How pairs of humans combine uncertain information

Imagine there are two referees who have different opinions about where a ball landed, in particular whether it went over some line. How can they cooperate to make a better decision than either one could individually?

We could flip a coin to decide which ref to believe. But this merely gives us a decision performance which is the average of the decision performance of each individual ref. So this is no good.

We could figure out which ref is better and then always believe that ref whenever the two refs disagree. But in that case the second ref isn’t contributing anything, so we may as well just have one ref.

However, if the two refs have some estimate of their uncertainty, then we can do better. If d1 represents where ref one thinks the ball landed, and d2 represents what ref 2 thinks, and if o1 and o2 represent each ref’s report of the standard deviation of their own estimate, then the optimal way to combine both refs’ guesses into a final guess is

(d1/o1^2 + d2/o2^2) / (1/o1^2 + 1/o2^2)

which gives better performance than either ref, individually. Since we only care about whether the ball went over the line, if the line is at d=0, then this rule simplifies to testing if

d1/(o1^2) + d2/(o2^2) > 0

Is that what humans do? No. Apparently, when put together in pairs and given the chance to communicate, humans communicate both their guesses and their uncertainties, and effectively use the following formula to make a cooperative guess:

d1/o1 + d2/o2 > 0

This rule provides better performance than individual decision-making provided the difference between o1 and o2 (the uncertainties of the refs) is less than 40%. If it is more than that, this rule is worse than just having the more reliable ref make all the decisions.

Why do humans use d1/o1 + d2/o2 instead of the optimal formula, d1/o1^2 + d2/o2^2? One hypothesis is that the former is unit-free, whereas the latter requires both refs to communicate with each other in terms of (matching) spatial units.


Marc O. Ernst. Decisions Made Better (27 August 2010)
Science 329 (5995), 1022.


Bahador Bahrami, Karsten Olsen, Peter E. Latham, Andreas Roepstorff, Geraint Rees, and Chris D. Frith. Optimally Interacting Minds (27 August 2010)
Science 329 (5995), 1081.

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