Bilinear utility and a threshold structure for nontransitive proferences
Let \({\mathbb{P}}\) be a set of probability measures defined on a set of oucomes and P be an asymmetric preference relation on \({\mathbb{P}}\). Since the work of von Neumann and Morgenstern, most decision theorists have centered their attention around expected utility theory, i.e. a numerical representation of P such that for all \(p,q\in {\mathbb{P}}\), pPq iff \(u(p)>u(q)\), this numerical representation being linear in probabilities, i.e., \(u(\alpha p+(1-\alpha)q)=\alpha u(p)+(1-\alpha)u(q)\) for all \(\alpha\in [0,1]\), where \(\alpha\) p-(1-\(\alpha)\)q denotes the convex combination of p and q. P. C. Fishburn recently proposed and axiomatized a far-reaching generalization of expected utility theory leading to a skew-symmetric bilinear representation (SSB) such that: pPq iff \(\Phi (p,q)>0\) with \(\Phi (p,q)=-\Phi (q,p)\) and \(\Phi\) being linear in both of its arguments. This paper proposes and axiomatizes a straightforward generalization of the SSB representation replacing the skew-symmetry condition by \(\Phi\) (p,p)\(\leq 0\) and \(\Phi (p,q)>0\Rightarrow \Phi (q,p)<0\). As the author rightly notes, dropping skew-symmetry may be interpreted as admitting imprecise preference discriminability in the model. In spite of the complexity of the resulting model, this paper is a valuable addition to the literature on the SSB representation.
scientific article; zbMATH DE number 4133769
scientific article