Decision under security

In the context of decision theory , decisions under security are spoken of when the decision-maker knows with certainty the occurring environmental condition ( ) and can therefore predict all the consequences of an action. Decisions with multiple objectives ( multi-criteria decision problems ) play the most important role. ${\displaystyle w_{j}=1}$

General

The assumption that all the consequences of an action are known in advance intuitively seems completely unrealistic. The importance of decision rules under security is nevertheless very great. For example, certain one-dimensional decision problems in decisions under uncertainty can be converted into a decision situation under certainty with several objectives.

Examples

• The goal actually strived for cannot be measured directly, and the influence of certain action parameters on this goal is not known with certainty.
• The complexity of a decision based on a global goal ( profit maximization ) is no longer manageable, so that individual sub-goals are defined whose influence on the global goal is known at least as a tendency .

For example, a company will usually make a decision about a new company location with the aim of long-term profit maximization, but the respective influence of the possible locations on profit cannot be determined directly. However, the location properties such as infrastructure , wage costs , taxes , subsidies granted , construction costs etc. are known with certainty, and there is also knowledge of the relationship between these factors and the global goal. The one-dimensional decision under uncertainty thus becomes a multi-dimensional (multi-criteria) decision under security.

One-dimensional decision problems

A problem that is of little relevance is when only one goal is pursued and the target characteristics are known when choosing different alternatives. A distinction can be made between two cases:

• Unlimited objectives : The aim is to maximize or minimize the extent of the objectives. Example: profit maximization, risk reduction .
• Limited goal setting : Here, a goal should either be achieved precisely (fixation) or at least / at most ( satisfication ). Example: To catch a flight, you have to be at the airport at least 1 hour in advance, but it doesn't matter whether you get there earlier (satisfication). Above a certain amount of sweets, the more sweets make you feel bad, there is an optimal amount at which deviations up and down are bad (fixation).

Multi-criteria decision problems

See also : Multi Criteria Analysis

A decision (that is, the selection of an alternative course of action from several available alternatives) usually has consequences for several goals, so that a multi-criteria decision problem is present. A target system exists when all the consequences (target values) and the characteristics of the individual desired targets (preference relation) are known for each available alternative course of action. The goals of this target system can be related to one another in various ways.

Target system

The target values ​​in a target system show which consequences a decision maker attaches importance to his alternative courses of action and, by evaluating the alternative courses of action with regard to one consequence, form the yardstick for assessing the alternative. The target system depends on the individual decision maker.

example

For the way to work, you can choose to travel by car or local public transport . The target system of the decision maker looks like this, for example:

${\displaystyle {\begin{array}{c|ccc}&z_{\boldsymbol {1}}&z_{\boldsymbol {2}}&z_{\boldsymbol {3}}&\\\hline a_{\boldsymbol {1}}&40&5&8&\\a_{\boldsymbol {2}}&45&4&4&\\\end{array}}}$

With to the various destinations (here: = journey time in minutes, = cost per journey in euros, = convenience on a scale from 1 to 10) and = journey by car and = journey by public transport. ${\displaystyle z_{1}}$ ${\displaystyle z_{3}}$ ${\displaystyle z_{1}}$ ${\displaystyle z_{2}}$ ${\displaystyle z_{3}}$ ${\displaystyle a_{1}}$ ${\displaystyle a_{2}}$

Another decision maker in a different location in the city can have a different target system, for example:

${\displaystyle {\begin{array}{c|ccc}&z_{\boldsymbol {1}}&z_{\boldsymbol {2}}&z_{\boldsymbol {3}}&\\\hline a_{\boldsymbol {1}}&30&4&-5&\\a_{\boldsymbol {2}}&60&6&-2&\\\end{array}}}$

This decision maker could not care about the convenience at all, so that for him it is an expression of the ecological consequences of the respective means of transport. ${\displaystyle z_{3}}$

With regard to the desired target characteristics, the above applies: Maximization, minimization, fixation or satisfaction are possible.

Target relationships

Goals can have different relationships to one another (Laux, 2003, p. 67ff.):

• Goal indifference or goal neutrality : The achievement of one goal is not influenced by the other goal, the decision problem can be broken down into one-dimensional sub-problems.
• Goal complementarity : Achieving one goal makes it easier to achieve the other goal. Example: knowledge of English and vacation in England. If goal 1 is to be able to speak English as well as possible and goal 2 is to spend as much vacation as possible in England, then a high achievement of goal 2 (a lot of vacation in England) automatically improves the achievement of goal 1.
• Conflicts of goals or competition between goals : The actually problematic situation arises when goals are conflicting with one another, i.e. the achievement of goal 1 has a negative effect on goal 2. Example: Earning money and free time: The more free time you want, the less you can work, the less money you earn.

Decision rules for multi-criteria decision problems

The dominance principle

To simplify the decision problem, those alternatives that are dominated by other alternatives should not be considered. An alternative is dominated if there is at least one other alternative that performs at least as well in all goals and is better in at least one goal. (Note: goals here do not denote the state, but are tied to the type of dominance)

Different types of dominance can occur, including absolute dominance, state dominance, and probability dominance. State dominance of an alternative action A compared to an alternative action B exists if the result value of A is at least the same in every state and is genuinely greater than that of B in at least one state. A is in absolute dominance over B when the worst result value of A is at least equal to the best result value of B across all states. Absolute dominance is the strictest criterion; H. it also implies state dominance as well as probability dominance.

Strict or strict dominance exists when the dominant alternative does better in all goals.

Example (state dominance)

${\displaystyle z}$= State of the environment = alternative course of action
${\displaystyle a}$

${\displaystyle {\begin{array}{c|ccc}&z_{\boldsymbol {1}}&z_{\boldsymbol {2}}&z_{\boldsymbol {3}}&\\\hline a_{\boldsymbol {1}}&10&5&8&\\a_{\boldsymbol {2}}&10&6&8&\\a_{\boldsymbol {3}}&7&7&7&\\\end{array}}}$

Alternative 1 is dominated by alternative 2 and no longer needs to be considered. Alternative 2 is better than alternative 3 in state 1 and state 3, but not in state 2, so that alternative 3 is not dominated.

Lexicographical order

In this process, the goals are ranked . Initially, only the most important target is viewed and assessed, which is why the process is also known as target suppression. If you do not come to a result because more than one alternative is equivalent with regard to the most important goal, then the next most important goal is looked at and so on. This can lead to implausible results.

example

(Goal 1 is most important before goal 2 before goal 3)

${\displaystyle {\begin{array}{c|ccc}&z_{\boldsymbol {1}}&z_{\boldsymbol {2}}&z_{\boldsymbol {3}}&\\\hline a_{\boldsymbol {1}}&101&0&0&\\a_{\boldsymbol {2}}&100&100&100&\\\end{array}}}$

Although alternative 2 scores only slightly worse in objective 1, but significantly better in the other two objectives, alternative 1 would be chosen according to the lexicographical order.

See also : section 'Application in microeconomics' in the article 'Lexicographical order'  and section 'Lexicographical order of preference' in the article 'Preference relation'

Target weighting

When weighting goals, the goals are ranked, but a weighting factor must be determined for each goal . When making a decision, the various goals are multiplied and added up for each alternative with the respective weighting factor. The alternative that receives the highest value is selected. In contrast to the lexicographical order, all target values ​​are taken into account for every alternative, i.e. H. a particularly high score for the second most important goal can compensate for a low score for the most important goal.

Körth rule

The aim is to maximize the minimum (relative) degree of target achievement. For this purpose, the maximum value of a target is searched for in all alternatives and all values ​​of the target value in the column are divided by this value. In the utility matrix, the values ​​are now normalized to the interval [0..1], so the target value is no longer specified, but the relative target achievement compared to the possible maximum. Each alternative (line) is evaluated according to the minimum relative degree of target achievement (the minimum is sought line by line). The alternative that has the highest value here is chosen

${\displaystyle \max _{i}\left(\min _{p}\left({\frac {u_{ip}}{\max _{h}u_{hp}}}\right)\right)}$

where the utility of the alternative is in relation to goal . ${\displaystyle u_{jk}}$ ${\displaystyle j}$ ${\displaystyle k}$

example

${\displaystyle {\begin{array}{c|ccc}&z_{\boldsymbol {1}}&z_{\boldsymbol {2}}&z_{\boldsymbol {3}}&\\\hline a_{\boldsymbol {1}}&16&20&5&\\a_{\boldsymbol {2}}&4&10&10&\\a_{\boldsymbol {3}}&8&8&8&\\\hline {\boldsymbol {Maximum}}&{\boldsymbol {16}}&{\boldsymbol {20}}&{\boldsymbol {10}}&\\\end{array}}}$

This matrix is ​​now transformed:

${\displaystyle {\begin{array}{c|ccc|c}&z_{\boldsymbol {1}}&z_{\boldsymbol {2}}&z_{\boldsymbol {3}}&{\boldsymbol {Minimum}}\\\hline a_{\boldsymbol {1}}&1&1&0,5&{\boldsymbol {0,5}}\\a_{\boldsymbol {2}}&0,25&0,5&1&{\boldsymbol {0,25}}\\a_{\boldsymbol {3}}&0,5&0,4&0,8&{\boldsymbol {0,4}}\\\end{array}}}$

This results in a preference order: alternative 1 (0.5) better than alternative 3 (0.4) better than alternative 2 (0.25).

Utility analysis

→ Main article : Benefit analysis

The utility analysis is also called point evaluation or scoring model. Here, each target criterion is assigned a point value on a point scale. These are then weighted and added so that the alternative with the highest score is the best.

Target programming

The method of goal programming is also called goal programming . One tries to minimize the sum of the weighted deviations. A value to be achieved is determined for a target ( flexible target programming ) - another variant simply takes the respective maximum as the target value ( rigid target programming ). The deviations of the alternatives are then weighted (also different weights up and down). These values ​​can then be raised to the power and are added up at the end. The smallest sum wins.

${\displaystyle z_{i}=z_{i}^{*}-z_{i}(x)}$with as a target and as a target function value of the alternative i ${\displaystyle z_{i}^{*}}$ ${\displaystyle z_{i}(x)}$

${\displaystyle \min {G=\left[\sum ({\underline {w}}_{i}{\underline {z}}_{i}^{p}+{\overline {w}}_{i}{\overline {z}}_{i}^{p})\right]^{\frac {1}{p}}}}$

in which

1. ${\displaystyle {\underline {z}}_{i}}$ the downward deviation of the target i,
2. ${\displaystyle {\underline {w}}_{i}}$ the weighting of the downward deviation of the target i,
3. ${\displaystyle {\overline {z}}_{i}}$ the deviation upwards,
4. ${\displaystyle {\overline {w}}_{i}}$ the weighting of the deviation of the target i upwards and
5. ${\displaystyle p}$ is the deviation factor (usually = 1).

Mostly there are , which does not distinguish between the deviation upwards and downwards. ${\displaystyle {\overline {w}}_{i}={\underline {w}}_{i}}$

Analytic hierarchy process

→ Main article : Analytic Hierarchy Process

The Analytic Hierarchy Process (AHP), also known as the Saaty method , offers support for a hierarchical target system and is mathematically more demanding, but also more precise.

See also

• Decision under risk
• Decision under uncertainty

literature

• W. v. Zwehl: Decision rules. In: Concise dictionary of business administration. Volume 1, 5th edition. Schäffer-Poeschel, 1993.
• H. Laux: Decision Theory. 5th edition. Springer, Berlin a. a. 2003.
• G. Bamberg, AG Coenenberg : Business decision-making. 14th edition. Verlag Vahlen, Munich 2008, pp. 41-66.

Individual evidence

1. ↑ Helmut Laux, Decision Theory , 2003, pp. 65 ff.