## A note on the scaling of desirability

In The Logic of Decision, Richard Jeffrey pointed out that the desirability (or "news value") of a proposition can be usefully understood as a weighted average of the desirability of different ways in which the proposition can be true, weighted by their respective probability. That is, if A and B are incompatible propositions, then

(1) Des(AvB) = Des(A)P(A/AvB) + Des(B)P(B/AvB).

So desirabilities are affected by probabilities. If you prefer A over B and just found out that conditional on their disjunction, A is more likely then B, then the desirability of the disjunction goes up. That seems right.

To factor out the effect of an agent's beliefs on her desirabilities, we can look at the desirability of maximally specific propositions ("possible worlds"): if a proposition can't be realized in several ways, its desirability doesn't depend on the relative probability of these ways. Let's call an agent's desirability distribution V over possible worlds her basic values. Assuming for simplicity that the number of worlds is finite, equation (1) tells us how basic values and probabilities together determine the desirability of arbitrary propositions (with positive probability):

(2) Des(A) = \sum_w V(w)P(w/A).

Jeffrey also proposed to normalize the desirability scale so that the desirability of any tautology T is zero:

(3) Des(T) = 0.

Again that makes intuitive sense. Propositions that carry good news have positive desirability, and propositions that carry bad news have negative desirability.

However, a problem arises when we look at conditional beliefs. If P is a probability function and B a proposition, let P_B be the function P conditionalized on B. Intuitively, P_B(A) is the probability of A after learning or supposing B. Similarly, let's define Des_B(A) as the desirability of A after learning or supposing B, assuming that the basic values remain the same. By equation (2),

(4) Des_B(A) = \sum_v V(w)P_B(w/A) = \sum_w V(w)P(w/A&B) = Des(A&B).

So the desirability of A given B equals the desirability of A&B. As a special case, the desirability of A given A equals the desirability of A:

(5) Des_A(A) = \sum_v V(w)P_A(w/A) = \sum_w V(w)P(w/A) = Des(A).

In the literature on "desire as belief", (5) is known as the Invariance assumption.

In an interesting recent paper (Stefansson 2014), Orri Stefansson argues that the Invariance assumption is false, and indeed inconsistent with Jeffrey's account of desirability. Since we've just derived the assumption from Jeffrey's account, this would imply that the account is inconsistent.

Orri's argument is quite simple: Any proposition that is known or supposed has probability 1, relative to the updated probability function. The supposed proposition is no longer "news" and shouldn't have either positive or negative news value. More specifically, if the normalization (3) holds for the updated desirability measure Des_A, then Des_A(A) must equal 0, since P_A(w/A) = P_A(w/T) and so

(6) Des_A(A) = \sum_w V(w)P_A(w/A) = \sum_w V(w)P_A(w/T) = Des_A(T) = 0.

Combining (6) with the Invariance assumption (5), we could infer that Des(A) = 0 for all propositions A with positive probability, which is clearly absurd. What went wrong?

When we derived (5), we assumed that the basic values remain constant as we move from Des to Des_A. However, this is incompatible with the assumption we used to derive (6), that Des_A(T) = 0, except in the special case where Des(A) = 0.

Consider a toy example. There are two possible worlds, w1 and w2, with basic values V(w1) = +1 and V(w2) = -1. The two worlds have equal probability, so the desirability of their disjunction is 0. Now the agent learns { w2 }. What is the new desirability of { w2 }? If basic values are preserved, it is +1. If tautologies have desirability 0, it is 0, for Des(T) = \sum_w V(w)P(w) = V(w2).

So we have to choose: either the basic values are preserved, or the neutrality of tautologies is preserved. We can't have both.

It would be wrong to think that one choice is right and the other wrong. As Jeffrey pointed out in Jeffrey 1977, the difference is a difference of scaling. If we hold fixed the basic values, we measure the new desirabilities on a scale whose zero is fixed by the old desirabilities. If instead we hold fixed the neutrality of tautologies, we measure the new desirabilities on a new scale in which the zero is moved to the previous desirability of the supposed proposition.

The problem resembles a well-known problem arising in semantics. If 'today' always denotes the present day, will it be true tomorrow that it is today? In one sense yes, in another sense no. Since the operator 'tomorrow' shifts the time of reference a day into the future, we can evaluate 'today' either with respect to the shifted time (call this the monstrous interpretation, after Kaplan) or with respect to the unshifted time of utterance (call this the tame interpretation). Both interpretations are intelligible and can be useful.

That said, in general I think it is advisable to not switch between different scales. So I'd recommend the tame reading of Des_A and Des_B on which Des_A(A) = Des(A).

On the monstrous reading, Des_A(A) = 0. More generally, since rescaling the desirabilities puts the new zero at the point previously occupied by the supposed proposition,

(7) Des_B(A) = Des(A&B) - Des(B).

The measure (7) for conditional desirability is defended in Bradley 1999 by an argument along the following lines. Suppose A and B are independent and each is worth $5 to you, so that the truth of A&B is worth$10. How much should you be willing to pay for A on the supposition that B is already true? Plausibly $5, not$10.

I guess that shows that if we want to use the heuristic of measuring desirability by how much you'd be willing to spend, then there's something intuitive about the monstrous reading that validates (7) rather than (4). Other heuristics favour the tame reading. For example, suppose we gauge the desirability of a state of affairs by how happy you would feel upon learning that the state obtains. Then the desirability of A on the supposition B plausibly equals the desirability of A&B.

In any case, I think it is wrongheaded to argue over the correctness of the monstrous or the tame reading. We don't have a clear enough pre-theoretic notion of conditional desirability to settle the question. Indeed, orthodox decision theory assumes that desirabilities are only defined up to positive linear transformation. But even if we assume a more substantively realist attitude towards desirability, there shouldn't be any debate once we see that the difference between the two accounts is just a matter of scaling. It's like arguing whether physical temperature is in Celsius or in Fahrenheit.

To some extent the fact that we can't preserve both basic values and the neutrality of tautologies across conditionalization actually supports the orthodox decision-theoretic stance, on which any choice of a zero and a unit is merely conventional. We should take care not to draw substantive conclusions from the choice of a scale.

(Jeffrey pretty much said all that in his 1977 paper, but it seems worth repeating, since the point is not obvious and people don't seem read that paper any more.)

PS. Causal decision theorists distinguish desirability from choiceworthiness. As James Joyce argues in his Foundations of Causal Decision Theory, both measures arise from the same basic value measure V. The difference is that desirability is computed in terms of ordinary conditional probabilities P(w/A), while choiceworthiness is computed in terms of subjunctive conditional probabilities P(w//A); that is,

(8) Cho(A) = \sum_w V(w)P(w//A).

Does Choiceworthiness satisfy Invariance? That is, does Cho_A(A) always coincide with Cho(A)? The answer is no. For note that P_A(w//A) = P(w/A). So

(9) Cho_A(A) = \sum_w V(w)P(w/A) = Des_A(A) = Des(A).

If Cho_A(A) = Cho(A), it would follow that choiceworthiness coincides with desirability.

On the other hand, choiceworthiness satisfies a subjunctive form of invariance: let Cho^B(A) be the choiceworthiness of A on the subjunctive supposition that B. Then

(10) Cho^A(A) = \sum_w V(w)P^A(w//A) = \sum_w V(w)P(w//A) = Cho(A).

Orri also argues against (10). He writes:

When I suppose that contrary to fact, A is true, I imagine myself to be in a counterfactual situation where I believe A to be true. But relative to that situation, A is neither more nor less desirable or valuable ... than the tautology.

The argument, I take it, is that since (i) P^A(A) = 1 and (ii) if A is certain, then Cho(A) = \sum_w V(w)P(w//A) = \sum_w V(w)P(w) = Des(T) = 0, it follows that the choiceworthiness of A relative to P^A is 0.

The argument again turns on the assumption that a different choiceworthiness scale should be used for Cho^A than for Cho. If we hold fixed the basic values and thus the scale of choiceworthiness, then A can have non-zero choiceworthiness relative to P^A even though it is certain according to P^A.