Two Reasons to Make Aggregated Probability Forecasts More Extreme

When aggregating the probability estimates of many individuals to form a consensus probability estimate of an uncertain future event, it is common to combine them using a simple weighted average. Such aggregated probabilities correspond more closely to the real world if they are transformed by pushing them closer to 0 or 1. We explain the need for such transformations in terms of two distorting factors: The first factor is the compression of the probability scale at the two ends, so that random error tends to push the average probability toward 0.5. This effect does not occur for the median forecast, or, arguably, for the mean of the log odds of individual forecasts. The second factor—which affects mean, median, and mean of log odds—is the result of forecasters taking into account their individual ignorance of the total body of information available. Individual confidence in the direction of a probability judgment (high/low) thus fails to take into account the wisdom of crowds that results from combining different evidence available to different judges. We show that the same transformation function can approximately eliminate both distorting effects with different parameters for the mean and the median. And we show how, in principle, use of the median can help distinguish the two effects.