What do you mean by ‘average’ I can think of at least four possibilities.
The standard of fairness “both players expect to make the same profit on average” implies that the average to use here is the arithmetic mean of the two probabilities, (p+q)/2. That’s what gives each person the same expected value according to their own beliefs.
This is easy to verify in the example in the post. Stakes of 13.28 and 2.72 involve a betting probability of 83%, which is halfway between the 99% and 67% that the two characters had.
You can also do a little arithmetic from the equal expected values to derive that this is how it works in general: the ratio of the amounts at stake should be (p+q):(2-p-q), which is the ratio mean(p,q):[1-mean(p,q)]. Using the arithmetic mean to set the betting odds gives both parties the same subjective expected value.
If there’s a simple intuitive sketch of why this has to be true, I don’t have it.
The extra work involved in the method in the post is to make it incentive-compatible for each person to state their true probability by allowing the stakes to vary.
So: To make the bet fair (equal EV), the betting odds should be based on the arithmetic mean of the probabilities. If you need to make it strategy-proof, then you can use the algorithm in this post to decide how much money to bet at those odds.
The standard of fairness “both players expect to make the same profit on average” implies that the average to use here is the arithmetic mean of the two probabilities, (p+q)/2. That’s what gives each person the same expected value according to their own beliefs.
This is easy to verify in the example in the post. Stakes of 13.28 and 2.72 involve a betting probability of 83%, which is halfway between the 99% and 67% that the two characters had.
You can also do a little arithmetic from the equal expected values to derive that this is how it works in general: the ratio of the amounts at stake should be (p+q):(2-p-q), which is the ratio mean(p,q):[1-mean(p,q)]. Using the arithmetic mean to set the betting odds gives both parties the same subjective expected value.
If there’s a simple intuitive sketch of why this has to be true, I don’t have it.
The extra work involved in the method in the post is to make it incentive-compatible for each person to state their true probability by allowing the stakes to vary.
So: To make the bet fair (equal EV), the betting odds should be based on the arithmetic mean of the probabilities. If you need to make it strategy-proof, then you can use the algorithm in this post to decide how much money to bet at those odds.