I claim the bet is fair if both players expect to make the same profit on average.
I like this idea. As you say, it’s not the only way to define it but it does seem like a very reasonable way. The two players have come upon a situation which seems profitable to both of them and they simply agree to “split the profit”.
In order to give the players incentives to be honest, the algorithm seems to “use up” some of the total potential profit. For example, in the OP, the players are instructed to bet $2.72 and $13.28 when each was actually willing to bet up to $25. I think this also means that this method of coming up with bet amounts is not strategy proof if players are able to lie about their maximum bet amounts.
A further problem with this mechanism: since the potential profit isn’t actually used up before the bet is resolved (i.e., each player still has the money they didn’t put on the table), the players can always make another bet with each other using their remaining money. But this knowledge changes the players’ incentives when they’re choosing the probabilities to output.
I’m guessing that in order to design a mechanism that is actually strategy proof, we’d need to make the cost more explicit, for example by having a third party collect some sort of fee based on the expressed probabilities.
I thought the rapidity with which the third person was described to have taken the funds in escrow and declared, “You wait” indicated that she (and you) had seen the problem pointed out by Wei_Dai and moved to block it. (Silly me?)
I wasn’t trying to indicate anything with the third person. Sorry for the confusion. I just thought that it would be hard to end the story without describing where the money goes, since the prediction I was referencing was not going to be answered for a year.
Oh, I see. You probably already understood that, but I’ll write it up for anyone else who didn’t initially grok the process (like me).
Intuitively, the original algorithm incentivises people to post their true estimates by scaling up the opponents investment with your given odds, so that it doesnt pay for you to artificially lower your estimate. The possible wins will be much lower; disproportionately to your investment, if you underestimate your odds. Conversely, the possible losses will not be covered by increased wins if you overestimate your chances.
It does not work if you scale the bets. If A believes he wins the bet half the time, and B believes it will be 90%, with the assumption of B being honest and both players setting the limit at 1 (for ease of calculation):
With A declaring 50%, the investment ratios would be:
A: 0.24
B: 0.56
With the original amount calculation that gives the expected value of
E(A) = (0.5 * 0.56 - 0.5 * 24) = 0.16
Whereas with scaled up bets A puts in 0.43 while B gives 1:
E'(A) = (0.5 * 1 - 0.5 * 0.43) = 0.285
With A declaring 20%, the numbers are:
A: 0.03
B: 0.17
E(A) = 0.5 * (0.17 - 0.03) = 0.07
While with scaled bets (B = 1, A = 0.18)
E'(A) = 0.5 * 1 - 0.5 * 0.18 = 0.41
Note how E(A) goes down if A lies, but E’(A) went way up.
I like this idea. As you say, it’s not the only way to define it but it does seem like a very reasonable way. The two players have come upon a situation which seems profitable to both of them and they simply agree to “split the profit”.
In order to give the players incentives to be honest, the algorithm seems to “use up” some of the total potential profit. For example, in the OP, the players are instructed to bet $2.72 and $13.28 when each was actually willing to bet up to $25. I think this also means that this method of coming up with bet amounts is not strategy proof if players are able to lie about their maximum bet amounts.
Yes. That is correct. It is only strategy proof if the max bet amount is fixed.
As for the used up potential, that is common in mechanism design. It is very common to have to choose between Pareto optimality and strategy proof.
A further problem with this mechanism: since the potential profit isn’t actually used up before the bet is resolved (i.e., each player still has the money they didn’t put on the table), the players can always make another bet with each other using their remaining money. But this knowledge changes the players’ incentives when they’re choosing the probabilities to output.
I’m guessing that in order to design a mechanism that is actually strategy proof, we’d need to make the cost more explicit, for example by having a third party collect some sort of fee based on the expressed probabilities.
From the OP:
The players had money left over. They only put in 16 of 50 dollars.
I thought the rapidity with which the third person was described to have taken the funds in escrow and declared, “You wait” indicated that she (and you) had seen the problem pointed out by Wei_Dai and moved to block it. (Silly me?)
Here I thought it was just a joke about someone stealing the stakes.
I wasn’t trying to indicate anything with the third person. Sorry for the confusion. I just thought that it would be hard to end the story without describing where the money goes, since the prediction I was referencing was not going to be answered for a year.
What’s wrong with just using this algorithm to establish ratios between bets, then scaling up to meet whichever limit is hit first?
In your example, it’d be scaled up to 5.12 against 25.
That is not strategy proof
Oh, I see. You probably already understood that, but I’ll write it up for anyone else who didn’t initially grok the process (like me).
Intuitively, the original algorithm incentivises people to post their true estimates by scaling up the opponents investment with your given odds, so that it doesnt pay for you to artificially lower your estimate. The possible wins will be much lower; disproportionately to your investment, if you underestimate your odds. Conversely, the possible losses will not be covered by increased wins if you overestimate your chances.
It does not work if you scale the bets. If A believes he wins the bet half the time, and B believes it will be 90%, with the assumption of B being honest and both players setting the limit at 1 (for ease of calculation):
With A declaring 50%, the investment ratios would be:
With the original amount calculation that gives the expected value of
Whereas with scaled up bets A puts in 0.43 while B gives 1:
With A declaring 20%, the numbers are:
While with scaled bets (B = 1, A = 0.18)
Note how E(A) goes down if A lies, but E’(A) went way up.