Friday, June 10, 2011

The Death of Pascal's Wager

Pascal's Wager (see also these entries) holds an unusual position within theological discourse: it's not argument for the existence of God, but rather an argument specifically for belief in the existence of God. It was formulated by the 17th-century philosopher Blaise Pascal, and it goes roughly as follows:
If you believe in God and you're right, you gain everything (heaven). If you disbelieve in God and you're wrong, you lose everything (hell). Therefore, any rational person should choose to believe in God.
Though it may seem like an effective argument, the Wager is in fact riddled with problems. The most basic of these is that we don't choose what we believe. Sure, some people do come to believe in God, but this happens as a result of some perceived evidence or a powerful wave of emotion, not through sheer force of will. I cannot simply choose to believe to believe in God any more than I can choose to believe that two and two make five. This one fact is enough to completely demolish the Wager as most people describe it.

Even if we could just choose to
believe one of these horses will
win, how would we know which
one to pick?
The next problem is that even if we could choose to believe, it's not at all clear which god we should choose. Pascal was arguing for the Catholic version, as though the choices were Catholicism and atheism. But what about the gods of Islam or Mormonism or any of the countless other mutually exclusive, infinite-stakes religions? We don't even have to limit ourselves to the gods proposed by major religions, so the possibilities are literally endless. How, then, could we even begin to choose?

Perhaps we can still salvage the Wager by reformulating it to take these issues into account. So we can't just choose to believe, and in any case there are limitless candidate gods to choose from. But the possibilities of infinite reward and punishment are still there, and it seems like we should respond to them somehow. Maybe we can instead reach the following conclusion, which I'll call Pascal's Imperative:
In order to seek infinite reward and avoid infinite punishment, a rational person should spend every possible moment of their lives seeking sufficient evidence to genuinely believe in whichever god is most likely.
This formulation looks very promising at first: it promotes genuine belief and doesn't create a false dilemma by referring to only one possible god. But the Imperative runs into serious problems of its own. One possibility is that the most likely god is one that sees the Imperative as cowardly, one who punishes us for living in constant fear of such an unlikely circumstance as hell. This god seems at least as likely to me as any of the standard ones—perhaps even more so.

Other objections have to do with the counterintuitive nature of infinity. First we have the strong atheist's objection: maybe we can say based on logical disproofs that the probability of a god existing is either zero or infinitesimal. This would cancel out the infinite reward and punishment; multiplying ∞ by 0 or 1/∞ means that our expected value for believing is undefined and thus unknowable. (Although we should consider meta-probabilities as well: it's quite possible that the strong atheist is mistaken about the soundness of his disproofs and so is underestimating the probability of a god.)

Second, there's the problem of mixed strategies. Suppose we follow the Imperative and end up believing in some god Q, and our expected value is infinite. Now, what if instead we flipped a coin, and only chose to follow the Imperative which would lead us to Q if it came up heads? One half times infinity is still infinity, so our expected value remains the same. And in fact, it turns out that we could basically do anything—even explicitly try not to believe in Q—and our expected value would still be infinite. So what's the point of following the Imperative at all?

Finally, there's a similar issue stemming from what's called the St. Petersberg paradox. Imagine that you have the option of participating in a game where you have a 1/2 chance of earning $2, a 1/4 chance of earning $4, a 1/8 chance of earning $8, and so on. Your expected return would be:
(1/2 x $2) + (1/4 x $4) + (1/8 x $8) + ... = $1 + $1 + $1 + ... = $∞
If it's always rational to maximize your expected return, you should be willing to pay any finite sum of money to participate in this game. But in reality, virtually no one would want to pay more than $30 or so. It seems that decision theory may break down somehow in extreme cases like these, and the same logic applies to Pascal's arguments.

Pascal's Wager fails on multiple levels, and it still fails even when we try to resurrect it in the form of Pascal's Imperative. As tempting as the logic may be, we're not justified in using the infinite incentives of heaven and hell to draw definite conclusions about how to approach theological questions.

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