Are group decisions better than individual decisions?
The Condorcet Jury Theorem
One of the earliest justifications for the superiority of group decisions over individual decisions is due to Condorcet (1785/1976). The idea of the Condorcet Jury Theorem is that whilst individuals may make judgemental mistakes or have individual biases, if there are a sufficient number of individuals, these mistakes and biases tend to 'cancel each other out', producing more accurate decisions than would be achieved by an individual decision maker. One possible conclusion of this result is that group decisions and aggregation may preferable to individual legal decision making.
More formally, suppose that there is a group of n jurors or legal decision makers (where n is odd) choosing between two alternatives (for example, the guilt or innocence of a defendant), each of which have equal likelihood of being correct, a priori. Suppose that each juror has
the same probability p of making the correct decision, with 1 < p < 1, and that jurors make decisions independently and sincerely. Then the probability that the group makes the correct decision using simple majority rule of aggregation is:7
This probability pn approaches unity as п^ш. That is, as the group becomes larger, the group decision becomes more accurate than the individual decision of any of its individual members. This result is proved in the Appendix to this Chapter.
To see the intuition behind this result, suppose that there are three jurors: A, B and C. Suppose that the correct decision is that the defendant
Figure 2.6.1 If p > 1 , the probability that a group of three jurors will make the correct decision exceeds P
is guilty. Let p > 1 be the probability that each juror believes that the defendant is guilty. Finally, suppose that each juror votes sincerely and
that decisions are made by simple majority rule. The probability that the defendant is found guilty is the probability that at least two of the jurors will vote to convict. This can happen if either A and B vote guilty, A and C vote guilty, B and C vote guilty, or A, B and C vote guilty. The probability that exactly two jurors will vote guilty is:
whilst the probability that exactly three decision makers will vote guilty is simply:
Hence, the probability that the defendant will be found guilty by the group is:
which corresponds to equation (2.11) above. As Figure 2.6.1 shows, even for a group of three jurors, the quantity p3, which is the probability
Figure 2.6.2 If 0 < p < —, the probability that a group of three jurors will make the correct decision is less than p
that the group of three jurors will make the correct decision (the thick solid curved line), exceeds p, the probability that any single juror will make the correct decision (the thin straight line). In other words, even for small numbers of jurors, the group has a better chance of making the correct decision than any single individual.
Note that there are several key assumptions that are involved in this result:
- • There are only two alternatives from which to choose. If there are three or more alternatives, the result breaks down.
- • The result assumes that p> 1 for each juror. But if p <1, so that
- 2 и 2
each juror's belief is biased towards making the wrong decision, then lim pn = 0 and adding more jurors makes the incorrect decision even more likely. This happens even if p is only slightly lower than —. Moreover, as Figure 2.6.2 shows, even if there is a small group 2 1
of jurors, when p < ^ the probability that the group will make the correct decision is less than the probability that any individual juror
Figure 2.6.3 Larger groups are more accurate than smaller groups if p> —, but 1 2
are less accurate if 0 < p <
will make the correct decision. Figure 2.6.3 shows that the extent of the inaccuracy gets worse, rather than better, as more individuals are added to the group. In this case, collective decision making is inferior to individual decision making.
- • The result assumes that decisions are taken by majority rule, rather than by unanimity. What about other voting procedures?
- • The result assumes that voting decisions are independent. But what if they are correlated?
- • The result assumes that voting decisions are sincere. But some or all voters may not have an incentive to vote sincerely if not doing so makes them better off.