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Condorcet's jury theorem

Political science theorem / From Wikipedia, the free encyclopedia

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Condorcet's jury theorem is a political science theorem about the relative probability of a given group of individuals arriving at a correct decision. The theorem was first expressed by the Marquis de Condorcet in his 1785 work Essay on the Application of Analysis to the Probability of Majority Decisions.[1]

Two-dimensional line graph showing that a group vote overperforms individual judgement – has higher probability of success – when the individuals’ chance of being right is greater than half.
Graph of cumulative probabilities of success (x axis) for a few binomial distributions with given individual chance of success (y axis) and number of “jurors” (color).

The assumptions of the theorem are that a group wishes to reach a decision by majority vote. One of the two outcomes of the vote is correct, and each voter has an independent probability p of voting for the correct decision. The theorem asks how many voters we should include in the group. The result depends on whether p is greater than or less than 1/2:

  • If p is greater than 1/2 (each voter is more likely to vote correctly), then adding more voters increases the probability that the majority decision is correct. In the limit, the probability that the majority votes correctly approaches 1 as the number of voters increases.
  • On the other hand, if p is less than 1/2 (each voter is more likely to vote incorrectly), then adding more voters makes things worse: the optimal jury consists of a single voter.

Since Condorcet, many other researchers have proved various other jury theorems, relaxing some or all of Condorcet's assumptions.