Epistemic rationality and Pascal's Wager

Pascal’s Wager is an argument that it is prudentially rational to engage in theistic belief promotion practices (TBPP), namely practices apt to promote one’s belief in God.

My interest this post is the standard epistemic rationality objection to the Wager, that engaging in TBPPs is irrational—a kind of brainwashing of oneself. Let’s think about the objection with a bit more care. Consider a specific TBPP Q, say Pascal’s example of going to Mass. If Q is indeed a TBPP, one expects engagement in TBPP to make it more likely that one believes in God. But how does one expect Q to achieve that goal? There are two possibilities. Either Q is expected to achieve that goal by providing one with evidence for theism or in some non-evidential way (or a combination of the two).

Suppose Q is expected to work evidentially. Then we already have the expectation of a higher credence given Q. This expectation is either rational or not. If it is not rational, then we don’t actually have good reason to engage in Q. If it is rational, however, then we should rationally raise our credence in theism right now, without having to bother engaging in Q, and for reasons having nothing to do with any wager. But if Q promotes belief in God non-rationally, then we should not engage in Q for the sake of such promotion of belief—we should not aim to non-rationally promote beliefs.

Let me make the first horn of the dilemma—namely, that the expectation of a higher credence is rational—a bit more precise. We can distinguish two (not mutually exclusive) ways in which a practice rationally increases one’s credence in a hypothesis H. One way is purely epistemic, by uncovering facts about reality. This is the usual way. But if that’s the way we expect to increase our credence in theism by engaging in Q, then we already have evidence that there are such theism-indicating facts to be discovered, and so we should already increase our credence in Q. The other way is practical, by promoting the hypothesis H in a way that shows up to us. The second way is a bit unusual, but here is an example: one way to increase your credence that you will not die of heart disease is to live a healthy life. For if you live a healthy life, you are less likely to die of heart disease, and since you will notice signs of improved cardiac health (e.g., lower resting heart rate, less huffing and puffing on stairs, etc.), your credence that you won’t die of heart diseases will also increase. But this practical way of increasing credence is utterly irrelevant in the case of theism, since nothing we can do can make God more or less likely to exist! So the only rational way that remains is the evidence-based way, and evidence-of-evidence is already evidence.

I used to be quite impressed by the worry that Pascal’s Wager leads to self-deception. I am less impressed. Here is why. There is a serious technical flaw in the argument for the first horn of the dilemma. A simple model for the relation of credence and belief is that you believe a proposition if and only if your credence is above some threshold β. This model might be false, but an analogue of what I will say should apply on more sophisticated models as well.

Here is the point. Consider a case where one is thinking about observing (and suppose this is a simple non-Newcombian observation that does not affect the hypothesis) whether some event E evidentially relevant to H has obtained. Then one’s expected posterior credence is:

1. C(E)C(H∣E) + C(∼E)C(H∣∼E),

where C is one’s credence function. But if one is a good Bayesian reasoner, then by total probability the value in (1) is simply equal to one’s prior C(H). Thus the value of one’s credence has no expectation of change upon observation when one is being rational. This seems to support the idea that if you expect your credence to go up, you should already raise it.

But in fact it’s not so simple. For even though the expected posterior credence equals one’s current credence, it could well be that it is more likely that the expected posterior credence exceeds the threshold β if you make the observation than if you don’t. Indeed, cases are obvious. Suppose the belief threshold β is 0.9, and you tossed a coin out of my sight. Suppose I have a prior credence 0.5 that this coin is fair and a prior credence 0.5 that it is double-headed. Then currently I don’t believe (or disbelieve) that the coin is fair. But if I look at the coin and I see tails, I will believe that it is fair—indeed, I will have posterior credence 1 in its fairness. But if I don’t look at the coin, I am not going to get any evidence, and I will continue not to believe that the coin is fair. If I look at the coin, the probability that I will see tails is 0.25 (I have credence 0.5 that it’s fair, and if it’s fair, the chance of tails is 0.5), and so the probability that I will believe if I look is 0.25 (since if I look and see tails, my posterior will be 1 which is bigger than β = 0.9), and the probability that I will believe if I don’t look is 0. Of course, if I don’t see tails, my credence that the coin is fair will go down. But while it will go down, it won’t affect whether I believe that the coin is fair—for I already don’t believe it (in the sense of not-believe, rather than in the sense of believe-not). And there is no irrationality of any sort in looking at the coin in this case.

In other words, the point is that while one’s rational credence has no positive or negative rational expectation of change upon observation, whether one’s rational credence is above a threshold certainly can have a positive or negative rational expection of change.

How could this work in a Pascal’s Wager situation? Let’s talk through one possibility. Take Pascal’s example of going regularly to Mass. Suppose, as Pascal says, your current credence in God is 0.5. You might think that if God exists, going to Mass has a decent chance, say 0.2, of resulting in an evident radical transformation E of your life, so evident and radical that updating your credence in theism on E will push your credence in God to above the threshold β. Of course, you might go to Mass it might not produce any such evident radical transformation (this is true even if God always improves the hearts of people who go to Mass, since he might do so more gradually or less evidently), and in that case your rational credence in God will go below 0.5. But going below 0.5 won’t affect whether you believe in God, since 0.5 is already, I assume, far below the belief threshold β. On the other hand, maybe if you don’t go to Mass, the probability that you will get evidence that will push your credence in theism above β is pretty small—smaller than 0.2. Very likely, your credence will just oscillate a little around 0.5 in the non-Mass-going case. Thus if there is a payoff you get for your credence exceeding the threshold β, it will be worth going to Mass, without there being any epistemic irrationality in the reasoning.

Thought of in this way, we get some practical guidance as to which TBPPs the agnostic or atheist should engage in. They should look for practices that, if God exists, have a decent chance of producing evidence for theism sufficient to push them above β.

I am a bit doubtful that Pascal meant us to think in the above way. He may well have been recommending TBPPs on the grounds of their non-rational effect on belief. My argument above does not defend that.

Two ways of evaluating rationality

Suppose that I am reliably informed that I am about to be cloned, with all my memories and personality. Tomorrow, I and my clone will both have apparent memories of having been so informed, though of course my clone will be wrong—my clone will not have been informed of the impending cloning, since he didn’t exist prior to the cloning.

After the cloning, what probability should I assign to the hypothesis that I am the original Alexander Pruss? It seems obvious that it should be 1/2. My evidence is the same as my clone’s, and exactly one of us is right, and so at this time it seems rational to assign 1/2.

But things look different from the forward-looking point of view. Suppose that after being informed that I am about to be cloned, and before the cloning is done, I have the ability to adopt any future epistemic strategy I wish, including the ability to unshakeably force myself to think that I am the original Alexander Pruss. The catch, of course, is that my clone will unshakeably think it’s the original, too. This may cause various inconveniences to me, and is unfortunate for the clone as one of its central beliefs will be wrong. But when one considers what is epistemically rational, one only considers what is epistemically good for oneself. And it is clear that I will have more of the truth if both I and the clone each thing ourselves to be the original person than if we both are sceptical. It thus seems that I ought to adopt the strategy of thinking myself to be the original, come what may.

Of course, after the cloning, I and my clone will both have apparent memories of having adopted that strategy. We may be impressed by the argument that we should assign probability 1/2, and each of us may struggle to suspend judgment on being the original, but in the end we will be stuck with the unshakeable belief—true in my case and false in his.

If my suggestions above are right, then a lesson of the story is that we need be very cautious about inferring what is rational to think now from what was a rational policy to have adopted. This should, for instance, make one cautious about arguments for Bayesian conditionalization on the grounds that such conditionalization is the optimal policy to adopt.

Prudential rationality

Prudential rationality is about what an agent should do in the light of what is good or bad for the agent. Prudential or self-interested rationality is a major philosophical topic and considered fairly philosophically fundamental. Why? There are many (infinitely many) other categories of goods and bads, and for each category it makes sense to ask what one should do in the light of that category. For instance, family rationality is concerned with what an agent should do in the light of what is good or bad for people in the agent's family; leftward rationality is concerned with the good and bad for the people located to the agent's left; nearest-neighbor rationality with the good or bad for the person other than the agent whose center of mass is closest to the agent's center of mass; green-eye rationality with the good or bad for green-eyed people; and descendant rationality with the good or bad for one's descendants. Why should prudential rationality get singled out as a topic?

It's true that in terms of agent-relative categories, the agent is particularly natural. But the agent's descendants is also a quite natural agent-relative category.

This question reminds me of this thought (inspired by Nancy Cartwright's work). Physicists study things that don't exist. They study the motion of objects in isolated gravitational systems, in isolated quantum systems, and so on. But there are no isolated systems, and any system includes a number of other forces. It is, however, sometimes useful to study the influences that particular forces would have on their own.

However, in the end what we want to predict in physics is how real things move. And they move in the light of all the forces. And likewise in action theory we want to figure out how real people should act. And they should act in the light of all the goods and bads. We get useful insight into how and why real things move by studying how they would move if they were isolated or if only one force was relevant. We likewise get useful insight into how and why real people should act by studying what actions would be appropriate if they were isolated or if only one set of considerations were relevant. As a result we have people who study prudential rationality and people who study epistemic rationality.

It is nonetheless crucial not to forget that the study of how one should act in the light of a subset of the goods and bads is not a study of how one should act, but only a study of how one would need to act if that subset were all that's relevant, just as the study of gravitational systems is not a study of how things move, but only of how things would move if gravity were all that's relevant.

That said, I am not sure prudential rationality is actually that useful to study. Its main value is that it restricts the goods and bads to one person, thereby avoiding the difficult problem of balancing goods and bads between persons (and maybe even non-persons). But that value can be had by studying not prudential or self-interested rationality, but one-recipient rationality, where one studies how one should act in the light of the goods and bads to a single recipient, whether that recipient is or is not the agent.

It might seem innocent to make the simplifying assumption that the single recipient is the agent. But I think that doing this has a tendency to hide important questions that become clearer when we do not make this assumption. For instance, when one studies risk-averseness, we lose sight of the crucially important question of whose risk-averseness is relevant: the agent's or the recipient's? Presumably both, but they need to interact in a subtle and important way. To study risk-averseness in the special case where the recipient is the agent risks losing sight of something crucial in the phenomenon, just as one loses a lot of structure when instead of studying a mathematical function of two variables, say, f(x,y)=sin x cos y, one studies merely how that function behaves in the special case where the variables are equal. Although one does simplify by not studying the interaction between the agent's and the recipient's risk-averseness, one does so at the cost of confusing the two and not knowing which aspect of one's results is due to the risk-averseness of the person qua agent and which part is due to the risk-averseness of the person qua recipient.

Similarly, when one is interested--as decision theorists centrally are--in decision-making under conditions of uncertainty, it is important to distinguish between the relevance of the uncertainty of the person qua agent and the uncertainty of the person qua recipient. When we do that, we might discover a structure that was hidden in the special case where the agent and recipient are the same. For instance, we may discover that with respect to means the agent's uncertainty is much more important than the recipient's, but with respect to ends the recipient's uncertainty is very important.

To go back to the gravitational analogy, it's very useful to consider the gravitational interaction between particles x and y. But we lose an enormous amount of structure when we restrict our attention to the case where x=y. We would do better to make the simplifying assumption that we're considering two different particles, and then think of the one-particle case as a limiting case. Likewise for rationality. While we do need to study simplified cases, we need to choose the cases in a way that does not lose too much structure.

Of course, if we have an Aristotelian theory on which all one's actions are fundamentally aimed at one's own good, then what I say above will be unhelpful. For in that case, prudential rationality does capture the central structure of rationality. But such theories are simply false.

Scoring rules and epistemic rationality

Scoring rules measures the inaccuracy of one's credences. Roughly, when p is true, and one assigns credence r to p, then a scoring rule measures the distance between r and 1, while when p is false, the scoring rule measures the distance between r and 0. The smaller the score, the better.

Some scoring rules are better than others. Let's suppose some scoring rules are right. Then this thesis seems to be implicit in some applications of scoring rules (e.g., here):

1. If S is the right scoring rule, then a credence-assignment policy is epistemically rational only if following the policy minimizes expected total or average S-scores.

(And there will be a debate about whether we should have "total" or "average"—see link.)

But (1) is false. Here's a simple counterexample that works for most reasonable scoring rules. Consider a situation like this: A fair coin is flipped. If you assign credence 0.51 to heads, a mindreader who knows your credence assignments will immediately reveal to you how the coin landed. Otherwise, you will never have any information on how the coin landed.

Obviously, the epistemically rational thing to do is to assign 0.5 to heads. But this leads to higher expected total and average scores on most reasonable scoring rules. For if you assign 0.51, then once the mindreader tells you how the coin landed, you will update your credence to be very close to 0 or 1, and your score will be very low. And the only cost of this scenario is the slight inoptimality from briefly having score 0.51 instead of the optimal score of 0.5. So the epistemically rational policy for dealing with situations like this, namely assigning 0.5, does less well in expected scores than the epistemically irrational policy of assigning 0.51.

The case may seem farfetched. But there are real-life cases that may be similar. It may be that for psychological reasons when you are a bit more sure, or a bit less sure (depending on your character and the thesis), of a thesis than rationality calls for, you will be better able to investigate whether the thesis is true. Thus it may be better for your long term epistemic score that you do what is epistemically irrational.