision maker would be willing to pay any amount, no matter how large, - i.e. his entire assets - to participate in the lottery, since the expected profit is infinitely large. In reality, however, hardly anyone is willing to swap their entire fortune for participation in the Saint Petersburg lottery.

The μ-σ rule

In the μ-σ-rule or the expectation-variance-principle and therefore actually the μ-σ²-rule , the risk attitude of the decision maker is taken into account in that the standard deviation is also taken into account. In the case of risk-neutral decision-makers, it corresponds to Bayes’s rule; for risk-averse (risk-averse) decision-makers, the attractiveness of an alternative decreases with increasing standard deviation. For decision-makers who are willing to take risks, however, the attractiveness increases.

The decision maker chooses the alternative that maximizes his preference function:

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

describes the risk aversion parameter.

  • For : The decision maker is willing to take risks, an alternative with a higher σ is preferred to an alternative with the same expected value but lower σ.
  • The following applies: The decision maker is averse to risk, an alternative with a lower expected value is preferred to an alternative with the same expected value, but a higher one .
  • For the rule corresponds to Bayes' rule , the decision maker is risk-neutral, the standard deviation has no influence on the evaluation of the alternatives.

Bernoulli principle

The Bernoulli principle was proposed by Daniel Bernoulli to resolve the Saint Petersburg paradox . Under certain assumptions it is considered a rational decision criterion.

The possible outcomes are first converted into utility values. This requires a utility function (also a risk utility function ). This individual utility function already contains the decision-maker's risk attitude:

However, it is also possible for the utility function to have both concave and convex areas. This is a good empirically observable fact. For example, people play the lottery (taking risks) and also take out insurance (risk aversion).

The alternative that maximizes the expected value of the utility function is chosen:

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 ( ). The decision maker reckons with a probability of that the share price will rise, with a probability of he reckons with a decrease in the stock price and with a probability of that the price will remain unchanged.

The utility function is assumed for the decision maker .





120 80 100
100 100 100

When applying the Bernoulli principle, one obtains the highest utility value of at . This alternative should therefore be selected. The form of the utility function is concave, so the decision-maker’s attitude towards risk is risk-averse.

Relation to the classic decision criteria

With a linear utility function of the form , the Bernoulli principle corresponds to Bayes' rule.

The μ-σ rule is generally not compatible with Bernoulli's principle, i.e. H. A preference function in the sense of the μ-σ rule cannot be mapped in all cases by an equivalent utility function and vice versa. This is possible e.g. B. with a quadratic utility function of the form , which leads to a preference function of the form , or with normally distributed future returns also in other cases.

See also

literature

  • Helmut Laux, Robert M. Gillenkirch, Heike Y. Schenk-Mathes: Decision Theory . 9th edition. Springer Gabler, 2014, doi : 10.1007 / 978-3-642-55258-8 .

Web links

Individual evidence

  1. Laux (2014), chap. 4.6
  2. a b Laux (2014), p. 105 f.
  3. a b Werner Gothein: Evaluation of investment strategies . Springer Fachmedien, Wiesbaden 1995, p. 30 , doi : 10.1007 / 978-3-663-08484-6 ( limited preview in Google book search).
  4. Laux (2014), chap. 5.4