I posted this in SMP.. But guess it probably belongs here:

Say there are two gamblers - for sake of argument, suppose they each have $1000 to bet. They decide they want to bet against eachother on whether some sports team will win a game (no ties). The first gambler believes the win probability is p%, the second believes it is q%. They haven't told each other... How would they determine the "most fair" betting odds?

Original thread: http://forumserver.twoplustwo.com/sh...php?p=45254296

Suppose I think the Spurs are 65% to win and you think they're 55%.

I'll bet the spurs, you'll bet the other team, and we need to solve for the payouts x and y. The payout odds I give you are the multiplicative inverse of those you'll give me.

In my mind, my EV is .65x - .35
In your mind, your EV is .45y - .55

Set those equations equal, and let x = 1/y

We end up with a quadratic: .45y^2 - .2y - .65 = 0

y = 13/9
x = 9/13

Plugging those back in, we each think we have a 10% edge (and that's the difference between our percentage estimates).

Let's see if that works in the following situation: I think spurs are 55%, you think they're 48%.

.55/y - .45 = .52y - .48
.52y^2 - .03y - .55 = 0
y = 55/52

We each think we have a 7% edge.


Edit: so actually, knowing that the final result should be a perceived edge equal to the difference of our estimates, it doesn't have to be a quadratic.

For the first one: .65x - .35 = .1
x = 9/13
y = 13/9

Wait, they haven't told each other their estimates? Then I don't know. It would explain why people invoked fancy stuff in the SMP thread when all I used here was kiddie algebra.

Yeah well it still seems to be correct that they decide on some formula (like you did) based on their estimates, then write down the estimates separately and share... But I'm wondering if there is a "best" formula. There seem to be several alternatives (when considering both EV and risk).

And i guess there could be some extra game involved if they have an idea about what the other - or a random "other" - might be thinking.

The next part of this question is this: if I know nothing about the game, and hold an equally respectable opinion about the two bettors in question - how should I estimate the probability?

Here is one idea.

Each gambler writes down a price at which he is willing to either buy or sell a security that pays $100 if team A wins.

The gamblers exchange papers, and each one elects to either buy or sell at the offered price.

Take heehaww's example, and start by assuming the gamblers play straightforwardly. Heehaww writes down $65 and pocketzeroes writes $55. Heehaww buys at $55, pocketzeroes sells at $65; so heehaww pays pocketzeroes $120 and gets $200 if the Spurs win.

Now assume heehaww knows that pocketzeroes is less enamored of the Spurs and figures he can save a little money by writing down $60. Well, that's part of the game and pocketzeroes can do the same. There's no reason to think that the outcome of the metagame is unfair.

Suppose someone misjudges, say Heehaww writes down $40 but pocketzeroes plays it straight at $55. Now pocketzeroes buys at $40, so heehaww has to pay him $15 and no money changes hands based on the sporting event. That sounds fair to me, and they two can try again if they want to bet.

you've assumed both players are being honest with their estimates when it not in their best financial interest

This question touches upon fair divisions, revealed preferences, private valuations, auction mechanisms, etc. In short, there is no easy answer.