Holding observations fixed but not updating on them is simply a misapplication of UDT.
A misapplication, strictly speaking, but not “simply”. Without restricting your attention to particular situations, while ignoring other situations, you won’t be able to consider any thought experiments. For any thought experiment I show you, you’ll say that you have to compute expected utility over all possible thought experiments, and that would be end of it.
So in applying UDT in real life, it’s necessary to stipulate the problem statement, the boundary event in which all relevant possibilities are contained, and over which we compute expected utility. You, too, introduced such an event, you just did it a step earlier than what’s given in the problem statement, by paying attention to the term “observation” attached to the calculator, and the fact that all other elements of the problem are observations also.
(On unrelated note, I have doubts about correctness of your work with that broader event too, see this comment.)
So in applying UDT in real life, it’s necessary to stipulate the problem statement, the boundary event in which all relevant possibilities are contained, and over which we compute expected utility.
Yes, of course. But you perform normal Bayesian updates for everything else (everything you hold fixed). Holding something fixed and not updating leads to errors.
Simple example: An urn with either 90% red and 10% blue balls or 90% blue and 10% red balls (0.5 prior for either). You have drawn a red ball and put it back. What’s the updateless expected utility of drawing another ball, assuming you get 1 util for drawing a ball in the same color and −2 utils for drawing a ball in a different color? Calculating as getting 1 util for red balls and −2 for blue, but not updating on the observation of having drawn a red ball suggests that it’s −0.5, when in fact it’s 0.46.
EDIT: miscalculated the utilities, but the general thrust is the same.
Holding something fixed and not updating leads to errors.
No, controlling something and updating it away leads to errors. Fixed terms in expected utility don’t influence optimality, you just lose ability to consider the influence of various strategies on them. Here, the strategies under considerations don’t have any relevant effects outside the problem statement.
I admit that I did not anticipate you replying in this way and even though I think I understand what you are saying I still don’t understand why. This is the main source of my uncertainty on whether I’m right at this point. It seems increasingly clear that at least one of us doesn’t properly understand UDT. I hope we can clear this up and if it turns out the misunderstanding was on my part I commit to upvoting all comments by you that contributed to enlightening me about that.
Unless I completely misunderstand you that’s a completely different context for/meaning of “fixed term” and while true not at all relevant here. I mean fixed in the sense of knowing the utilities of red and blue balls in the example I gave.
No, controlling something and updating it away leads to errors.
Also leads to errors, obviously. And I’m not doing that anyway. Something leading to errors is extremely weak evidence against something else also leading to error, so how is this relevant?
Correct (if you mean to say that all errors apparently caused by lack of updating can also be framed as being caused by wrongly holding something fixed) for a sufficiently wide sense of not fixed. The fact that you are considering to replace odd results in counterfactual worlds with even results and not the other way round, or the fact that the utility of drawing a red ball is 1 and for a blue ball −2 in my example (did you get around to taking a look at it?) both have to be considered not fixed in that sense.
Basically in the terminology of this comment you can consider anything in X1 fixed and avoid the error I’m talking about by updating. Or you can avoid that error by not holding it fixed in the first place. The same holds for anything in X2 for which the decision will never have any consequences anywhere it’s not true (or at least all its implications fully carry over), though that’s obviously more dangerous (and has the side effect of splitting the agent into different versions in different environments).
The error you’re talking about (the very error which UDT is correction for) is holding something in X2 fixed and updating when it does have outside consequences. Sometimes the error will only manifest when you actually update and only holding fixed gives results equivalent to the correct ones.
The test to see whether it’s allowable to update on x is to check whether the update results in the same answers as an updateless analysis that does not hold x fixed. If an analysis with update on x and one that holds x fixed but does not update disagree the problem is not always with the analysis with update. In fact in all problems CDT and UDT agree (most boring problems) the version with update should be correct and the version that only holds fixed might not be.
A misapplication, strictly speaking, but not “simply”. Without restricting your attention to particular situations, while ignoring other situations, you won’t be able to consider any thought experiments. For any thought experiment I show you, you’ll say that you have to compute expected utility over all possible thought experiments, and that would be end of it.
So in applying UDT in real life, it’s necessary to stipulate the problem statement, the boundary event in which all relevant possibilities are contained, and over which we compute expected utility. You, too, introduced such an event, you just did it a step earlier than what’s given in the problem statement, by paying attention to the term “observation” attached to the calculator, and the fact that all other elements of the problem are observations also.
(On unrelated note, I have doubts about correctness of your work with that broader event too, see this comment.)
Yes, of course. But you perform normal Bayesian updates for everything else (everything you hold fixed). Holding something fixed and not updating leads to errors.
Simple example: An urn with either 90% red and 10% blue balls or 90% blue and 10% red balls (0.5 prior for either). You have drawn a red ball and put it back. What’s the updateless expected utility of drawing another ball, assuming you get 1 util for drawing a ball in the same color and −2 utils for drawing a ball in a different color? Calculating as getting 1 util for red balls and −2 for blue, but not updating on the observation of having drawn a red ball suggests that it’s −0.5, when in fact it’s 0.46.
EDIT: miscalculated the utilities, but the general thrust is the same.
P(RedU)=P(BlueU)=P(red)=P(blue)=0.5
P(red|RedU)=P(RedU|red)=P(blue|BlueU)=P(BlueU|blue)=0.9
P(blue|RedU)=P(RedU|blue)=P(BlueU|red)=P(Red|BlueU)=0.1
U_updating=P(RedU|red)*P(red|RedU)*1 + P(BlueU|red)*Pred(|BlueU)*1 - P(RedU|red)*P(blue|RedU)*2 - P(BlueU|red)*P(blue|BlueU)*2 = 0.9*0.9+0.1*0.1-0.9*0.1*2*2= 0.46
U_semi_updateless=P(red)*1-P(blue)*2=-0.5
U_updateless= P(red)(P(RedU|red)*P(red|RedU)*1 + P(BlueU|red)*Pred(|BlueU)*1 - P(RedU|red)*P(blue|RedU)*2 - P(BlueU|red)*P(blue|BlueU)*2) +P(blue)(P(BlueU|blue)*P(blue|BlueU)*1 + P(RedU|blue)*P(blue|RedU)*1 - P(BlueU|blue)*P(red|BlueU)*2 - P(RedU|blue)*P(red|RedU)*2) =0.5*(0.9*0.9+0.1*0.1-0.9*0.1*2*2)+0.5* (0.9*0.9+0.1*0.1-0.9*0.1*2*2)=0.46
(though normally you’d probably come up with U_updateless in a differently factored form)
EDIT3: More sensible/readable factorization of U_updateless:
P(RedU)((P(red|RedU)(P(red|RedU)*1-P(blue|RedU)*2)+(P(blue|RedU)(P(blue|RedU)*1-P(red|RedU)*2)) + P(BlueU)((P(blue|BlueU)(P(blue|BlueU)*1-P(red|BlueU)*2)+(P(red|BlueU)(P(red|BlueU)*1-P(blue|BlueU)*2))
No, controlling something and updating it away leads to errors. Fixed terms in expected utility don’t influence optimality, you just lose ability to consider the influence of various strategies on them. Here, the strategies under considerations don’t have any relevant effects outside the problem statement.
(I’ll look into your example another time.)
Just to make sure: You mean something like updating on the box being empty in transparent Newcomb’s here, right? Not relevant as far as I can see.
I admit that I did not anticipate you replying in this way and even though I think I understand what you are saying I still don’t understand why. This is the main source of my uncertainty on whether I’m right at this point. It seems increasingly clear that at least one of us doesn’t properly understand UDT. I hope we can clear this up and if it turns out the misunderstanding was on my part I commit to upvoting all comments by you that contributed to enlightening me about that.
Unless I completely misunderstand you that’s a completely different context for/meaning of “fixed term” and while true not at all relevant here. I mean fixed in the sense of knowing the utilities of red and blue balls in the example I gave.
Also leads to errors, obviously. And I’m not doing that anyway. Something leading to errors is extremely weak evidence against something else also leading to error, so how is this relevant?
This is the very error which UDT (at least, this aspect of it) is correction for.
That still doesn’t make it evidence for something different not being an error. (and formal UDT is not the only way to avoid that error)
Not updating never leads to errors. Holding fixed what isn’t can.
Correct (if you mean to say that all errors apparently caused by lack of updating can also be framed as being caused by wrongly holding something fixed) for a sufficiently wide sense of not fixed. The fact that you are considering to replace odd results in counterfactual worlds with even results and not the other way round, or the fact that the utility of drawing a red ball is 1 and for a blue ball −2 in my example (did you get around to taking a look at it?) both have to be considered not fixed in that sense.
Basically in the terminology of this comment you can consider anything in X1 fixed and avoid the error I’m talking about by updating. Or you can avoid that error by not holding it fixed in the first place. The same holds for anything in X2 for which the decision will never have any consequences anywhere it’s not true (or at least all its implications fully carry over), though that’s obviously more dangerous (and has the side effect of splitting the agent into different versions in different environments).
The error you’re talking about (the very error which UDT is correction for) is holding something in X2 fixed and updating when it does have outside consequences. Sometimes the error will only manifest when you actually update and only holding fixed gives results equivalent to the correct ones.
The test to see whether it’s allowable to update on x is to check whether the update results in the same answers as an updateless analysis that does not hold x fixed. If an analysis with update on x and one that holds x fixed but does not update disagree the problem is not always with the analysis with update. In fact in all problems CDT and UDT agree (most boring problems) the version with update should be correct and the version that only holds fixed might not be.