The result there is sometimes abbreviated as UDT=CDT+SIA, although UDT⊂CDT+SIA is more accurate, because the optimal UDT policies are a subset of the policies which CDT+SIA can follow. This is because UDT has self-coordination power which CDT+SIA lacks.
I feel like this understates the way in which CDT+SIA is philosophically/intuitively crazy/implausible. Consider a variant of AMD where U(A)=1, U(B)=0, U(C)=2. Obviously one should select CONT with probability 1 in order to reach C, but “EXIT with probability 1” seems to be another CDT+SIA solution. The CDT+SIA reasoning there (translated from math to English) is: Suppose my policy is “EXIT with probability 1″. Then I’m at X with probability 1. Should I deviate from this policy? If I do CONT instead, I’m still at X with probability 1 and the copy of me at Y will still do EXIT with probability 1 so I’ll end up at B for sure with utility 0, therefore I should not deviate. Isn’t this just obviously crazy (assuming I didn’t misunderstand something)?
we could say that UDT1.0 = CDT+SIA
But UDT1.0 already gives a unique and correct solution to the problem above.
Caspar Oesterheld commented on that post with an analogous EDT+SSA result.
I tried to understand Caspar’s EDT+SSA but was unable to figure it out. Can someone show how to apply it to an example like the AMD to help illustrate it?
> I tried to understand Caspar’s EDT+SSA but was unable to figure it out. Can someone show how to apply it to an example like the AMD to help illustrate it?
Sorry about that! I’ll try to explain it some more. Let’s take the original AMD. Here, the agent only faces a single type of choice—whether to EXIT or CONTINUE. Hence, in place of a policy we can just condition on p when computing our SSA probabilities. Now, when using EDT+SSA, we assign probabilities to being a specific instance in a specific possible history of the world. For example, we assign probabilities of the form PSSA(X in XYB∣p), which denotes the probability that given I choose to CONTINUE with probability p, history XYB (a.k.a. CONTINUE, EXIT) is actual and that I am the instance intersection X (i.e., the first intersection). Since we’re using SSA, these probabilities are computed as follows:
PSSA(X in XYB∣p)=P(XYB∣p)⋅PSSA(X∣XYB)=P(XYB∣p)⋅12.
That is, we first compute the probability that the history itself is actual (given p). Then we multiply it by the probability that within that history I am the instance at X, which is just 1 divided by the number of instances of myself in that history, i.e. 2.
Now, the expected value according to EDT + SSA given p can be computed by just summing over all possible situations, i.e. over all combinations of a history and a position within that history and multiplying the probability of that situation with the utility given that situation:
PSSA(X in XYB∣p)⋅4+PSSA(X in XYC∣p)⋅1+PSSA(X in XA∣p)⋅0+PSSA(Y in XYB∣p)⋅4+PSSA(Y in XYC∣p)⋅1=12PSSA(XYB∣p)⋅4+12PSSA(XYC∣p)⋅1+12PSSA(XYB∣p)⋅4+12PSSA(XYC∣p)⋅1=PSSA(XYB∣p)⋅4+PSSA(XYC∣p)⋅1
And that’s exactly the ex ante expected value (or UDT-expected value, I suppose) of continuing with probability p. Hence, EDT+SSA’s recommendation in AMD is the ex ante optimal policy (or UDT’s recommendation, I suppose). This realization is not original to myself (though I came up with it independently in collaboration with Johannes Treutlein) -- the following papers make the same point:
Thanks, I think I understand now, and made some observations about EDT+SSA at the old thread. At this point I’d say this quote from the OP is clearly wrong:
So, we could say that CDT+SIA = EDT+SSA = UDT1.0; or, CDT=EDT=UDT for short.
I feel like this understates the way in which CDT+SIA is philosophically/intuitively crazy/implausible. Consider a variant of AMD where U(A)=1, U(B)=0, U(C)=2. Obviously one should select CONT with probability 1 in order to reach C, but “EXIT with probability 1” seems to be another CDT+SIA solution. The CDT+SIA reasoning there (translated from math to English) is: Suppose my policy is “EXIT with probability 1″. Then I’m at X with probability 1. Should I deviate from this policy? If I do CONT instead, I’m still at X with probability 1 and the copy of me at Y will still do EXIT with probability 1 so I’ll end up at B for sure with utility 0, therefore I should not deviate. Isn’t this just obviously crazy (assuming I didn’t misunderstand something)?
But UDT1.0 already gives a unique and correct solution to the problem above.
I tried to understand Caspar’s EDT+SSA but was unable to figure it out. Can someone show how to apply it to an example like the AMD to help illustrate it?
> I tried to understand Caspar’s EDT+SSA but was unable to figure it out. Can someone show how to apply it to an example like the AMD to help illustrate it?
Sorry about that! I’ll try to explain it some more. Let’s take the original AMD. Here, the agent only faces a single type of choice—whether to EXIT or CONTINUE. Hence, in place of a policy we can just condition on p when computing our SSA probabilities. Now, when using EDT+SSA, we assign probabilities to being a specific instance in a specific possible history of the world. For example, we assign probabilities of the form PSSA(X in XYB∣p), which denotes the probability that given I choose to CONTINUE with probability p, history XYB (a.k.a. CONTINUE, EXIT) is actual and that I am the instance intersection X (i.e., the first intersection). Since we’re using SSA, these probabilities are computed as follows:
That is, we first compute the probability that the history itself is actual (given p). Then we multiply it by the probability that within that history I am the instance at X, which is just 1 divided by the number of instances of myself in that history, i.e. 2.
Now, the expected value according to EDT + SSA given p can be computed by just summing over all possible situations, i.e. over all combinations of a history and a position within that history and multiplying the probability of that situation with the utility given that situation:
And that’s exactly the ex ante expected value (or UDT-expected value, I suppose) of continuing with probability p. Hence, EDT+SSA’s recommendation in AMD is the ex ante optimal policy (or UDT’s recommendation, I suppose). This realization is not original to myself (though I came up with it independently in collaboration with Johannes Treutlein) -- the following papers make the same point:
Rachael Briggs (2010): Putting a value on Beauty. In Tamar Szabo Gendler and John Hawthorne, editors, Oxford Studies in Epistemology: Volume 3, pages 3–34. Oxford University Press, 2010. http://joelvelasco.net/teaching/3865/briggs10-puttingavalueonbeauty.pdf
Wolfgang Schwarz (2015): Lost memories and useless coins: revisiting the absentminded driver. In: Synthese. https://www.umsu.de/papers/driver-2011.pdf
My comment generalizes these results a bit to include cases in which the agent faces multiple different decisions.
Thanks, I think I understand now, and made some observations about EDT+SSA at the old thread. At this point I’d say this quote from the OP is clearly wrong:
In fact UDT1.0 > EDT+SSA > CDT+SIA, because CDT+SIA is not even able to coordinate agents making the same observation, while EDT+SSA can do that but not coordinate agents making different observations, and UDT1.0 can (probably) coordinate agents making different observations (but seemingly at least some of them require UDT1.1 to coordinate).
Agreed. I’ll at least edit the post to point to this comment.