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.
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.