Wednesday, May 21, 2008

Rational decision theory

Suppose that I assign a non-zero probability to some religion, and this religion tells me that a certain action decreases the probability of an infinitely valuable outcome (e.g., eternity in heaven, or avoiding an eternity in hell). If there is no non-zero probability hypothesis on which the action increases the probability of an infinitely valuable outcome (or decreases the probability of an infinitely disvaluable outcome, but I shall count the avoiding of an infinitely disvaluable outcome itself an infinitely valuable outcome for simplicity), it is plain that in prudential rationality I ought to avoid the action.

Suppose that a number of religions have non-zero probability. Then if A is any action such that at least one of the religions claims that A increases the probability of an infinitely valuable outcome (IVO), and none of the religions claim that A decreases this probability, and, further, there are no non-religious hypotheses of non-zero probability that would make A decrease the probability of an IVO, again I ought to refrain from doing A. Now, sometimes there will be a genuine conflict between religions, where one religion tells me that some action increases the likelihood of an IVO and another tells me that it decreases it. In that case, I need to get my hands dirty with probabilistic calculations. I need to compare the IVOs (not all IVOs are equal; eternity in heaven with daily apple pie is not quite as good as eternity in heaven with daily apple pie à la mode); I need to compare the degree to which according to the respective religions A contributes to the likelihood of the respective IVOs, and finally I need to compare the probabilities I assign to the respective religions. These computations involve comparisons of infinities, but that's not at all a big deal—I just use an appropriate non-standard arithmetical model of infinite values.

Except in the rare cases where things balance out precisely, say when there are only two religions of non-zero probability, and they have equal probability, and one says that A increases the chance of an IVO by 0.2 and other says that it decreases it by 0.2, and the IVO is the same, or in the somewhat less rare, but still rare, cases where no religion of non-zero probability says anything relevant about A, these religious considerations will trump all other self-interested considerations. After all, only religious claims involve IVOs, and any change in the likelihood of an IVO trumps any change in the likelihood of something of finite value.[note 1]

Unless the number of religions that I assign non-zero probability to is very small (say, 0, 1 or 2), or there is a lot of similarity between the religions I assign non-zero probability to, taking these considerations into account will lead to a rather unappealing way of life, since there will be a lot of restrictions on one's actions, as, typically, any action that at least one of the religions forbids will be forbidden to one, as it will be relatively rare that an action forbidden by one religion will be positively required by another.

I think this is a reductio of rational decision theory, whether of a self-interested variety or of a utilitarian stripe. (After all, in utilitarian expected value calculations, I will need to take into account any IVOs for me or for others that have non-zero probability.)

3 comments:

Mike Almeida said...

Suppose that I assign a non-zero probability to some religion, and this religion tells me that a certain action decreases the probability of an infinitely valuable outcome (e.g., eternity in heaven, or avoiding an eternity in hell). If there is no non-zero probability hypothesis on which the action increases the probability of an infinitely valuable outcome. . . it is plain that in prudential rationality I ought to avoid the action.

I'm pretty sure that's not true. Suppose ~A (say, not praying)decreases the probability that I have the beatific vision and this is infinitely valuable. Clearly then my having such a vision is higher on performing A than it is on ~A. But, in terms of practical rationality, it does not matter whether I pray or not. No matter what I do, there is also some non-zero probability that I pray. If I go to the zoo, there is a non-zero probability that I pray. If spend the day doing cartwheels, there is again a non-zero probability that I pray. As long as there is a non-sero probability that I pray, the expected utility of what I do is infinitely high. I have a small chance of realizing an infnitely valuable outcome no matter what I do. But then it does not matter what I do, it's expected utility is infinite. Al Hajek argues analogously against the need to believe theism in Pascal's Wager. All you need is some small chance that you will believe.

Heath White said...

Two thoughts.

1. How do these considerations affect the choice of, say, a religiously dedicated life (a monk, a priest) vs. a secular but religiously faithful life? Wouldn't they show that we ought to undertake the most extreme religious rigors possible?

2. I think there is a lot of unclarity on what counts as an IVO. In particular, I think you are making the assumption that a finite life has finite value. But if that is all the life you have, as there is a non-zero probability is true, it's not clear that maximizing the value of that finite life is irrational.

For (imperfect) comparison: I once heard Alasdair MacIntyre remark, apropos of the fact that poor people spend a large proportion of their income on state lotteries, that he had once simply thought these people irrational. But then he realized that, for them, winning the lottery was not just winning money, the expected value of which could be calculated in the standard way, it was a ticket to a whole different kind of life. For a poor person, the state lottery is not unlike Pascal's Wager: a small chance of a huge benefit should rationally be wagered on.

Anyway, the materialist who thinks it most probable that she has only a single finite life to live, should probably not make utility calculations which treat the whole of her life in the same way she would compare smaller chunks of it.

Alexander R Pruss said...

Mike:

Going by expected values where there is an infinite payoff only makes sense when one uses some form of Nonstandard Analysis (NSA). Without NSA, one gets the result you mention, and that is absurd. Here are two ways of seeing it as absurd.

1. Suppose you are on your way to play a lottery where the first prize has infinite value, and the chance of winning is one in a billion. On your way, you meet your omniscient friend George who offers to tell you the winning numbers. By the lottery rules, there is nothing wrong with accepting George's offer. Clearly, self-interested rationality calls for you to do that. (And if the lottery is one where multiple winners are possible, then one does not need a solely self-interested rationality here.) A certainty of a positive outcome is better than a small probability of the positive outcome. This is true even if the outcome is infinite. (Consider it in the limit. The greater the outcome, the more rational it is to opt for the certainty over the uncertainty. If the outcome has very small value, it matters little.)

2. Consider two creators, George and Sally, neither of whom has middle knowledge, and neither of whom is capable of intervening in creation once creation has begun. George creates a world containing ten persons, each of whom is certain to have an infinite life of beatitude. Sally creates a world containing ten persons, each of whom has a random one in a billion chance of an infinite life of beatitude (by "random", I mean that it's not dependent on the choices of the persons), and otherwise will get a very miserable life of 100 years. Clearly, George has done something better than Sally, even though the expected outcome is plus infinity for each created person in both scenarios, as long as we don't use NSA.

So we had better use NSA. Fix a particular infinity to represent an IVO, and then do our calculations. And if we do that, then standard arithmetic is right. If I is infinite, I is bigger than I/2 and so on.

Heath:

1. If one of the religions hold that the life of the monk increases the probability of salvation, and none of the others hold that it decreases it, then one should be a monk on these grounds. But I suspect that the more rigorous one makes the life of the monk, the more likely that one of the other religions will object to it.

2. Suppose a materialist comes across two scientists, one of whom offers a 1/2 chance of extending her lifetime to 200 years, and the other offers a 1/2 chance of extending her lifetime to 2000 years. Both, moreover, assure her that the life will be a good one. It seems that our materialist, to be rational, needs to take these offers into account, as long as the probabilities that they are telingthe truth are non-zero. Suppose now a third scientist shows up. She says that her calculations show that there is a 1/10 chance we live in a steady-state universe, and that there is a method by which (e.g., by transfering our minds into a computational system made out of a system of blackholes) by which we have a 1/2 chance of surviving blissfully forever if the steady-state hypothesis is correct. Clearly, if our materialist thinks there is a 1/10 chance that the scientist is right, she needs to take the resulting 1/200 chance of endless bliss into account on self-interested rationality theories. Likewise, she needs to take religious hypotheses into account if she thinks their probability is non-zero.