Tuesday, June 24, 2008

Pascal’s Fallacy

Slopping thinking by the great mathematician has been enshrined for decades in the undergraduate economics and statistics curriculum. You know the story: the problem of whether to believe in the Christian god and follow church teachings can be formulated as one of expected value, the cost in pleasure of being religious times the likelihood that there is no such god compared with the cost in eternal damnation times the probability that the Bearded One actually exists. The small likelihood of divine existence is outweighed by the even greater imbalance in the value cofactor. Except that it’s all wrong.



There are different ways to approach the issue, but the simplest error is that of assuming only two possibilities, no god or the one particular god promulgated by Pascal’s church. Logically, the case for a Christian deity is no stronger than for many other pretenders. Pascal could follow all the church precepts, die and come face to face with a Hopi god, for example, or maybe a god no human group had latched onto. Perhaps the “godly” practices Pascal had adhered to would just annoy the true, ex post god. Or maybe, if there is a god, this omnipotent character has no self esteem problem to speak of and doesn’t care whether lowly mortals love, hate or ignore him/her/it. And I’m leaving out a wide swath of other possibilities, from multiple godheads to extra-dimensional beings we can’t begin to conceive of.

The point is that the expected value formula works only if you calculate over the full set of possible states or outcomes; the p’s have to add up to 1.

I’m reminded of this because of the column by Peter Bernstein in the Sunday NY Times, which repeats the Pascal chestnut even as it invokes the spirit (but not the name) of Nassim Nicholas Taleb to argue that we should anticipate the unexpected.

12 comments:

kevin quinn said...

And if there were a God, why would She be impressed with a belief in Her adopted for the reasons Pascal urges?

Anonymous said...

It's especially interesting that a brilliant mind like Pascal still fell into this trap (seeing only the currently possibilities). This shows how basically inept we are at thinking about problems where we have such strong emotional interests.

Lee said...

Even when the choice isn't binary, Pascal's decision is still rational. The crux is the imbalance between a finite benefit (higher value attributable to a life without religious constraints) and an infinite detriment (eternal damnation). Even when the likelihood of the detriment is small, it's infinite magnitude outweighs the benefit.

No disagreement, though, that the choice of Christianity is no more probable of being the right one than Hopi beliefs (though given the community he lived in, choosing Christianity was likely to carry the lowest costs for Pascal).

Anonymous said...

Belief isn't really something that can be willed into existence either.

Sandwichman said...

Belief isn't really something that can be willed into existence either.

How do you know?

Headache said...

Nobody of sufficient age and sanity can really will themselves to believe that the moon is made of green cheese. We all have our limits.

(I'm the second anon, but not the first.)

Jack said...

How does one believe something without the will to believe that something? Even if one's belief is the result of an educational process initiated early in life, one still must exercise the will to continue to adhere to that early learning.

reason said...

I think Lee actually got to the crux of Pascals argument - he was trying to illustrate a point about investment theory under uncertainty! He chose a lousy example.

Anonymous said...

Saved!
copyright Patricia M. Shannon 1992

I have an acquaintance named Sue
who believes all religions are true,
because she can't tell
which will send her to hell
if she doesn't sit in its pew.

synykyl said...

"Even when the choice isn't binary, Pascal's decision is still rational. The crux is the imbalance between a finite benefit (higher value attributable to a life without religious constraints) and an infinite detriment (eternal damnation). Even when the likelihood of the detriment is small, it's infinite magnitude outweighs the benefit."

So, you believe in every religion that threatens you with eternal damnation? ;-)

Sandwichman said...

The crux is the imbalance between a finite benefit... and an infinite detriment...

That would be the case if uncertainty could be unambiguosly expressed as probabalistic risk. It can't. See Ellsberg, "Risk, ambiguity and the Savage axioms." In an uncertain universe there might even be a small chance of being condemned to eternal suffering for NOT having lived a life free from religious constraints. Who is to say God is not a secular humanist?

rosserjb@jmu.edu said...

Well, of course more recent lit on this stuff (sorry, no cites or links), drags the subjective discount rate into this. So, sinners are just stupid people with high discount rates. Anyway, not surprising that church attendance tends to rises as death nears.

BTW, one can think what one wants of it, but we do have substantial evidence regarding "near death" experiences. Of course, given that the people involved did not actually die, all of that can be dismissed, but if it does in fact show what goes on as most people die, it is mostly rather comforting, all those tunnels of light and figures of light and warm and glowy experiences of oneness and all that.

What has been much less advertised is that not eveybody has those. A small minority (5-15% depending on who is counting), in fact have awful experiences, empty alienating darkness. Ugh. What is interesting is that there seems to be nearly zero correlation with religious belief or lack thereof regarding peoples' outcomes on this. Regarding behavior the story is more mixed, with at least some people who had these experiences feeling when they returned that they were associated with their having done bad things in their lives they had not undone. What are those bad things? Oh, the most basic morality stuff: murdering, stealing, lying about other people, being mean to other people, that sort of thing. Duh.
(But then some other researchers say there is no correlation even with behavior regarding these "hell" experiences, goody.)

Barkley