Entropy-Related Measures of the Utility of Gambling

The first author has known Peter for a very long time, dating back some 45 years to when we met at a colloquium he gave at the University of Pennsylvania. After that our paths crossed fairly often. For example, in the early 1970s, he spent a year at the Institute for Advanced Study where Luce spent three years until the attempt to establish a program in scientific social science was abandoned for a more literary approach favored by the humanists and, surprisingly, the mathematicians then at the Institute. The second author has learnt a tremendous amount about both substantive and technical issues from Peter’s work, beginning with Peter’s book “Utility Theory for Decision Making” (Fishburn, 1970), which he reviewed for Contemporary Psychology (see Marley, 1972). Peter’s volume on interval orders (Fishburn, 1985) was a marvelous development of various ideas related to the algebra of imperfect discrimination that elaborated the first author’s initial work on semiorders (Luce, 1956). Beginning in 1988, Peter made a major contribution in his integrative book “Nonlinear Preference and Utility Theory.” And in the first half of the 1990s, Fishburn and Luce collaborated on three efforts to understand better the rank-dependent generalizations of expected utility that had attracted considerable notice in the 1980s (Fishburn & Luce, 1995; Luce & Fishburn, 1991, 1995). It was here that we first came up with the so-called p-additive form for the utility of joint receipts. All of that played a major role in Luce’s (2000) attempt to pull together many of the results about utility, both experimental and theoretical, of the period starting in 1979.