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.
Trying to make this more intuitive: consider a prediction market which is currently priced at x, where each share will pay out $1 if it resolves as True.
If you think it's underpriced because your probability is y, where y>x, then your subjective EV from buying a share is y-x. e.g., If it's priced at $0.70 and you think p=0.8, your subjective EV from buying a share is $0.10.
If you think it's overpriced because your probability is z, where z<x, then your subjective EV from selling a share is x-z. e.g., If it's priced at $0.70 and you think p=0.56, your... (read more)
(Cross-posted on my personal blog, which has LaTeX, and is easier to read.)
Let's say that you are are at your local less wrong meet up and someone makes some strong claim and seems very sure of himself, "blah blah blah resurrected blah blah alicorn princess blah blah 99 percent sure." You think he is probably correct, you estimate a 67 percent chance, but you think he is way over confident. "Wanna bet?" You ask.
"Sure," he responds, and you both check your wallets and have 25 dollars each. "Okay," he says, "now you pick some betting odds, and I'll choose which side I want to pick."
"That's crazy," you say, "I am going to pick the odds so that I cannot be taken advantage of, which means that I will be indifferent between which of the two options you pick, which means that I will expect to gain 0 dollars from this transaction. I wont take it. It is not fair!"
"Okay," he says, annoyed with you. "We will both write down the probability we think that I am correct, average them together, and that will determine the betting odds. We'll bet as much as we can of our 25 dollars with those odds."
"What do you mean by 'average' I can think of at least four possibilities. Also, since I know your probability is high, I will just choose a high probability that is still less than it to maximize the odds in my favor regardless of my actual belief. Your proposition is not strategy proof."
"Fine, what do you suggest?"
You take out some paper, solve some differential equations, and explain how the bet should go.
Satisfied with your math, you share your probability, he puts 13.28 on the table, and you put 2.72 on the table.
"Now what?" He asks.
A third meet up member takes quickly takes the 16 dollars from the table and answers, "You wait."
I will now derive a general algorithm for determining a bet from two probabilities and a maximum amount of money that people are willing to bet. This algorithm is both strategy proof and fair. The solution turns out to be simple, so if you like, you can skip to the last paragraph, and use it next time you want to make a friendly bet. If you want to try to derive the solution on your own, you might want to stop reading now.
First, we have to be clear about what we mean by strategy proof and fair. "Strategy proof" is clear. Our algorithm should ensure that neither person believes that they can increase their expected profit by lying about their probabilities. "Fair" will be a little harder to define. There is more than one way to define "fair" in this context, but there is one way which I think is probably the best. When the players make the bet, they both will expect to make some profit. They will not both be correct, but they will both believe they are expected to make profit. I claim the bet is fair if both players expect to make the same profit on average.
Now, lets formalize the problem:
Alice believes S is true with probability p. Bob believes S is false with probability q. Both players are willing to bet up to d dollars. Without loss of generality, assume p+q>1. Our betting algorithm will output a dollar amount, f(p,q), for Alice to put on the table and a dollar amount, g(p,q) for Bob to put on the table. Then if S is true, Alice gets all the money, and if S is false, Bob gets all the money.
From Alice's point of view, her expected profit for Alice will be p(g(p,q))+(1-p)(-f(p,q)).
From Bob's point of view, his expected profit for Bob will be q(f(p,q))+(1-q)(-g(p,q)).
Setting these two values equal, and simplifying, we get that (1+p-q)g(p,q)=(1+q-p)f(p,q), which is the condition that the betting algorithm is fair.
For convenience of notation, we will define h(p,q) by h(p,q)=g(p,q)/(1+q-p)=f(p,q)/(1+p-q).
Now, we want to look at what will happen if Alice lies about her probability. If instead of saying p, Alice were to say that her probability was r, then her expected profit would be p(g(r,q))+(1-p)(-f(r,q)), which equals p(1+q-r)h(r,q)+(1-p)(-(1+r-q)h(r,q))=(2p-1-r+q)h(r,q).
We want this value as a function of r to be maximized when r=p, which means that -h+(2r-1-r+q) (dh/dr)=0.
Separation of variables gives us (1/h)dh=1/(-1+r+q)dr,
which integrates to ln(h)=C+ln(-1+r+q) at r=p,
which simplifies to h=e^C(-1+r+q)=e^C(-1+p+q).
This gives the solution f(p,q)=e^C(-1+p+q)(1+p-q)=e^C(p^2-(1-q)^2) and g(p,q)=e^C(-1+p+q)(1+q-p)=e^C(q^2-(1-p)^2).
It is quick to verify that this solution is actually fair, and both players' expected profit is maximized by honest reporting of beliefs.
The value of the constant multiplied out in front can be anything, and the most either player could ever have to put on the table is equal to this constant. Therefore, if both players are willing to bet up to d dollars, we should define e^C=d.
Alice and Bob are willing to bet up to d dollars, Alice thinks S is true with probability p, and Bob thinks S is false with probability q. Assuming p+q>1, Alice should put in d(p^2-(1-q)^2), while Bob should put in d(q^2-(1-p)^2). I suggest you use this algorithm next time you want to have a friendly wager (with a rational person), and I suggest you set d to 25 dollars and require both players to say an odd integer percent to ensure a whole number of cents.