Social choice theory

from Wikipedia, the free encyclopedia

The social choice theory ( Engl. Social choice theory ), and theory of collective decisions (Engl. Theory of collective choice ) called, deals with group decision making by aggregating individual preferences / choices to a collective preference / choice in terms of voting and elections and with it emerging problems and paradoxes and their avoidance, probability and solution.

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

Social choice theory is an interdisciplinary and “homeless” field of research that v. a. is operated by representatives of mathematics , economics , political science , psychology , philosophy and law . Social choice theory is sometimes confused with or incorrectly equated with the theory of rational decision ; there are also overlaps with the New Political Economy .


The economists Kenneth Arrow and Duncan Black are considered to be the main founders and pioneers of social choice theory in the mid-20th century . The later Nobel Prize winner Arrow proved mathematically in his Arrow theorem that there is no such thing as a “perfect” democratic rule of aggregation based on orders of preference. Black discovered in his research independently of Arrow historical predecessors who had dealt with problems in voting procedures. So he presented the forgotten works of Jean Charles Borda , Marquis de Condorcet and Charles Lutwidge Dodgson .

Other researchers found that as early as the Middle Ages, analytical studies on electoral processes and voting rules were undertaken. a. by Ramon Llull and Nikolaus von Kues .

Throughout the 19th and early 20th centuries, v. a. Legal scholar with aggregation procedures, especially in the extremely lively discussion about the voting method in colleges of judges ("total vote" or "vote according to reasons") and in the introduction and design of proportional representation .


In social choice theory, an analytical, mathematically formal language and method are used; Relationships have an important meaning here. This is often done with assumptions and simplifications, v. a. in modeling individual preferences.

The limitations of social election theory are based on the one hand on the fact that it does not take sufficient account of coalition formation and strategic voting behavior, which are widespread in elections. Instead, the - unrealistic - assumption is usually assumed that those involved express their attitudes “sincere” when casting their votes (see the section on “ herestheticsbelow ).

Introduction and simple insights

Importance of the aggregation rule

A simple finding of social choice theory is that the result of elections and voting also depends on the aggregation rule used. Different aggregation methods can result in very different election results 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 can lose to all others in a paired election method ( Condorcet method ) and thus take last place.

Choice example

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

first preference a a b b c c
second preference b c a c a b
third preference c b c a b a
Order of preference of x people 6th 0 5 2 5 3

Explanation: 6 people have the preference: a before b, a before c and b before c. (The lower case of the letters indicates individual preferences.)

In this example, the election result is particularly dependent on the voting method:

  • With the simple majority method (pluralism), candidate C wins with 8 votes. Candidate B receives 7 and candidate A 6 votes. Election result : C before B before A.
  • In the paired voting method (Condorcet method), candidate A wins against every other candidate. Candidate C loses to everyone else. Election result : A before B before C.
  • The following election result arises from the Borda election. Candidate B receives 44 votes, candidate A 43 and candidate C 39 votes. Election result : B before A before C.

If, however, the formation of coalitions is included in the analysis, the result is that an existing Condorcet winner will prevail in all electoral processes in which the parties involved have equal votes. The prerequisite for this, however, is that the participants know the preferences of the other participants and vote in such a way that the result they prefer comes out.

General aggregation problems


Put simply, aggregation problems and paradoxes can arise 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 procedures (see the list of social election procedures below ). Social choice theory has developed a number of criteria with the help of which the advantages and disadvantages of individual processes can be characterized. The most important are:

Non-dictatorship The social decision does not depend on the preferences of a single individual. All participants have equal rights.
completeness The procedure allows any number of decision alternatives and any number of participants. The individual preference arrangements (a is preferred over d and d over f) of the individual participants are not subject to any restrictions.
Independence from irrelevant alternatives The ranking of two alternatives is independent of other alternatives and their evaluation.
Independence from clone alternatives The result does not change if the same alternative is available multiple times (cloned) or if clones are removed. Clone alternatives are those between which no participant classifies any other option.
Majority criterion If an absolute majority wants a certain alternative, then it will prevail.
Consistency criterion If the list of decision alternatives including the results is divided (arbitrarily) and one alternative is ranked best in all sublists, then this alternative is also ranked best in the overall list.
Condorcet criterion If a certain alternative is preferred in a pairwise comparison over all other alternatives, then this alternative is also ranked best in the overall list.
Weak Pareto principle If all individuals prefer an alternative d over alternative f, this also applies to the collective preference.
Condorcet loser criterion If a certain alternative is rejected in a pairwise comparison against all other alternatives, then this alternative is also ranked worst in the overall list.
Transitivity criterion If a is preferred over d and d again over f, then it follows: a is preferred over f.

Not all of these criteria are independent or equally strong. So it follows e.g. B. from the fulfillment of the Condorcet criterion the fulfillment of the majority criterion directly, the reverse is not the case. In addition, for all preference systems, compliance with the Condorcet criterion results in a violation of the consistency criterion, and vice versa.

List / most important properties of the social election process

  • The majority vote or majority voting: Each participant gives his or her vote to a single alternative. He cannot express his preferences more finely.
→ There is no independence from irrelevant alternatives.
  • The preferred choice (voting preferential, ranked voting): Each participant arranges the alternatives according to their individual preferences into a sequence. This is a finer gradation than with the majority vote, but the participant has no way of expressing the intensity of his preferences.
Examples would be: Borda election , Condorcet method , Coombs election , Instant Runoff Voting (IRV), Ranked Pairs , Schulze method , Bucklin election , and others.
→ The restrictions of Arrow's impossibility theorem or the Gibbard-Satterthwaite theorem apply to all methods of the preferred choice .
  • The choice of evaluation (range voting, rated voting): Each participant evaluates all alternatives with points from a given interval. This allows the participant to express the ranking and intensity of his preference for the respective alternative.
Examples of this would be: evaluation voting , voting by consent and majority judgment .

Heresthetics: The Art of Political "Manipulation"

Unfulfilled quality criteria (see above) can lead to voters not expressing their “true” individual decision, but rather following “election tactical” considerations in order to achieve a certain effect (see Gibbard-Satterthwaite theorem). So this is "tactical / strategic" voting.

Unfulfilled quality criteria also allow legal procedures and methods to influence and “manipulate” the election result. Examples would be the introduction of further alternative choices if independence from irrelevant alternatives is not given, or control over the order of the elections, especially in pair comparisons, if the Condorcet criteria are not met.

The political scientist William Harrison Riker called this “art of political manipulation” (by legal means) heresthetic or heresthetics . The classic example of "manipulation" of a vote will take place at the Roman writer Pliny the Younger in his letters (eighth book, 14th letter).


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 -François Laslier, Richard McKelvey, Bernard Monjardet, Herve Moulin, Richard Niemi, Hannu Nurmi, Peter Ordeshook, Prasanta Pattanaik , Charles Plott , Douglas Rae, William H. Riker , Donald Saari, Mark Satterthwaite, Norman Schofield, Amartya Sen .

Individual evidence

  1. ^ The Augsburg Web Edition of Llull's Electoral Writings
  2. Epistulae VIII, 14 (Commons: Pliny Minor). German translation at


  • Kenneth J. Arrow: Social Choice and Individual Values. 2nd Edition. Wiley, New York 1963, ISBN 0-300-01363-9 .
  • Kenneth J. Arrow, Amartya K. Sen, Kotaro Suzumura (Eds.): Handbook of Social Choice and Welfare. Vol. 1, Elsevier Science / North-Holland, Amsterdam 2002, ISBN 0-444-82914-8 .
  • Konstantin Beck: The probability of paradoxical voting results. Lang, Bern 1993, ISBN 3-906750-28-0 .
  • Duncan Black: The Theory of Committees and Elections. Cambridge University Press, London / New York 1958.
  • Walter Bossert, Frank Stehling: Theory of collective decisions. An introduction. Springer, Berlin 1990, ISBN 3-540-53029-0 .
  • John Craven: Social Choice: A Framework for Collective Decisions and Individual Judgments. Cambridge University Press, Cambridge 1992, ISBN 0-521-31051-2 .
  • Wulf Gaertner: Domain Conditions in Social Choice Theory. Cambridge University Press, Cambridge 2001, ISBN 0-521-79102-2 .
  • Wulf Gaertner: A Primer in Social Choice Theory. Oxford University Press, Oxford 2006, ISBN 0-19-929751-7 .
  • Wulf Gaertner: Social choice theory. In: Stefan Gosepath, Wilfried Hinsch, Beate Rössler (eds.): Handbook of Political Philosophy and Social Philosophy. Volume 2, Walter Gruyter, Berlin / New York 2008, ISBN 978-3-11-017408-3 , pp. 1248-1254.
  • Jonathan K. Hodge, Richard E. Klima: The Mathematics of Voting and Elections: A Hands-On Approach. American Mathematical Society, Providence, RI 2005, ISBN 0-8218-3798-2 .
  • Lucian Kern, Julian Nida-Rümelin: Logic of collective decisions. Oldenbourg, Munich / Vienna 1994, ISBN 3-486-21016-5 .
  • Iain McLean, Arnold B. Urken (Eds.): Classics of Social Choice. University of Michigan Press, Ann Arbor 1995, ISBN 0-472-10450-0 .
  • Hannu Nurmi: Voting Paradoxes and How to Deal with Them. Springer, Berlin 1999, ISBN 3-540-66236-7 .
  • William H. Riker: Liberalism Against Populism: A Confrontation Between the Theory of Democracy and the Theory of Social Choice. Freeman, San Francisco 1982, ISBN 0-88133-367-0 .
  • William H. Riker: The Art of Political Manipulation. Yale University Press, New Haven / London 1986, ISBN 0-300-03591-8 .
  • Donald G. Saari: Basic Geometry of Voting. Springer, Berlin 1995, ISBN 3-540-60064-7 .
  • Stephan Schulz: Collective decisions in the stock corporation. A social choice theoretical analysis of selected problems of company law. German Univ.-Verlag, Wiesbaden 2005, ISBN 3-8350-0064-0 .
  • Amartya K. Sen: Collective Choice and Social Welfare. Holden-Day, San Francisco 1970, ISBN 0-8162-7765-6 .
  • George G. Szpiro: The darn mathematics of democracy. Springer, Berlin 2011, ISBN 978-3-642-12890-5 , doi: 10.1007 / 978-3-642-12891-2 .
  • Wolfgang Ernst: Small voting primer. Guide to the Congregation. Book publisher Neue Zürcher Zeitung, Zurich 2011, ISBN 978-3-03823-717-4 .

Web links