CDT Dutch Book
In a previous post, I speculated that you might be able to Dutch-Book CDT agents if their counterfactual expectations differed from the conditional expectations of EDT. The answer turns out to be yes.
I’m going to make this a short note rather than being very rigorous about the set of decision problems for which this works.
(This is an edited version of an email, and benefits from correspondence with Caspar Oesterheld, Gerard Roth, and Alex Appel. In particular, Caspar Oesterheld is working on similar ideas. My views on how to interpret the situation have changed since I originally wrote these words, but I’ll save that for a future post.)
Suppose a CDT agent has causal expectations which differ from its evidential expectations, in a specific decision.
We can modify the decision by allowing an agent to bet on outcomes in the same act. Because the bet is made simultaneously with the decision, the CDT agent uses causal expected value, and will bet accordingly.
Then, immediately after (before any new observations come in), we offer a new bet about the outcome. The agent will now bet based on its evidential expectations, since the causal intervention has already been made.
For example, take a CDT agent in Death in Damascus. A CDT agent will take each action with 50% probability, and its causal expectations expect to escape death with 50% probability. We can expand the set of possible actions from (stay, run) to (stay, run, stay and make side bet, run and make side bet). The side bet could cost 1 util and pay out 3 utils if the agent doesn’t die. Then, immediately after taking the action but before anything else happens, we offer another deal: the agent can get .5 util in exchange for −3 util conditional on not dying. We offer the new bet regardless of whether the agent agrees to the first bet.
The CDT agent will happily make the bet, since the expected utility is calculated along with the intervention. Then, it will happily sell the bet back, because after taking its action, it sees no chance of the 3 util payout.
The CDT agent makes the initial bet even though it knows it will later reverse the transaction at a cost to itself, because we offer the second transaction whether the agent agrees to the first or not. So, from the perspective of the initial decision, taking the bet is still +.5 expected utils. If it could stop itself from later taking the reverse bet, that would be even better, but we suppose that it can’t.
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. Actions with very low probability will imply negligible monetary loss. Still, in terms of classical Dutch-Book-ability, CDT is Dutch-Bookable.
Both CDT and EDT have dynamic inconsistencies, but only CDT may be Dutch-booked in this way. I’m not sure how persuasive this should be as an argument—how special a status should Dutch-book arguments have?
ETA: The formalization of this is now a question.