## Marlboro Man

18 March 2009I've been taking another run at Subjective Probability: The Real Thing (2004) by Richard C. Jeffrey. I'd started reading it a while back, but got distracted. Anyway, Jeffrey was an important subjectivist — someone who argued that probability is a measure of belief, and that *any* degree of belief that does not violate certain rationality constraints is permitted. (As I have noted earlier, the *subjectivism* here is in the assignment of quantities not specifically required by objective criteria. The subjectivists believe either that quantity by reason *must* be assigned, albeït often arbitrarily, or that Ockham's Razor is not a binding constraint.) And the posthumous Subjective Probability was his final statement.

At some point, I encountered the following entry in the index:

Nozick, Robert, 119, 123

which entry was almost immediately annoying. Page 119 is in the References

section, and indeed has the references for Nozick, but that's a pretty *punk* thing to drop in an index. Even more *punk* would be an index entry that refers to *itself*; and, indeed, page 123 is in the index, and it is on that page that one finds Nozick, Robert, 119, 123

.

Well, actually, I'd forgot something about this book, which is probably an artefact of its being *posthumous*: Most or all of the index entries are *off by ten pages*, such that one ought to *translate* Nozick, Robert, 119, 123

to Nozick, Robert, 109, 113

. And, yes, there are references to Nozick on those pages (which are part of a discussion of Newcomb's Problem and of related puzzles). It was just chance-coïncidence that ten pages later one found the listings in the references and in the index.

In decision theory, there are propositions call independence

axiomata. The first such proposition to be explicitly advanced for discussion (in an article by Paul Anthony Samuelson) is the *Strong* Independence Axiom, the gist of which is that the value of a reälized outcome is independent of the probability that it had before it was reälized. Say that we had a lottery of possible outcomes `X _{1}`,

`X`,…

_{2}`X`, each

_{n}`X`having associated probability

_{i}`p`. If we assert that the expected value of this lottery were

_{i}Σ[p·_{i}u(X)]_{i}

where `u`( ) is some utility function, then (amongst other things) we've accepted an independence proposition. Otherwise, we may have to assert something such as that the expect value were

Σ[p·_{i}u(X,_{i}p)]_{i}

to account for such things as people taking an *unlikely* million dollars to be somehow *better* than a likely million dollars.

Anyway, there's another proposition which to most of us doesn't *look* like the Strong Independence Axiom, and yet is pretty much the same thing, the Sure Thing

Principle, which is associated with Leonard Jimmie Savage (an important subjectivist, whom I much admire, and with whom I markèdly disagree). Formally, it's thus:

{[(A∧B) prefC] ∧ [(A∧ ¬B) prefC]} ⇒ (AprefC)

Less formally,

If the combination ofAandBis preferred toC, and the combination ofAwithoutBis preferred toC, thenAis just plain preferred toC, regardless ofB.

Savage gives us the example of a businessman trying to decide whether to buy a piece of property with an election coming-up. He thinks-through whether he would be better off with the property if a Democrat is elected, and decides that he would prefer that he had bought the property in that case. He thinks-through whether he would be better off with the property if a Republican is elected, and decides that he would prefer that he had bought the property in that case. So he buys the property. This seems *very* reasonable.

But there is a famous class of counter-examples, presented by Jeffrey in the form of the case of the Marlboro Man

. The hypothetical Marlboro Man is trying to decide whether to *smoke*. He considers that, if he should live a *long* life, he would wish at its end that he had enjoyed the pleasure of smoking. He considers that, if he should live a short life, he would wish at its end that he had enjoyed the pleasure of smoking. So he smokes. That doesn't seem nearly so reasonable.

There is an underlying difference between our two examples. The businessman would not normally expect his choice to affect the outcome of the election; the Marlboro Man ought to expect his choice to affect the length of his life. Jeffrey asserts that Savage only meant the Sure Thing Principle to hold in cases where the probability of `B` were independent of `A`.

But what makes the discussion *poignant* is this: Jeffrey, dying of surfeit of Pall Malls

, wrote this book *as* his last, and passed-away from lung cancer on 9 November 2002.

Tags: decision theory, probability

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