Take the minimum. If you require that both parties are required to have the amount of money they are willing to bet, then taking the minimum and then running the algorithm is strategy proof and fair. The optimal strategy is to put in as much money as you can, then report honest preferences.
Yeah, but you might be able to find a better strategy. Perhaps you can get it so Alice and Bob both have a higher expected benefit. Perhaps you can make Alice only go down slightly, but Bob go up significantly.
I think I can prove that you cannot increase the expected benefit for both without knowing something about their probabilities ahead of time. You can sacrifice fairness, but I do not think Alice can go up much more than Bob goes down, and I do not want to.
I feel like if Bob has enough money, he ought to just be able to raise the stakes if he has enough money. In your example, if Alice is 50% sure, then she will never bet more than a quarter of her money, regardless of how certain and rich Bob is.
Even if I do have to let Bob go down a little, I still feel like it could be worth it. I don’t know before-hand if I’m going to be Alice or Bob, so it’s automatically fair. I want to maximize the sum of their expected profits.
Take the minimum. If you require that both parties are required to have the amount of money they are willing to bet, then taking the minimum and then running the algorithm is strategy proof and fair. The optimal strategy is to put in as much money as you can, then report honest preferences.
Yeah, but you might be able to find a better strategy. Perhaps you can get it so Alice and Bob both have a higher expected benefit. Perhaps you can make Alice only go down slightly, but Bob go up significantly.
I think I can prove that you cannot increase the expected benefit for both without knowing something about their probabilities ahead of time. You can sacrifice fairness, but I do not think Alice can go up much more than Bob goes down, and I do not want to.
I feel like if Bob has enough money, he ought to just be able to raise the stakes if he has enough money. In your example, if Alice is 50% sure, then she will never bet more than a quarter of her money, regardless of how certain and rich Bob is.
Even if I do have to let Bob go down a little, I still feel like it could be worth it. I don’t know before-hand if I’m going to be Alice or Bob, so it’s automatically fair. I want to maximize the sum of their expected profits.