I think your definition of “dominates” is a little too strict. In your “don’t pick the wrong integer” game, “don’t bet” isn’t dominated simply because it is possible to lose the bet, regardless of how good the odds are and how good the relative payoffs are. Min/max-ing strategies (do the thing that minimizes how bad things are in the worst case) aren’t dominated by strategies that are willing to tolerate risk, but they do leave free money on the table if they have less than perfect certainty that the table isn’t booby trapped.
I don’t know how to say “if you don’t maximize expected utility you won’t have the maximum expected utility” without turning it into a tautology, but a strategy that turns down every good bet (for a reasonable definition of “good”) simply because losing is possible seems to be doing something wrong.
I agree it might be too ambitious to look at all nondominated strategy.
I went for “nondominated” as a condition because it was, in my eyes, the best formal translation of the initial intuitive claim I was trying to test. Besides, that’s what is used in the complete class theorem.
There might be interesting variations of the conjecture with stricter requirements on the strategy. But I also think it would be very hard to give a non-tautological result that uses this notion of “no matter the odds”.
The very notion that there are odds to discuss is what we are trying to prove.
I think your definition of “dominates” is a little too strict. In your “don’t pick the wrong integer” game, “don’t bet” isn’t dominated simply because it is possible to lose the bet, regardless of how good the odds are and how good the relative payoffs are. Min/max-ing strategies (do the thing that minimizes how bad things are in the worst case) aren’t dominated by strategies that are willing to tolerate risk, but they do leave free money on the table if they have less than perfect certainty that the table isn’t booby trapped.
I don’t know how to say “if you don’t maximize expected utility you won’t have the maximum expected utility” without turning it into a tautology, but a strategy that turns down every good bet (for a reasonable definition of “good”) simply because losing is possible seems to be doing something wrong.
I agree it might be too ambitious to look at all nondominated strategy. I went for “nondominated” as a condition because it was, in my eyes, the best formal translation of the initial intuitive claim I was trying to test. Besides, that’s what is used in the complete class theorem.
There might be interesting variations of the conjecture with stricter requirements on the strategy. But I also think it would be very hard to give a non-tautological result that uses this notion of “no matter the odds”. The very notion that there are odds to discuss is what we are trying to prove.