weighting function

[dollyprane]

Hi everybody,

I am currently writing a master's thesis in probabilistic insurance, for which i am using prospect theory (derived by Kahneman and Tversky in 1979). .
Just as a preamble: this is not a probability problem, rather the study of a function.
I have an intuition but I cannot prove the result, and i have been working the whole day on it!

Let me explain simply:

Let \(\displaystyle \pi\) be a probabililty weighting function such that, for all \(\displaystyle p \in [0,1]\) we have:

*****
\(\displaystyle \pi(0) = 0\) et \(\displaystyle \pi(1)=1\)

***** Overestimation of small probabilities and underestimation of large probabilities
\(\displaystyle p <\bar{p} \ , \quad \pi(p) > p\)
\(\displaystyle p >\bar{p} \ , \quad \pi(p) < p\)

where \(\displaystyle \bar{p}\) is a fix point of the function \(\displaystyle \pi\)


***** Subadditivity of small probabilities
\(\displaystyle p < \bar{p}, \ \pi(rp) > r \pi(p)\) for any \(\displaystyle 0<r<1\)


***** Subcertainty
\(\displaystyle \pi(p) + \pi(1-p) < 1\) for all \(\displaystyle 0<p<1\)


***** Subproportionality
\(\displaystyle \frac{\pi(pq)}{\pi(p)}\leq\frac{\pi(pqr)}{\pi(pr)}\) for all \(\displaystyle 0<p,q,r \leq 1\)


A generic graphical representation of this function is to be found below:

My question:​

Thanks to functions plotting softwares i have the impression that:
for all \(\displaystyle p_1, p_2 < \bar{p}\) :

\(\displaystyle \pi(p_1) \pi(p_2) < \pi(p_1 p_2) \left[ \pi(p_1) \pi(p_2) + \pi(p_1) \pi(1-p_2) + \pi(1-p_1) \right]\)

But i cannot manage to prove it...
We have clearly: \(\displaystyle \pi(p_1) \ \pi(p_2)< \pi(p_1 \ p_2)\) thanks to the subproportionality criterium, and we have that \(\displaystyle \left[ \pi(p_1) \pi(p_2) + \pi(p_1) \pi(1-p_2) + \pi(1-p_1) \right] < 1 \). But even when I try to group terms or use any criteria above, I am stuck...
I know this result is not valid on [0,1], but i think it is the case for small values of p (\(\displaystyle p < \bar{p} \)).

I would be so grateful if you could help me out to confirm or invalidate my intuition. any help is appreciated!
thanks in advance.
Guillaume