▁a ▁wind fall . ▁ ▁Form ally , ▁optim izing ▁for ▁robust ness ▁or ▁optim izing ▁for ▁opportun eness ▁yields ▁a ▁preference ▁relation ▁on ▁the ▁set ▁of ▁dec isions , ▁and ▁the ▁decision ▁rule ▁is ▁the ▁" optim ize ▁with ▁respect ▁to ▁this ▁preference ". ▁ ▁In ▁the ▁below , ▁let ▁ ▁be ▁the ▁set ▁of ▁all ▁available ▁or ▁feas ible ▁decision ▁vectors ▁. ▁ ▁Rob ust - s atisf icing ▁ ▁The ▁robust ness ▁function ▁generates ▁robust - s atisf icing ▁prefer ences ▁on ▁the ▁options : ▁dec isions ▁are ▁ranked ▁in ▁increasing ▁order ▁of ▁robust ness , ▁for ▁a ▁given ▁critical ▁reward , ▁i . e ., ▁by ▁ ▁value , ▁meaning ▁ ▁if ▁▁ ▁A ▁robust - s atisf icing ▁decision ▁is ▁one ▁which ▁maxim izes ▁the ▁robust ness ▁and ▁satisf ices ▁the ▁performance ▁at ▁the ▁critical ▁level ▁. ▁ ▁Den ote ▁the ▁maximum ▁robust ness ▁by ▁ ▁( form ally ▁ ▁for ▁the ▁maximum ▁robust ness ▁for ▁a ▁given ▁critical ▁reward ), ▁and ▁the ▁corresponding ▁decision ▁( or ▁dec isions ) ▁by ▁ ▁( form ally , ▁ ▁the ▁critical ▁optim izing ▁action ▁for ▁a ▁given ▁level ▁of ▁critical ▁reward ): ▁ ▁Usually , ▁though ▁not ▁invari ably , ▁the ▁robust - s atisf icing ▁action ▁ ▁depends ▁on ▁the ▁critical ▁reward ▁. ▁ ▁O pport une - wind fall ing ▁ ▁Con vers ely , ▁one ▁may ▁optimize ▁opportun eness : ▁the ▁opportun eness ▁function ▁generates ▁opport une - wind fall ing ▁prefer ences ▁on ▁the ▁options :