Decision-making mechanism

As a decision-making mechanism in the context of be game theory refers to decisions that represent a conscious choice between alternatives or between several different variants. This is done on the basis of certain preferences and the information available from several rational players. The goal of decision-making is to achieve or avoid the player's own anticipated desired or undesired consequences. Often there are specific rules as to how such decisions are made and which majorities are required to reach a decision. János von Neumann showed that every player can calculate the rational decision-making behavior in certain conflict situations.

Areas of application and problems

In the context of game theory, decision-making can be assumed in almost all areas (from society, politics to business and private family or friends) in which several game members are involved, such as:

In the family or among friends it is about dividing a household budget or determining how to spend your free time.
Entrepreneurial committees decide on investment projects , production programs or advertising strategies with limited economic and ecological resources.
Political bodies on questions of financial policy , social policy or education policy. In political elections, parties or projects correspond to the alternatives.

However, making decisions in everyday life is not always easy. Particularly in the context of a company, when making collective decisions, one is faced with the problem of combining the divergent individual values of different stakeholders into a group value concept .

Selected procedures with example

As mentioned above, an important question here is how a company or an enterprise should choose between various alternative interests of its shareholders. If a group does not want to or cannot come to a decision cooperatively in a decision-making process, this decision (based on different goals and preferences of the group members) must be made in a vote.

Voting rules are methods that combine the order of preference of the individual group members into a order of preference for the entire group. The question is: Can a combination of individual preferences into social preferences be a conceivable solution so that the procedure meets certain standards of internal consistency, efficiency, but also democracy? Example:

A general partnership consists of eight partners. There are five different expansion projects to choose from for future corporate development. In a first step, each partner put the projects in a sequence (best project: 1, worst: 5). Now the best project for the whole group should be determined with the help of voting rules.

	Project 1	Project 2	Project 3	Project 4	Project 5
Shareholder 1	1	2	3	4th	5
Shareholder 2	5	2	1	4th	3
Shareholder 3	1	2	4th	3	5
Shareholder 4	1	2	5	4th	3
Shareholder 5	3	5	2	4th	1
Shareholder 6	4th	1	2	5	3
Shareholder 7	3	5	1	2	4th
Shareholder 8	1	4th	2	3	5

Simple majority vote

A simple majority vote (also known as the single-vote rule ) means that the alternative that receives the most votes is preferred. As an evaluation rule, one point is given for each first place awarded.

The result is shown in the following table:

Simple majority vote
	Project 1	Project 2	Project 3	Project 4	Project 5
Points	4th	1	2	0	1

This means that project 1 should be proposed before project 3, then projects 2 and 5 follow before project 4.
But: If only project 1 and project 3 were to be voted on, project 1 and 3 would be voted 4 4 votes result in a tie.

Voting in pairs
Shareholder ${\ displaystyle \ rightarrow}$	G 1	G 2	G 3	G 4	G 5	G 6	G 7	G 8	Number of votes
Project 1	1	0	1	1	0	0	0	1	4th
Project 3	0	1	0	0	1	1	1	0	4th

Absolute majority

Rule of the absolute majority : The absolute majority is achieved when an expansion plan can unite more than 50% of the votes of the voters. This means that in this case the absolute majority should consist of 5 or more votes with 8 voters. However, none of the expansion projects achieve this (see second figure).

This problem is referred to in the literature as the Condorcet cycles . This means that more decision-makers do not automatically ensure constancy and stability, but in the case of certain individual preference orders there can be at least three alternatives. In such a cycle, the alternative that initially prevailed with a majority over another is always beaten by a third.

Qualified majority

A qualified majority is achieved when the expansion plan receives a freely definable, pre-determined percentage, which is usually 50% of the votes of the voters. An absolute majority is therefore also a qualified majority.

Double-election process

The double-election procedure consists of two ballots, the first of which corresponds to the absolute majority method . If an alternative achieves an absolute majority (> 50%) in the first ballot, this is elected and the decision-making process is ended. If this is not the case, the two alternatives will be voted again with the greatest approval. With this procedure it can happen that the alternatives with the second and third most votes receive the same number of votes and therefore no further procedure is defined. Therefore, the use is often limited to decision-making problems with a very high number of decision-makers, such as presidential elections. On the other hand, further regulations can also be added to force further action.

Double-election procedure: 1st ballot
Shareholder ${\ displaystyle \ rightarrow}$	G 1	G 2	G 3	G 4	G 5	G 6	G 7	G 8	Total of the 1st vote
Project 1	1	0	1	1	0	0	0	1	${\ displaystyle 4 \ triangleq 50 \, \%}$
Project 2	0	0	0	0	0	1	0	0	${\ displaystyle 1 \ triangleq 12 {,} 5 \, \%}$
Project 3	0	1	0	0	0	0	1	0	${\ displaystyle 2 \ triangleq 25 \, \%}$
Project 4	0	0	0	0	0	0	0	0	${\ displaystyle 0 \ triangleq 0 \, \%}$
Project 5	0	0	0	0	1	0	0	0	${\ displaystyle 1 \ triangleq 12 {,} 5 \, \%}$

Only the two alternatives with the highest approval - i.e. project 1 and project 3 - are considered in the second ballot.

Double-election procedure: 2nd ballot
Shareholder ${\ displaystyle \ rightarrow}$	G 1	G 2	G 3	G 4	G 5	G 6	G 7	G 8	Total of the 2nd vote
Project 1	1	0	1	1	0	0	0	1	${\ displaystyle 4 \ triangleq 50 \, \%}$
Project 3	0	1	0	0	1	1	1	0	${\ displaystyle 4 \ triangleq 50 \, \%}$

No decision is made in this example based on the double-election procedure.

Double-vote procedure

Each decision maker has two votes, which are distributed to the two alternatives with the highest order of preference. The alternative with the greatest approval wins.

Double-vote procedure
Shareholder ${\ displaystyle \ rightarrow}$	G 1	G 2	G 3	G 4	G 5	G 6	G 7	G 8	total
Project 1	1	0	1	1	0	0	0	1	5
Project 2	1	1	1	1	0	1	0	0	5
Project 3	0	1	0	0	1	1	1	1	5
Project 4	0	0	0	0	0	0	1	0	1
Project 5	0	0	0	0	1	0	0	0	1

The decision maker is indifferent in his decision between the alternatives project 1, project 2, project 3.

Borda rule

To avoid the problem mentioned in the last chapter, the so-called Borda rule is suggested in the literature. In the Borda rule, alternatives are selected in that each group member gives his most preferred alternative votes, the second most preferred alternative votes, etc. Thus, the positions of the alternatives in the individual preference systems are also included in the decision-making process. The votes are added over the individual alternatives and the alternative with the most votes is selected. The Borda rule leads to a complete order of preference for the group over all alternatives. ${\ displaystyle a}$ ${\ displaystyle a}$ ${\ displaystyle a-1}$

Evaluation according to Borda rule
Shareholder ${\ displaystyle \ rightarrow}$	G 1	G 2	G 3	G 4	G 5	G 6	G 7	G 8	total
Project 1	5	1	5	5	3	2	3	5	${\ displaystyle {\ underline {29}}}$
Project 2	4th	4th	2	4th	1	5	1	2	23
Project 3	3	5	4th	1	4th	4th	5	4th	${\ displaystyle {\ underline {30}}}$
Project 4	2	2	3	2	2	1	4th	3	19th
Project 5	1	3	1	3	5	3	2	1	19th

In contrast to the previous experiments, Project 3 is the best alternative here. But the Borda rule is also problematic. Although the Borda rule always leads to a transitive order, the result of the Borda rule can depend on "irrelevant alternatives".

In the example, project 4 can be such an irrelevant alternative because it does not have a direct initial vote by shareholders. If project 4 is no longer available, we have the following result:

Dependence on irrelevant alternatives
Shareholder ${\ displaystyle \ rightarrow}$	G 1	G 2	G 3	G 4	G 5	G 6	G 7	G 8	total
Project 1	4th	1	4th	4th	2	1	4th	3	${\ displaystyle {\ underline {23}}}$
Project 2	3	3	3	3	1	4th	1	2	20th
Project 3	2	4th	2	1	3	3	4th	3	${\ displaystyle {\ underline {22}}}$
Project 4	0	0	0	0	0	0	0	0	0
Project 5	1	2	1	2	4th	2	2	1	15th

Without project 4, the assessment point is calculated from 1 to 4 (previously from 1–5). As shown in the figure, project 1 (not project 3) should now be proposed. As a result, the Borda rule can easily be manipulated in practice by additionally including irrelevant alternatives in the decision.

Procedure by Nason

In Nanson's method , the alternatives with the highest order of preference of the respective decision maker are assigned the number of points , with the number of possible alternatives and the assigned value being reduced by 1 with decreasing preference until the alternative with the lowest order of preference subsequently has the value 0 receives. Then the votes for the individual alternatives are added up and the alternative with the most votes is selected (1st ballot). The alternatives that did not receive more than the average number of points are then no longer considered. ${\ displaystyle a_ {i}}$ ${\ displaystyle n-1}$ ${\ displaystyle n}$ ${\ displaystyle {\ overline {x}}}$

{\ displaystyle {\ overline {x}} = m \ cdot {\ frac {n-1} {2}}}

{\ displaystyle m: {\ text {Number of decision makers}}}

{\ displaystyle n: {\ text {number of alternatives}}}

{\ displaystyle {\ overline {x}}: {\ text {Average number of points}}}

Assessment according to Nason 1st ballot
Shareholder ${\ displaystyle \ rightarrow}$	G 1	G 2	G 3	G 4	G 5	G 6	G 7	G 8	total
Project 1	4th	0	4th	4th	2	1	2	4th	${\ displaystyle {\ underline {21}}}$
Project 2	3	3	1	3	0	4th	0	1	15th
Project 3	2	4th	3	0	3	3	4th	3	${\ displaystyle {\ underline {22}}}$
Project 4	1	1	2	1	1	0	3	2	11
Project 5	0	2	0	2	4th	2	1	0	11

{\ displaystyle {\ overline {x}} = 8 \ cdot {\ frac {5-1} {2}} = 16}

The result is an average score of 16. Only alternatives that are greater than this value are considered in the second round - i.e. project 1 and project 3.

Assessment according to Nason 2nd ballot
Shareholder ${\ displaystyle \ rightarrow}$	G 1	G 2	G 3	G 4	G 5	G 6	G 7	G 8	total
Project 1	1	0	1	1	0	0	0	1	${\ displaystyle {\ underline {4}}}$
Project 3	0	1	0	0	1	1	1	0	${\ displaystyle {\ underline {4}}}$

According to the Nason procedure, the decision between project 1 and project 3 is indifferent.

If a third round of voting is necessary to bring about a decision, the number of remaining alternatives is used instead of the number of alternatives when calculating the average of the number of points . ${\ displaystyle m}$ ${\ displaystyle k}$

{\ displaystyle {\ overline {x}} = m \ cdot {\ frac {k-1} {2}}}

{\ displaystyle k \ colon {\ text {number of remaining alternatives}}}

Pairwise comparison

In the Paired Comparison (Method of Individual Election) two randomly selected alternatives are put to the vote. The inferior alternative is eliminated and the chosen alternative competes against an alternative selected at random from the remaining alternatives. The process is over when, in the optimal case, there is only one alternative left or until all alternatives were available once to choose. If there is an alternative that wins against all others, it is called the Condorcet winner or the Condorcet alternative. If this is not the case, the order may influence the result.

{\ displaystyle {\ text {Project}} \ 2> {\ text {Project}} \ 5 \ (6 \ to \ 2)}

Project 5 is eliminated.

{\ displaystyle {\ text {Project}} \ 2 <{\ text {Project}} \ 3 \ (3 \ to \ 5)}

Project 2 is canceled.

{\ displaystyle {\ text {Project}} \ 3> {\ text {Project}} \ 4 \ (7 \ to \ 1)}

Project 4 is eliminated.

{\ displaystyle {\ text {Project}} \ 3 = {\ text {Project}} \ 4 \ (4 \ to \ 4)}

The decision between project 3 and 4 is indifferent.

Arrow theorem

The Arrow theorem named by the Nobel Prize winner Kenneth Arrow shows that it is then possible to always derive a clear preference of the group from the preferences of the individuals in a group, if this derivation is to meet some of the four ethical and methodological conditions at the same time :

Completeness and transitivity
Independence from irrelevant alternatives
Pareto principle
Exclusion of a dictator

According to Arrow's theorem, however, there is not a single social decision-making mechanism that meets all four requirements. All collective decisions that satisfy axioms 1 through 3 inevitably violate the condition of non-dictatorship. Accordingly, the result shows that there cannot be a perfect decision-making mechanism, so that compromises have to be made in one direction or the other.

literature

Avinash K. Dixit: Game theory for beginners: Strategic know-how for winners . 1997
Christian Rieck: Game Theory - An Introduction . Rieck, Eschborn 2007
Hüftle: Group decisions and game theory . 2006
Guillermo Owen: Game Theory . Academic Press, San Diego 1995
János von Neumann: Theory of Games and Economic Behavior . 1944
John von Neumann, Oscar Morgenstern: Theory of Games and Economic Behavior . University Press, Princeton NJ 1944, 2004

Web links

Uni Halle accessed December 6, 2008
bibb.de (PDF; 120 kB) accessed January 9, 2008

Individual evidence

↑ Hüftle: Group decisions and game theory . 2006, p. 2.
^ János von Neumann: Theory of Games and Economic Behavior . 1944, p. 233.
^ Otto: Decision making in organizations . 2005, p. 3.
↑ Hüftle: Group decisions and game theory . 2005, p. 7.
↑ Helmut Laux, Robert M. Gillenkirch, Heike Y. Schenk-Mathes: Decision theory. 9th edition. Springer, p. 515.
↑ Kenneth A. Shepsle, Mark Bonchek: Analyzing Politics. 1997, pp. 49-55.
↑ Roswitha Meyer: Decision Theory: A Text and Work Book. 2nd Edition. Gabler, Wiesbaden 2000, p. 139.
↑ Roswitha Meyer: Decision Theory: A Text and Work Book. 2nd Edition. Gabler, Wiesbaden 2000, pp. 140-141.
↑ ^a ^b Klaus M. Schmidt: Script Microeconomics . 2006, Chapter 2005, p. 6.
↑ Roswitha Meyer: Decision Theory: A Text and Work Book. 2nd Edition. Gabler, Wiesbaden 2000, p. 141.
↑ Roswitha Meyer: Decision Theory: A Text and Work Book. 2nd Edition. Gabler, Wiesbaden 2000, p. 142.

[1] Hüftle: Group decisions and game theory . 2006, p. 2.

[2] János von Neumann: Theory of Games and Economic Behavior . 1944, p. 233.

[3] Otto: Decision making in organizations . 2005, p. 3.

[4] Hüftle: Group decisions and game theory . 2005, p. 7.

[5] Helmut Laux, Robert M. Gillenkirch, Heike Y. Schenk-Mathes: Decision theory. 9th edition. Springer, p. 515.

[6] Kenneth A. Shepsle, Mark Bonchek: Analyzing Politics. 1997, pp. 49-55.

[7] Roswitha Meyer: Decision Theory: A Text and Work Book. 2nd Edition. Gabler, Wiesbaden 2000, p. 139.

[8] Roswitha Meyer: Decision Theory: A Text and Work Book. 2nd Edition. Gabler, Wiesbaden 2000, pp. 140-141.

[Schmidt.6-9] Klaus M. Schmidt: Script Microeconomics . 2006, Chapter 2005, p. 6.

[10] Roswitha Meyer: Decision Theory: A Text and Work Book. 2nd Edition. Gabler, Wiesbaden 2000, p. 141.

[11] Roswitha Meyer: Decision Theory: A Text and Work Book. 2nd Edition. Gabler, Wiesbaden 2000, p. 142.