Neel, just because a hack solves a problem doesn’t mean that that particular hack is The Solution and The Only Solution.
The problem here is that McGee is summing an infinite series and getting a value of −1 when the sums of the finite series approach positive infinity.
If you’re willing to concede this in the first place—and a lot of mathematicians will simply refuse to sum the series in the first place—then the obvious approach is to say that infinite decisions don’t have to approach the limit of finite decisions.
If you introduce an artificial bound on the utility function, as a hack, then you are simply implementing the wrong morality: A morality that will—at some point—trade off a 50% probability of having a decillion units of fun against a 49.99% probability of having 3^^^3 units of fun, which seems simply stupid to me.
All I need to do is say, “In infinite decisions whose payoffs are not the limit of the payoffs of a series of finite decisions, I’m going to keep my utility function (that is, make decisions according to my actual morality instead of some other morality) but I’m not necessarily going to make infinite decisions that look like the limit of my decisions in the finite cases.”
McGee’s dilemma does not have an “optimal” solution. No matter how many of McGee’s bets you take, you can always take one more bet and expect an even higher payoff. It’s like asking for the largest integer. There isn’t one, and there isn’t an optimal plan in McGee’s dilemma.
On day 1 I give you a dollar, but offer to trade it for $2 on day 2. On day 2 I offer to trade the $2 for $3 on day 3. If I let you continue this situation forever, then there is no maximal plan, and the limit of the plans in any finite case never gets the money. So you can’t maximize; just pick a number.
Don’t tell me the solution is to bound my utility function. That doesn’t even solve anything. If your utility function has an upper bound at $1 then I can offer to trade you $.50 for $.75 the next day, and $.75 for $.875 the day after, and so on, and you’ve got exactly the same problem: the limit of the behavior for the best finite plans, does not yield good behavior in the infinite plan.
Neel, just because a hack solves a problem doesn’t mean that that particular hack is The Solution and The Only Solution.
The problem here is that McGee is summing an infinite series and getting a value of −1 when the sums of the finite series approach positive infinity.
If you’re willing to concede this in the first place—and a lot of mathematicians will simply refuse to sum the series in the first place—then the obvious approach is to say that infinite decisions don’t have to approach the limit of finite decisions.
If you introduce an artificial bound on the utility function, as a hack, then you are simply implementing the wrong morality: A morality that will—at some point—trade off a 50% probability of having a decillion units of fun against a 49.99% probability of having 3^^^3 units of fun, which seems simply stupid to me.
All I need to do is say, “In infinite decisions whose payoffs are not the limit of the payoffs of a series of finite decisions, I’m going to keep my utility function (that is, make decisions according to my actual morality instead of some other morality) but I’m not necessarily going to make infinite decisions that look like the limit of my decisions in the finite cases.”
McGee’s dilemma does not have an “optimal” solution. No matter how many of McGee’s bets you take, you can always take one more bet and expect an even higher payoff. It’s like asking for the largest integer. There isn’t one, and there isn’t an optimal plan in McGee’s dilemma.
On day 1 I give you a dollar, but offer to trade it for $2 on day 2. On day 2 I offer to trade the $2 for $3 on day 3. If I let you continue this situation forever, then there is no maximal plan, and the limit of the plans in any finite case never gets the money. So you can’t maximize; just pick a number.
Don’t tell me the solution is to bound my utility function. That doesn’t even solve anything. If your utility function has an upper bound at $1 then I can offer to trade you $.50 for $.75 the next day, and $.75 for $.875 the day after, and so on, and you’ve got exactly the same problem: the limit of the behavior for the best finite plans, does not yield good behavior in the infinite plan.