Ok, I don’t like gnomes making current decisions based on their future values.
For the selfish case, we can easily get around this by defining the gnome’s utility function to be the amount of $ in the cell. If we stipulate that this can only change through humans buying lottery tickets (and winning lotteries) and that humans cannot leave the cells, the gnome’s utility function coincides with the human’s. Similarly, we can define the gnome’s utility function to be the amount of $ in all cells (the average amount of $ in those cells inhabited by humans) in the total (average) utilitarian case.
This seems to be a much neater way of using the gnome heuristic than the one I used in the original post, since the gnome’s utility function is now unchanging and unconditional. The only issue seems to be that before the humans are created, the gnome’s utility function is undefined in the average utilitarian case (“0/0”). However, this is more a problem of average utilitarianism than of the heuristic per se. We can get around it by defining the utility to be 0 if there aren’t any humans around yet.
The incubator always creates two people, but in the heads world, the second person can never gain (nor lose) anything, no matter what they agree to: any deal is nullified. This seems a gnome setup without the gnomes. If everyone is an average utilitarian, then they will behave exactly as the total utilitarians would (since population is equal anyway) and buy the ticket for x<$2/3. So this setup has changed the outcome for average utilitarians. If its the same as the gnome setup (and it seems to be) then the gnome setup is interfering with the decisions in cases we know about. The fact that the number of gnomes is fixed is the likely cause.
I don’t follow. As I should have written in the original post, total/average utilitarianism includes of course the wellbeing and population of humans only, not of gnomes. Otherwise, it’s trivial that the presence of gnomes affects the conclusions. That the presence of an additional human affects the conclusion for average utilitarians is not surprising, since in contrast to the presence of gnomes, an additional human changes the relevant population.
Incidentally, one reason for the selfish=average utilitarian is that I sometimes model selfish as the average between total utilitarian incubator and anti-incubator (where the two copies hate each other in the tail world). 50%-50% on total utilitarian vs hatred seems to be a good model of selfishness, and gives the x<$1/2 answer.
Hm, so basically one could argue as follows against my conclusion that both selfish and total utilitarians pay up to $2/3: A hater wouldn’t pay anything for a ticket that pays $1 in the tails world. Since selfishness is a mixture of total utilitarianism and hating, a selfish person certainly cannot have the same maximal price as a total utilitarian.
However, I feel like “caring about the other person in the tail world in a total utilitarian sense” and “hating the other person in the tail world” are not exactly mirror images of each other. The difference is that total utilitarianism is lexicality-independent, while “hating the other person” isn’t. My claim is: However you formalize “hating the person in the other room in the tail world” and “being a total utilitarian”, the statements “a total utilitarian pays up to $2/3″ and “selfishness is a mixture of total utilitarianism and hating” and “a hater would not pay more than $0 for the ticket” are never simultaneously true.
Imagine that the human formally writes down their utility function in order to apply the “if there were a gnome in my room, what maximal prize to pay would it advise me after asking itself what advice it would have precommited to?” heuristic. We introduce the variables ‘vh’ and ‘vo’ for “$-value in this/the other room”. These are 0 if there’s no human, -x after buying a ticket after head, and 1-x after buying a ticket after tail. We also introduce a variable ‘t’ which is 1 after tail and 0 after head.
We can then write down the following utility functions with their respective expectation values (from the point of view of the gnome before the coin flip):
egoist: vh ⇒ 1⁄4 * (-x+0+(1-x)+(1-x))
total ut.: vh + t vo ⇒ 1⁄4 (-x+0+2 (1-x)+2 (1-x))
hate: vh—t vo ⇒ 1⁄4 (-x+0+0+0)
Here, we see that egoism is indeed a mixture of total utilitarianism and hating, the egoist pays up to 2⁄3, and the hater pays nothing. However, according to this definition of total utilitarianism, a t.u. should pay up to 4⁄5. Its utility function is lexicality-dependent (the variable t enters only the utility coming from the other person), in contrast to true total utilitarianism.
In order to write down a lexicality-independent utility function, we introduce new variables ‘nh’ and ‘no’, the number of people here and in the other room (0 or 1). Then, we could make the following definitions:
egoist: nh vh total ut.: nh vh + no vo hate: nh vh—no * vo
(The ‘nh’ and ‘no’ factors are actually redundant, since ‘vh’ is defined to be zero if ‘nh’ is.)
With these definitions, both an egoist and a t.u. pay up to 2⁄3 and egoism is a mixture of t.u. and hating. However, the expected utility of a hater is now 0 independent of x, such that there is no longer a contradiction. The reason is that we now count the winnings of the single head-human one time positively (if ze is in our room) and one time negatively (if ze is in the other room). This isn’t what we meant by hating, so we could modify the utility function of the hater as follows:
hate: nh (vh—no vo)
This reproduces again what we mean by hating (it is equivalent to the old definition ‘vh—t * vo’), but now egoism is no longer a combination of hating and t.u..
In conclusion, it doesn’t seem to be possible to derive a contradiction between “a hater wouldn’t pay anything for a lottery ticket” and “both egoists and total utilitarians would pay up to $2/3″.
The broader question is “does bringing in gnomes in this way leave the initial situation invariant”? And I don’t think it does. The gnomes follow their own anthropic setup (though not their own preferences), and their advice seems to reflect this fact (consider what happens when the heads world has 1, 2 or 50 gnomes, while the tails world has 2).
I also don’t see your indexical objection. The sleeping beauty could perfectly have an indexical version of total utilitarianism (“I value my personal utility, plus that of the sleeping beauty in the other room, if they exist”). If you want to proceed further, you seem to have to argue that indexical total utilitarianism gives different decisions than standard total utilitarianism.
This is odd, as it seems a total utilitarian would not object to having their utility replaced with the indexical version, and vice-versa.
The broader question is “does bringing in gnomes in this way leave the initial situation invariant”? And I don’t think it does. The gnomes follow their own anthropic setup (though not their own preferences), and their advice seems to reflect this fact (consider what happens when the heads world has 1, 2 or 50 gnomes, while the tails world has 2).
As I wrote (after your comment) here, I think it is prima facie very plausible for a selfish agent to follow the gnome’s advice if a) conditional on the agent existing, the gnome’s utility function agrees with the agent’s and b) conditional on the agent not existing, the gnome’s utility function is a constant. (I didn’t have condition b) explicitly in mind, but your example showed that it’s necessary.) Having the number of gnomes depend upon the coin flip invalidates their purpose. The very point of the gnomes is that from their perspective, the problem is not “anthropic”, but a decision problem that can be solved using UDT.
I also don’t see your indexical objection. The sleeping beauty could perfectly have an indexical version of total utilitarianism (“I value my personal utility, plus that of the sleeping beauty in the other room, if they exist”). If you want to proceed further, you seem to have to argue that indexical total utilitarianism gives different decisions than standard total utilitarianism.
That’s what I tried in the parent comment. To be clear, I did not mean “indexical total utilitarianism” to be a meaningful concept, but rather a wrong way of thinking, a trap one can fall into. Very roughly, it corresponds to thinking of total utilitarianism as “I care for myself plus any other people that might exist” instead of “I care for all people that exist”. What’s the difference, you ask? A minimal non-anthropic example that illustrates the difference would be very much like the incubator, but without people being created. Imagine 1000 total utilitarians with identical decision algorithms waiting in separate rooms. After the coin flip, either one or two of them are offered to buy a ticket that pays $1 after heads. When being asked, the agents can correctly perform a non-anthropic Bayesian update to conclude that the probability of tails is 2⁄3. An indexical total utilitarian reasons: “If the coin has shown tails, another agent will pay the same amount $x that I pay and win the same $1, while if the coin has shown heads, I’m the only one who pays $x. The expected utility of paying $x is thus 1⁄3 (-x) + 2⁄3 2 * (1-x).” This leads to the incorrect conclusion that one should pay up to $4/5. The correct (UDT-) way to think about the problem is that after tails, one’s decision algorithm is called twice. There’s only one factor of 2, not two of them. This is all very similar to this post.
To put this again into context: You argued that selfishness is a 50⁄50 mixture of hating the other person, if another person exists, and total utilitarianism. My reply was that this is only true if one understands total utilitarianism in the incorrect, indexical way. I formalized this as follows: Let the utility function of a hater be vh—h vo (here, vh is the agent’s own utility, vo the other person’s utility, and h is 1 if the other person exists and 0 otherwise). Selfishness would be a 50⁄50 mixture of hating and total utilitarianism if the utility function of a total utilitarian were vh + h vo. However, this is exactly the wrong way of formalizing total utilitarianism. It leads, again, to the conclusion that a total utilitarian should pay up to $4/5.
A minimal non-anthropic example that illustrates the difference
The decision you describe in not stable under pre-commitments. Ahead of time, all agents would pre-commit to the $2/3. Yet they seem to change their mind when presented with the decision. You seem to be double counting, using the Bayesian updating once and the fact that their own decision is responsible for the other agent’s decision as well.
In the terminology of paper http://www.fhi.ox.ac.uk/anthropics-why-probability-isnt-enough.pdf , your agents are altruists using linked decisions with total responsibility and no precommitments, which is a foolish thing to do. If they were altruists using linked decisions with divided responsibility (or if they used precommitments), everything would be fine (I don’t like or use that old terminology—UDT does it better—but it seems relevant here).
But that’s detracting from the main point: still don’t see any difference between indexical and non-indexical total utilitarianism. I don’t see why a non-indexical total utilitarian can’t follow the wrong reasoning you used in your example just as well as an indexical one, if either of them can—and similarly for the right reasoning.
The decision you describe in not stable under pre-commitments. Ahead of time, all agents would pre-commit to the $2/3. Yet they seem to change their mind when presented with the decision. You seem to be double counting, using the Bayesian updating once and the fact that their own decision is responsible for the other agent’s decision as well.
Yes, this is exactly the point I was trying to make—I was pointing out a fallacy. I never intended “lexicality-dependent utilitarianism” to be a meaningful concept, it’s only a name for thinking in this fallacious way.
Thanks for your reply.
For the selfish case, we can easily get around this by defining the gnome’s utility function to be the amount of $ in the cell. If we stipulate that this can only change through humans buying lottery tickets (and winning lotteries) and that humans cannot leave the cells, the gnome’s utility function coincides with the human’s. Similarly, we can define the gnome’s utility function to be the amount of $ in all cells (the average amount of $ in those cells inhabited by humans) in the total (average) utilitarian case.
This seems to be a much neater way of using the gnome heuristic than the one I used in the original post, since the gnome’s utility function is now unchanging and unconditional. The only issue seems to be that before the humans are created, the gnome’s utility function is undefined in the average utilitarian case (“0/0”). However, this is more a problem of average utilitarianism than of the heuristic per se. We can get around it by defining the utility to be 0 if there aren’t any humans around yet.
I don’t follow. As I should have written in the original post, total/average utilitarianism includes of course the wellbeing and population of humans only, not of gnomes. Otherwise, it’s trivial that the presence of gnomes affects the conclusions. That the presence of an additional human affects the conclusion for average utilitarians is not surprising, since in contrast to the presence of gnomes, an additional human changes the relevant population.
Hm, so basically one could argue as follows against my conclusion that both selfish and total utilitarians pay up to $2/3: A hater wouldn’t pay anything for a ticket that pays $1 in the tails world. Since selfishness is a mixture of total utilitarianism and hating, a selfish person certainly cannot have the same maximal price as a total utilitarian.
However, I feel like “caring about the other person in the tail world in a total utilitarian sense” and “hating the other person in the tail world” are not exactly mirror images of each other. The difference is that total utilitarianism is lexicality-independent, while “hating the other person” isn’t. My claim is: However you formalize “hating the person in the other room in the tail world” and “being a total utilitarian”, the statements “a total utilitarian pays up to $2/3″ and “selfishness is a mixture of total utilitarianism and hating” and “a hater would not pay more than $0 for the ticket” are never simultaneously true.
Imagine that the human formally writes down their utility function in order to apply the “if there were a gnome in my room, what maximal prize to pay would it advise me after asking itself what advice it would have precommited to?” heuristic. We introduce the variables ‘vh’ and ‘vo’ for “$-value in this/the other room”. These are 0 if there’s no human, -x after buying a ticket after head, and 1-x after buying a ticket after tail. We also introduce a variable ‘t’ which is 1 after tail and 0 after head.
We can then write down the following utility functions with their respective expectation values (from the point of view of the gnome before the coin flip):
egoist: vh ⇒ 1⁄4 * (-x+0+(1-x)+(1-x))
total ut.: vh + t vo ⇒ 1⁄4 (-x+0+2 (1-x)+2 (1-x))
hate: vh—t vo ⇒ 1⁄4 (-x+0+0+0)
Here, we see that egoism is indeed a mixture of total utilitarianism and hating, the egoist pays up to 2⁄3, and the hater pays nothing. However, according to this definition of total utilitarianism, a t.u. should pay up to 4⁄5. Its utility function is lexicality-dependent (the variable t enters only the utility coming from the other person), in contrast to true total utilitarianism.
In order to write down a lexicality-independent utility function, we introduce new variables ‘nh’ and ‘no’, the number of people here and in the other room (0 or 1). Then, we could make the following definitions:
egoist: nh vh
total ut.: nh vh + no vo
hate: nh vh—no * vo
(The ‘nh’ and ‘no’ factors are actually redundant, since ‘vh’ is defined to be zero if ‘nh’ is.)
With these definitions, both an egoist and a t.u. pay up to 2⁄3 and egoism is a mixture of t.u. and hating. However, the expected utility of a hater is now 0 independent of x, such that there is no longer a contradiction. The reason is that we now count the winnings of the single head-human one time positively (if ze is in our room) and one time negatively (if ze is in the other room). This isn’t what we meant by hating, so we could modify the utility function of the hater as follows:
hate: nh (vh—no vo)
This reproduces again what we mean by hating (it is equivalent to the old definition ‘vh—t * vo’), but now egoism is no longer a combination of hating and t.u..
In conclusion, it doesn’t seem to be possible to derive a contradiction between “a hater wouldn’t pay anything for a lottery ticket” and “both egoists and total utilitarians would pay up to $2/3″.
The broader question is “does bringing in gnomes in this way leave the initial situation invariant”? And I don’t think it does. The gnomes follow their own anthropic setup (though not their own preferences), and their advice seems to reflect this fact (consider what happens when the heads world has 1, 2 or 50 gnomes, while the tails world has 2).
I also don’t see your indexical objection. The sleeping beauty could perfectly have an indexical version of total utilitarianism (“I value my personal utility, plus that of the sleeping beauty in the other room, if they exist”). If you want to proceed further, you seem to have to argue that indexical total utilitarianism gives different decisions than standard total utilitarianism.
This is odd, as it seems a total utilitarian would not object to having their utility replaced with the indexical version, and vice-versa.
As I wrote (after your comment) here, I think it is prima facie very plausible for a selfish agent to follow the gnome’s advice if a) conditional on the agent existing, the gnome’s utility function agrees with the agent’s and b) conditional on the agent not existing, the gnome’s utility function is a constant. (I didn’t have condition b) explicitly in mind, but your example showed that it’s necessary.) Having the number of gnomes depend upon the coin flip invalidates their purpose. The very point of the gnomes is that from their perspective, the problem is not “anthropic”, but a decision problem that can be solved using UDT.
That’s what I tried in the parent comment. To be clear, I did not mean “indexical total utilitarianism” to be a meaningful concept, but rather a wrong way of thinking, a trap one can fall into. Very roughly, it corresponds to thinking of total utilitarianism as “I care for myself plus any other people that might exist” instead of “I care for all people that exist”. What’s the difference, you ask? A minimal non-anthropic example that illustrates the difference would be very much like the incubator, but without people being created. Imagine 1000 total utilitarians with identical decision algorithms waiting in separate rooms. After the coin flip, either one or two of them are offered to buy a ticket that pays $1 after heads. When being asked, the agents can correctly perform a non-anthropic Bayesian update to conclude that the probability of tails is 2⁄3. An indexical total utilitarian reasons: “If the coin has shown tails, another agent will pay the same amount $x that I pay and win the same $1, while if the coin has shown heads, I’m the only one who pays $x. The expected utility of paying $x is thus 1⁄3 (-x) + 2⁄3 2 * (1-x).” This leads to the incorrect conclusion that one should pay up to $4/5. The correct (UDT-) way to think about the problem is that after tails, one’s decision algorithm is called twice. There’s only one factor of 2, not two of them. This is all very similar to this post.
To put this again into context: You argued that selfishness is a 50⁄50 mixture of hating the other person, if another person exists, and total utilitarianism. My reply was that this is only true if one understands total utilitarianism in the incorrect, indexical way. I formalized this as follows: Let the utility function of a hater be vh—h vo (here, vh is the agent’s own utility, vo the other person’s utility, and h is 1 if the other person exists and 0 otherwise). Selfishness would be a 50⁄50 mixture of hating and total utilitarianism if the utility function of a total utilitarian were vh + h vo. However, this is exactly the wrong way of formalizing total utilitarianism. It leads, again, to the conclusion that a total utilitarian should pay up to $4/5.
The decision you describe in not stable under pre-commitments. Ahead of time, all agents would pre-commit to the $2/3. Yet they seem to change their mind when presented with the decision. You seem to be double counting, using the Bayesian updating once and the fact that their own decision is responsible for the other agent’s decision as well.
In the terminology of paper http://www.fhi.ox.ac.uk/anthropics-why-probability-isnt-enough.pdf , your agents are altruists using linked decisions with total responsibility and no precommitments, which is a foolish thing to do. If they were altruists using linked decisions with divided responsibility (or if they used precommitments), everything would be fine (I don’t like or use that old terminology—UDT does it better—but it seems relevant here).
But that’s detracting from the main point: still don’t see any difference between indexical and non-indexical total utilitarianism. I don’t see why a non-indexical total utilitarian can’t follow the wrong reasoning you used in your example just as well as an indexical one, if either of them can—and similarly for the right reasoning.
Yes, this is exactly the point I was trying to make—I was pointing out a fallacy. I never intended “lexicality-dependent utilitarianism” to be a meaningful concept, it’s only a name for thinking in this fallacious way.