# What is decision theory

## Decision theory

Deals with the explanation of decisions (descriptive decision theory) and with norms for decisions (normative decision theory). Descriptive decision theory tries to find answers to the question of why decisions are made one way and not another. It uses sociological and, above all, psychological knowledge. The normative decision theory assumes a person who acts economically rationally in every situation (Homo oeconomicus), systematizes decision-making situations and derives rules for an optimal decision. In particular, decision-making situations are examined in which the result is not yet determined by the decision alone, but different environmental developments, e.g. through reactions of the opponent, lead to different results. Normative decision theory uses statistical models. These models become very complex if realistic decision-making situations are used as a basis.

Decision theory focuses on making decisions about an alternative course of action, which is to be selected from a large number of alternatives when one or more objectives are met. The prerequisite is that the alternative found has the best result of the possible alternatives with regard to the decision situation. With the help of the mathematical decision models, the determination of an optimal alternative course of action is made easier. With a high degree of complexity, low determinability and an imperfect degree of information of the quantities that go into the model, the informative value of the mathematical models is limited.

(engl. decision theory) Decision theory examines the decision-making behavior in decision-making situations. Here, on the one hand, the question can be asked how the decision should be made (normative decision theory), and on the other hand how a decision is made (descriptive decision theory). In the following, only the first direction, i.e. normative decision theory, will be discussed. In a decision-making situation, the decision-maker has the choice between various alternatives (options for action, actions), taking into account his objectives. The set of all actions forms the action space or decision space. Of course, the decision maker must also take into account the various possible environmental states or, in short, states when making his choice. The set of these states is also referred to as the state space. With regard to the information available to the decision maker about the state space, different decision situations are distinguished. In a security situation there is only one possible environmental condition and this is known to the decision maker. This means that for each alternative, the result that your choice leads to is also certain. In the risk situation, the decision maker not only knows all future environmental conditions, but can also assign a probability of occurrence to them. In the case of a decision situation under uncertainty, all possible states are also known, but no probabilities of occurrence can be assigned to them. There is also the game theory situation, in which the result of every alternative action taken by a decision maker depends on the actions of an opponent. Each alternative course of action leads to a result depending on the possible environmental conditions. In order to choose between the alternatives, each environmental status must be assessed according to the objectives of the decision maker. This process is formally described by a utility function (utility) of the decision-maker, with which a so-called utility value is assigned to each event, i. H. a measure of the degree of target achievement when the relevant event occurs. A decision situation characterized as above can be represented by a table of the following type, which is also called a decision matrix: The possible actions are here with a1, ..., a ,,, and the elements of the state space with zi, ..., z ,, denotes. A table element u; j stands for the utility value that results from the choice of the alternative a and the occurrence of the environmental status z.

In the case of a decision-making situation under security, the table has only one column. It is up to a single-purpose decision maker to choose the action that appears in the row with the highest utility value. In a risk situation, the decision matrix can be expanded to include the occurrence probabilities p, the environmental conditions z. A decision rule for the risk situation is the Bayes principle or also the expected value principle. In this case, the expected value, the benefit value, has to be calculated for each alternative course of action, i.e. That is, first of all, the utility values ​​are to be multiplied by the probability of the associated states and the resulting products are to be added:

According to Bayes' principle, that alternative is to be chosen which has the highest expected value. What is critical about this decision rule is that it is based only on the expected value and does not take into account the fact that the utility values ​​can under certain circumstances deviate very strongly from the expected value, i.e. H. strongly do scatter the expected value. The standard deviation is a measure of this variation. It is calculated as the square root of the sum of the squared deviations from the mean, weighted with the probability of occurrence. For the standard deviation o; the alternative i therefore applies:

The lack of Bayes' principle is countered in the approach of the o criterion by forming a function (; o) in which the risk attitude of the decision maker (risk aversion, risk neutrality, risk sympathy) is reflected. For example, the function cp (; n) = 5 o is based on a risk-averse setting, because the function value decreases with increasing variance, i.e. increasing risk. Risk neutrality is expressed, for example, in the function cp (p; o) =, because the variance is not an argument of the function here. It thus delivers the same decision as the Bayesian criterion. With the function cp (; o) = + o risk sympathy can be expressed, because its function value increases with the risk. In the o criterion, first the expected value and the standard deviation and then with these the function value of cp (; o) are to be determined for each alternative. The alternative with the highest function value is to be chosen.

Another decision criterion, the Bernoulli principle, is based on a slightly different approach. Here, too, as with the creation of the decision matrix, the events are evaluated with the help of a utility function, the Bernoulli utility function. In addition to other preferences of a decision maker, this also reflects his or her subjective risk appetite. Bernoulli showed that under certain behavioral assumptions, such a function exists for every decision maker. However, there are no instructions for the construction of such a utility function. This is a difficulty in applying the Bernoulli Principle. After a Bernoulli benefit has been assigned to each event value, the expected value for each alternative must be determined, the expected benefit value. Finally, the alternative with the highest expected utility value is to be selected.

In the case of decisions made under uncertainty, no probabilities of occurrence can be assigned to the possible environmental conditions. The Laplace's leech, or rule of insufficient reason, ties in directly to this fact by assuming all environmental conditions to be equally probable and then proceeding as with the Bayesian leech. An expected value is formed for each alternative and the one with the highest value is chosen. To a certain extent, the Laplace egel also implies a risk-neutral attitude on the part of the decision-maker.

The Maximax egel assumes an optimistic attitude on the part of the decision-maker by recommending the alternative that has the highest utility value over all environmental conditions. The Maximin leech or forest leech suggests an opposite approach. According to it, the smallest utility value over all states must first be determined for each alternative (line minimum). The alternative with the largest line minimum must then be selected. With this decision, the best of all the worst cases is chosen. In this sense, the Maximin egel assumes a pessimistic decision maker.

The Hurwicz egel or pessimism optimism egel tries to balance the two extreme positions by introducing an optimism parameter 7v, which can have at least the value 0 and a maximum of 1. For each alternative, the highest utility value Max, and (as with the Maximin rule) the smallest utility value Min; certainly. Then the value Max, X, + Min, (1 X) is determined (for each alternative). The alternative with the highest value is to be selected. The choice of the parameter X. reflects the risk attitude of the decision-maker: the larger the parameter, the more optimistic the decision-maker is (for 7v = 1, the maximum rule results again); the smaller a, the more pessimistic the decision maker is (for X, = 0 the Maximin rule results). The Savage Niehaus egel (also the rule of the smallest regret) is also based on a pessimistic basic attitude of the decision-maker. First of all, the maximum Max (zi) is determined over all alternatives in each column, i.e. for each environmental condition. Then the difference to the column maximum Max (zi) uii is calculated for each utility value of a column. In terms of content, this difference indicates how great the disadvantage is for the decision maker when the corresponding state i has occurred and he has decided on an alternative other than that belonging to Max (zi). The maximum difference is then noted for each alternative. The alternative is then to be chosen for which this value is the lowest. Compared to all other alternatives, this alternative has the property that with it the deviation from the maximum possible utility value due to the occurrence of another environmental condition (and then regret about it) is the smallest.

The game situation differs from the decision-making situations considered so far in that the possible environmental conditions are defined by the action of a rationally acting opponent. It is assumed that the opponent is acting in his own interest that opposes the decision-maker. In addition to these so-called two-person games, game theory also examines multi-person games. The goal of game theory decision models is to determine what are known as optimal strategies that ensure maximum profit for the decision maker.

In each of the decision-making situations presented so far, a decision had to be made by the decision-maker. A generalization are multi-period decision models. They are intended to map decision-making situations in which a decision-maker has to make decisions at several successive points in time. To illustrate such situations, the representation in the form of a decision tree is often chosen. When examining decision-making models, it should be noted that only those aspects of a decision-making situation can be mapped that can be evaluated, i.e. H. can be expressed by numbers.

examines economic and corporate policy decisions as an economic sub-discipline, i. H. the targeted selection from several alternative courses of action. In the context of decision-theoretical investigations, what is of less interest are the influencing factors that are decisive for the occurrence of alternative courses of action and their consequences (e.g. macroeconomic factors such as exchange rates, wage structure, tax system, etc., or microeconomic determinants such as sales prices, wage rates, market conditions, etc.); these aspects of economic decisions are analyzed in other economic sub-disciplines. Rather, decision theory examines the problem of targeted alternative selection in a general way, with the selection process as such moving into the center of interest. In its descriptive (empirical-realistic) orientation, decision theory examines how decisions in economic life are actually made. In particular, the sequence of individual activities, known as the decision-making process, which are connected with the preparation and implementation of decisions, is analyzed. On the one hand, the decision-making behavior of individuals is examined. In the earlier approaches of this kind, an attempt was first made to use models of prescriptive decision theory at the same time to explain actually observable decision-making behavior. In the further development and overcoming of these generally unsuccessful attempts, the categories of psychology were used to an ever greater extent. The location of such approaches in the context of economics is therefore controversial. A second focus of the investigation is the actual course of decision-making processes in organizations whose members each pursue different individual goals. Decision theory in its prescriptive (practical-normative) orientation, on the other hand, aims to provide orientation aids for the practical handling of corporate or economic policy decisions through the logical penetration of the implications associated with various forms of decision-making. Corresponding logical decision-making analyzes are predominantly carried out using various forms of decision-making models. In contrast to the descriptive approaches, it is assumed that • a clear target system is known, • a complete catalog of mutually exclusive alternative courses of action is given and • the resulting possible results are known and can be expressed by the values ​​of one or more result variables . The summary presentation of alternative courses of action and the resulting possible results is usually done in the form of a decision matrix. With regard to the level of information about the possible results, decision-making situations are usually divided into two categories: Decisions relating to security are characterized by the fact that each alternative course of action can be assigned clear result values. In the case of decisions under uncertainty, on the other hand, the results of the alternative courses of action can no longer be clearly stated; rather, each alternative is identified by a whole set of alternative possible result values. Which of the alternative possible result values ​​actually occurs when the corresponding action alternative is chosen, depends on the development of exogenous environmental conditions that cannot be controlled by the decision subject. Depending on whether or not probabilities of occurrence can be specified for these environmental conditions, a further distinction is made between risk situations and situations of uncertainty. The focus of prescriptive decision theory is to develop different decision rules for each of the types of different decision situations mentioned and to analyze these with regard to the implications associated with them. In prescriptive decision theory it is assumed that the environmental conditions that are decisive for the result resulting from the choice of an alternative are beyond the control of the decision maker and come about independently of the decision problem under consideration and the choice of an alternative. In contrast, situations are conceivable in which the environmental conditions that are decisive for the action results of a primary decision-making subject result from the decisions of other rationally acting persons, the results of which are in turn influenced by the decisions of the primary decision-making subject. In such cases it becomes necessary to consider the interdependencies between the decisions of the "environment" and those of the primary decision-making subject. Such analyzes take place within the framework of game theory. Literature: Bamberg, GJQoenenberg, A., Business Economics Decision Making, 7th Edition, Munich 1992. Bitz, M., Decision Theory, Munich 1981. Laux, H., Decision Theory, Basics, Berlin et al. 1982.

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