My confidence bounds were 75% and 98% for defect, so my estimate was diametrically opposed to yours. If the admittedly low sample size of these comments is any indication, we were both way off.
Why do you think most would cooperate? I would expect this demographic to do a consequentialist calculation, and find that an isolated cooperation has almost no effect on expected value, whereas an isolated defection almost quadruples expected value.
Nice job on the survey. I loved the cooperate/defect problem, with calibration questions.
I defected, since a quick expected value calculation makes it the overwhelmingly obvious choice (assuming no communcation between players, which I am explicitly violating right now). Judging from comments, it looks like my calibration lower bound is going to be way off.
I agree that the statement is not crystal clear. It makes it possible to confuse the (change in the average) with the (average of the change).
Mathematically speaking, we represent our beliefs as a probability distribution on the possible outcomes, and change it upon seeing the result of a test (possibly for every outcome). The statement is that “if we average the possible posterior probability distributions weighted by how likely they are, we will end up with our original probability distribution.”
If that were not the case, it would imply that we were failing to make use of all of the prior information we have in our original distribution.
A misunderstood reading of the statement is that “the average of the absolute change in the probability distribution on measurement is zero.” This is not the case, as you rightly point out. It would imply that we expect the test to yield no information.
For the moment, I’m going to strike the comment from the post. I don’t want to ascribe a viewpoint to VincentYu that he doesn’t actually hold.
I added a section called “Deciding how to decide” that (hopefully) deals with this issue appropriately. I also amended the conclusion, and added you as an acknowledgement.
I’m not sure why it got moved: maybe not central to the thesis of LW, or maybe not high enough quality. I’m going to add some discussion of counter-arguments to the limit method. Maybe that will make a difference.
I noticed that the discussion picked up when it got moved, and I learned some useful stuff from it, so I’m not complaining.
Ok, I think I’ve got it. I’m not familiar with VNM utility, and I’ll make sure to educate myself.
I’m going to edit the post to reflect this issue, but it may take me some time. It is clear (now that you point it out) that we can think of the ill-posedness coming from our insistence that the solution conform to aggregative utilitarianism, and it may be possible to sidestep the paradox if we choose another paradigm of decision theory. Still, I think it’s worth working as an example, because, as you say, AU is a good general standard, and many readers will be familiar with it. At the minimum, this would be an interesting finite AU decision problem.
Thanks for all the time you’ve put into this.
I would like to include this issue in the post, but I want to make sure I understand it first. Tell me if this is right:
It is possible mathematically to represent a countably infinite number of immortal people, as well as the process of moving them between spheres. Further, we should not expect a priori that a problem involving such infinities would have a solution equivalent to those solutions reached by taking infinite limits of an analogous finite problem. Some confusion arises when we introduce the concept of “utility” to determine which of the two choices is better, since utility only serves as a basis on which to make decision for finite problems.
If that’s what you’re saying, I have a couple of questions.
Do you view the paradox as therefore unresolvable as stated, or would you claim that a different resolution is correct?
If I carefully restricted my claim about ill-posedness to the question of which choice is better from a utilitarian sense, would you agree with it?
The final section has been edited to reflect the concerns of some of the commenters.
Thanks to whomever moved this to Discussion. From the FAQ, I wasn’t sure where to put it. This is better, in retrospect.
Thanks! Do you guys want to copy edit my journal papers? ;)
You’re completely right! As stated, the problem is ill posed, i.e. it has no unique solution, so we didn’t solve it.
Instead, we solved a similar problem by introducing a new parameter, \alpha. It was useful because we gained a mathematical description that works for very large n and s, and which matches our intuition about the problem.
It is important to recognize, as you point out, that that taking limits does not solve the problem. It just elucidates why we can’t solve it as stated.
I agree that it’s a lot to cover, but I wanted to work a full example. We talk a lot on LW about decision analysis and paradoxes in the abstract, but I’m coming from a math/physics background, and it’s much more helpful for me to see concrete examples. I assume some other people feel the same way.
Self-referential problems would be an interesting area to study, but I’m not familiar with the techniques. I suspect you’re right, though.
Fixed. Thanks for reading so closely. It’s amazing how many little mistakes can survive after 10 read-throughs.
By the way, are you talking about this meme, or is there another problem with monkeys and bananas?
Great problem, thanks for mentioning it!
I think the answer to “how many balls did you put in the vase as T->\infty” and “How many balls have been destroyed as T->\infty” both have well defined answers. It’s just a fallacy to assume that the “total number of balls in the vase as T->\infty” is equal to the difference between these quantities in their limits.
My parents stopped me from skipping a grade, and apart from a few math tricks, we didn’t work on additional material at home. I fell into a trap of “minimum effort for maximum grade,” and got really good at guessing the teacher’s password. The story didn’t change until graduate school, when I was unable to meet the minimum requirements without working, and that eventually led me to seek out fun challenges on my own.
I now have a young son of my own, and will not make the same mistake. I’m going to make sure he expects to fail sometimes, and that I praise his efforts to go beyond what’s required. No idea if it will work.
MrMind explains in better language below.
The plots were done in Mathematica 9, and then I added the annotations in PowerPoint, including the dashed lines. I had to combine two color functions for the density plot, since I wanted to highlight the fact that the line s=n represented indifference. Here’s the code:
r = 1; ua = 1;ub = −1; f1[n, s] := (ns—s^2r ) (ua—ub); Show[DensityPlot[-f1[n, s], {n, 0, 20}, {s, 0, 20}, ColorFunction → “CherryTones”, Frame → False, PlotRange → {-1000, 0}], DensityPlot[f1[n, s], {n, 0, 20}, {s, 0, 20}, ColorFunction → “BeachColors”, Frame → False, PlotRange → {-1000, 0}]]
It’s true: if you’re optimizing for altruism, cooperation is clearly better.
I guess it’s not really a “dilemma” as such, since the optimal solution doesn’t depend at all on what anyone else does. If you’re trying to maximize EV, defect. If you’re trying to maximize other people’s EV, cooperate.