This post does not prove Maximal Lottery Lotteries exist. Instead, it redefines MLL to be equivalent to the Nash bargaining solution (in a way that is obscured by using the same language as the MLL proposal), and then claims that under the new definition MLL exist (because the Nash bargaining solution exists).
I like Nash bargaining, and I don’t like majoritarianism, but the MLL proposal is supposed to be a steelman of majoritarianism, and Nash bargaining is not only not MLL, but it is not even majoritarian. (If a majority of voters have the same favorite candidate, this is not sufficient to make this candidate win in the Nash bargaining solution.)
Scott Garrabrant
Karma: 8,316
I agree with this.
More like illuminating ontologies than great predictions, but yeah.
I think Chris Langan and the CTMU are very interesting, and I there is an interesting and important challenge for LW readers to figure out how (and whether) to learn from Chris. Here are some things I think are true about Chris (and about me) and relevant to this challenge. (I do not feel ready to talk about the object level CTMU here, I am mostly just talking about Chris Langan.)
Chris has a legitimate claim of being approximately the smartest man alive according to IQ tests.
Chris wrote papers/books that make up a bunch of words there are defined circularly, and are difficult to follow. It is easy to mistake him for a complete crackpot.
Chris claims to have proven the existence of God.
Chris has been something-sort-of-like-canceled for a long time. (In the way that seems predictable when “World’s Smartest Man Proves Existence of God.”)
Chris has some followers that I think don’t really understand him. (In the way that seems predictable when “World’s Smartest Man Proves Existence of God.”)
Chris acts socially in a very nonstandard way that seems like a natural consequence of having much higher IQ than anyone else he has ever met. In particular, I think this manifests in part as an extreme lack of humility.
Chris is actually very pleasant to talk to if (like me) it does not bother you that he acts like he is much smarter than you.
I personally think the proof of the existence of God is kid of boring. It reads to me as kind of like “I am going to define God to be everything. Notice how this meets a bunch of the criteria people normally attribute to God. In the CTMU, the universe is mind-like. Notice how this meets a bunch more criteria people normally attribute to God.”
While the proof of the existence of God feels kind of mundane to me, Chris is the kind of person who chooses to interpret it as a proof of the existence of God. Further, he also has other more concrete supernatural-like and conspiracy-theory-like beliefs, that I expect most people here would want to bet against.
I find the CTMU in general interesting, (but I do not claim to understand it).
I have noticed many thoughts that come naturally to me that do not seem to come naturally to other people (e.g. about time or identity), where it appears to me that Chris Langan just gets it (as in he is independently generating it all).
For years, I have partially depended on a proxy when judging other people (e.g. for recommending funding) that is is something like “Do I, Scott, like where my own thoughts go in contact with the other person?” Chris Langan is approximately at the top according to this proxy.
I believe I and others here probably have a lot to learn from Chris, and arguments of the form “Chris confidently believes false thing X,” are not really a crux for me about this.
IQ is not the real think-oomph (and I think Chris agrees), but Chris is very smart, and one should be wary of clever arguers, especially when trying to learn from someone with much higher IQ than you.
I feel like I am spending (a small amount of) social credit in this comment, in that when I imagine a typical LWer thinking “oh, Scott semi-endorses Chris, maybe I should look into Chris,” I imagine the most likely result is that they will reach the conclusion is that Chris is a crackpot, and that Scott’s semi-endorsements should be trusted less.
So, I am trying to talk about the preferences of the couple, not the preferences of either individual. You might reject that the couple is capable of having preference, if so I am curious if you think Bob is capable of having preferences, but not the couple, and if so, why?
I agree if you can do arbitrary utility transfers between Alice and Bob at a given exchange rate, then they should maximize the sum of their utilities (at that exchange rate), and do a side transfer. However, I am assuming here that efficient compensation is not possible. I specifically made it a relatively big decision, so that compensation would not obviously be possible.
Here are the most interesting things about these objects to me that I think this post does not capture.
Given a distribution over non-negative non-identically-zero infrafunctions, up to a positive scalar multiple, the pointwise geometric expectation exists, and is an infra function (up to a positive scalar multiple).
(I am not going to give all the math and be careful here, but hopefully this comment will provide enough of a pointer if someone wants to investigate this.)
This is a bit of a miracle. Compare this with arithmetic expectation of utility functions. This is not always well defined. For example, if you have a sequence of utility functions U_n, each with weight 2^{-n}, but which alternate in which of two outcomes they prefer, and each utility function gets an internal weighting to cancel out their small weight an then some, the expected utility will not exist. There will be a series of larger and larger utility monsters canceling each other out, and the limit will not exist. You could fix this requiring your utility functions are bounded, as is standard for dealing with utility monsters, but it is really interesting that in the case of infra functions and geometric expectation, you don’t have to.
If you try to do a similar trick with infra functions, up to a positive scalar multiple, geometric expectation will go to infinity, but you can renormalize everything since you are only working up to a scalar multiple, to make things well defined.
We needed the geometric expectation to only be working up to a scalar multiple, and you cant expect a utility function if you take a geometric expectation of utility functions. (but you do get an infrafunction!)
If you start with utility functions, and then merge them geometrically, the resulting infrafunction will be maximized at the Nash bargaining solution, but the entire infrafunction can be thought of as an extended preference over lotteries of the pair of utility functions, where as Nash bargaining only told you the maximum. In this way geometric merging of infrafunctions is starting with an input more general than the utility functions of Nash bargaining, and giving an output more structured than the output of Nash bargaining, and so can be thought of as a way of making Nash bargaining more compositional. (Since the input and output are now the same type, so you can stack them on top of each other.)
For these two reasons (utility monster resistance and extending Nash bargaining), I am very interested in the mathematical object that is non-negative non-identically-zero infrafunctions defined only up to a positive scalar multiple, and more specifically, I am interested in the set of such functions as a convex set where mixing is interpreted as pointwise geometric expectation.
I have been thinking about this same mathematical object (although with a different orientation/motivation) as where I want to go with a weaker replacement for utility functions.
I get the impression that for Diffractor/Vanessa, the heart of a concave-value-function-on-lotteries is that it represents the worst case utility over some set of possible utility functions. For me, on the other hand, a concave value function represents the capacity for compromise—if I get at least half the good if I get what I want with 50% probability, then I have the capacity to merge/compromise with others using tools like Nash bargaining.
This brings us to the same mathematical object, but it feels like I am using the definition of convex set related to the line segment connecting any two points in the set is also in the set, where Diffractor/Vanessa is using the definition of convex set related to being an intersection of half planes.I think this pattern where I am more interested in merging, and Diffractor and Vanessa are more interested in guarantees, but we end up looking at the same math is a pattern, and I think the dual definitions of convex set in part explains (or at least rhymes with) this pattern.
Then it is equivalent to the thing I call B2 in edit 2 in the post (Assuming A1-A3).
In this case, your modified B2 is my B2, and your B3 is my A4, which follows from A5 assuming A1-A3 and B2, so your suspicion that these imply C4 is stronger than my Q6, which is false, as I argue here.However, without A5, it is actually much easier to see that this doesn’t work. The counterexample here satisfies my A1-A3, your weaker version of B2, your B3, and violates C4.
Your B3 is equivalent to A4 (assuming A1-3).
Your B2 is going to rule out a bunch of concave functions. I was hoping to only use axioms consistent with all (continuous) concave functions.
I am skeptical that it will be possible to salvage any nice VNM-like theorem here that makes it all the way to concavity. It seems like the jump necessary to fix this counterexample will be hard to express in terms of only a preference relation.
The answers to Q3, Q4 and Q6 are all no. I will give a sketchy argument here.
Consider the one dimensional case, where the lotteries are represented by real numbers in the interval , and consider the function given by . Let be the preference order given by if and only if .
is continuous and quasi-concave, which means is going to satisfy A1, A2, A3, A4, and B2. Further, since is monotonically increasing up to the unique argmax, and then monotonically decreasing, is going to satisfy A5.
is not concave, but we need to show there is not another concave function giving the same preference relation as . The only way to keep the same preference relation is to compose with a strictly monotonic function , so ).
If is smooth, we have a problem, since . However, since, must be on some , but concavity would require to be decreasing.
In order to remove the inflection point at , we need to flatten it out with some that has infinite slope at . For example, we could take . However, any f that removes the inflection point at , will end up adding an inflection point at , which will have a infinite negate slope. This newly created inflection point will cause a problem for similar reasons.
You can also think of A5 in terms of its contrapositive: For all , if , then for all
This is basically just the strict version of A4. I probably should have written it that way instead. I wanted to use instead of , because it is closer to the base definition, but that is not how I was natively thinking about it, and I probably should have written it the way I think about it.
Alex’s counterexample as stated is not a counterexample to Q4, since it is in fact concave.
I believe your counterexample violates A5, taking , , and .
That does not rule out your counterexample. The condition is never met in your counterexample.
The answer to Q1 is no, using the same counter example here. However, the spirit of my original question lives on in Q4 (and Q6).
Claim: A1, A2, A3, A5, and B2 imply A4.
Proof: Assume we have a preference ordering that satisfies A1, A2, A3, A5, and B2, and consider lotteries , and , with . Let . It suffices to show . Assume not, for the purpose of contradiction. Then (by axiom A1), . Thus by axiom B2 there exists a such that . By axiom A3, we may assume for some . Observe that where . is positive, since otherwise . Thus, we can apply A5 to get that since , we have . Thus , a contradiction.
Oh, nvm, that is fine, maybe it works.
Oh, no, I made a mistake, this counterexample violates A3. However, the proposed fix still doesn’t work, because you just need a function that is decreasing in probability of , but does not hit 0, and then jumps to 0 when probability of is 1.
I proposed this same voting system here: https://www.lesswrong.com/s/gnAaZtdwjDBBRpDmw
It is not strategy proof. If it were, that would violate https://en.wikipedia.org/wiki/Gibbard–Satterthwaite_theorem [Edit: I think, for some version of the theorem. It might not literally violate it, but I also believe you can make a small example that demonstrates it is not strategy proof. This is because the equilibrium sometimes extracts all the value from a voter until they are indifferent, and if they lie about their preferences less value can be extracted.]
Further, it is not obviously well defined. Because of the discontinuities around ties, you cannot take advantage of the compactness of the space of distributions, so it is not clear that Nash equilibria exist. (It is also not clear that they don’t exist. My best guess is that they do.)