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.