Aequationes Math. 76 (2008) 281–304 0001-9054/08/030281-24 DOI 10.1007/s00010-008-2931-0 c  Birkh¨ auser Verlag, Basel, 2008 Aequationes Mathematicae On the utility of gambling: extending the approach of Meginniss (1976) Che Tat Ng, R. Duncan Luce and Anthony A. J. Marley Summary. J. R. Meginniss modified expected utility to accommodate a concept of the utility of gambling that led to a representation composed of a utility expectation term plus an entropy of degree κ term. He imposed several apparently strong assumptions. One of these is that a number of unknown generating functions are identical. A second is that he assumed he was working with given probabilities. Here we follow his general framework but weaken considerably those assumptions. Our problem is reduced to solving some functional equations induced by gamble decomposition. From the solutions, we obtain the representation of the utility function. Further axiomatic restrictions are imposed that lead ultimately to Meginniss’ earlier result. Mathematics Subject Classification (2000). 94A17, 91B16, 39B22. Keywords. Entropy, expected utility, functional equations, gamble decomposition, utility of gambling. 1. Background Within the axiomatic theories of utility developed over the past sixty years, men- tion has been made of the problem of the utility of gambling itself. For example, both seminal articles of Ramsey [16] and of von Neumann and Morgenstern [18] mention it as a problem. But beyond a few, more-or-less discursive treatments of the problem, little has been done on it. Some of the attempts to address the problem are summarized by Luce and Marley in [7]; but, for the most part, they have not been axiomatic in nature. A notable, but largely ignored 1 , attempt is a short article by Meginniss [12] in which he arrived, in a partially axiomatic fashion, at two possible forms for the utility of gambling. Our realization of the power of his approach has led to the current attempt to generalize it. We have completed three further articles, aimed more at an economic and 1 One person, David Wolpert, when alerted to Meginniss [12] said “... it’s amazing! He derives Tsallis entropy over a decade before Tsallis popularized it. Typewritten, with penciled-in corrections; it’s like finding a hieroglyphic tablet saying ‘E = mc 2 ’.” (personal communication, October 28, 2004). As was reported in the book of Acz´ el, & Dar´ oczy [1], what has been called the Tsallis entropy was first arrived at by Havrda, & Charv´ at [4].