Confusion about Newcomb is confusion about counterfactuals

(This is the first, and most new­comer-ac­cessible, post in a planned se­quence.)

New­comb’s Prob­lem:

Joe walks out onto the square. As he walks, a ma­jes­tic be­ing flies by Joe’s head with a box la­beled “brain scan­ner”, drops two boxes on the ground, and de­parts the scene. A passerby, known to be trust­wor­thy, comes over and ex­plains...

If Joe aims to get the most money, should Joe take one box or two?

What are we ask­ing when we ask what Joe “should” do? It is com­mon to cash out “should” claims as coun­ter­fac­tu­als: “If Joe were to one-box, he would make more money”. This method of trans­lat­ing “should” ques­tions does seem to cap­ture some­thing of what we mean: we do seem to be ask­ing how much money Joe can ex­pect to make “if he one-boxes” vs. “if he two-boxes”. The trou­ble with this trans­la­tion, how­ever, is that it is not clear what world “if Joe were to one-box” should re­fer to—and, there­fore, it is not clear how much money we should say Joe would make, “if he were to one-box”. After all, Joe is a de­ter­minis­tic phys­i­cal sys­tem; his cur­rent state (to­gether with the state of his fu­ture self’s past light-cone) fully de­ter­mines what Joe’s fu­ture ac­tion will be. There is no Phys­i­cally Irre­ducible Mo­ment of Choice, where this same Joe, with his own ex­act ac­tual past, “can” go one way or the other.

To restate the situ­a­tion more clearly: let us sup­pose that this Joe, stand­ing here, is poised to two-box. In or­der to de­ter­mine how much money Joe “would have made if he had one-boxed”, let us say that we imag­ine reach­ing in, with a mag­i­cal sort of world-surgery, and al­ter­ing the world so that Joe one-boxes in­stead. We then watch to see how much money Joe re­ceives, in this sur­gi­cally al­tered world.

The ques­tion be­fore us, then, is what sort of mag­i­cal world-surgery to ex­e­cute, be­fore we watch to see how much money Joe “would have made if he had one-boxed”. And the difficulty in New­comb’s prob­lem is that there is not one but two ob­vi­ous world-surg­eries to con­sider. First, we might sur­gi­cally reach in, af­ter Omega’s de­par­ture, and al­ter Joe’s box-tak­ing only—leav­ing Omega’s pre­dic­tion about Joe un­touched. Un­der this sort of world-surgery, Joe will do bet­ter by two-box­ing:

Ex­pected value ( Joe’s earn­ings if he two-boxes | some un­changed prob­a­bil­ity dis­tri­bu­tion on Omega’s pre­dic­tion ) >
Ex­pected value ( Joe’s earn­ings if he one-boxes | the same un­changed prob­a­bil­ity dis­tri­bu­tion on Omega’s pre­dic­tion ).

Se­cond, we might sur­gi­cally reach in, af­ter Omega’s de­par­ture, and si­mul­ta­neously al­ter both Joe’s box-tak­ing and Omega’s pre­dic­tion con­cern­ing Joe’s box-tak­ing. (Equiv­a­lently, we might reach in be­fore Omega’s de­par­ture, and sur­gi­cally al­ter the in­sides of Joe brain—and, thereby, al­ter both Joe’s be­hav­ior and Omega’s pre­dic­tion of Joe’s be­hav­ior.) Un­der this sort of world-surgery, Joe will do bet­ter by one-box­ing:

Ex­pected value ( Joe’s earn­ings if he one-boxes | Omega pre­dicts Joe ac­cu­rately) >
Ex­pected value ( Joe’s earn­ings if he two-boxes | Omega pre­dicts Joe ac­cu­rately).

The point: New­comb’s prob­lem—the prob­lem of what Joe “should” do, to earn most money—is the prob­lem which type of world-surgery best cashes out the ques­tion “Should Joe take one box or two?”. Disagree­ment about New­comb’s prob­lem is dis­agree­ment about what sort of world-surgery we should con­sider, when we try to figure out what ac­tion Joe should take.