This struck me as interesting since it relates the Sleeping Beauty problem to a choice of utility function. Is Beauty a selfish utility maximizer with very high discount rate, or a selfish utility maximizer with low discount rate, or a total utility maximizer, or an average utility maximizer? The type of function affects what betting odds Beauty should accept.
Incidentally, one thing that is not usually spelled out in the story (but really should be) is whether there are other sentient people in the universe apart from Beauty, and how many of them there are. Also, does Beauty have any/many experiences outside the context of the coin-toss and awakening? These things make a difference to SSA (or to Bostrom’s SSSA).
The last time I had an anthropic principle discussion on Less Wrong I was pointed at the following paper
While that work is interesting, knowing how to get probabilities means we can basically just ignore it :P Just assume Beauty is an ordinary utility-maximizer.
These things make a difference to SSA (or to Bostrom’s SSSA).
They make a difference if those things are considered as mysterious processes that output correct probabilities. But we already know how to get correct probabilities—you just follow the basic rules, or, in the equivalent formulation used in this post, follow the information. If SSA is used in any other way than as a set of starting information, it becomes an ad hoc method, not worth much consideration.
Not sure I follow that… what did you mean by an “ordinary” utility maximizer”? Is it a selfish or a selfless utility function, and if selfish what is the discount rate? The point about Armstrong’s paper is that really does matter.
Most of the utility functions do give the 2⁄3 answer, though for the “average utilitarian” this is only true if there are lots of people outside the Sleeping Beauty story (or if Beauty herself has lots of experiences outside the story).
I’m a bit wary about using an indifference principle to get “the one true answer”, because in the limit it suffers from the Presumptuous Philosopher problem. Imagine that Beauty (or Beauty clones) is woken a trillion times after a Tails toss. Then the indifference principle means that Beauty will be very near certain that the head fell Tails. Even if she is shown sworn affidavits and video recordings of the coin falling Heads, she’ll believe that they were faked.
Not sure I follow that… what did you mean by an “ordinary” utility maximizer”? Is it a selfish or a selfless utility function, and if selfish what is the discount rate? The point about Armstrong’s paper is that really does matter.
So you have this utility function U, and it’s a function of different outcomes, which we can label by a bunch of different numbers “x”. And then you pick the option that maximizes the sum of U(x) * P(x | all your information).
There are two ways this can fail and need to be extended—either there’s an outcome you don’t have a utility for, or there’s an outcome you don’t have a probability for. Stuart’s paper is what you can do if you don’t have some probabilities. My post is how to get those probabilities.
If something is unintuitive, ask why it is unintuitive. Eventually either you’ll reach something wrong with the problem (does it neglect model uncertainty?), or you’ll reach something wrong with human intuitions (what is going on in peoples’ heads when they get the monty hall problem wrong?). In the meanwhile, I still think you should follow the math—unintuitiveness is a poor signal in situations that humans don’t usually find themselves in.
This looks like what Armstrong calls a “selfless” utility function i.e. it has no explicit term for Beauty’s welfare here/now or at any other point in time.. The important point here is that if Beauty bets tails, and the coin fell Tails, then there are two increments to U, whereas if the coin fell Heads then there is only one decrement to U. This leads to a 2⁄3 betting probability.
In the trillion Beauty case, the betting probability may depend on the shape of U and whether it is bounded (e.g. whether winning 1 trillion bets really is a trillion times better than winning one).
This looks like what Armstrong calls a “selfless” utility function i.e. it has no explicit term for Beauty’s welfare here/now or at any other point in time.
Stuart’s terms are a bit misleading because they’re about decision-making by counting utilities, which is not the same as decision-making by maximizing expected utility. His terms like “selfish” and “selfless” and so on are only names for counting rules for utilities, and have no direct counterpart in expected utility maximizers.
So U can contain terms like “I eat a candy bar. +1 utility.” Or it could only contain terms like “a sentient life-form eats a candy bar. +1 utility.” It doesn’t actually change what process Sleeping Beauty uses to make decisions in anthropic situations, because those ideas only applied to decision-making by counting utilities. Additionally, Sleeping Beauty makes identical decisions in anthropic and non-anthropic situations, if the utilities and the probabilities are the same.
OK, I think this is clearer. The main point is that whatever this “ordinary” U is scoring (and it could be more or less anything) then winning the tails bet scores +2 whereas losing the tails bet scores −1. This leads to 2⁄3 betting probability. If subjective probabilities are identical to betting probabilities (a common position for Bayesians) then the subjective probability of tails has to be 2⁄3.
The point about alternative utility functions though is that this property doesn’t always hold i.e. two Beauties winning doesn’t have to be twice as good as one Beauty winning. And that’s especially true for a trillion Beauties winning.
Finally, if you adopt a relative frequency interpretation (the coin-toss is repeated multiple times, and take limit to infinity) then there are obviously two relative frequencies of interest. Half the coins fall Tails, but two thirds of Beauty awakenings are after Tails. Either of these can be interpreted as a probability.
If subjective probabilities are identical to betting probabilities (a common position for Bayesians)
If we start with an expected utility maximizer, what does it do when deciding whether to take a bet on, say, a coin flip? Expected utility is the utility times the probability, so it checks whether P(heads) U(heads) > P(tails) U(tails). So betting can only tell you the probability if you know the utilities. And changing the utility function around is enough to get really interesting behavior, but it doesn’t mean you changed the probabilities.
Half the coins fall Tails, but two thirds of Beauty awakenings are after Tails. Either of these can be interpreted as a probability.
What sort of questions, given what sorts of information, would give you these two probabilities? :D
For the first question: if I observe multiple coin-tosses and count what fraction of them are tails, then what should I expect that fraction to be? (Answer one half). Clearly “I” here is anyone other than Beauty herself, who never observes the coin-toss.
For the second question: if I interview Beauty on multiple days (as the story is repeated) and then ask her courtiers (who did see the toss) whether it was heads or tails, then what fraction of the time will they tell me tails? (Answer two thirds.)
What information is needed for this? None except what is defined in the original problem, though with the stipulation that the story is repeated often enough to get convergence.
Incidentally, these questions and answers aren’t framed as bets, though I could use them to decide whether to make side-bets.
I haven’t read the paper, but it seems like one could just invent payoff schemes customized for her utility function and give her arbitrary dilemmas that way, right?
The last time I had an anthropic principle discussion on Less Wrong I was pointed at the following paper: http://arxiv.org/abs/1110.6437 (See http://lesswrong.com/lw/9ma/selfindication_assumption_still_doomed/5sbv)
This struck me as interesting since it relates the Sleeping Beauty problem to a choice of utility function. Is Beauty a selfish utility maximizer with very high discount rate, or a selfish utility maximizer with low discount rate, or a total utility maximizer, or an average utility maximizer? The type of function affects what betting odds Beauty should accept.
Incidentally, one thing that is not usually spelled out in the story (but really should be) is whether there are other sentient people in the universe apart from Beauty, and how many of them there are. Also, does Beauty have any/many experiences outside the context of the coin-toss and awakening? These things make a difference to SSA (or to Bostrom’s SSSA).
While that work is interesting, knowing how to get probabilities means we can basically just ignore it :P Just assume Beauty is an ordinary utility-maximizer.
They make a difference if those things are considered as mysterious processes that output correct probabilities. But we already know how to get correct probabilities—you just follow the basic rules, or, in the equivalent formulation used in this post, follow the information. If SSA is used in any other way than as a set of starting information, it becomes an ad hoc method, not worth much consideration.
Not sure I follow that… what did you mean by an “ordinary” utility maximizer”? Is it a selfish or a selfless utility function, and if selfish what is the discount rate? The point about Armstrong’s paper is that really does matter.
Most of the utility functions do give the 2⁄3 answer, though for the “average utilitarian” this is only true if there are lots of people outside the Sleeping Beauty story (or if Beauty herself has lots of experiences outside the story).
I’m a bit wary about using an indifference principle to get “the one true answer”, because in the limit it suffers from the Presumptuous Philosopher problem. Imagine that Beauty (or Beauty clones) is woken a trillion times after a Tails toss. Then the indifference principle means that Beauty will be very near certain that the head fell Tails. Even if she is shown sworn affidavits and video recordings of the coin falling Heads, she’ll believe that they were faked.
So you have this utility function U, and it’s a function of different outcomes, which we can label by a bunch of different numbers “x”. And then you pick the option that maximizes the sum of U(x) * P(x | all your information).
There are two ways this can fail and need to be extended—either there’s an outcome you don’t have a utility for, or there’s an outcome you don’t have a probability for. Stuart’s paper is what you can do if you don’t have some probabilities. My post is how to get those probabilities.
If something is unintuitive, ask why it is unintuitive. Eventually either you’ll reach something wrong with the problem (does it neglect model uncertainty?), or you’ll reach something wrong with human intuitions (what is going on in peoples’ heads when they get the monty hall problem wrong?). In the meanwhile, I still think you should follow the math—unintuitiveness is a poor signal in situations that humans don’t usually find themselves in.
This looks like what Armstrong calls a “selfless” utility function i.e. it has no explicit term for Beauty’s welfare here/now or at any other point in time.. The important point here is that if Beauty bets tails, and the coin fell Tails, then there are two increments to U, whereas if the coin fell Heads then there is only one decrement to U. This leads to a 2⁄3 betting probability.
In the trillion Beauty case, the betting probability may depend on the shape of U and whether it is bounded (e.g. whether winning 1 trillion bets really is a trillion times better than winning one).
Stuart’s terms are a bit misleading because they’re about decision-making by counting utilities, which is not the same as decision-making by maximizing expected utility. His terms like “selfish” and “selfless” and so on are only names for counting rules for utilities, and have no direct counterpart in expected utility maximizers.
So U can contain terms like “I eat a candy bar. +1 utility.” Or it could only contain terms like “a sentient life-form eats a candy bar. +1 utility.” It doesn’t actually change what process Sleeping Beauty uses to make decisions in anthropic situations, because those ideas only applied to decision-making by counting utilities. Additionally, Sleeping Beauty makes identical decisions in anthropic and non-anthropic situations, if the utilities and the probabilities are the same.
OK, I think this is clearer. The main point is that whatever this “ordinary” U is scoring (and it could be more or less anything) then winning the tails bet scores +2 whereas losing the tails bet scores −1. This leads to 2⁄3 betting probability. If subjective probabilities are identical to betting probabilities (a common position for Bayesians) then the subjective probability of tails has to be 2⁄3.
The point about alternative utility functions though is that this property doesn’t always hold i.e. two Beauties winning doesn’t have to be twice as good as one Beauty winning. And that’s especially true for a trillion Beauties winning.
Finally, if you adopt a relative frequency interpretation (the coin-toss is repeated multiple times, and take limit to infinity) then there are obviously two relative frequencies of interest. Half the coins fall Tails, but two thirds of Beauty awakenings are after Tails. Either of these can be interpreted as a probability.
If we start with an expected utility maximizer, what does it do when deciding whether to take a bet on, say, a coin flip? Expected utility is the utility times the probability, so it checks whether P(heads) U(heads) > P(tails) U(tails). So betting can only tell you the probability if you know the utilities. And changing the utility function around is enough to get really interesting behavior, but it doesn’t mean you changed the probabilities.
What sort of questions, given what sorts of information, would give you these two probabilities? :D
For the first question: if I observe multiple coin-tosses and count what fraction of them are tails, then what should I expect that fraction to be? (Answer one half). Clearly “I” here is anyone other than Beauty herself, who never observes the coin-toss.
For the second question: if I interview Beauty on multiple days (as the story is repeated) and then ask her courtiers (who did see the toss) whether it was heads or tails, then what fraction of the time will they tell me tails? (Answer two thirds.)
What information is needed for this? None except what is defined in the original problem, though with the stipulation that the story is repeated often enough to get convergence.
Incidentally, these questions and answers aren’t framed as bets, though I could use them to decide whether to make side-bets.
I haven’t read the paper, but it seems like one could just invent payoff schemes customized for her utility function and give her arbitrary dilemmas that way, right?