Partial Preference Information and First Order Differential Optimality: An Illustration

Abstract

Suppose that objective vectors $y \in {{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{R}}^{M}}$ are ranked by a value function v(y|θ) involving an unknown or partially known parameter θ. The set of values to which θ is restricted by prior information will be denoted ⊝. If y and z are objective vectors, say that y is dominated by z under ⊝, and write

$$ y{ < _{\theta }}z\quad if\;v(y\left| {\theta ) < v(z} \right|\theta )\quad for\,all\;\theta \in \theta $$

.