Saturday, August 20, 2011

Commonplaces and Comments 5

I'm working on a very long post on a topic that's sure to be of local interest. In the meantime, I've found a copy of Richard Jeffrey's Logic of Decision1, in which his generalization of Bayesian conditioning, probability kinematics, was first published. Decision theory is one of the greatest applications of Bayesian instruments, so I surprise myself in not having discussed it earlier. To motivate interest, I'll discuss what is probably the most famous decision-theoretic argument of all time, Pascal's wager.

As those familiar with the argument know, Pascal's wager is an argument for the strategy of belief in Christian theology, not an argument for its truth. Hitchens dismisses this approach as mere hucksterism, but it need not be so crude. Pascal did not require that we flick on the `believe in God' switch in our heads, nor that we attempt to fool God through insincere belief. Rather, the intended conclusion of the argument is this: that one should by the best strategy available attempt to arrive at sincere Christian belief.

Pascal's argument is often presented using a decision-matrix (see the previous link). If we accept his options - Christianity or atheism - as being exhaustive, the payoff of correctly wagering on Christianity (infinite) is greater than the payoff of correctly wagering against it (finite), just as it is greater than the payoff of incorrect wagers with respect to Christianity (finite). If we denote the payoff of wagering correctly on atheism by x and wagering correctly on Christianity by y, the expected utility of wagering on Christianity is
prob(Christianity)*y+prob(Atheism)*x. So as long as the probability of Christianity is greater than zero, the payoff is infinite. Similarly, wagering against Christianity has `infinitely negative' expected utility. So even if one ascribes prob(Atheism)=1-1/[Graham's number], the infinite payoffs of correct Christian belief swamp the expected utilities.

The most common objection to Pascal's wager - which I think sound - is that the payoff matrix is not comprehensive, i.e. the argument is unsound. So long as we can imagine differing possible `infinite' payoffs against Christian belief, no best strategy follows. But there is another, more interesting objection - which I discovered via Alan Hájek - which disputes the validity of the argument, not the soundness.

To see why, recall that the intended conclusion of the argument is to convince skeptics that a strategy maximizing the likelihood of their arrival at sincere belief is the wisest strategy. Call such a strategy S, say living amongst worshippers and partaking in Christian missionary activities. Let's say the probability that adopting S results in Christian belief is 0.99. Now let's take another strategy, T, which does not likely result in Christian belief, say having a beer, where the probability of `success' is 0.0001. Check the following to your own satisfaction: the utilities of these strategies are the same. In fact, any `mixed strategy' is equally good, so long as there is a non-zero probability that Christian belief will result.

Cool. That's what happens when we play around with infinite utilities, and not just for Christian belief. This is a much more interesting and illuminating objection to Pascal's wager than dry statements about the decision-matrix. See why I think that philosophy of religion is a great `gateway drug' for the rest of philosophy? Whenever I see the Wager being discussed in the future, I'll try boring people with decision theory.

Onto The Logic of Decision, then.

Dominance and Disarmament

Dominance is an introductory topic in The Logic of Decision, like that which appears in Pascal's wager. His more secular example is that of nuclear disarmament (p8). Consider the general 2x2 decision array:

Let the first column correspond to an outbreak of war, and the second to the maintenance of peace. Similarly, let the first row correspond to the outcome of nuclear armament in the first column and disarmament in the second. Then d1 is the utility of nuclear weapons in the event of war, d2 is the utility of nuclear weapons in times of peace, e1 is the utility of disarmament in times of war, and e2 is the utility of disarmament in times of peace.

To illustrate, let's suppose that these utilities correspond to a naively pacifistic outlook (p2):


As Jeffrey notes, disarmament is a very complicated example, but proponents may argue that the values of the utilities are unimportant, so long as e1>d1 and e2>d2. Given the matrix, this condition does not appear to be overly controversial.2

Yet the argument is fallacious, and not because any utility in question purports to be infinite.

Where the `superdominance' of Pascal's decision-matrix allowed for virtually any strategy to be of equal merit, the `dominance' of this decision matrix excludes certain strategies but not all mixed strategies. As Jeffrey elaborates, deterrence is a sound counter-example to the argument:
The assumption that the dominant act is the better is correct if an extra premise is introduced, i.e., that the probabilities of the conditions are the same, no matter which act is performed. [Emphasis in original] (p9)
A disarmament advocate could preserve the validity of the argument by conceding that the deterrence strategy reduces the probability of war, but that this benefit is countered by the increased probability of accidental war (p10). But the need for that additional premise remains. In general, what is needed for the argument for disarmament and other arguments is a decision-theoretic framework which allows for the effects of the strategies themselves on the probabilities of the outcomes.

And for that, dear reader, you'll have to wait. There are too many goodies in this book to cram into a single blog post. :D

1. Richard C. Jeffrey. The Logic of Decision (2nd Edition). UC Press: Chicago, 1983.

2. This is how I have often argued in the past, inspired by such beautiful entreaties as the Russell-Einstein Manifesto. Of course, the REM is not simply an argument for nuclear disarmament, but for the abolition of war. As the decision matrix does not account for, the likely event that disarmed countries will manufacture weapons means that in all probability, disarmament is only useful for the temporary reduction of tensions and the avoidance of horrible accidents. So though I nevertheless support it, it is not with a `golden age' in view.

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