Imagine two people must bet on the timing of the occurrence of an event, which we know will happen within the next 12 months, but we don't know in which month. Each person has 12 dollars and can bet in units of 1 USD, resolution monthly.
It's 1 January, and Person B thinks the event can happen with equal probability in any of the the next 12 months, and subsequently bets 1 USD per month for each month.
Person A has very high confidence that the event will occur in October, and bets all 12 USD in that month.
The "pool" is thus 24 USD. How do I fairly allocate the pool if the event does happen in October, and how do I allocate it if it doesn't?
The issue here is that Person A is taking much more risk. In some way there seems to be much more "information content" in their bet, than in Person B's.
How do I fairly reward this risk/information content?
Which branch of statistics or probability theory will help me to model such problems?
EDIT 1 - different sized bets
Please also indicate if in fact to solve this problem fairly, the bet sizes need to be different for Person A and Person B, and in which case, how does one calculate the ratios?
EDIT 2 - probability distribution
@usεr11852 points out that if the distribution is uniform then Person A is getting robbed. The point is that the the distribution is NOT known in advance, and we're trying to reward people fairly who correctly predict the future observed timing, in advance, with more reward for more information content. It's entirely possible that Person A knows something that others don't with high confidence, and we want to reward that person for disclosing that information via the information-rich distribution they choose to bet on, relative to less information-rich distributions such as uniform or gamma distribution. For simplicity, in this I have illustrated the question with a blocky histogram; I assume the principles are the same for continuous distributions.
Here is a slightly more realistic example adding a third bet:
Person B hasically has a view that the event can occur anytime. Person A very strongly believes it will happen in October, Person C thinks it will happen in H2 with a peak probability in September and October. Each of persons C, A have increasing amounts of confidence in the timing of the event, and are prepared to backup that confidence with bet concentration. Person B thinks they're clueless and taking too much risk, thinks the distribution is uniform, so wants to win if they're wrong.
EDIT 3 - response to @Davey
- Each player does NOT know what the others are betting. Everybody bets blind, in a single round at the same time on 1 Jan.
- The house takes zero margin.
- The players do not know how many others are playing.
- Fairness means that on 1 Jan, each player's
payout_potential / information_content
ratio is the same.
So since I do not know the distribution up front, I need to be able measure the information provided by each participant, and pay them out according to how much information they provided, fairly (in the event that they are correct, of course). Obviously, if the event happens in October, Person A needs to get a lot of payout, whereas if it happens in say, May, Person B needs to get some (much smaller) payout. What are those relative payouts?
If this question is better analysed using continuous distribution functions and entropy, feel free!