How does this actually work mechanically? You don’t actually make an even-money bet about a probability, you make a weighted bet about an outcome, right? There’s no such thing as a $2 bet that 66% is correct. There’s a $3-against-$2 bet that the thing happens. (that is, the person who says “more likely than 66%” wins $2 if it happens, and loses $3 if it doesn’t happen. The person saying “less likely than 66%” is the exact opposite.)
I’m guessing the “max wager” is what each wants their maximum loss to be, and if you distribute that evenly across the range, it ends up exactly equivalent of a single bet at the midway-point between the stated beliefs.
And that means, assuming you know the direction of disagreement (if you’re A and give something a 60% probability, and you know B thinks it’s more probable), you’re incented to OVERSTATE your difference from your opponent (A is going to win if the event doesn’t happen, so wants to get better odds on more bets, so claims a 0% probability, and gets a way better distribution of actual wagers).
By $2 bet at 66% odds, I mean that the Yes position costs $2*66% and the No position costs $2*34%.
You’re right that “max wager” is meant to be maximum loss. I think you’re picking up on the fact that I made a mistake in calculating loss for each player. I was calculating the potential loss for “$2 bet at 66%” as 2 dollars for both players, but that’s obviously wrong, and no reason afaik that the players should have the same maximum loss. Thanks for the observation.
I don’t understand your observation about the incentive to overstate.
Let’s say A gives event E 60% odds and B gives E 90% odds. For a bet at even odds:
EV_A(YES) = .4 * -.5 + .6 * .5 = .1
EV_A(NO) = .6 * -.5 + .4 * .5 = -.1
From A’s perspective the no position on the 50⁄50 bet (or any bet where the no position costs more than 40 cents on the dollar) is negative EV. So if A submitted 0% odds, they’d be forcing themselves to take a lot of negative EV bets.
In fact, I misunderstood your proposal—I am incorrect in saying that lying helps. You’d get more bets on the same side as your “good” bets, but at unfavorable payouts so you’d lose too much when you lose.
How does this actually work mechanically? You don’t actually make an even-money bet about a probability, you make a weighted bet about an outcome, right? There’s no such thing as a $2 bet that 66% is correct. There’s a $3-against-$2 bet that the thing happens. (that is, the person who says “more likely than 66%” wins $2 if it happens, and loses $3 if it doesn’t happen. The person saying “less likely than 66%” is the exact opposite.)
I’m guessing the “max wager” is what each wants their maximum loss to be, and if you distribute that evenly across the range, it ends up exactly equivalent of a single bet at the midway-point between the stated beliefs.
And that means, assuming you know the direction of disagreement (if you’re A and give something a 60% probability, and you know B thinks it’s more probable), you’re incented to OVERSTATE your difference from your opponent (A is going to win if the event doesn’t happen, so wants to get better odds on more bets, so claims a 0% probability, and gets a way better distribution of actual wagers).
By $2 bet at 66% odds, I mean that the Yes position costs $2*66% and the No position costs $2*34%.
You’re right that “max wager” is meant to be maximum loss. I think you’re picking up on the fact that I made a mistake in calculating loss for each player. I was calculating the potential loss for “$2 bet at 66%” as 2 dollars for both players, but that’s obviously wrong, and no reason afaik that the players should have the same maximum loss. Thanks for the observation.
I don’t understand your observation about the incentive to overstate.
Let’s say A gives event E 60% odds and B gives E 90% odds. For a bet at even odds:
EV_A(YES) = .4 * -.5 + .6 * .5 = .1
EV_A(NO) = .6 * -.5 + .4 * .5 = -.1
From A’s perspective the no position on the 50⁄50 bet (or any bet where the no position costs more than 40 cents on the dollar) is negative EV. So if A submitted 0% odds, they’d be forcing themselves to take a lot of negative EV bets.
In fact, I misunderstood your proposal—I am incorrect in saying that lying helps. You’d get more bets on the same side as your “good” bets, but at unfavorable payouts so you’d lose too much when you lose.