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.
@article{1176350378,
author = {Fishburn, Peter C. and LaValle, Irving H.},
title = {A Nonlinear, Nontransitive and Additive-Probability Model for Decisions Under Uncertainty},
journal = {Ann. Statist.},
volume = {15},
number = {1},
year = {1987},
pages = { 830-844},
language = {en},
url = {http://dml.mathdoc.fr/item/1176350378}
}
Fishburn, Peter C.; LaValle, Irving H. A Nonlinear, Nontransitive and Additive-Probability Model for Decisions Under Uncertainty. Ann. Statist., Tome 15 (1987) no. 1, pp. 830-844. http://gdmltest.u-ga.fr/item/1176350378/