Using experts' noisy quantile judgments to quantify risks: Theory and application to agribusiness

Motivated by a unique agribusiness setting, this paper develops an optimization- based approach to estimate the mean and standard deviation of probability distributions from noisy quantile judgments provided by experts. The approach estimates the mean and standard deviations as weighted linear combinations of quantile judgments, where theweights are explicit functions of the expert's judgmental errors. The approach is analytically tractable, and provides flexibility to elicit any set of quantiles from an expert. The approach also establishes that using an expert's quantile judgments to deduce the distribution parameters is equivalent to collecting data with a specific sample size and enables combining the expert's judgments with those of other experts. It also shows analytically that the weights for the mean add up to one and the weights for the standard deviation add up to zero-these properties have been observed numerically in the literature in the last 30 years, but without a systematic explanation. The theory has been in use at Dow AgroSciences for two years for making an annual decision worth $800 million. The use of the approach has resulted in the following monetary benefits: (i) firm's annual production investment has reduced by 6%-7% and (ii) profit has increased by 2%-3%.We discuss the implementation at the firm, and provide practical guidelines for using expert judgment for operational uncertainties in industrial settings.