Consider an action where failure implies you get F(<0) and success S(>0), you think the probability of success is p.. You think the average is:
Failure Success Average F S (1-p)F + pS = p(S-F) + F
Now there is someone willing to bet, he believes that the probability is infact q. He offers compensation if it fails C, but he wants βC if it succeeds.
Failure Success Average You F+C S-βC (1-p)(F+C) + p(S-βC) = p(S-F-C(1+β)) + F + C Gambler -C βC -(1-q)C + qβC = qC(1+β) - C
Now You will accept if you will do better than earlier;
p(S-F-C(1+β)) + F + C ≥ p(S-F) + F
-pC(1+β) + C ≥ 0
β ≤ 1/p -1
So the payout of success and failure; S, F dont matter! In general can add whatever you want and it doesnt matter. It is just a bet? So why do just bet for stuff like insurance?
Three reasons come to mind:
Our utility(wellbeing) is not linear with the amount of money we own. U(F) and U(S) are much further apart than F and S. But U(F+C) and U(S-βC).. Infact we can approximate;
U(F+C) - U(S-βC) ≈ U'(F+C)⋅(F + C - S + βC)
This aswel makes this calculation still work as the approximation is linear and the factor doesnt matter for the analysis. So the bet is in a sense a side issue.
The mechanisms in which someone else bets that way indicates someone else has trust in -for instance- a vendor.
You are particularly aware of the risks and deals and their probability.
Of course, in the current world, insurers are often large institutions and their many bets even out.
It can also be investment, where C is given beforehand. This implies that it can be used as stake. After all, if a contract(like RANDAO) requires ethers as stake, and a gambler trusts you would not lose that stake, he can make a deal. If you fail, his belief in you will decrease; that would be the what is at stake for you.
The gambler side
The gambler, assume he wants some profit; E
qC(1+β) - C ≥ E
β ≥ (E/C +1)/q -1 = 1/q' - 1
But then we notice that the profit motive shows itself as an apparent probability; q' = q/(E/C + 1) as the gambler can change the two accordingly, there is not point to the freedom; we choose E=0 and note that probability is only apparent.
A deal is possible if: 1/p -1 ≥ β ≥ 1/q' -1
From many gamblers, the best deal is from one with the highest (Apparent)probability, he will be able to demand the smallest payment. Perhaps, however, he doesnt have enough money, or doesnt want to risk at those probabilities that much money. Here this would be again because of nonlinearity of utility, but also because the probability of failure may be dependent on the amount.
If we assume we have the comprehensive list of gamblers, after a gambler is depleted, you can go to the next one with a lower q.
We presumably can find these gamblers within a system in Ethereum. Mind that finding them being possibly client-side massively increases the ability to search without using gas. You search client-side, and then just tell the contracts where to go to convince it.
Note that the estimation of probability may not also feel that way to the user. For instance, if the bet is about the ability of a web shop to deliver, the user might see prices based on what the bets are wrt insuring the transaction. The user simply bets by accepting the price.
Stabilizing gambler income
The gambler expects on average factor on the input ethers per unit time. However depending on the number of bets and their dependence, actual results can vary. He could insure himself against it, or predict some growth use a Contract for a difference where people may bet it is more.
Of course, the insurance and contract for a difference may largely be the same thing.
In relation to reputation networks
Actions can be deals between people, or even a single person. The estimated probability can depend on what those people are like.
So there is a relation to reputation, this can two ways;
The reputation network is used to estimate the probabilities. And people handling it take their consequences based on failing deals.
Working backward from the probabilities to reputation. Oppositely, the insurance builds the reputation system. This one is likely much harder.
Of course, there is much more to be said about this topic.