Condorcet's jury theorem

The Condorcet Jury Theorem is named after Marie Jean Antoine Nicolas Caritat, Marquis de Condorcet. It addresses the question of under what circumstances a binary decision group has higher quality, so really more likely to fail, as the decision of a single member.

Representation

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 members of k, where k > 2 and odd is.
• Each member of the jury was in a position, with probability q to select the better decision; q is thus the conditional probability that a member chooses A if A is better than B, and B, if B is better than A.
• The jury decides by an absolute majority of its members.

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

• Q (k, q) > q;
• Q (k, q) increases with k;
• If k tends to infinity, then Q (k, q) goes against one.

For the case q <0.5 the opposite is true: the fewer the members vote, the better. If q ', however, equal to 0, 0.5 or 1, then applies Q (k, q) = q.

Importance

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 provides the TV quiz show "Millionaire? ". If the candidate himself did not know the answer, he ( and others) choose between the audience and the Joker Telefonjoker. Selects the candidate the phone Joker, a previously designated person will be called. Not infrequently, the candidate attaches to the called party to a high expertise in the relevant field of knowledge. When selecting the audience Jokers viewers are allowed to vote in the studio. This is likely to be a happy accident, should be located and experts for the required field of knowledge among them.

In the above notation introduced so shall normally 1> qt > q > 0, where qt the competence parameters of the telephone Partners and q denotes the modeled average studio audience. After the Condorcet jury theorem can still > qt > q be possible Q (k, q). In this case, the aggregated decision of the k studio audience would be better than that of the experts on the phone. His higher competence would then be more than offset 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 correlation between their decisions are excluded. In practice, players in large groups, however, are equipped with different skills. Moreover, they could influence each other, or their decisions could be based on mutually correlated information. The main statements of the theorem, however, have been confirmed theoretically and heterogeneous juries and for the case of correlated decisions, see Mountain (1993) and Ladha / Krishna (1992).

A more stringent assumption is the absence of strategic interaction. Select the jury members " naive ", they give their vote according to their convictions from. If one assumes, however, as in the economic game theory common strategic interaction between rational actors, they could individual jury members have an interest in distorting their true belief by delivering a different vote. In this modified game, the statements of the theorem would no longer fully apply, so Feddersen / Pesendorfer (1998).