Condorcet jury theorem

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The Condorcet Jury Theorem is named after Marie Jean Antoine Nicolas Caritat, Marquis de Condorcet . It deals with the question under which circumstances a binary group decision is of higher quality, i.e. is more likely to be correct, than the decision of an individual member.

presentation

In its basic form, the Condorcet jury theorem is based on the following assumptions:

  • A jury had to choose between two options A and B.
  • The jury consists of k members, where k> 2 and is odd.
  • Every member of the jury is able to choose the better decision with probability q; So q is the conditional probability that a member will choose A if A is better than B and for B if B is better than A.
  • The jury decides with an absolute majority of its members.

Now let Q (k, q) denote the (conditional) probability of a correct jury decision. If 0.5 <q <1, then the following three statements apply under the above assumptions:

  • Q (k, q)> q;
  • Q (k, q) increases with k;
  • If k approaches infinity, then Q (k, q) approaches one.

In the case of 0 <q <0.5, the opposite applies: the fewer members vote, the better. On the other hand, if q is 0, 0.5 or 1, then Q (k, q) = q.

meaning

The jury - Theorem has significance for the comparison between representative and direct democracy, between federal and centralized systems, or between steep or flat hierarchies in organizations.

A popular application of the theorem is offered by the television quiz “ Who Wants to Be a Millionaire? ". If the candidate does not know the answer himself, he can choose (among other things) between the public joker and the telephone joker. If the candidate chooses the telephone joker, a previously named person is called. It is not uncommon for the candidate to attribute a high level of technical competence to the person called in the field of knowledge in question. If the audience joker is chosen, the audience can vote in the studio. It should be a lucky coincidence here, should experts for the required field of knowledge be among them.

In the notation introduced above, 1> q t > q> 0 usually applies , where q t denotes the competence parameter of the telephone partner and q models the average studio viewer. According to the Condorcet jury theorem , Q (k, q)> q t > q can still be possible. In this case, the aggregated decision of the k spectators in the studio would be better than that of the expert on the phone. His higher competence would then be overcompensated by the sheer number of (less competent) viewers.

Modifications and additions

The theorem is based on strict assumptions. In particular, the jury members should be homogeneous and there should be no correlation between their decisions. In practice, however, actors in large groups have different levels of competence. They could also influence each other, or their decisions could be based on correlated information. The main statements of the theorem, however, have also been theoretically confirmed for heterogeneous juries and for the case of correlated decisions, see Berg (1993) and Ladha / Krishna (1992).

Another strict assumption is the lack of strategic interaction. The jury members vote "naively", they cast their vote according to their convictions. However, if one assumes, as is usual in economic game theory , strategic interaction between rational actors, then individual jury members could have an interest in distorting their true convictions by giving a dissenting vote. In this modified game, the statements of the theorem would no longer apply without restrictions, according to Feddersen / Pesendorfer (1998).

See also

Literature and web links

  • Jean-Antoine-Nicolas de Caritat Condorcet, marquis de: Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix . Imprimerie royale, Paris 1785 ( full text in the Google book search).
  • Berend, Daniel / Paroush, Jacob 1998: When is Condorcet's Jury Theorem valid? In: Social Choice and Welfare 15 (4), 481-488.
  • Berg, Sven 1993: Condorcet's Jury Theorem, Dependency Among Voters. In: Social Choice and Welfare 10, 87-96.
  • Berg, Sven 1996: Condorcet's Jury Theorem and the Reliability of Majority Voting. In: Group Decisions and Negotiation 5, 229-238.
  • Boland, Philip J. 1989: Majority Systems and the Condorcet Jury Theorem. In: The Statistician 38 (3), 181-189.
  • Feddersen, Timothy / Pesendorfer, Wolfgang 1998: Convicting the Innocent: The Inferiority of Unanimous Jury Verdicts under Strategic Voting. In: American Political Science Review 92 (1), 23-35.
  • Kirstein, R. 2006: The Condorcet Jury-Theorem with Two Independent Error-Probabilities. Center for the Study of Law and Economics Discussion Paper 2006-03, Saarbrücken. abstract file (PDF file; 190 kB)
  • Ladha, Krishna K. 1992: The Condorcet Jury Theorem, Free Speech, and Correlated Votes. In: American Journal of Political Science 36 (3), 617-634.
  • List, Christian / Goodin, Robert E. 2001: Epistemic Democracy: Generalizing the Condorcet Jury Theorem. In: The Journal of Political Philosophy 9 (3), 277-306.
  • Surowiecki, James 2004: The Wisdom of Crowds. Why the Many Are Smarter than the Few. Reprint 2005, Abacus, London.