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Beyond Pascal's Wager

David Cowles

Mar 1, 2023

“Once we get past skyscrapers and suspension bridges, we really have no idea what’s going on, do we?”

A Hasidic folk tale, possibly dating back to the Baal Shem Tov himself, tells of a certain Rabbi on his way home from Schule who comes across an elderly widow crawling on her hands and knees under a streetlamp. Not wanting to play the part of passer-by in some future production of The Good Samaritan and eager to perform one last mitzvah before going to bed for the night, the Rabbi offers to assist.  

"I've lost my shiny penny, the only thing I have left in the world," the widow wails.

"And is this where you lost it?" the Rabbi inquires innocently.

"No," she replies. "I lost it over there."

The quizzical Rabbi, still hoping to be helpful, asks, "Then why are you looking for it here?" 

"Because the light is better...," the exasperated widow sighs as she continues her search. But under her breath, she mutters with all the derision she can muster, "... Obviously!" 

This story is often presented as a commentary on the widow's folly, but in fact, the widow is the hero of the story. She defies conventional wisdom. Why look for a lost penny in pitch darkness where there's no hope of finding anything? 

Pennies can be funny; they have a tendency to bounce and roll. They often end up in the places you'd least expect to find them. Possibly, the widow's penny rolled under the lamp, probably it did not. In any event, it would make no sense to look for it anywhere else because, even if it were there, we would have zero chance of finding it. 

Let’s recap. It is unlikely that the woman and the Rabbi, now both on hands and knees, will find the penny under the streetlamp; but it’s possible. On the other hand, were they to look in the probability-rich darkness, they would have no chance whatsoever of finding the coin. Real searching is not just idle gazing; searching presupposes a target and a method of getting to that target.

I’m making myself a bowl of cereal and I need milk. I open the refrigerator door and look in…for several seconds; then I call out, as we all knew I would, “Are we out of milk?” At which point, my wife gets off the couch, walks to the fridge and pulls out a brand new, unopened half gallon. I gazed; she searched.

We look for rainbows; we search for intelligent life…extra-terrestrial and otherwise. 

This story reminds me of ‘Pascal's Wager’ (1650 CE). Faced with the God Question, the French mathematician, Blaise Pascal, employed a forerunner of what is today called ‘Game Theory’. 

Conceptually, Pascal divided a sheet of paper into four quadrants. Across the top of the page, he wrote, “I believe that God exists;” down the left-hand margin, he wrote, “God does exist.”  Then he filled it in: (N/N), (N/Y), (Y/N) and (Y/Y). 

Got it? Of course not. Would it be easier to understand if we expressed it in the language of computers, i.e., binary numbers? (0/0), (0/1), (1/0) and (1/1). No? Ok then, (0*0), (0*1), (1*0), (1*1). Now? Ok!

Pascal looked at the ‘God Question’ hyper-objectively. He reasoned that one of four things must be true: (1) I do not believe that God exists, and he doesn’t; (2) I do not believe that God exists, but he does; (3) I believe that God exists, but he doesn’t; (4) I believe that God exists, and he does. 

Flash back: I’m standing at the craps table (Bellagio, Las Vegas) and I’m holding one $25 chip in my hand. It’s all I have left after a week of heavy drinking and gourmet dining. This is my very last chance to recoup.

The ‘felt’ offers me more than a dozen possible wagers, but I need to know my best bet, objectively speaking. Where do I put my last $25 to give myself the best chance of winning? (It’s not as easy as it looks. Different bets come with different odds but also with different payouts.)

Back home: (Don’t ask me what happened in Vegas, that stays in Vegas.) I’m still looking to optimize my chances, but this time I’m playing a different game; it’s called the Game of Eternal Life.  My options:

  1. God does not exist, and I never thought he did: Winner!

  2. I believed in God but, as it turns out, I was wrong: Loooser!

  3. I did not believe in God, but I was wrong; God does exist: Loooser!

  4. God exists and I knew it all along: Winner!  

Defining Terms - Belief in this context does not refer merely to our intellectual assent to a certain proposition; we are not talking about some sort of creedal affirmation here. Far from it!

Believing X means living life as if X were true. So, if you choose one of the “I believe that God exists” options, you are committing yourself, not just to the recitation of a creed but to the adoption of a lifestyle, even if it turns out that you’ve guessed wrong.

How about God? An old man in a white robe living on a cloud? Maybe…but probably not. Then, what does it mean to say, “I believe in God?” 

What I call the God Hypothesis runs something like this: (A) an objective universe, more or less consistent with experience, exists, and embodies or reflects values that transcend* it; and (B) those values are normative for the way I live my life and the way I live my life, ‘makes a difference’: somewhere, somehow, on some level.

*A value is transcendent if and only if it is valid throughout the universe, in all places, at all times, regardless of circumstances, and would be valid in any possible universe (don’t leave home without it) and would still be valid even if there was no universe to reflect or embody it.

And finally, universe. Here we must make a distinction. By ‘universe’, I mean what science means: the sum of all events that occur relative to me in spacetime. But we also need to talk about universe, an entity in its own right, not just the totalization of other entities. 

Everything that happens in the universe happens in spacetime; but universe itself does not. Universe per se transcends spacetime. Nothing limits it. Nothing is beyond it. Therefore, in its own way, it is infinite and eternal.

Payoff: So, two winners, two losers. I should break even over time, right? I guess it all comes down to a coin toss after all, which is what I thought in the first place. But not so fast:  Remember, different bets come with different payouts.

Option (1), “God does not exist, and I never thought he did”, is a winner… Cue confetti! What’s my payout? Zero… Did you just say, “Zero?” What a rip-off!

I suppose I have the satisfaction of knowing that 75 years of depression and anxiety have been warranted…but that still leaves me $2.95 short of a Grande at Starbucks. Cold comfort on a winter’s morning. 

Options (2) and (3) are obvious losers, so “no payout for me – you loooser!”

That leaves option (4). Like option (1), (4) is a winner…but this time, hold the confetti until we check the payoff. I’m expecting the dealer to put a few chips down in front of me (I’m hoping for black ones). Instead, he hands me a sheaf of papers. What the heck? – OMG, it’s the deed to the casino.

The Bellagio. I’ll confess I’ve always wanted to own the Bellagio, but that’s not this. This isn’t the deed to the Bellagio…it’s the deed to the entire Universe:

“Winner, winner, chicken dinner!” Who needs confetti?

You’re not convinced. You think you’ve found an error in my reasoning: “Your calculations assume that all 4 outcomes are equally probable: P = 25% across the board. But that may not be the case.” 

“What if the probability that God does not exist is a 99.99999% and the probability that God does exist is only 0.00001%? What then?

Nothing then, that’s what! Odds don’t matter! (1), (2) and (3) payout nothing – it doesn’t matter the odds. Only (4) offers a positive payout, so who cares what the odds are? Is P(4) > 0? (Few would assert otherwise.) 0.00001 is a pretty lousy average payout, but hey, it’s better than 0.00000. So, there is nothing to do but to go ‘all in’ on (4)…and pray. Since (4) pays out ‘the entirety’ and the other three options pay out ‘nothing’, I really have no choice, do I? And that, my friends, is Pascal’s Wager…or is it?

Face the facts: our world is incomprehensible and frankly absurd. Sure, we can hit a rock, hurtling through deep space at breakneck speed, with a rocket launched years earlier from Earth. But that’s the equivalent of rearranging deck chairs on the Titanic

On the surface, reality is somewhat intelligible and highly manipulable, but only on the surface. Once we get past skyscrapers and suspension bridges, we really have no idea what’s going on, do we? 

Could the world have been created by a super-intelligence? I suppose so. Could it have just arbitrarily sprung into being? So they tell me. Will anyone ever know for sure? Probably not. Will anyone ever know for sure that they’ll never know for sure? Again, probably not!

“Daddy, you know everything: why is the sky blue? Why is water wet? How many seconds are there is a year? Daddy, who am I, really?” I was doing so well; I was on a role…and then I hit the wall. We can answer a million questions but the ones we can’t answer are the only ones worth answering. 

So what to do? Lapse back into agnosticism, skepticism, or nihilism? We are confronted with an epistemological barrier: we can’t use deductive reasoning to prove existential propositions; neither can we rely on induction. But we can use Game Theory to our advantage…and we just did. What if the implications of Game Theory could take us ‘beyond Pascal’s Wager’?

Let’s revisit our base equations. Options (1), (2) and (3) pay out nothing; that’s certain. There’s no phenomenological difference between options that pay zero and options that don’t exist at all. Option 123 is equivalent to null-universe.  

Now let’s take a look at option (4). We’ve called the payout ‘1’ and so it is… in our two value number system. Therefore, the expected payout is just 0.000001. But what is the meaning of ‘1’? It’s not ‘half of 2’ because there is no 2; and half of 1 is still 1. There is only one and none, and one is any x where x > 0. 

But we’ve seen above that one is everything (Universe) and Universe (everything) is infinite (eternal). In a bizarre mathematical twist, if P(4) > 0, no matter how small P(4) is, then P(4)*∞ = ∞. This is the ultimate ‘participation trophy’. Just playing the game guarantees you an infinite payout…as long as you put your paltry pile of chips on #4. Winner, winner, cosmic dinner!

Now, that’s what I’m talkin’ about, Willis!  


David Cowles is the founder and editor-in-chief of Aletheia Today Magazine. He lives with his family in Massachusetts where he studies and writes about philosophy, science, theology, and scripture. He can be reached at


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