I conclude from this that CDT should equal EDT (hence, causality must account for logical correlations, IE include logical causality). By “CDT” I really mean any approach at all to counterfactual reasoning; counterfactual expectations should equal evidential expectations.
As with most of my CDT=EDT arguments, this only provides an argument that the expectations should be equal for actions taken with nonzero probability. In fact, the amount lost to Dutch Book will be proportional to the probability of the action in question. So, differing counterfactual and evidential expectations are smoothly more and more tenable as actions become less and less probable.
I’m having a little trouble following the terminology here (despite the disclaimer).
One particular thing that confuses me—you say, “the expectations should be equal for actions taken with nonzero probability” and also “differing counterfactual and evidential expectations are smoothly more and more tenable as actions become less and less probable”, but I’m having trouble understanding how they could both be true. How does, “they’re equal for nonzero probability” match with “they move further and further apart the closer the probability gets to zero”? (Or are those incorrect paraphrases?)
It seems to me that if you have two functions that are equal whenever then input (the probability of an action) is nonzero, then they can’t also get closer and closer together as the input increases from zero—they’re already equal as soon as the input does not equal zero! I assume that I have misunderstood something, but I’m not sure which part.
“The expectations should be equal for actions with nonzero probability”—this means a CDT agent should have equal causal expectations for any action taken with nonzero probability, and EDT agents should similarly have equal evidential expectations. Actually, I should revise my statement to be more careful: in the case of epsilon-exploring agents, the condition is >epsilon rather than >0. In any case, my statement there isn’t about evidential and causal expectations being equal to each other, but rather about one of them being conversant across (sufficiently probable) actions.
“differing counterfactual and evidential expectations are smoothly more and more tenable as actions become less and less probable”—this means that the amount we can take from a CDT agent through a Dutch Book, for an action which is given a different casual expectation than evidential expectation, smoothly reduces as the probability of an action goes to zero. In that statement, I was assuming you hold the difference between evidential and causal expectations constant add you reduce the probability of the action. Otherwise it’s not necessarily true.
I’m having a little trouble following the terminology here (despite the disclaimer).
One particular thing that confuses me—you say, “the expectations should be equal for actions taken with nonzero probability” and also “differing counterfactual and evidential expectations are smoothly more and more tenable as actions become less and less probable”, but I’m having trouble understanding how they could both be true. How does, “they’re equal for nonzero probability” match with “they move further and further apart the closer the probability gets to zero”? (Or are those incorrect paraphrases?)
It seems to me that if you have two functions that are equal whenever then input (the probability of an action) is nonzero, then they can’t also get closer and closer together as the input increases from zero—they’re already equal as soon as the input does not equal zero! I assume that I have misunderstood something, but I’m not sure which part.
“The expectations should be equal for actions with nonzero probability”—this means a CDT agent should have equal causal expectations for any action taken with nonzero probability, and EDT agents should similarly have equal evidential expectations. Actually, I should revise my statement to be more careful: in the case of epsilon-exploring agents, the condition is >epsilon rather than >0. In any case, my statement there isn’t about evidential and causal expectations being equal to each other, but rather about one of them being conversant across (sufficiently probable) actions.
“differing counterfactual and evidential expectations are smoothly more and more tenable as actions become less and less probable”—this means that the amount we can take from a CDT agent through a Dutch Book, for an action which is given a different casual expectation than evidential expectation, smoothly reduces as the probability of an action goes to zero. In that statement, I was assuming you hold the difference between evidential and causal expectations constant add you reduce the probability of the action. Otherwise it’s not necessarily true.