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Don’t bet on Pascal’s wager

Pascal only pretended to be a gambling man.



~ Pascal, Blaise

Blaise Pascal (1623 - 1662) made important contributions to mathematics in general and the theory of probability in particular. Without him, our current understanding of judgment and decision making under risk and uncertainty might not be what it is. During these intellectual adventures, Pascal seemed to remain troubled by his unresolved relationship to Catholicism. In 1654 he reverted to his ancestral faith and abandoned much of his mathematical activity. In his ruminations on faith, the Pensées, however, he landed one last grand stroke: his famous wager, which became a foundation of modern decision theory.

This idea is quite simple. You cross-tabulate two choice alternatives, such as ‘yea; and ‘nay' with two true states of nature, ‘true' and ‘false.' As a result you get four outcomes: yea-true is a Hit, yea-false is a False Alarm, nay-true is a Miss, and nay-false is a Correct Rejection. Next, you affix values to each of the four outcomes, and - if at all possible - a probability for ‘true' to be true, and off you go. Now you know all you need to know in order to make a rational decision, a decision that maximizes the value to you.

Here goes Pascal: His ‘yea' is to believe in the existence of god; his ‘nay' is to not. His ‘true' is that god exists; his ‘false' is that he does not. Of the four outcomes, the setting an infinitely positive value for the Hit (eternal bliss for the believers) is critical. As long as there is no other infinitely positive value and as long as the probability of god's existence is not considered zero, the only rational choice is to believe in god or act as if one did. From this, it follows that anyone who does not believe in god, or who acts as if, must be stupid. Hence the wager creates a new puzzle. Why do so many reasonable people not believe in god, or act . . . If Pascal's intent was to make some room for reason in questions of faith, he failed.

It is crucial to remember that Pascal cooked up his wager after he had returned to Catholic orthodoxy. He was a fervent believer. Why did he not settle on being satisfied with the irrational foundations of faith? Why did he try to rationalize it? I cannot answer these questions because I am not his biographer of psychoanalyst. I am concerned, however, with the fact that his famous wager was a mere exercise in foregone conclusions. It leads me to ask, what might the wager have looked like had it not been written by someone with a pre-commitment to belief.

A decision maker without religious pre-commitment might wonder about the role of intensity and duration in the outcome states that are being considered. The notion of eternal bliss is actually quite weird if eternity means there is no limit in duration and if bliss means there is no limit in how good you would feel. Suppose there were limits on one but not the other. What would you choose: Feeling a 9 out of 10 forever or feeling 10 out of 10 forever? If you prefer the latter, then the former is not an infinite value. Conversely, would you prefer infinite bliss for a year over infinite bliss for a day? If so, time matters in your valuation.

What if god only promised really good but limited feelings for a really long but finite time? If so, the a priori probability of god's existence would become critical for rational choice. That is, conditions would exist under which reasonable people could decide not to believe. At the same time, the wager would become meaningless as an exercise in theology because the prior probability of god's existence would be positively related to the posterior probability, i.e., the subjective probability that god exists given the fact that one has chosen to believe he does.

In the intellectual history of the West, Pascal's wager is seen as progress over the medieval attempts to prove the existence of god. Pascal seemed to admit that no such proof is possible and that reason can only serve to make the right or prudent choice. Again, however, if reason is strapped in a straight-jacket, its operation is meaningless. Pascal's wager is not a proof of god - and was not intended to be one - but a mere rationalization of existing belief. By succeeding in a technical sense, the wager accomplishes nothing. Perhaps there is some consolation in the fact that the earlier putative proofs did not accomplish anything either. That may be in the best interest of believers. If these proofs had succeeded, god would be accepted by reason, and faith would be unnecessary - even undesirable.