So, I read

this. It's a system for automatically determining the odds that ought to be laid on a bet given two players' probability estimates of the outcomes, such that neither one can game the system given knowledge of the other's estimate.

Seems like a great way to automatically set up bets - no negotiations. You just say, "I will accept auto-oddsed bets with total bet scale = S, based on my probability estimate of P%"

Then someone can reply with "I'll accept X of that bet with my probability estimate Q." Which you may optionally follow up with "This works out to your betting D to my E." or you can leave it to me to calculate (which I will anyway since the formula is simple).

You just opine on a subject, other players can offer a different opinion on the same subject, and bam you've got a fair bet.

What is that formula? To borrow a summary by DavidS, "Each puts in the square of their surprise, then swap." with surprise being the probability that you had assigned to the outcome not occurring.

At greater length:

S = what that link calls 'max bet' but I'd call 'bet scale', since the actual bet amount will generally be much smaller than it...

P = probability that player one assigns to the proposition being true

Q = probability that player two assigns to the proposition being

falseNOTE: to find your real max bet given S and the P that you gave, take the larger of S*P^2 and S*(1-P)^2. If you want to bet around 25, say, and had a probability of 80%, then set a scale of 35).

If P < 1-Q, swap the phrasing of the bet so that P -> 1-P and Q -> 1-Q. This keeps the bet offers positive.

player 1 puts in S * (P*P - (1-Q)*(1-Q))

player 2 puts in S * (Q*Q - (1-P)*(1-P))

You can try messing around with the probabilities and find out that you get a bigger payoff but bigger risks for making more confident bets, in an exactly balanced way, so that you get the

most expected payoff by being honest with your estimates. By letting both players limit the scale of the bet we don't open up the method of abuse described in the discussion of this method, either.

Example:

I say '80% chance that Banhammer did not just disband the gwiffon, bet scale 100'

Bob says 'I take 50 of that bet, with 60% chance he did just disband the gwiffon'

So,

I put in 50 * (0.8*0.8 - 0.4*0.4) = 24

Bob puts in 50 * (0.6*0.6 - 0.2*0.2) = 16

I expect to win 80% * 16 - 20% * 24 = 8

Bob expects to win 60% * 24 - 40% * 16 = 8

So it's fair - we each expect to win 8. Given this example I'd end up paying out 24 to Bob.

After this bet, I still have 50 bet scale free to be taken by someone else at any probability they want.

Note that if you set a probability of 50%, the most you can end up paying out is quarter of the bet scale. It goes higher as you approach the extremes.

Example 2:

Bob says "I put a 50q scale auto-oddsed bet on a my 60% chance that Banhammer didn't just disband that gwiffon"

I say "Only 60%? I'll take that - 80% that he didn't!"

First we swap the sense of Bob's statement so that he's putting 40% chance of Banhammer not disbanding the gwiffon, and I'm putting 80% that he did.

So he puts in 50*(0.4*0.4 - 0.2*0.2) = 6

I put in 50*(0.8*0.8 - 0.6*0.6) = 14

Fairness check:

I expect to win 0.8*6 - 0.2*14 = 2

Bob expects to win 0.4*14 - 0.6*6 = 2

Yup, works out!

And again given this bet, I'd end up paying out to Bob, but less - 14 - because his estimate wasn't so different from mine as last time.

With that in mind, I put 50 bet scale on each of the following probabilities:

85% that Scrofula does not end up adopting Lord Crush's plan (at least in substantial part) without a push from a third party (even one on his side)

75% that Scrofula dies during 'Lord Crush' updates.

96% that Lord Crush is alive or decrypted-alive at some time past the end of the 'Lord Crush' updates.

also, scale 100 on:

45% that Squashcourt falls to Bullyclub during 'Lord Crush' updates

(All of these expired with no takers right after I offered them. I got the two higher-probability ones 'right' and was also 'right' on the 55% that Squashcourt wouldn't fall to Bullyclub. Screwed up Scrofula croaking, though)