Anti-comonotone random variables and Anti-monotone risk aversion

This paper focuses on the study of decision making under risk. We, first, recall some model-free definitions of risk aversion and increase in risk. We propose a new form of behavior under risk that we call anti-monotone risk aversion (hereafter referred to as ARA) related to the concept of anti-comonotony a concept investigated in Abouda, Aouani and Chateauneuf (2008). Note that many research has already been done in this field e.g. through the theory of comonotonicity. We give relationships between comonotone, strict comonotone, anti-comonotone and strict anti-comonotone random variables. Then, after the motivation of ARA, we show that this new aversion is weaker than monotone risk aversion while stronger than weak risk aversion.