Electronic Supplementary Material for “Decision accuracy in complex
environments is often maximized by small group sizes”
A. Fraction of environments in the two-cue scenario that allows for the wisdom of
crowds
Here we calculate, for the two-cue scenario, the proportion of environments in which the
wisdom of crowds is observed for the maximizing, matching, and random strategies. In
general, in order to guarantee the wisdom of crowds, a large group needs more than half
of its members to utilize correct, low correlation information. For the maximizing
strategy, the wisdom of crowds is observed only if the low correlation cue is more
reliable than the high correlation cue. Because a priori we do not expect one cue to be
more reliable than the other, the probability that this occurs is 50%. For the matching
strategy, the probability that an individual uses the low correlation cue is rL/(rL + rH), and
the probability that it uses the high correlation cue is rH/(rL + rH). Because the wisdom of
crowds is observed only if more than half of the group receives correct, low correlation
information, we need rL2/(rL + rH) > 1/2, or rH < 2rL2 – rL. If we assume that rL and rH
have uniform distributions between 0.5 and 1, then 18% of that space satisfies the above
condition. For the random strategy, only half of the group uses the low correlation cue,
and therefore the proportion of the group that receives correct, low correlation
information is rL/2. Since rL is generally less than 1, the proportion is less than 1/2 and
the wisdom of crowds is never observed.
B. Fraction of environments in scenarios with arbitrarily many cues that allows for
the wisdom of crowds
Similarly, we calculate, for the many-cue scenario, the proportion of environments in
which the wisdom of crowds is observed for the maximizing, matching, and random
strategies, as well as the majority strategy. For the maximizing strategy, the wisdom of
crowds is observed only if the most reliable cue is a low correlation cue. Since we
assume that the reliabilities of all of the cues are drawn from the same uniform
distribution, the probability that a low correlation cue is the most reliable is ML/M. For
the matching strategy, the probability of using a cue is r/ΣMr, where ΣM is the sum over
all cues. We need more than half of the group to receive correct, low correlation
information, so we need ΣL(r2)/ΣMr > 1/2, where ΣL is the sum over all low correlation
cues. Since ΣMr = M<r>M and ΣLr2 = ML<r2>L (where <>M is the mean across all of the
cues, and <>L is the mean across the low correlation cues), the condition for the wisdom
of crowds is ML/M > <r>M/(2<r2>L). For the random strategy, all cues are equally likely
to be used. In order for more than half of the group to receive correct, low correlation
information, we need ΣL(r)/M > 1/2. Since ΣLr = ML<r>L, the condition for the wisdom of
crowds is ML/M > 1/(2<r>L). Finally, for the majority strategy, an individual favors the
option that the majority of the cues that it observes indicates. Here a majority of
individuals needs to receive a majority of cues that are correct and have low correlation.
The logic is the same as in the random strategy case, and the condition for the wisdom of
crowds is again ML/M > 1/(2<r>L).