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June, 1987 A Nonlinear, Nontransitive and Additive-Probability Model for Decisions Under Uncertainty
Peter C. Fishburn, Irving H. LaValle
Ann. Statist. 15(2): 830-844 (June, 1987). DOI: 10.1214/aos/1176350378


Let $\succ$ denote a preference relation on a set $F$ of lottery acts. Each $f$ in $F$ maps a state space $S$ into a set $P$ of lotteries on decision outcomes. The paper discusses axioms for $\succ$ on $F$ which imply the existence of an SSB (skew-symmetric bilinear) functional $\phi$ on $P \times P$ and a finitely additive probability measure $\pi$ on $2^S$ such that, for all $f$ and $g$ in $F$, $f \succ g \Leftrightarrow \int_S \phi(f(s), g(s)) d\pi(s) > 0.$ This $S^3B$ (states SSB) model generalizes the traditional Ramsey-Savage model in which $\phi$ decomposes as $\phi(p, q) = u(p) - u(q)$, where $u$ is a linear functional on $P$. The $S^3B$ model preserves the probability structure of the Ramsey-Savage model while weakening their assumptions of transitivity and independence.


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Peter C. Fishburn. Irving H. LaValle. "A Nonlinear, Nontransitive and Additive-Probability Model for Decisions Under Uncertainty." Ann. Statist. 15 (2) 830 - 844, June, 1987.


Published: June, 1987
First available in Project Euclid: 12 April 2007

zbMATH: 0633.90002
MathSciNet: MR888443
Digital Object Identifier: 10.1214/aos/1176350378

Primary: 90A05
Secondary: 06A99, 90A06, 90A10

Rights: Copyright © 1987 Institute of Mathematical Statistics


Vol.15 • No. 2 • June, 1987
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