General Solution of an Inequality Containing Several Unknown Functions, with Applications to the Generalized Problem of “How to Keep the Expert Honest”

Abstract

The following is a generalization of the problem rhetorically called “how to keep the expert (or forecaster) honest” (see, e.g., McCarthy 1956, Marschak 1959, Good 1952, 1954, Aczél-Pfanzagl 1966, Fischer 1972, Aczél- Ostrowski 1973, Aczél 1973, 1974, Aczél-Daróczy 1975, Walter 1976). Let the events x₁,...,x_n be results of an experiment (market situation, weather, etc.). We are interested in their probabilities, so we ask an expert. He may know the true probabilities (or, at least, have subjective probabilities) p₁, p₂,...,p_n, but tells us q₁, q₂,...,q_n instead. Till now everything is very realistic. Now we make the somewhat idealistic assumption that the expert agrees to be paid the amount f_k(q_k) after one (and only one) of the events, x_k, happened. So his expected gain is

$$\sum\limits_{k = 1}^n {{p_k}{f_k}\left( {{q_k}} \right)}$$

.