A Nonlinear, Nontransitive and Additive-Probability Model for Decisions Under Uncertainty

Abstract

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.