Wednesday, September 21, 2016

Pascal’s wager and the wages of motivated cognition

Now Eli has on occasion been a betting bunny and he has a long term one going with one Blaise Pascal, one that sooner than the Bunny would enjoy is going to pay off or not.  Still it has been a good time so far but it does pay to consult the experts and Rabett Run has called in a philosopher, a well known racehorse and investment company, all going by the name of Kelso to help cook the argument, and so it goes

Pascal’s wager makes a famous argument for motivated cognition. While there is a good reply that doesn’t address this aspect of the wager, a response that goes straight for the jugular is more illuminating, providing a wider lesson about the relation between beliefs, preferences and rational choice. The wager is also a great example of how short, vivid and easily taught arguments in philosophy can have much broader implications.

Pascal asks non-believers to consider a choice between two options: believing in God and disbelieving. He assumes that belief has a modest net cost if God does not exist, since belief requires at least some sacrifice of time and effort. But disbelief has an infinite net cost if God does exist: the loss of a blissful eternity in heaven. The upshot is clear: at any odds, the expected value of believing will exceed the expected value of disbelief.

Pascal understands that we can’t just choose to believe—so the real choice of the wager is between trying to become a believer and not trying. Pascal recommended going through the motions of religious belief, making religious ritual and engagement a regular part of life, in the hope that belief will follow. The first reply takes advantage of the other side of this challenge: we can’t choose once and for all not to believe, either. No matter how committed you may be to your agnosticism or atheism, you just might have a sudden conversion. But adding this possibility to the calculation balances the scales—both alternatives now provide a finite chance of an infinite return: the expected values are equal after all, so Pascal’s argument fails.

But the argument’s focus on belief in God is a distraction, bringing in a lot of background noise. Many do believe in a God who rewards believers and punishes unbelievers. Others reject the suggestion that a good God, if she exists, would be so intent on primping in the mirror of believers’ faith as to enact such a policy. But this back-and-forth misses the real point. There’s a general puzzle here that has nothing to do with theism. Suppose the required belief was any belief we have no evidence for, such that having the belief at the end of your life would be infinitely rewarded if it were true. What belief might that be? Here’s a template:

B: If B is true and I believe B at the time of my death, then I will be infinitely rewarded. Two questions come up for any belief like this:

  1.  Should you try to acquire the belief? 
  2.  If you decide to try, how do you go about it?

The reply given above points out that there’s no guarantee that you will fail to have the belief if you don’t try. This equalizes the expected values, so there’s no reason to try.

But I prefer a second reply: if beliefs aren't based on evidence, Pascal's method for deciding what it's rational to do collapses, and the argument fails again, but in a more general and illuminating way. This reply has more heft: it targets the legitimacy of motivated beliefs, drawing on Pascal’s own model of rational choice to argue that choosing beliefs using Pascal’s wager-type arguments undermines the rationality of the appeal to expected values.

Rational choice in gambling combines beliefs about the probability of various outcomes given each alternative action with valuations of those outcomes to determine which action to choose. Like other early probability theorists, Pascal could calculate probabilities in games of chance when most professional gamblers couldn’t. But if we choose our beliefs based on whether we think having those beliefs will lead to better outcomes, Pascal’s method becomes an ouroboros: choices like that break the link between the beliefs we adopt and the actual probabilities/reliability of those beliefs. If our choices aren’t based on reliable judgements of probability, they can’t do the job Pascal’s account of rational choice needs them to do.

It’s the job of beliefs to be true and of probability assignments to reliably reflect ratios of outcomes in like cases. Dodging philosophical worries about ‘truth’ and focusing exclusively on the pragmatics we can say that to be useful, beliefs need to be a reliable basis for expectations about the consequences of our choices. (Similarly, it’s the business of evaluations of outcomes to reflect their real value to us.) When these conditions are met and we apply Pascal’s method our choices will be good ones, though of course we can still be unlucky. When the conditions aren’t met (think of professional gamblers fleeced by early probability theorists (probably frequentists - ER) or someone actively seeking an outcome they later regret) our choices are bad even if we’re lucky and things to turn out well. So like motivated cognition in general, denialism is a recipe for bad outcomes, well-earned: flying on a wing and a prayer may sound like fun, but it’s not likely to pay off.

This raises an obvious question: if the rationality (reliability) of beliefs is essential to making good choices, why do so many people reason poorly and have irrational beliefs? It doesn’t require a trip into real issues to show this (though those issues are what we’re really after here)—Daniel Kahneman and Amos Tversky showed long ago that people make similar mistakes in very simple cases (a story told elegantly in Kahneman’s Thinking Fast, Thinking Slow). Their suggestion is that our psychology combines a capacity for careful, reliable reasoning with a quicker, more reflexive system that ‘cuts to the chase’ but often gets things wrong. This makes a lot of sense: rationality is a lot of work (consider Kepler’s travails in calculating by hand how well Brahe’s observations fit with the hypothesis of elliptical orbits obeying his three laws). Sense perception (a more basic evolutionary heritage) is quicker and easier—and so is guessing. Even though it can be misleading, sometimes making quick calls is more important than consistently making the right call.

The point is that rationality is not something nature built into us, but a difficult, piecemeal, always-incomplete accomplishment. It demands that we think hard, apply critical reflection and evaluate our reasoning carefully rather than leap to conclusions. These habits don’t come without effort, and even once we’ve learned them, it can be hard to resist jumping to conclusions. Science is built on conclusions that have been tested carefully to ensure they provide a reliable basis for evaluating both new information and the possible consequences of our actions. There’s no guarantee that it’s always right. But any conclusion that survives scientific examination has shown itself to be reliable in a range of applications and circumstances. And (at least for a pragmatist) that’s about the best we can expect.


Florifulgurator said...

BTW, Pascal's wager was already used by the Buddha, some 2000 years earlier. It is found at least twice in the Pali canon: Kalama sutta (AN 3.65), Apannaka sutta (MN 60).

snarkrates said...

And of course Florifulgurator's comment illustrates another fallacy inherent in Pascal's wager--the idea that the choice is binary. Would Pascal's god react kindly to a person opting for the Buddhist path to enlightenment rather than belief in talking snakes? Does belief entail eschewing shellfish, pork and tatoos or simple belief in a savior. Which god? Which path to that god?

It all reminds me of the Emo Philips joke:

magmacc said...

There is a weighting problem with Pascal's wager, in that even an infinitesimal probability multiplied by infinity is still infinite.

And there's the nontrivial problem of identifying the correct god(s) or sect in advance.

Bryson said...

A response to the binary challenge was given by William James, in "The Will to Believe". He frames it in terms of what beliefs are (in some sense) 'available' to you: if Xianity is something you could see yourself accepting, while other religions (with their own Pascal-like doctrines on the importance of belief to your prospects in the after-life) are not, James argued that it could be justifiable for you to make the effort Pascal recommends wrt Xianity...

Florifulgurator said...

... which reminds me of "Homer Simpson's wager" who famously said, "What if we've picked the wrong religion? Every week we're just making God madder and madder?". The answer is of course to choose a religion without god. :-)

8c7793aa-15b2-11e5-898a-67ca934bd1df said...

The religion without god that I choose is science, of course.

I think people live with making bad decisions because they are poor, living in (relative) poverty, possibly hungry, and therefore anxious and possibly desperate. Hunger for knowledge is another problem for me. Oppression and persecution is such a deleterious state is the clincher for well motivated, spontaneous decision making, good or bad.

Therefore, I advocate rehabilitation over incarceration. Another meme I have developed is verbal incivility as non-violent protest, as opposed to say, the Mahatma Gandhi paradigm, which is now obsolete.

William Connolley said...

> But disbelief has an infinite net cost if God does exist: the loss of a blissful eternity in heaven.

It isn't obviously an infinite cost. If you use any non-zero discount rate, even infinite life has a finite value (assuming the value at any given time doesn't grow; but presumably it doesn't, since heaven is timeless or some such bollox).

If the cost isn't infinite then the argument falls apart of its own, since "infinitesimal times finite" can then be very small indeed.

Mitch Golden said...

I am afraid I don't understand the argument in the first part of the post. In Pascal's argument there are the following relevant quantities:

Pg = Probability that God exists
Pbt = Probability that you will die a believer if you try to believe
Pbd = Probability that you will die a believer if you don't try to believe
Vbg = Present value of dieing a believer if God exists
Vdg = Present value of dieing a disbeliever if God exists
Vbn = Present value of dieing a believer if God does not exist
Vdn = Present value of dieing a disbeliever if God does not exist

If we make the make the assumption that the probability that God exists is independent of one's choice of trying to believe or not (which is more or less a statement that God's existence is an objective reality), then the expected present value if you try to believe is

Pg*[Pbt*Vbg + (1-Pbt)*Vdg] + (1-Pg)*[Pbt*Vbn + (1-Pbt)*Vdn]

and the expected present value if you don't believe is

Pg*[Pbd*Vbg + (1-Pbd)*Vdg] + (1-Pg)*[Pbd*Vbn + (1-Pbd)*Vdn]

Subtracting the second from the first yields

Pg*[(Pbt-Pbn)*Vbg + (Pbd-Pbt)*Vdg] + (1-Pg)*[(Pbt-Pbd)*Vbn + (Pbd-Pbt)*Vdn]

or, rearranging:

(Pbt-Pbn) * [Pg*(Vbg-Vdg) + (1-Pg)*(Vbn-Vdn)]

The argument that Pascal is making is that Vbg is a large (possibly infinite, but if so, the only meaningful way to understand any of this is to take a limit) positive number, and Vdg is large and negative. He also believes that the present values of the God-nonexistence scenarios, Vbn and Vdn, are not large, or at least, not large enough to be bigger than Vbg-Vdg multiplied by Pg, the probability of God's existing.

Now it's true that he has without thinking set Pbd to 0. But that's a quibble in practice. It's perfectly reasonable to assess that the factor in front is positive - that is, that one is more likely to die a believer by trying than not trying.

Consider the other flaws in Pascal's argument, already discussed by others: God's existence shouldn't be presented as a binary choice, or that there's no a priori reason not to believe that God might hate it if you believe in him and flip the signs of Vbg and Vdg. If you are willing to put all of that aside, this new point has almost no effect - it's totally intuitively reasonable. It certainly doesn't make the argument fail, as claimed.