Saturday, October 27, 2012

Myriad Problems with Pascal's Wager.

For those of you who don't know, Pascal was a 17th century French philosopher, mathematician, and physicist.    For most of his early life he was ambivalent about religion, taking it up only to drop it again.  Then at the age of 31 he had an intense religious experience late one night in bed, after which he became devoutly Catholic.  Still, as a philosopher he realized there was a problem.  In the absence of such an experience  what compelling reason would there be to believe. Rigorous skepticism, with which Pascal was intimately familiar as a scientist, would suggest not believing.  One must expect therefore a considerable number of people "so made as they cannot believe" on perfectly reasonable grounds.  What of them?  As a probability theorist, Pascal attempted to solve the problem mathematically as a wager in the following terms.
  1. "God is, or He is not"
  2. A Game is being played... where heads or tails will turn up.
  3. According to reason, you can defend either of the propositions.
  4. You must wager. (It's not optional.)
  5. Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing.
  6. Wager, then, without hesitation that He is. (...) There is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite. And so our proposition is of infinite force, when there is the finite to stake in a game where there are equal risks of gain and of loss, and the infinite to gain.

In short,
If you bet on the truth of atheism and are wrong, you suffer an infinite infinity of misery in hell.
If you bet on the truth of atheism and are right, it has little to no effect on your outcomes.

If you bet on the truth of Christian doctrine and are wrong, it has little to no effect on your outcomes.

If you bet on the truth of Christian doctrine and are right, you enjoy an infinity of infinite happiness in heaven.

Therefore, Pascal argues, even if the truth of the Christian Doctrine is extremely unlikely, one should still choose to believe in it to maximize your expected outcomes. Why? Because of the math.
The expected outcome of any binary wager is the sum of the two outcomes weighted by their respective probabilities.  Say the probability that Christian doctrine is true is ten million to one, or 0.0000001.  That sets  the probability that atheism is true at 0.9999999.  What happens if you bet on atheism.  If atheism turns out to be right as the odds suggest, you have simply ended up with some fixed quantity of benefit from your one-and-only life.  If you were fortunate it was pretty good even if it seemed a little too short.  On the other hand it may have sucked but at least its over now, que sera.  In either case its a finite quantity, lets call it X.  If atheism turns out to be wrong though, you get an eternity of infinite suffering tacked on after death, which we express as the quantity minus infinity (-inf).  The expected return E of a bet on atheism is then  negative infinity.

E=0.0000001*(-inf) + 0.9999999*X = -inf

Because even 1 tenmillionth of negative infinity is a negative infinity, and it remains so even if you add 9,999,999 ten millionths of a finite quantity to it.  The presence of an infinite quantity overwhelms the finite probabilities and the finite outcome X.

Flip it around and for exactly the same reasons, a bet in favor of the Christian doctrine where the reward for being right is an eternity of infinite happiness, or plus infinity (+inf), gives an expectation value of

E=0.0000001*(+inf) + 0.9999999*X = +inf

Therefore, says Pascal, using the same logic you would apply to any game of chance, or an investment portfolio, bet on the Christian Doctrine.

That all seems straightforward, right? .  Obviously one should believe in the Christian Doctrine.

Well, no.

Notice first how Pascal drastically and artificially limits the possibilities.  Only Atheism or the Christian doctrine are permitted.  Then the associated outcomes after death limited to three possibilities: oblivion, infinite eternal torment, or infinite eternal happiness.  These are added on to whatever one's worldly life amounts to, which only counts, in Pascal's scheme if Atheism is correct, because then your one and only worldly life is all you get.

Pascal gets away with only two options because most of us spend our lives embedded in a society with one dominant religion. Having one option to unbelief seems natural.  But there and have been thousands of religions based on myriad gods, with incompatible dogma and doctrines. This means that at least most of them have to be wrong.  Pascal just excludes these out of hand, but we have to take all comers.  I would argue that as the minimum number of candidate religions that must be wrong grows, the odds that all of them are probably wrong goes up too.  Even if we can be sure that there is a correct choice, we should expect to get it wrong given the sheer number of options.

We also must ask what constitutes an acceptable wager on Christian Doctrine under Pascal's conditions. When Pascal tells us to just go ahead and believe, what does that mean? How does that go?  Starting from a point of unbelief, a lot depends on what holds is necessary and sufficient for Christian salvation.  If you have to genuinely believe, then Pascal is assuming that for the right reward (instead of a convincing case) we can, of our own volition, create true belief; spinning it fully formed out of a sliver of doubt in our atheism.  On the other hand, if true belief is not required, Pascal would appear to be telling us that we should simply act as if we believe. This runs right up against the debate about whether Christian salvation is a matter of what you really believe, or what you say or appear to believe, or dependent on act commensurate with the doctrine.

Neither seems very plausible. Belief isn't a matter of volition. I cannot arbitrarily decide that I believe the the Earth is a flat disc; I KNOW it's a sphere. Is it plausible that merely for some reward (say $2,000,000), I could actually stop thinking that? No! That is not how true belief and knowledge work. On the other hand it's even less plausible that feigned belief would pass as a hedge bet by a true unbeliever. It implies a God that accepts phony Bad Faith, which seems a bit sleazy.

It's also no accident that Pascal needs the stakes to be so exaggerated, otherwise there would be nothing to offset the utter lack of evidence on which to base a decision. This is simple game theory. Who would buy a $1 Lotto ticket at 47 million:1 odds if the max payout was $15? No one. The stakes HAVE to be outrageous.  This is the heart of Pascal's argument.: Given the unlimited size of the purported "Jackpot" isn't even a remote chance worth the gamble?

Well, No. Exagerating the stakes in cases where the odds against are astronomically high is merely an exploitation ploy against human psychology.  This is how people are duped into buying Lotto tickets, or penny stocks.  Most of us lack the intuition to grasp really large and really small values. When outrageous claims are in play, it becomes almost certain that someone is being played for a sap on that basis.

The last thing wrong with the wager is it's faulty assessment of the cost of belief.  Pascal asserted that the cost was essentially nil. If there is no god, no heaven, the true believer simply dies and ceases to be regardless of his belief. In this case, since he is not even alive to be disappointed, Pascal sees this as a wash.

But is this true? True Belief spawns large and small changes in one's behavior. A sizable portion of these changes serve to benefit the faith as an institution much more than the believer  as an individual. True Believers invest significant portions of their money and time, and cede a lot of their autonomy to the church and its observances.  For the private unbeliever, who is merely trying to hedge his bet per Pascal's recommendation the situation is worse.   He is still compelled by his strategy to act out as if he were a true believer.  To cover remote chance that salvation will be needed, he must make the best show of belief he can muster.  If what he really thinks doesn't matter, the quality of his pretense is all that's left.  This is the most insidious hidden cost ignored by Pascal.  It is physically and emotionally painful to break the integrity between how we truly think and feel and how we speak and act.  It feels good to be honest with and about ourselves, both privately and publicly, whereas we suffer when we are false.  There are certainly exceptions, and our ability to tolerate a certain amount of ambiguity is one of our better traits.  Still, the evidence is that we are more miserable, perhaps most miserable, when we deny true selves.

This is not to deny value of the community, solidarity and moral teamwork one finds in church involvement. This can offset some of the real cost of belief, but that's the point. Such things have to be counted because Pascal is wrong in his basic assumption. Belief, true or pretended, comes with a real bill which is often high.