I disagree with both of these methods. If EY were 100% sure, and NB were 50% sure, then I think the entire 20 should go to EY, and neither of the two methods have this property. I am very interested in trying to figure out what the best formula for this situation is, but I do not yet know. Here is a proposal:
Take the least amount of evidence so that you can shift both predictions by this amount of evidence, to make them the same, and split according to this probability.
Is this algorithm good?
Thanks. It was never off on my browser, but I think I may have fixed it. Am I right?
I fully agree that you could have more information that tells you how to combine the probabilities, but we don’t always have that information, and we need to make a decision anyway. Maybe this means the problem does not have a definitive answer, but I am still trying to decide what I would do.
I think that when p+q=1, the answer is clearly 1⁄2 due to symmetry. How did you get p+q?
Method 3 chooses the unique f such that updating p to f and updating q to f require the same amount of information.
Before reading GreedyAlgorithm’s post, I decided independently that I support method 3, although I may find another answer I like better. Methods 1 and 2 I do not like, because if you let p=1 and q=1/2, they do not give all the money to EY. Method 4 I do not like, because I think that f(p,p) should equal p. However, I have no argument for 3 other than the fact that it feels right, and it meets all the criteria I can think of in degenerate cases.
I think to apply Aumann, you have to assume that both people have consistent probability distributions, which I think is an unreasonable assumption. People are not perfect Bayesianists.
Giving the extra to the driver doesn’t work if p>q, because the sum of the values each person believes they are entitled will be more than $20.00
This is valid. However, for the reasons I am interested in the problem (which I am not going to describe now) I don’t get any to consider those probabilities. Pretend that I am stubborn, and refuse to consider anything other than p and q, but still would like advice on how to combine them.
While this is what would normally happen in this situation, I don’t think that answers or takes away from the deeper question of what is the most fair way to distribute the money.
Reading that, I believe his reason for endorsing #4 is that he made the assumption that neither of their claims relies on any evidence relied on by the other. If we assume this, I agree that #4 is the right answer, but I do not think that is a valid assumption.
Do you stand behind that decision, even though it does not give the entire 20 to EY when he is 100% sure he deserves it?
It shouldn’t go all the way to 0:1 at p=0 or 1:0 at p=1
I am surprised you feel this way. If EY says, “Oh, I put that extra 20 in there,” and NB says he has no idea, then I think EY should get the 20 back.
The geometric mean gives 0 an advantage over 1, which makes it not symmetric in general. If we trust EY and NB the same, then f(p,1-q) should equal 1-f(q,1-p) (in both cases, one person thinks they get p of the dollar and the other thinks they get q.) Your variant of 2 does not satisfy this, which I think is the feature I think is the most important.
As promised, here is my analysis.
First of all, I think that the g and f functions should be the same. My reason is that to be completely satisfied with who gets the 20, both people should update their probabilities to the same value, so f(p,q) seems like the probability that EY and NB should both walk away with saying is the probability that EY deserved the 20. Since EY and NB trust each other, this should be the same if there is a single person with both beliefs learning they are equivalent. Some of the features I want in f are easier to justify in the g example, which might have caused some of my mistakes.
I think this question is inherently about how much we treat p and q as coming from independent sources of information. If we say that the sources are independent, then #4 is the only reasonable answer. However, the dependency of the evidence is not known.
I think f should have the following properties:
A: f(p,p)=p—If we wanted f to be a general rule for how to take two probabilities and output a probability that is an agreement of the two, then there is a danger in setting g(p,p) to anything other than p. In that we can repeatedly apply our rule and get different answers. We update p and q to g(p,q) and g(p,q), but then these are our new probabilities, so we will update them both to g(g(p,q),g(p,q)). Our new answer is not consistent under reflection. Therefore, I think g(p,p) should be p, and this makes me believe that f(p,p) should also be p. (Maybe I am working by analogy, but I still believe this.)
B: f(p,q)=1-f(1-q,1-p) -- This is just saying that the answer is symmetric with respect to swapping EY and NB.
C: f(p,q)=f(q,p) -- This is saying that the answer is symmetric with respect to swapping who says what. This property seems really necessary in the g problem.
D: f(1,q)=1 -- If EY knows he put the 20 in, he should get it back. In the g problem, if A is a theorem, and we learn B if and only if A, then we can prove B also.
E: p1>=p2, then f(p1,q)>=f(p2,q) -- If EY is more sure he should get the money, he shouldn’t get less money.
F: f(f(p,q),f(r,s))=f(f(p,r),f(q,s)) -- This relation doesn’t mean much for f, but for g, it is saying that the order in which we learn conjectures are equivalent shouldn’t change the final answer.
G: f is continuous—A small change in probability shouldn’t have a huge effect on the decision.
I think that most people would agree with B, C, and E, but looking at the comments, A and D are more controversial. F doesn’t make any sense unless you believe that f=g. I am not sure how people will feel about G. Notice that if B, C, and D together imply that f(1,0) is not defined, because it would have to be both 1 and 0, I think this is okay. You can hardly expect EY and NB to continue trusting each other after something like this, but it is necessary to say to be mathematically correct.
Now, to critique the proposed solutions.
1 Violates C and D (I didn’t check F)
2 Violates D
4 Violates A
3 Does not violate any of my features.
I did not do a good job of describing where #3 came from, so let me do better. #3 chooses the unique value f, such that if we wanted to update EYs probability to f and update NBs probability to f, this would take the same amount of evidence. It satisfies A because if they are both already the same, then it doesn’t take any evidence. It satisfies D, because no finite amount of evidence can bring you back from certainty.
We do not have enough information to say that #3 is the unique solution. If we were to try to, it would look like roughly like this:
If we think about the problem by looking at p =log(p/1-p). Then #3 just finds the f =(p +q )/2. E and A together tell us that f should be somewhere between p and q but it is not immediately clear that the arithmetic mean is the best compromise. However, I believe this implies that we can apply some monotone function h, such that h(f ) is the always the arithmetic mean of h(p ) and h(q ). B tells us that this monotone function must be an odd function (h(-x)=-h(x)).
From here, #3 assumes that h(x)=x, but if we were to take h(x)=x^3 for example, we would still meet all of our properties. We have freedom from the fact that we can weigh probabilities with different distances from 1⁄2 differently.
I meant that if EY is 100% sure, while NB is 50% that EY should get the entire 20. I don’t think that I see anything you don’t on this. To me, it seems that if EY knows something, and NB trusts him, that NB should update to know it too, but it looks like you disagree. Perhaps I am working by analogy. I think that the equivalent property for the g function is more clearly true. If A is a theorem, and B is a conjecture, and we prove they are equivalent, then B as a theorem as well.
However, it at least seems that EY should get close to all of it. If there is 60 in the pot If NB just threw money in and has no idea whether he put 20 or 40 in, while EY knows (99.99%) that he put exactly 40 into the pot, EY should get a lot more than 15$
Sure! You start with a likelihood ratio of 1:1 of the 20 belonging to EY or NB. EY says that it is his with probability 20%. This is evidence with a ratio of 20:80 or 1:4. After you update to account for EYs claim, you assign a 20% chance that it belongs to EY. Then, you have NBs claim, which is evidence in the ratio of 85:15 or 17:3. You multiply the 1 by 17 and the 4 by 3 to get 17:12. This means that EY should get 17⁄29 of the money, which is $11.72.
This is using the notation of likelihood ratios, which is easier to work with. Trying to attack it with Bayes Theorem directly is a more confusing. The reason is because we are trusting EY and NBs claims as evidence, without actually specifying some event that caused that evidence. A good way to think about it is that we start by taking EY into account and thinking that the probabiltiy of A is 20%. Then we say that NB has a sensor that tells him whether or not the 20 is his and gets it right 85% of the time. We let B be the event that NBs sensor tells him that the 20 belongs to EY. Then, P(B|A)=85%, and B(B)=85% of 20% plus 15% of 80%, which is 29%. Therefore, we get P(A|B)=P(A)P(B|A)/P(B)=20%85%/(29%)=17/29.
Does that make sense?
There is a question of how independent their estimates are, and I think that the algorithm should be consistent under being repeatedly applied. If EY and NB update their probabilities to the same thing, and then try to update again, their estimates should not change.
In my opinion, the question should not about how to apply the Aumann agreement theorem, but how to compromise. That is the spirit of #2 and #3. They attempt to find the average value. (The difference is that one measures thinks that the scale that should be used is p, and the other thinks it is log(p/(1-p)).)
I do not think that this question has a unique solution. Probability doesn’t give us an answer. We are trying to determine what is fair. I think that my position is that the fair thing to do is to follow the result of the g question. The g question tells us how to combine probabilities without information of how independent they are if the goals and beliefs belong to a single person. If we have two people who trust each other and do not want more than their share, then they should adopt the same probability as if they were one person.
For the question on the g function, it is not about what is fair, and instead about what is safe. If I am going to prescribe a general rule on how to combine these estimates that does not know how independent they are, I want it to be consistent under repeated application so I don’t send all of my probabilities off to 1 when I shouldn’t.
Hello Less Wrong! I am Scott Garrabrant, a 23 year old math PhD student at UCLA, studying combinatorics. I discovered Less Wrong about 4 months ago. After reading MoR and a few sequences, I decided to go back and read every blog post. (I just finished all Eliezer’s OB posts) I was going to wait and start posting after I got completely caught up, but then I started attending weekly meetups 2 months ago, and now I need to earn enough karma to make meetup announcements.
I have been interested in meta-thinking for a long time. I have spent a lot of time thinking about the nature of rationality, purely out of curiosity, and have independently made many of the same conclusions I have found on this blog. I believe that I realized that decision/probability theory was the correct language to talk about rationality in high school about 6 years ago. It has made me very happy to learn that there are so many like-minded people.
However, there has been one mistake I have been making for a long time. I have been giving other people too much respect in their rationality. I have been treating other people as almost rational agents with different utility functions and very different prior probabilities. This blog has taught me how wrong that view was, which is causing me to rethink some of my prior views.
One thing would like some help in deciding right now is about Unitarian Universalism. I would love it if any rationalists who know anything about Unitarianism (or who don’t) could help me out. I am agnostic (If you define the god hypothesis to include the simulation hypothesis, atheist otherwise). I believe that most of the bad parts of religion and theism come from the fact that they tend to encourage irrationality. So far, my picture of the average Unitarian is above average rationality, but not great. The main thing that attracts me to the group is that they (at least claim to) promote “a free and responsible search for truth and meaning.” Their search algorithms could really use some work, but they both view truth as a goal and understand that they have not attained it completely. In looking for a local community to provide “brownies and babysitters,” it seems to be the best I have found. Also, although I do not have a “god shaped hole” that needs to be filled, I understand that many people do, and so I can see that it might be good to support an organization that will help to allow those people to fill that hole with something that does not encourage irrationality. On the other hand, sometimes I feel like Unitarians care a lot a lot more about the “free” part of “free and responsible search for truth and meaning” than the “responsible” part. I am worried that they like to discuss their individual beliefs as they would discuss their favorite colors, and never actually change. Maybe with our current messed up society, the first step is for people to feel free to believe what they want, and then learn how to be critical.
In attending Unitarian churches, I have repeatedly enjoyed myself, thought about interesting philosophy (even though I often disagree with the sermon), had sufficiently strong emotional responses from the music (e.g. “imagine”), and been encouraged by how much people were willing to help each other. I already know that I enjoy experience. What I am trying to decide is morally if I should be willing to support this organization. For the future, I am also trying to decide if I should be worried that being around this kind of thinking might be bad for my future kids.