Social choice theory

The social choice theory (English social choice theory ), and theory of collective decisions (English theory of collective choice) called, deals with group decisions by aggregating individual preferences / decisions to a collective preference / decision in the form of votes and elections and with this emerging problems and paradoxes and their prevention, probability and solution.

The "problem of cyclical majorities " ( Condorcet paradox) and the "method of pairwise voting " ( Condorcet method ) are usually used as an introduction to social choice theory; other known examples are the choice Borda, the Ostrogorski paradox and the paradox of liberalism.

The social choice theory is an interdisciplinary and " homeless " research field, which is operated esp. of representatives of mathematics, economics, political science, psychology, philosophy and law. The social choice theory is sometimes confused with the theory of rational decision or wrongly equated; In addition, there are overlaps between public choice theory.

• 4.1 Requirements
• 4.2 Quality criteria

Historical

The main founders and pioneers of social choice theory in the mid-20th century, the economists Kenneth Arrow and Duncan Black apply. The Nobel laureate Arrow proved mathematically in his Arrow 's theorem that there is no " perfect " democratic aggregation rule based on preference orderings. Black discovered independently by Arrow historical predecessors who had dealt with problems in electoral processes in his research. He presented the forgotten works of Jean Charles Borda, Marquis de Condorcet and Charles Lutwidge Dodgson.

Other researchers found that analytical studies on election procedures and election rules have been made since the Middle Ages, including by Ramon Llull and Nicholas of Cusa.

Throughout the 19th and early 20th century dealt v. a legal scholar with aggregation process, especially in the very lively discussions on the tuning method to judge colleges ( "total vote " or " vote for reasons " ) and in the introduction and refinement of proportional representation.

Methods

In social choice theory is an analytical, mathematically formal language and method for use; Relations here have an important meaning. It is frequently performed with assumptions and simplifications esp. in the modeling of individual preferences.

• The axiomatic social choice theory studies properties of electoral processes and provides conditions ( eg restrictions of preferences ), under which no choice problems. These theorems they tried to mathematically prove, inter alia, with the help of logic and set theory. The most famous and important theorems of social choice theory is Arrow 's theorem and the Gibbard - Satterthwaite the theorem.
• The probabilistic social choice theory attempts to determine the probability of selection problems and paradoxes by means of probability theory, combinatorics and geometry.

The limitations of social choice theory is based on the one that they considered coalition building and strategic voting behavior, which are widely used in elections only insufficiently. Instead, most of the - assumed assumption that the parties " sincerely " express settings in the voting process (see the section on " heresthetics " ) - unrealistic.

Introduction and Simple insights

Importance of the aggregation rule

A simple realization of social choice theory is that the outcome of elections and voting also depends on the aggregation rule used. So can have very different election results result different aggregation methods with identical (individual) preferences. For example, in an election with more than two candidates, the candidate who is victorious in an election with a relative majority in a paired choice method ( Condorcet method ) lose to everyone else and they occupy the last place.

Selection Example

Given a group of n = 21 people who = 3 candidates { A, B, C} select m a chairman. Members of the group have the following preferences.

Explanation: 6 people have the preference: a before b, a before c and b before c. ( The lowercase letters indicate individual preferences. )

The election result is particularly dependent on the selection method in this example:

• When a simple majority (plurality choice) method candidate C wins by 8 votes. Candidate B reaches 7 and Candidate A 6 votes. Election result: C before B before A.
• When the pairwise voting ( Condorcet method ) method candidate A wins against any other candidate. Candidate C loses against each other. Election result: A before B before C.
• In the Borda count following election result is produced. Candidate B reaches 44 votes, Candidate A and Candidate C 43 39 votes. Election result: B before A before C.

If, however, the formation of coalitions involving in the analysis, it is clear that an existing Condorcet winner interspersed in all electoral processes, in which the parties have equal voting power. However, this requires that the parties know the preferences of other stakeholders and vote so that their preferred result comes out.

General aggregation problems

Requirements

In simplified terms, aggregation problems and paradoxes can occur under the following conditions:

• There are more than two candidates / alternatives for election / voting
• The individual preferences are not homogeneous and
• No candidate or no alternative has an absolute majority.

Quality criteria

There are numerous aggregation method (see below the list of social choice procedure). The social choice theory has developed a number of criteria, the advantages and disadvantages of each method are characterized by their help. The most important are:

Not all of these criteria are independent and equally strong. So, for example, follows from the validity of the Condorcet criterion directly the validity of the majority criterion, the reverse is not the case. At the same follows for all preferential voting systems from the validity of the Condorcet criterion, the violation of the consistency criterion, and vice -versa.

List / important properties of social choice procedures

• The majority vote or a majority decision ( Plurality voting ): Each participant gives his voice a single alternative. He can not express his more fine-grained preferences.

• The preferred choice ( preferential voting, ranked voting ): Each participant ranks the alternatives according to their individual preferences in order. This is a finer gradation than the majority vote, but the user has no way to express the intensity of his preferences.

• The valuation option (range voting, rated voting ): Each participant rated all alternatives with points from a given interval. This allows the participants, sequence and intensity of his preference of alternative express.

Heresthetik: The art of political "manipulation"

Unfulfilled quality criteria (see above) can cause the voters do not bring their "true" individual decision expressed, but " electoral " follow considerations in order to achieve a certain effect (see Gibbard - Satterthwaite theorem). It therefore is " tactical / strategic " Select.

Unfulfilled quality criteria also allow legal procedures and methods for influencing and "manipulation" of the election results. Examples would be the introduction of more choice alternatives if the independence of irrelevant alternatives is not given, or control over the order of the elections, especially in pair comparisons, if the Condorcet criteria are not met.

This " art of political manipulation " ( by legal means ) described the political scientist William Harrison Riker heresthetic or heresthetics. The classic example of "manipulation" of a vote can be found in the Roman writer Pliny the Younger in his letters ( Book 8, 14 letter ).

Researcher

Well-known and important representatives and researchers of social choice theory are: Kenneth Arrow, Duncan Black, Sven Berg, Steven Brams, Donald Campbell, Robin Farquharson, Peter Fishburn, Wulf Gaertner, William Gehrlein, Allan Gibbard, Bernard Grofman, Melvin Hinich, Jerry Kelly, Jean Laslier -François, Richard McKelvey, Bernard Monjardet, Hervé Moulin, Richard Niemi, Hannu Nurmi, Peter Ordeshook, Prasanta Pattanaik, Charles Plott, Douglas Rae, William Harrison Riker, Donald Saari, Mark Satterthwaite, Norman Schofield, Amartya Sen.