I see, I misparsed the terms of the argument, I thought it was doubling my current utilons, you’re positing I have a 90% chance of doubling my currently expected utility over my entire life.
The reason I bring up the terms in my utility function, is that they reference concrete objects, people, time passing, and so on. So, measuring expected utility, for me, involves projecting the course of the world, and my place in it.
So, assuming I follow the suggested course of action, and keep drawing cards until I die, to fulfill the terms, Omega must either give me all the utilons before I die, or somehow compress the things I value into something that can be achieved in between drawing cards as fast as I can. This either involves massive changes to reality, which I can verify instantly, or some sort of orthogonal life I get to lead while simultaneously drawing cards, so I guess that’s fine.
Otherwise, given the certainty that I will die essentially immediately, I certainly don’t recognize that I’m getting a 90% chance of doubled expected utility, as my expectations certainly include whether or not I will draw a card.
I don’t think “current utilons” makes that much sense. Utilons should be for a utility function, which is equivalent to a decision function, and the purpose of decisions is probably to influence the future. So utility has to be about the whole future course of the world. “Currently expected utilons” means what you expect to happen, averaged over your uncertainty and actual randomness, and this is what the dilemma should be about.
“Current hedons” certainly does make sense, at least because hedons haven’t been specified as well.
Like Douglas_Knight, I don’t think current utilons are a useful unit.
Suppose your utility function behaves as you describe. If you play once (and win, with 90% probability), Omega will modify the universe in a way that all the concrete things you derive utility from will bring you twice as much utility, over the course of the infinite future. You’ll live out your life with twice as much of all the things you value. So it makes sense to play this once, by the terms of your utility function.
You don’t know, when you play your first game, whether or not you’ll ever play again; your future includes both options. You can decide, for yourself, that you’ll play once but never again. It’s a free decision both now and later.
And now a second has passed and Omega is offering a second game. You remember your decision. But what place do decisions have in a utility function? You’re free to choose to play again if you wish, and the logic for playing is the same as the first time around...
Now, you could bind yourself to your promise (after the first game). Maybe you have a way to hardwire your own decision procedure to force something like this. But how do you decide (in advance) after how many games to stop? Why one and not, say, ten?
OTOH, if you decide not to play at all—would you really forgo a one-time 90% chance of doubling your lifelong future utility? How about a 99.999% chance? The probability of death in any one round of the game can be made as small as you like, as long as it’s finite and fixed for all future rounds. Is there no probability at which you’d take the risk for one round?
Why on earth wouldn’t I consider whether or not I would play again? Am I barred from doing so?
If I know that the card game will continue to be available, and that Omega can truly double my expected utility every draw, either it’s a relatively insignificant increase of expected utility over the next few minutes it takes me to die, in which case it’s a foolish bet, compared to my expected utility over the decades I have left, conservatively, or Omega can somehow change the whole world in the radical fashion needed for my expected utility over the next few minutes it takes me to die to dwarf my expected utility right now.
This paradox seems to depend on the idea that the card game is somehow excepted from the 90% likely doubling of expected utility. As I mentioned before, my expected utility certainly includes the decisions I’m likely to make, and it’s easy to see that continuing to draw cards will result in my death. So, it depends on what you mean. If it’s just doubling expected utility over my expected life IF I don’t die in the card game, then it’s a foolish decision to draw the first or any number of cards. If it’s doubling expected utility in all cases, then I draw cards until I die, happily forcing Omega to make verifiable changes to the universe and myself.
Now, there are terms at which I would take the one round, IF you don’t die in the card game version of the gamble, but it would probably depend on how it’s implemented. I don’t have a way of accessing my utility function directly, and my ability to appreciate maximizing it is indirect at best. So I would be very concerned about the way Omega plans to double my expected utility, and how I’m meant to experience it.
In practice, of course, any possible doubt that it’s not Omega giving you this gamble far outweighs any possibility of such lofty returns, but the thought experiment has some interesting complexities.
You’re free to choose to play again if you wish, and the logic for playing is the same as the first time around
This, again, depends on what you mean by “utility”. Here’s a way of framing the problem such that the logic can change.
Assume that we have some function V(x) that maps world histories into (non-negative*) real-valued “valutilons”, and that, with no intervention from Omega, the world history that will play out is valued at V(status quo) = q.
Then Omega turns up and offers you the card deal, with a deck as described above: 90% stars, 10% skulls. Stars give you double V(star)=2c, where c is the value of whatever history is currently slated to play (so c=q when the deal is first offered, but could be higher than that if you’ve played and won before). Skulls give you death: V(skull)=d, and d < q.
If our choices obey the vNM axioms, there will be some function f(x), such that our choices correspond to maximising E[f(x)]. It seems reasonable to assume that f(x) must be (weakly) increasing in V(x). A few questions present themselves:
Is there a function, f(x), such that, for some values of q and d, we should take cards every time this bet is offered?
Yes. f(x)=V(x) gives this result for all d<q.
Is there a function, f(x), such that, for some values of q and d, we should never take the bet?
Yes. Set d=0, q=1000, and f(x) = ln(V(x)+1). The offer gives vNM utility of 0.9ln(2001)~6.8, which is less than ln(1001)~6.9.
Is there a function, f(x), such that, for some values of q and d, we should take cards for some finite number of offers, and then stop?
Yes. Set d=0, q=1, and f(x) = ln(V(x)+1). The first time you get the offer, it’s vNM utility is 0.9ln(3)~1 which is greater than ln(2)~0.7. But at the 10th time you play (assuming you’re still alive), c=512, and the vNM utility of the offer is now 0.9ln(1025)~6.239, which is less than ln(513)~6.240. So you play up until the 10th offer, then stop.
* This is just to ensure that doubling your valutilons cannot make you worse off, as would happen if they were negative. It should be possible to reframe the problem to avoid this, but let’s stick with this for now.
I see, I misparsed the terms of the argument, I thought it was doubling my current utilons, you’re positing I have a 90% chance of doubling my currently expected utility over my entire life.
The reason I bring up the terms in my utility function, is that they reference concrete objects, people, time passing, and so on. So, measuring expected utility, for me, involves projecting the course of the world, and my place in it.
So, assuming I follow the suggested course of action, and keep drawing cards until I die, to fulfill the terms, Omega must either give me all the utilons before I die, or somehow compress the things I value into something that can be achieved in between drawing cards as fast as I can. This either involves massive changes to reality, which I can verify instantly, or some sort of orthogonal life I get to lead while simultaneously drawing cards, so I guess that’s fine.
Otherwise, given the certainty that I will die essentially immediately, I certainly don’t recognize that I’m getting a 90% chance of doubled expected utility, as my expectations certainly include whether or not I will draw a card.
I don’t think “current utilons” makes that much sense. Utilons should be for a utility function, which is equivalent to a decision function, and the purpose of decisions is probably to influence the future. So utility has to be about the whole future course of the world. “Currently expected utilons” means what you expect to happen, averaged over your uncertainty and actual randomness, and this is what the dilemma should be about.
“Current hedons” certainly does make sense, at least because hedons haven’t been specified as well.
Like Douglas_Knight, I don’t think current utilons are a useful unit.
Suppose your utility function behaves as you describe. If you play once (and win, with 90% probability), Omega will modify the universe in a way that all the concrete things you derive utility from will bring you twice as much utility, over the course of the infinite future. You’ll live out your life with twice as much of all the things you value. So it makes sense to play this once, by the terms of your utility function.
You don’t know, when you play your first game, whether or not you’ll ever play again; your future includes both options. You can decide, for yourself, that you’ll play once but never again. It’s a free decision both now and later.
And now a second has passed and Omega is offering a second game. You remember your decision. But what place do decisions have in a utility function? You’re free to choose to play again if you wish, and the logic for playing is the same as the first time around...
Now, you could bind yourself to your promise (after the first game). Maybe you have a way to hardwire your own decision procedure to force something like this. But how do you decide (in advance) after how many games to stop? Why one and not, say, ten?
OTOH, if you decide not to play at all—would you really forgo a one-time 90% chance of doubling your lifelong future utility? How about a 99.999% chance? The probability of death in any one round of the game can be made as small as you like, as long as it’s finite and fixed for all future rounds. Is there no probability at which you’d take the risk for one round?
Why on earth wouldn’t I consider whether or not I would play again? Am I barred from doing so?
If I know that the card game will continue to be available, and that Omega can truly double my expected utility every draw, either it’s a relatively insignificant increase of expected utility over the next few minutes it takes me to die, in which case it’s a foolish bet, compared to my expected utility over the decades I have left, conservatively, or Omega can somehow change the whole world in the radical fashion needed for my expected utility over the next few minutes it takes me to die to dwarf my expected utility right now.
This paradox seems to depend on the idea that the card game is somehow excepted from the 90% likely doubling of expected utility. As I mentioned before, my expected utility certainly includes the decisions I’m likely to make, and it’s easy to see that continuing to draw cards will result in my death. So, it depends on what you mean. If it’s just doubling expected utility over my expected life IF I don’t die in the card game, then it’s a foolish decision to draw the first or any number of cards. If it’s doubling expected utility in all cases, then I draw cards until I die, happily forcing Omega to make verifiable changes to the universe and myself.
Now, there are terms at which I would take the one round, IF you don’t die in the card game version of the gamble, but it would probably depend on how it’s implemented. I don’t have a way of accessing my utility function directly, and my ability to appreciate maximizing it is indirect at best. So I would be very concerned about the way Omega plans to double my expected utility, and how I’m meant to experience it.
In practice, of course, any possible doubt that it’s not Omega giving you this gamble far outweighs any possibility of such lofty returns, but the thought experiment has some interesting complexities.
This, again, depends on what you mean by “utility”. Here’s a way of framing the problem such that the logic can change.
Assume that we have some function V(x) that maps world histories into (non-negative*) real-valued “valutilons”, and that, with no intervention from Omega, the world history that will play out is valued at V(status quo) = q.
Then Omega turns up and offers you the card deal, with a deck as described above: 90% stars, 10% skulls. Stars give you double V(star)=2c, where c is the value of whatever history is currently slated to play (so c=q when the deal is first offered, but could be higher than that if you’ve played and won before). Skulls give you death: V(skull)=d, and d < q.
If our choices obey the vNM axioms, there will be some function f(x), such that our choices correspond to maximising E[f(x)]. It seems reasonable to assume that f(x) must be (weakly) increasing in V(x). A few questions present themselves:
Is there a function, f(x), such that, for some values of q and d, we should take cards every time this bet is offered?
Yes. f(x)=V(x) gives this result for all d<q.
Is there a function, f(x), such that, for some values of q and d, we should never take the bet?
Yes. Set d=0, q=1000, and f(x) = ln(V(x)+1). The offer gives vNM utility of 0.9ln(2001)~6.8, which is less than ln(1001)~6.9.
Is there a function, f(x), such that, for some values of q and d, we should take cards for some finite number of offers, and then stop?
Yes. Set d=0, q=1, and f(x) = ln(V(x)+1). The first time you get the offer, it’s vNM utility is 0.9ln(3)~1 which is greater than ln(2)~0.7. But at the 10th time you play (assuming you’re still alive), c=512, and the vNM utility of the offer is now 0.9ln(1025)~6.239, which is less than ln(513)~6.240. So you play up until the 10th offer, then stop.
* This is just to ensure that doubling your valutilons cannot make you worse off, as would happen if they were negative. It should be possible to reframe the problem to avoid this, but let’s stick with this for now.