I tried to cover what you’re talking about with my statement in brackets at the end of the first paragraph. Set the value for disagreeing too high and you’re rewarding it, in which case people start deliberately making randomised choices in order to disagree. Too low and they ought to be going out of their way to try and agree above all else—except there’s no way to do that in practice, and no way not to do it in the abstract analysis that assumes they think the same. A value of 9 though is actually in between these two cases—it’s exactly the average of the two agreement options, and it neither punishes nor rewards disagreement. It treats disagreement “fairly”, and in doing so entirely un-links the two agents. Which is exactly why I picked it, and why it simplifies the problem. Again I think I’m thinking of these values relatively while you’re thinking absolutely—a value of epsilon for disagreeing is not rewarding disagreeing slightly, it’s still punishing it severely relative to the other outcomes.
To me what it illustrates is that the linking between the two agents is something of an illusion in the first place. Punishing disagreement encourages the agents to collaborate on their vote, but the problem provides no explicit means for them to do so. Introducing an explicit means to co-operate, such as pre-commitment or having the agents run identical decision algorithms, would dissolve the problem into a clear solution (actually, explicitly identical algorithms makes it a version Newcomb’s Paradox, but that’s at least a well studied problem). It’s the ambiguity of how to co-operate combined with the strong motivation, lack of explicit means, and abundance of theoretical means to hand-wave agreement that creates the paradox.
As for the stuff you say about the probability and the bucket of coloured balls, I get all that. The original probability of the coin flip was 1⁄2 each way. The evidence that you’ve been asked to vote makes the subjective likelihood of tails 2⁄3. Also somehow the number 3⁄4 appears in the SSA solution to the Sleeping Beauty problem (which to me seems just flat-out wrong, and enough for me to write off that method unless I see a very good defence of it), which made me worry that somewhere out there was a method which somehow comes up with 3⁄4. So I covered my bases by saying “no method gives probability higher than 3/4”, which was the minimum neccesary requirement and what I figured was fairly safe statement. The reality is 2⁄3 is simply just correct for the subjective probability of tails, for reasons like you say, and maybe I just confuse things by mucking about trying to cover all possible bad solutions. It is I admit a little confusing to talk about whether anything is “more than 3/4″ when the only two values under serious consideration are the a-priori 1⁄2 and the subjective posterior 2⁄3.
Yeah, I didn’t know exactly what problem statement you were using (the most common formulation of the non-anthropic problem I know is this one), so I didn’t know “9” was particularly special.
Though since the point at which I think randomization becomes better than honesty depends on my P(heads) and on what choice I think is honest. So what value of the randomization-reward is special is fuzzy.
I guess I’m not seeing any middle ground between “be honest,” and “pick randomization as an action,” even for naive CDT where “be honest” gets the problem wrong.
which made me worry that somewhere out there was a method which somehow comes up with 3⁄4.
Somewhere in Stuart Armstrong’s bestiary of non-probabilistic decision procedures you can get an effective 3⁄4 on the sleeping beauty problem, but I wouldn’t worry about it—that bestiary is silly anyhow :P
I tried to cover what you’re talking about with my statement in brackets at the end of the first paragraph. Set the value for disagreeing too high and you’re rewarding it, in which case people start deliberately making randomised choices in order to disagree. Too low and they ought to be going out of their way to try and agree above all else—except there’s no way to do that in practice, and no way not to do it in the abstract analysis that assumes they think the same. A value of 9 though is actually in between these two cases—it’s exactly the average of the two agreement options, and it neither punishes nor rewards disagreement. It treats disagreement “fairly”, and in doing so entirely un-links the two agents. Which is exactly why I picked it, and why it simplifies the problem. Again I think I’m thinking of these values relatively while you’re thinking absolutely—a value of epsilon for disagreeing is not rewarding disagreeing slightly, it’s still punishing it severely relative to the other outcomes.
To me what it illustrates is that the linking between the two agents is something of an illusion in the first place. Punishing disagreement encourages the agents to collaborate on their vote, but the problem provides no explicit means for them to do so. Introducing an explicit means to co-operate, such as pre-commitment or having the agents run identical decision algorithms, would dissolve the problem into a clear solution (actually, explicitly identical algorithms makes it a version Newcomb’s Paradox, but that’s at least a well studied problem). It’s the ambiguity of how to co-operate combined with the strong motivation, lack of explicit means, and abundance of theoretical means to hand-wave agreement that creates the paradox.
As for the stuff you say about the probability and the bucket of coloured balls, I get all that. The original probability of the coin flip was 1⁄2 each way. The evidence that you’ve been asked to vote makes the subjective likelihood of tails 2⁄3. Also somehow the number 3⁄4 appears in the SSA solution to the Sleeping Beauty problem (which to me seems just flat-out wrong, and enough for me to write off that method unless I see a very good defence of it), which made me worry that somewhere out there was a method which somehow comes up with 3⁄4. So I covered my bases by saying “no method gives probability higher than 3/4”, which was the minimum neccesary requirement and what I figured was fairly safe statement. The reality is 2⁄3 is simply just correct for the subjective probability of tails, for reasons like you say, and maybe I just confuse things by mucking about trying to cover all possible bad solutions. It is I admit a little confusing to talk about whether anything is “more than 3/4″ when the only two values under serious consideration are the a-priori 1⁄2 and the subjective posterior 2⁄3.
Yeah, I didn’t know exactly what problem statement you were using (the most common formulation of the non-anthropic problem I know is this one), so I didn’t know “9” was particularly special.
Though since the point at which I think randomization becomes better than honesty depends on my P(heads) and on what choice I think is honest. So what value of the randomization-reward is special is fuzzy.
I guess I’m not seeing any middle ground between “be honest,” and “pick randomization as an action,” even for naive CDT where “be honest” gets the problem wrong.
Somewhere in Stuart Armstrong’s bestiary of non-probabilistic decision procedures you can get an effective 3⁄4 on the sleeping beauty problem, but I wouldn’t worry about it—that bestiary is silly anyhow :P