Alice and Bob are talking about the odds of some event E. Alice's odd's of E are 55% and Bob's are 90%. It becomes clear to them that they have different odds, and being good (and competitive) rationalists they decide to make a bet.
Essentially, bet construction can be seen as a bargaining problem, with the gap in odds as surplus value. Alice has positive EV on the "No" position for bets at >55% odds. Bob has neutral or better EV on the "Yes" position for bets at <90% odds.
Naive bet construction strategy: bet with 50/50 odds. Negative EV for Alice, so this bet doesn't work.
Less naive bet construction strategy: Alice and Bob negotiate over odds. The problem here, in my eyes, is that Alice and Bob have an incentive to strategically misrepresent their private odds of E in order to negotiate a better bet. If Alice is honest that her odds are 50%, and Bob lies that his odds are 70%, so they split the difference at 60%, Bob takes most of the surplus value.
If both were honest and bargaining equitably, they'd have split the difference at 72.5% instead. So I'll call 72.5% the "fair" odds for this bet.
A nicer and more rationalist aligned bet construction strategy wouldn't reward dishonesty! So, here it is.
1. Alice and Bob submit their maximum bets and their odds.
2. Take the minimum of the two maximum bets. Let's say its $198.
3. Construct 99 mini bets*; one at 1% odds of E, 2% odds of E... 99% odds of E. Each player automatically places 2$ on each mini bet that is favorable according to their odds ($198/99 = $2).
*99 chosen for simplicity. You could choose a much higher number for the sake of granularity.
So, in this case, Alice accepts the No position on all bets at =>55% odds, and Bob accepts the Yes position on all bets at =<90% odds, so they make 35 $2 bets, the average odds of which are 72.5%, which is the fair odds.
Observe that there is no incentive for either player to have misrepresented their odds. If Alice overrepresented her odds as 60%, she would just deny herself the ability to bet on bets at 56% through 59%, which have positive EV for her.
Note that Alice and Bob only bet $70 -- less than half of the maximum bet. If Bob wanted Alice to bet more money than she was really willing to risk, he might try to convince her that his odds were close to hers, such that a high maximum bet would still lead to a low actual bet. Does this seem like a problem to you? I think this method is still an improvement.
*The mini bets can be abbreviated analytically as one bet at average odds, I just like the mini bets concept for making the intuition clear.
I am not sure how exciting this method is to anyone. I like it because misrepresentation of value is a core problem in 2 player bargaining, and I realized betting is a bit of a special case. Other special cases exist too-- anything where players would be willing to accept uncertainty over the size of the trade (and are trading a continuous good such that that's even possible).