Weighting Function: Proving Intuition of Probabilistic Insurance

[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 [TEX]\pi[/TEX] be a probabililty weighting function such that, for all [TEX]p \in [0,1][/TEX] we have:

*****
[TEX]\pi(0) = 0[/TEX] and [TEX]\pi(1)=1[/TEX]

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

where [TEX]\bar{p}[/TEX] is a fix point of the function [TEX]\pi[/TEX]***** Subadditivity of small probabilities
[TEX]p < \bar{p}, \ \pi(rp) > r \pi(p)[/TEX] for any [TEX]0<r<1[/TEX]***** Subcertainty
[TEX]\pi(p) + \pi(1-p) < 1[/TEX] for all [TEX]0<p<1[/TEX]***** Subproportionality
[TEX]\frac{\pi(pq)}{\pi(p)}\leq\frac{\pi(pqr)}{\pi(pr)}[/TEX] for all [TEX]0<p,q,r \leq 1[/TEX]A generic graphical representation of this function is to be found below:View attachment 3272My question:

Thanks to functions plotting softwares i have the impression that:
for all [TEX]p_1, p_2 < \bar{p}[/TEX] :

[TEX]\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][/TEX]

But i cannot manage to prove it...
We have clearly: [TEX]\pi(p_1) \ \pi(p_2)< \pi(p_1 \ p_2)[/TEX] thanks to the subproportionality criterium, and we have that [TEX] \left[ \pi(p_1) \pi(p_2) + \pi(p_1) \pi(1-p_2) + \pi(1-p_1) \right] < 1 [/TEX]. 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 ([tex] p < \bar{p} [/tex]).

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

 

[jvicens]

Dear Guillaume,

Thank you for sharing your interesting research topic with us. From what you have described, it seems like you are working on a very complex and challenging problem. I am not familiar with prospect theory myself, but I will do my best to offer some insights and suggestions.

First of all, it is important to note that proving mathematical results can be a difficult and time-consuming process. It often requires a combination of logical reasoning, creativity, and persistence. Therefore, it is completely normal to spend a whole day or even longer on a single problem.

In order to prove your intuition, it would be helpful to break down the problem into smaller, more manageable parts. For example, you can start by looking at specific values of p_1 and p_2, and try to prove the inequality for those values. Then, you can try to generalize your proof to a larger range of values. It may also be helpful to consider special cases, such as p_1 = 0 or p_2 = 1, and see if the inequality holds in those cases.

Another approach could be to use mathematical induction, where you prove the inequality for a base case (e.g. p_1 = 0 or p_2 = 1) and then show that if it holds for a certain value, it also holds for the next value. This can help you to prove the inequality for a larger range of values.

Additionally, you can try to use the criteria you have mentioned, such as subadditivity and subproportionality, to your advantage. For example, you can try to manipulate the terms in the inequality using these criteria and see if it leads you to a proof.

Lastly, it may also be helpful to consult with other researchers or experts in the field of prospect theory. They may have insights or suggestions that can guide you in your proof.

Overall, I would encourage you to continue your efforts and not get discouraged by any setbacks. With persistence and a systematic approach, I am sure you will be able to prove your intuition. I wish you all the best in your research.