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?

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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:

enter image description here

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!

  • 2
    $\begingroup$ This is most likely a game theory problem. Fairness probably needs to be more fully specified. Are player's outcomes dependent upon the choices of other players? Can the house retain a portion of the pool to retain equity? Are players aware of how many other players are playing? Can players choose to play? $\endgroup$
    – Davey
    Nov 16, 2023 at 22:58
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    $\begingroup$ @SextusEmpiricus simple case Bernoulli with $P(X =1) = 0.9$ one person puts 1\$ on 1 the other splits his 1$ bet 1:9. The expected return for person 2 is 1.05\$ $\endgroup$ Nov 17, 2023 at 14:09
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    $\begingroup$ The way to embed risk-aversion in a betting framework is to use expected utilities. Very roughly, your setup seems to be one where a planner organizes a betting scheme to elicit certain types of bets from certain types of people while maximizing their expected utility. This falls under the purview of mechanism design. $\endgroup$ Nov 17, 2023 at 18:31
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    $\begingroup$ This is a very classic paper that may help you think through how one can incentivize information revelation in a strategic setting. brown.edu/Departments/Economics/Faculty/Glenn_Loury/… $\endgroup$ Nov 17, 2023 at 18:36
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    $\begingroup$ Personal 2 cents: Placing all bet at the first month is probably a bad idea. Doing it adaptively over time (monthly?) is what I see more optimal, where each person bet on if the event will be absent in the next month. If the event happens in the first month, the game ends, and person B will lose less if they enter the bet adaptively every month. I am inclined to think about this in the framework of survival analysis. $\endgroup$ Nov 20, 2023 at 22:46

2 Answers 2


Split up in smaller bets with dollars paired to each other

Consider it as twelve times paired bets of single dollars. With outcomes loss, win, or equal depending on whether one or both of the bets were right.

  • you win the other dollar if the month is right and the other wrong
  • you loose the dollar if the month is wrong and the other right
  • no loss and win if both are wrong or both are right

Then the one person wins 1 dollar when the month is not October and the other person wins 11 dollars when the month is October.

In the case of bets from two people it doesn't matter how you make the pairs. However in the case of multiple people, or when the amounts of money are unequal, then we can do the following.

For a prize pool of size $T$ dollars we can create pairs with $1/(T-1)$-th piece of every placed dollar. Each dollar is paired with every other dollar.

Let's define total bets of $x_i$ dollars on the winning month and total bets of $y_i$ on the loosing months, where $i$ is a subscript that refers to the person. The earnings for person $i$ are

$$\underbrace{\frac{1}{T-1} x_i \sum y_i}_{winnings} - \underbrace{\frac{1}{T-1} y_i \sum x_i }_{losses}$$

Each dollar on the winning month get's $\frac{1}{T-1}$ dollars from each dollar on a loosing month. And vice versa, each dollar on a loosing month gives $\frac{1}{T-1}$ dollars for each dollar on a winning month.

With this method of defining the prizes we can compute the expected winnings as follows.

  • Let's consider $x_k$ and $y_k$ the fractions of their bets by the two people X and Y in month $k$ (and in the multi person variant we can group all opponents as one single opponent).

  • Let the probability for an event on month $k$ be $p_k$.

  • Let the total prize money be $T$ and the money by person $X$ and $Y$ be $T_X$ and $T_Y$.

Then profit for person X when there's an event in month $k$ is

$$profit|k = (x_k-y_k) \frac{T_X T_Y}{T-1} $$

and the expected profit is

$$E[profit] = \frac{T_X T_Y}{T-1} \sum p_k(x_k-y_k)$$

When we consider the opponents bets, $y_k$, fixed or uncontrollable and consider our own winnings, then we need to maximize

$$\sum p_k x_k$$

This means that we should bet on the mode of the distribution $p_k$. Spreading the bets might reduce the variance in winnings and losses, but it comes at the cost of a decrease in the expected profit.

Winnings based on market share.

A different scheme is to share the entire prize money among the winning bids according to their market share.

For example if the event is in October then the one player gets 1/13-th of the 24 dollar prize money and the other player player gets 12/13-th of the 24 dollar prize money.

So if $x_k$ and $y_k$ are the fractions of placed bets and the total prize money is $T$, and $T_X$ and $T_Y$ the amounts of money paid by the players, then if the event happens on month $k$ the return is

$$profit|k = \frac{x_k}{x_k+y_k T_Y/T_X} T - T_X$$

In this scheme the players have an incentive to spread their bets according to the distribution that they believe the event will take place. (If the players have information about the other person's bets). Or at least, if two players have the same believes about the probability of the event $p_k$, then betting $x_k = y_k = p_k$ is a Pareto optimal solution, none of the players can improve their expected outcome by changing their bets. When we take for simplicity $T_X = T_Y = 1$ then the expected outcome for player X, conditional on the believe $p_k$ being the true distribution, is

$$E[profit] = - 1 + 2 \sum p_k \frac{x_k}{x_k+y_k} $$

And when the opponent bets $y_k = p_k$ then

$$\begin{array}{} E[profit] &=& - 1 + 2 \sum \frac{x_kp_k}{x_k+p_k} \\ &=& - 1 + 0.5 \sum \frac{(x_k+p_k)^2-(x_k-p_k)^2}{x_k+p_k}\\ &=& - 1 + 0.5 \sum (x_k + p_k) - 0.5 \sum \frac{(x_k-p_k)^2}{x_k+p_k} \\ &=& - 0.5 \sum \frac{(x_k-p_k)^2}{x_k+p_k} \end{array}$$

And that last sum is all non-negative terms and gets minimized when it is equal to zero which happens if $x_k = p_k$.

So if both players believe that the event happens with probability $p_k$ and if $y_k = p_k$ then player X also wants to bet with $x_k = p_k$. And if $x_k = p_k$ then player Y wants to bet with $y_k = p_k$.

  • $\begingroup$ An awesome, comprehensive, and thought provoking solution. $\endgroup$ Nov 19, 2023 at 9:46
  • $\begingroup$ Yes, but this doesn't answer the question. $\endgroup$
    – whuber
    Nov 20, 2023 at 14:01
  • $\begingroup$ The second solution talks about Pareto stability and betting strategies that get close to the true distribution or believes about the true distribution. But it is still that bets with the highest information content or closest distance in terms of KL divergence do not neccesarily have a higher expected profit. For this one would need a return that is equal to the $\log$ of the placed bet. The more you bet on the same horse, the less profit you get from it. $\endgroup$ Nov 20, 2023 at 14:52

An elementary argument solves this without relying on heavy theory...

The fact that each person has $12 is something of a red herring. Imagine that each person has 12 children; they hand out a dollar coin to each child. The children then bet, so that you (the tournament organiser) see 24 bets of 1 dollar each. You hand out rewards to the children who choose the correct month. The parents then gather up their children's winnings.

It would be bizarre if your payouts to the children depended on who their parent was. Indeed, intuitively speaking, you should not have access to that information. From your perspective, all you see is that 24 people bet $1 each and you need to pay each of them fairly.

So in the case you describe, 13 people correctly bet $1 on October. By symmetry, the only fair way to reward them is to split the reward pool between them, so that each receives $24 / 13.

A will then receive 12x that quantity, and B will receive 1x that quantity.

  • $\begingroup$ The division of 1:12 of the prize money seems fair. But why should the total prize money be $24. Why is that fair? In that respect different choices can be made. $\endgroup$ Nov 20, 2023 at 11:21
  • $\begingroup$ “By symmetry, the only fair way to reward them is to split the reward pool between them” this seems to be like a crucial step, but it is a bit handwaving. What symmetry is regarded? Why is it fair? $\endgroup$ Nov 20, 2023 at 11:28
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    $\begingroup$ @SextusEmpiricus The bit about total prize money is explicit in the q: "The "pool" is thus 24 USD. How do I fairly allocate the pool". $\endgroup$
    – Mohan
    Nov 20, 2023 at 13:57
  • $\begingroup$ @SextusEmpiricus The symmetry group you get by permuting the 'children'. Another way to look at this is that any other scheme must be sensitive to the difference between a) 1 person placing a $2 bet and b) two people placing $1 bets. But in that case people can arbitrage by getting friends to place bets for them. $\endgroup$
    – Mohan
    Nov 20, 2023 at 13:59
  • $\begingroup$ How about each of the 24 children each split up their dollar into 23 parts in order to make wages with the other 23 children. In that case you also have a symmetric situation that's not sensitive to the two different situations that you mention, but, you wil get a different result. The kids that bet on October probably won't agree that their prize is only 1/13 th size of the prize of a kid that bets on one of the other 11 months. $\endgroup$ Nov 20, 2023 at 14:42

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