  In this paper, we aims to state some proprieties of willingness to pay (WTP)
for partial risk reduction and links with insurance within the dual theory of
decision. In the case of partial reduction, we get as Langlais (2005) that a
risk-averse decision maker (DM) can have a willingness to pay small than this
of a neutral one. By decomposition the WTP as Courbage and al (2008), we get
that a strong averse DM is willing to give more for a reduction of a high
probability portion rather than a low probability one. The main result is that
in the dual theory, reducing probability of risk and supply insurance can be
complementary if the surplus is increasing in risk reduction.
