Decision Theory Notes

Newcomb’s Problem

(On the gene problem from Peterson, section 9.1, pg 202)

To paraphrase the problem, if you have the gene, then you will have the desire to read section 9.2 and have a miserable life. If you don’t have the gene, then you may or may not have any of the above. Also, if you read section 9.2, you will certainly pass; else, you will certainly fail. Finally, 99% of students taking the course are known to have the gene.

Should you read section 9.2 or not?

The straightforward reasoning is that if you have the gene and you read the section, you will pass the exam but have a miserable life. If you have the gene and don’t read, you will fail the exam and have a miserable life anyway. If you don’t have the gene, then if you read the section you will pass, else you will fail. In both cases, reading seems better than not reading. So, you should read the section.

But… if you have the desire to read the section, it’s evidence that you have the gene (and will have a miserable life)!

Hypothesis: The naive student seems to think that the gene causes one to actually read, and thus he can make himself not have the gene if he avoids reading. But the gene causes only your desire to read. If you intervene on your reading, you cut off any evidence it would have given you about the gene.

(But then, your intervention may have been caused by the gene. Maybe people without the gene always intervene to not read, and people with the gene don’t intervene.)

What is the analogy to Newcomb’s problem?

The choice we face there is whether to one-box or two-box. If Omega predicted that we would one-box and we one-box, then we get $1m. If it predicted that we would one-box and we two-box, we get $1m + $1k. If it predicted that we would two-box and we one-box, then we get $0. If it predicted that we would two-box and we two-box, we get $1k. For both predictions, two-boxing dominates one-boxing.

But… if you have the desire to two-box, it’s evidence that Omega would predict you would two-box (and thus give you just $1k).

What if you intervened on yourself to want to one-box?

Causal Models

Case 1: The gene causes only your initial desire (not your desire to intervene on yourself).

So, if the student has the desire to read but intervenes on himself to remove the desire to read, he will fail the exam and still have a miserable life, because he has the gene. Erasing the diagnosis on your medical report and writing “I’m perfectly healthy” doesn’t change whether you have cancer or not.

Case 2: The gene causes your final decision, whether the desire was natural or self-induced.

Your final decision will be “read” if you have the natural desire to read and you don’t avert it, or if you don’t have the natural desire but you induce it. Your final decision will be “don’t read” if you have the natural desire to read but avert it, or if you don’t have the natural desire and you don’t intervene.

For your gene to be the cause of your final decision, its presence must cause you to have the desire and not want to change it or must cause you to not have the natural desire but want to induce it. Its absence must do the opposite.

In this case, it seems like what you choose to do will give you evidence about whether or not you have the gene.

Is your Decision Evidence?

Hypothesis: The key question is whether your decision itself is evidence about the gene.

Remember that you cannot intervene on a system from the inside (by the definition of “intervene”). You can only do that from the outside.

Hypothesis: CDT assumes that the final decision is caused by your natural desire and your free will. The gene causes your natural desire, but your free will can override it. Thus, if you choose to exert your free will, then it doesn’t give you any evidence about the gene and thus you should do what is “best”.

Alternative theory: the final decision is caused by your natural desire and your desire to intervene. The gene causes both of these. So, if you “choose” to intervene on your natural desire, it is still evidence about whether you have the gene. Therefore, you should take that into account when deciding.

Corollary: In Newcomb’s problem, CDT assumes that your final decision to one-box or two-box is caused by your natural desire and your free will. So, if you choose to exert your free will, it doesn’t give you any evidence about whether Omega predicted you would one-box or two-box, and thus you should just pick what is “best” (i.e., two-boxing).

The alternative theory assumes that your final decision to one-box or two-box is caused by your natural desire and your desire to intervene. Omega can predict both of these. So, if you “choose” to intervene on your natural desire, it is still evidence about what Omega predicted. Therefore, you should take that into account when deciding.

Hypothesis: The key difference seems to be that CDT assumes you can intervene freely (i.e., that you have “free will” and can choose independently of the gene or the past state that Omega observed) and this alternative theory assumes that you cannot intervene freely (i.e., that you don’t have “free will” and will choose something firmly decided by the gene or the past state that Omega observed).