Pascal’s Wager, a Special Pony, and Belief

Blaise Pascal Seventeenth century French mathematician, physicist, and philosopher, Blaise Pascal famously argued that one should believe in God (presumably the Christian God) for the simple reason that doing so is the safest bet.  His argument has become known as Pascal’s Wager.  However, Pascal’s conclusion—that one should have faith in God—is predicated on several assumptions not based in evidence.  Thus, his conclusion itself is derived from faith.

Pascal’s Wager is often described using a “payout table” as shown below.  In this table, the columns represent the possible answers to the question, “Does God exist?”  The rows represent one’s choices with respect to having faith in God.  The numbers in the table represent the expected benefit, i.e., the “payout”, one would experience for the corresponding column/row permutation.

Payout Table

God Exists God Does Not Exist
Faith in God 0
No Faith in God -∞ 0

From this table, we can see that if God does not exist, then whether or not one has faith in God is irrelevant.  In either case the “payout” for one’s faith, or lack of faith, is zero.  But if God does exist, the payout for having faith is infinite—infinite bliss in heaven—while the payout for lack of faith is infinitely negative—an eternity in the tortures of Hell (or at least an eternity away from God, if you prefer a softer stance).

We can also see that the worst one can do by having faith in God (zero) is at least as good as the best one can do by not having faith in God (also zero).  Thus, if one accepts this framing of Pascal’s Wager, it is indeed rational to choose to have faith in God. But there are multiple problems with this analysis.

A Quick Word on Infinities

Infinities are abstract placeholders.  For Pascal’s Wager, we don’t really know how to assign actual numeric values, or even what the units would be.  So we use infinity to indicate the highest value we can imagine for some cell.

Further, we cannot know that the value of a particular cell is infinite (positive or negative). For example we don’t know that the experience of Heaven would actually be infinite.  We certainly would expected it to be extremely high, higher than anything we would experience elsewhere, but would God be capable of making the experience even better?  Presumably, since God is omnipotent.

So, when we see an infinite value in a cell, we interpret that as meaning “extremely large and potentially infinite”.

Pascal’s Wager Assumes Belief is a Choice

The most fundamental problem with Pascal’s Wager is the implicit assumption that belief or non-belief is a choice.  While this may be the case for some people in some situations, it is quite likely not universally so.  For many, belief is based on reason and evidence.  Propositions in harmony with reason and evidence are believable, but even then, belief is provisional.  Such beliefs change as reasoning and evidence change.  Such beliefs are the conclusions of a rational process, they are not concepts labeled “believable” as a matter of personal preference. Could you choose to believe the in the very special pony?

Consider, as an analogy, that your friend hands you a metal box that appears to be empty.  You are then told that the box is not empty, but rather contains a very special pony that is invisible, weightless, and immaterial.  Further, the pony is very strong and in wonderful health, and one day, you may be able to ride it, but only if you truly and honestly, in your heart of hearts, believe the pony is real.

Could you choose to believe in the pony?

You’re also able to do any sort of experiment or analysis you wish for as long as you wish, but none offer any compelling evidence of the pony.  When you appear to be skeptical, your friend provides you with a book containing many stories about the pony.  Your friend assures you the book is true and correct in every respect, and even produces many, many other people who will attest to the veracity of the book.

Would you then choose to believe.  Could you?

Pascal’s Wager Assumes One God

Another problem with Pascal’s Wager is that it assumes that there is exactly one god in which one does or does not have faith.  Pascal was presumably considering only the Christian God.  But what if there is some other god?  Or multiple gods?  Throughout human history, there have been so many gods, and even more religions.  Had Pascal taken the possibility of other gods seriously, he would have framed the analysis much differently.

By framing the analysis as he did, Pascal has already committed an act of faith—presuming that there is (or might be) at most one possible god.

Pascal’s Wager Assumes No Cost of Faith

Setting aside the (significant) problems already described, we still have the problem of cost.

To properly assess the attractiveness of any wager, one must weigh not only the payout, but also the cost.  If I asked you whether $100 was a good payout for predicting the role of a die, would you take that bet?  Clearly, that should depend on what it costs you to play.  How much is the ante?  Ante $1?  That’s a great bet. Ante $50? That’s a terrible bet.

One might think that an infinite payout should always be attractive, but what if the associated cost is also infinite?

It’s difficult to pin down a precise cost for either choice in Pascal’s Wager—to have faith or to not have faith.  However, living a life of faith precludes one from certain pursuits, specifically those that run counter to whatever faith dictates.  For example, some Christians, including United States Senator James Inhofe (R-OK), believe that the bible refutes the possibility of climate change.  In this case, the work of climate scientists is essentially precluded as it is incompatible with that faith.

But what if it turns out that Christian faith is completely wrong?  And what if all faith in gods is completely wrong for the reason that gods do not exist?  That instead, each of us comes into existence through purely natural processes, lives for a few short years, and then dies and is no more?

One might argue, in this case, that all life is worthless and it matters nothing at all what one does with his or her life since, in the course of eternity, each life is insignificant.  But one might argue quite differently; that in all the vast expanse of time, these few shorts years of life are extremely rare, extremely special.

And yet, here we are, privileged to be alive in this infinitesimal slice of eternity.  We have the opportunity to experience life, to explore and try to understand nature, to document reality such that our descendents are able to experience an even better life than ours.  Squandering, or even actively thwarting, these exceedingly rare opportunities as a result of a faith in the fictions of our ancestors strikes me as tragic waste of life.

There are even those whose faith not only predicts, but actually strives for, Armageddon—the end of not only one’s own life, but all lives.  It’s difficult for me to imagine a greater folly.

So what is the value of—and thus the cost of squandering—an infinitesimally rare resource, such as life?  This is a tricky question, but without some sort of cost model, we cannot really do a meaningful analysis of Pascal’s Wager.

Finally, faith might be nothing more than a private thought pondered by an individual who otherwise makes life choices based on reason and evidence.  But it might also be an organizing principle, driving life choices ranging from organizing prayer meetings to organizing genocides.  Clearly, the costs could be quite high.  But it could drive even greater harm.  Faith could lead people to ignore certain realities—climate change, virus mutations, for example—that, ultimately, could lead to the end of all life on Earth.  As already noted, it’s difficult to imagine a greater folly than bringing about the end of all life.  We can imagine the cost of this sort of faith to be “extremely large and potentially infinite”.

So, if we consider a “great folly” variant of faith [1], we can construct a simple cost table:

Cost Table

God Exists God Does Not Exist
Faith in God 0
No Faith in God 0 0

Now we can look at a more realistic payout table.  The following table shows the net payouts obtained by subtracting the associated costs for each payout:

Net Payout Table

God Exists God Does Not Exist
Faith in God -∞
No Faith in God -∞ 0

From this table, a rational choice is not so obvious.  Each row and each column offer the possibility of an infinitely bad outcome.  We’re not able to choose the column; that is the unknown on which we are wagering by our choice of row.  But on what should we base our choice of row?  It is reasonable to wish to simultaneously:

  1. maximize our potential upside; and
  2. minimize our potential downside.

However, that is not possible; the maximum upside is in the first row, while the minimum downside is in the second.  So we need some other decision strategy.  We might just rely on a gut feeling, a hunch, or even a coin flip.  But, it seems we should be able to do better than that.

I suggest the most rational strategy is to consider the probability of the columns.  And for this, we reason carefully about the evidence.  All of the evidence available to us.  And like all evidence-based reasoning, our conclusions should be provisional.  We should be willing to change in light of new evidence or more compelling reasoning.

As part of our reasoning, we should also acknowledge the second problem outlined above: there may be more than one god.  This implies that a more realistic table has many more than two columns, further confounding our attempt to choose the winning column.

Blaise Pascal was a towering intellect of the 17th century; there is no doubt about that. But with regard to his famous wager, there, I have my doubts.  Pascal’s Wager suffers from at least three separate problems, any one of which is sufficient to cast Pascal’s conclusions into serious doubt.  Or, as the folks at Rational Wiki wryly describe it,

“Pascal’s wager: Believing in and searching for Kryptonite on the off chance that Superman exists and wants to kill you.”


[1] One might counter that accepting a different variant of faith could result in a less than “extremely high and potentially infinite” cost of faith. This, in turn, would then support Pascal’s recommendation.  But this is circular reasoning.  Such a variant relies on reason and evidence to differentiate between variants in order to mitigate any potential grand folly of faith.  But in so doing, one has already not fully accepted Pascal’s argument.  At the very least, allowing variants of faith suggests that Pascal’s argument is flawed in that, minimally, the table requires additional rows.

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