## Desire Reflection

Bas van Fraassen's Reflection Principle says that your current beliefs should be in line with your current beliefs about your future beliefs. More precisely,

PRB:P_1(A | P_2(A)=x) = x.

P_1 is your credence at time 1, P_2 your credence at time 2. PRB says that conditional on the assumption that at time 2 you believe A to degree x, you should already believe A to degree x at time 1. For agents who believe that they will (or might) change their beliefs in irrational ways between the two times, PRB is not a reasonable demand: if you know that you will be hit on the head tomorrow and consequently believe that the Earth is flat, you shouldn't believe that the Earth is flat now. On the other hand, if you're certain you will not change your beliefs in any such irrational way between now and tomorrow, then PRB is reasonable: suppose tomorrow you will believe that the Earth is flat by rationally responding to some very surprising new information; then you can infer that there exists some such information strongly supporting that the Earth is flat. But the fact that there is evidence for P is of course itself evidence for P. Hence you should already believe today that the Earth is probably flat.

One can express a similar principle for degrees of desire: your current degrees of desire should be in line with your current beliefs about your future desires. That is,

PRD:D_1(A | D_2(A)=x) = x.

Again, this is only plausible if you assume that changes in degrees of desire between time 1 and time 2 will not come about by knocks on the head, but in response to relevant evidence. For instance, suppose you know that tomorrow you will find out something in response to which you will desperately want to travel to Hamilton next weekend. Then you should already want to travel to Hamilton next weekend now.

The two principles are closely related. In fact, under the no-knocks-on-the-head constraints on which both are plausible, PRB entails PRD. (Proof sketch: *suppose* D_2(A) = x, and that you know this will be the rational response to some new information. Let I be a partition of all possible evidence you might get between time 1 and time 2, and let E be the disjunction of all i from I for which D_1(A | i) = x. Since you know that you will have learned one of those i at time 2, P_1(P_2(E)=1) = 1. Hence by PRB, P_1(E) = 1. And by the definition of E, D_1(A | E) = x. The last two facts together entail that D_1(A) = x. Hence D_1(A) = x *on the assumption* that D_2(A) = x: D_1(A | D_2(A)=x) = x. QED.)

Frank Arntzenius, in his paper "No Regrets, or: Edith Piaf Revamps Decision Theory" (Erkenntnis 2008, PDF at Springer or here), argues that evidential decision theory is inadequate because it violates the Reflection Principle for Desire. This would be surprising, as then evidential decision theory could be expected to violate the Reflection Principle for Belief as well.

Arntzenius uses a slightly different version of the two principles. His Reflection Principle for Belief is

PRB': P_1(A) = E_1(P_2(A)),

where E_1(P_2(A)) is the expectation value of P_2(A), that is, the average of possible P_2(A)-values weighted by their P_1-probability. PRB' is entailed by PRB, but not conversely. Arntzenius's Principle for Desire is the parallel

PRD': D_1(A) = E_1(D_2(A)).

Here is his argument that evidential decision theory violates PRD'.

Mary faces a Newcomb problem: in front of her is a transparent box containing $1 and an opaque box. She can take either just the opaque box or both. The guy who set up the boxes has tried to predict what she will do and has put $10 in the opaque box iff he predicted her not to take both boxes. The success rate of his predictions is 90%. Using evidential decision theory, Mary calculates the desirability of taking one box as 0.9 * $10 + 0.1 * $0 = $9, and the desirability of taking both boxes as 0.9 * $1 + 0.1 * $11 = $2. (For simplicity, we measure desirability in dollars.)

Sometime later, Mary has forgotten what she ended up doing (or perhaps she never knew it because she delegated the decision to somebody else). At this point, she gets to see what is in the opaque box. There are two possibilities. Either the box contains $10 or it is empty. In the first case, Mary will then hope she chose both boxes: given her new information, the desirability of one-boxing is $10 and the desirability of two-boxing is $11. In the second case, if the box is empty, she will also hope she chose both boxes: the desirability of one-boxing is then $0 and the desirability of two-boxing $1. Finally, assume that Mary knew all along that she would eventually find out what's in the boxes. Then at the time before she made her decision, she preferred one-boxing over two-boxing even though she could be certain that afterwards she will prefer to have been a two-boxer.

To see how Mary violates PRD', consider Mary's initial expectation value for her future desirability of two-boxing, E_1(D_2(2B)). Suppose she is fairly certain that she will one-box. So she is confident that she will later find $10 in the opaque box, and thus give desirability $11 to being a two-boxer. So her expectation value for her future desirability of two-boxing is around $10, while her present desirability of two-boxing is $2.

I don't quite see why the Newcomb setup is relevant to this argument. Here is another version.

Mary faces a lottery. She can either buy a ticket for $1 or not. If she buys the ticket, there's a 50% chance that she will win $1000. If she doesn't buy it, the chance of winning (due to an administrative error) is 0.0001. She figures out that it is worth buying the ticket.

Sometime later, Mary has forgotten what she ended up doing (or perhaps she never knew it). At this point, she learns whether she won the lottery or not. In the first case, if she won, Mary will hope she didn't buy a ticket: given her new information, the desirability of not having bought a ticket is $1000, the desirability of having bought one is $999. In the second case, if she didn't win, she will also hope she didn't buy a ticket: the desirability of having bought one is $-1, the desirability of not having bought one is $0. Hence in the outset, Mary preferred buying a ticket over not buying one even though she could be certain that afterwards she will prefer to not have bought a ticket.

Again, we can consider Mary's initial expectation value for her future desirability of not buying a ticket, E_1(D_2(N)). If she is confident that she will buy a ticket, she will give credence 0.5 to winning the lottery and thus to her future desirability of not buying a ticket being $1000. So her expectation value for her future desirability of not buying a ticket is around $500, while her present desirability of not buying a ticket is close to $0.

I can't find any mistake in this reasoning, nor in Mary's attitudes or decisions. Her violation of PRD' is perfectly rational. This means that we should reject PRD' as a principle of rationality, even under the idealized conditions under which PRB' is okay.

What then about my argument that Desire Reflection is entailed by Belief Reflection under those conditions? The argument was that under those conditions, PRB entails PRD. But Arntzenius's Principle isn't PRD. It is PRD'. And while PRD follows from PRB, PRD' does not follow from PRB', nor from PRB. (That's because the law of alternatives, P(A) = \sum_X P(A | X) * X, where X partitions logical space, does not carry over to desirability: D(A) != \sum_X D(A | X) * X.)

Does Mary violate PRD? No. The desirability of A given B is just the desirability of A & B. Hence D_1(A | D_2(A)=x) = D_1(A & D_2(A)=x). In the lottery case, Mary knows that tomorrow her desirability of not buying a ticket will either be $1000 or $0, depending on whether she wins or not. In particular, D_2(N)=$1000 iff she wins. So N & D_2(N)=$1000 is the proposition that she wins without buying a ticket. And the desirability of that is $1000. So D_1(N | D_2(N)=$1000) = $1000. Mary satisfies PRD.

The upshot is that under the usual constraints, a Reflection Principle for Desire in the van Fraassen style (PRD) is quite plausible. A Reflection Principle for Desire in the Arntzenius style (PRD') is not. It is perfectly okay to do things of which one is certain that one will later hope that one didn't do them!

(I should thank Weng Hong Tang and Grant Reaber for discussions, and I should mention that I haven't read Arntzenius's paper very carefully: I might have completely missed his point. Anyway, I thought all this is kinda interesting.)

"suppose tomorrow you will believe that the Earth is flat by rationally responding to some very surprising new information; then you can infer that there exists some such information strongly supporting that the Earth is flat. But the fact that there is evidence for P is of course itself evidence for P. Hence you should already believe today that the Earth is probably flat."

I wonder how to read that reflection principle. Suppose I'm running the high hurdles today and the hurdles on the track are difficult to clear. Tomorrow I will be running the low hurdles and will believe that "the hurdles on the track are easy to clear". Since I know now that there will be convincing evidence that the hurdles are the track are easy to clear, should I believe now that the hurdles on the track are easy to clear?