The betting sort

Image credit: Paul Cézanne, The Card Players, 1894-95

You may be proud, wise, and fine, but death will wipe you off the face of the earth as though you were no more than mice burrowing under the floor, and your posterity, your history, your immortal geniuses will burn or freeze together with the earthly globe.

― Anton Chekhov, The Bet

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If you were French in the latter sixteenth century lucky enough to have time to pass (rather than little to none, as your subsistence was not all consuming), you basically had two options: playing cards or fighting religious wars. On this, historical accounts agree.

And if you were Catholic, the odds of winning were tilted in your favor if for no other reason then demographics. Only about seven to ten percent of the French population were Huguenots (Protestants)—maybe 1-2 million souls. Stacked against you was King Henry IV, who was a Huguenot and, as king ruling by divine right, had absolute power. Including the army. Until 1598, that is, when, in one of the most palliative spiritual rebirths in history, Henry converted to Catholicism (famously sniffing, “Paris is worth a Mass”) and issued the Edict of Nantes. Peace at last. 

Thus, by the early 17th century, your options for passing time were whittled down to playing cards.¹ As a consequence, the fair apportionment of the winnings became the subject of much cerebral coal burning by those with time to burn. Particularly perplexing was the apparently common situation wherein a game of a fixed number of rounds was interrupted by an unexpected event. How to fairly divide the winnings?²

A simple solution had been found³ and routinely employed, taking a retrospective approach: divide the winnings among the players based on the number of rounds won. Simple, easy to calculate, fair. But what if only one round had been played before the unexpected interruption? Award all the winnings to the lucky sod who won the first round? That didn’t seem … fair. And humans are all about fairness.

Various geniuses took a stab at improving on the simple solution, none entirely successful. Concern with this pressing matter spread from Italy to France, which, like so many European countries in the 17th century, had a surfeit of mathematicians with time on their hands. Among these were Blaise Pascal and Pierre de Fermat. Enter a social gadfly, who styled himself a chevalier but was not, Antoine Gombaud.⁴ Gombaud the problem to his young and mathematically gifted acquaintances, Blaise and Pierre, who got immediately to work.

Pascal turned the traditional solution on its head. Instead of taking a retrospective approach to its solution, he adopted a prospective path: What would likely have happened if the game had continued to its conclusion? After a great deal of coal burning, Pascal had a draft solution, but he was unsure of it. So he consulted a highly respected mathematician with whom he had been in correspondence on other pressing issues. In a letter dated Monday August 24, 1654—running to nearly 3,000 words—Pascal began:

I was not able to tell you my entire thoughts regarding the problem of points by the last post, and at the same time, I have a certain reluctance at doing it for fear lest this admirable harmony which obtains between us and which is so dear to me should begin to flag, for I am afraid that we may have different opinions on this subject. I wish to lay my whole reasoning before you, and to have you do me the favor to set me straight if I am in error or to indorse me if I am correct. I ask you this in all faith and sincerity for I am not certain even that you will be on my side.⁵

Between them, Pascal and Fermat “solved” the problem of points in an iterative process,⁶ and in so doing invented (or discovered) fundamental concepts in what was to become probability theory.⁷ As all solutions do, it had its limits, as a fair game was assumed: all players have an equal chance of winning and each round is independent of the outcome of previous rounds. How many games in life do you know that follow those rules?

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Pascal must have been pleased with the solution he and Fermat had reached, as he applied probability theory to what he considered the most consequential matter imaginable: a proof of God’s existence. Or at any rate, that’s what most people— likely you as well—think Pascal’s famous wager to have been about. And if that is what you think, you would be entirely wrong. His wager was not about proving God’s existence, but about proving that you ought to believe that God exists. And betting on God existing is the best bet available—statistically.

It’s all about decision theory, more specifically decisions under risk. Following is the text of Pascal’s wager; it’s a good bet you’ve never actually read it, more than maybe the last line, and to a near certainty not the entire pensée.

Yes; but you must wager. It is not optional. You are embarked. Which will you choose then? Let us see. Since you must choose, let us see which interests you least. You have two things to lose, the true and the good; and two things to stake, your reason and your will, your knowledge and your happiness; and your nature has two things to shun, error and misery. Your reason is no more shocked in choosing one rather than the other, since you must of necessity choose. This is one point settled. But your happiness? 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. Wager, then, without hesitation that He is.⁸

If you find Pascal’s prose somewhat opaque (and the more you read the more likely that outcome) here is the wager in a more approachable decision table:

	OUTCOME POSSIBILTIES
YOUR BET	GOD EXISTS	GOD DOES NOT EXIST
Wager for God	– Gain all –	– Status quo –
Wager against God	– Misery –	– Status quo –

Whichever wager you make, the outcome of betting on God’s existence is better than betting against, because you have everything to gain and nothing to lose. In technical terms, “Wagering for God superdominates wagering against God.”⁹ Bet for God’s existence and the worst outcome (status quo) is at least as good as the best outcome as betting against His existence (status quo).

And what, exactly, is this God you’re betting on? Pascal has a ready answer:

“If there is a God, He is infinitely incomprehensible, since, having neither parts nor limits, He has no affinity to us. We are then incapable of knowing either what He is or if He is.”¹⁰

This goes beyond apophasis: you cannot know what or if He is, but you can bet He has no affinity to you. Pretty grim. Nevertheless, “…you must wager. It is not optional. You are embarked.”

Except, maybe it’s complicated. “That is very fine. Yes, I must wager; but I may perhaps wager too much.”¹¹ (Maybe you do have something to lose….) Or, what if you assign the probability of God’s existence as 0? Nothing irrational in that. Or, oughtn’t there be more columns, as there are many gods (the Catholic god exists or not; the Huguenot god exists or not; the Islamic god exists or not, Zeus exists or not…)?

You may be unsurprised to learn that (having lived long enough to limit that emotion) any number of people have spent entire careers on Pascal’s wager (objecting, defending, modifying, exegeting, teaching, etc.). Paid careers!

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The one thing Pascal got right is that you are embarked and must wager, willy nilly. Comes with the territory, unless you’re the sort of person with more practical concerns and happy to leave such matters to your inherited ideology (religious, cultural, political, some combination). So maybe no wager is entailed.

On the unlikely assumption that you at least occasionally drop your golf clubs long enough to ask such questions as How should I behave? What ought I to endorse or condemn? How should I distinguish ought from is?, mightn’t it be better to start with what can be known? Time for a little cataphasis, even if only palliative?

Good. Another point of agreement with the mathematician, now based on what is known rather than the reverse, has to do with affinity. Or its lack. As far as can be known, the universe—containing God or not—has no affinity with you. In either case, this is of necessity.

What else is thought to be known? Oh, maybe the universe is something like 13 billion years old, started in a highly ordered state (low entropy), had a Big Bang introducing all kinds of busyness, and is moving toward its original tidy condition. (If the universe were created, God must be Germanic—all those orderly villages and tidy gardens).

Biological life seems to have appeared a little less than 4 billion years ago, mammals about 178 million years ago, primates 55 million, Homo sapiens about 260,000 years ago, and all-seeing gods about 12,000 years ago. More than comparatively recent, life seems also to be vanishingly rare, evanescent—and is certainly contingent. If the universe were capable of awareness, you would likely come and go so quickly and be of so little matter as to capture no attention.

Pascal had the benefit of none of that accumulated knowledge—some of it accumulated quite recently. If he had, he may have updated his credences and revised his wager. Perhaps proposing something more modest.

Or maybe not. Almost no one inhabiting the planet has or does (update credences). And that is the strange thing: it’s still happening. It? Oh, only the pursuit of pastimes like the French Religious Wars, the Thirty Years’ War etc. etc. Would that they took up cards instead.

Those with religious commitments will likely object, But those were not really religious wars; they were about political power. As if those two antecedents have ever been severable; rare and disputable exceptions prove the rule.

Even today actual people (not avatars) are being shunned, excommunicated, tortured, raped, thrown off towers, mutilated, bombed for the ideological bets laid, most of them theological. In the 21st century, more than four centuries past Pascal’s time…! Is that acceptable? Pascal seems to have thought so:

“The justice of God must be vast like His compassion. Now justice to the outcast is less vast, and ought less to offend our feelings than mercy towards the elect.”¹²

(Seems that Pascal did know something about his deity.)

Much of humanity today would endorse that view, if they could parse it. Someone else died in the fire, earthquake, flood, war, plague. You, having survived, likely find God’s mercy to you more vast than the trivial justice He meted out to the outcasts (who died). Must be, right? You’re still embarking….

Artificial General Intelligence may well emerge before general human intelligence. Ah Bartleby, ah humanity.

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What to do? Nothing? Not an option; you must wager. Well, at least you must exist, which comes with both requirements and constraints, held in considerable tension (as the latter may largely preclude the former).

Given the known givens, how you chose to live your life would seem to be entirely up to you (if you have a choice and within culturally tolerable limits). You can behave in any way you can get away with. You might choose to use your resources to help those less lucky, or you might pile up more money than you can possibly spend, perhaps occasionally doling out some fraction of your hoard in an (inevitably) attributed philanthropic gesture.

You might expend effort enough to think of others at least half as much as you think about yourself, or not bother. You might attempt to be kind even on a foul-weather day, or let your mood rip. The universe probably can’t care, and if it could it probably couldn’t care less.

If that’s something close to true, if the universe knows neither good or bad, on what basis might you do the better thing even when it’s the harder thing? Maybe, a different wager is in order, one more modest and in keeping with what is thought to be known. As you’re here, pull your horns in and try to construct a better bet.

Maybe something like: Pay the universe a compliment it may not deserve. After all, you have nothing to lose and possibly something to gain if it turns out that what is currently known is insufficient.

Not bad. That would seem to pass the decision-table test:

	OUTCOME POSSIBILTIES
YOUR BET	MORAL UNIVERSE	AMORAL UNIVERSE
Do the better thing	– Possible benefit –	– Status quo –
Do what you want	– Possible detriment –	– Status quo –

Oh, but there are likely to be objections, modifications, scoffings hung like millstones round the neck of your best and most sincere efforts. Exhausting to consider. Maybe just give it up and surrender to the likely eventuality that you and all your efforts, like coal in your cerebral furnace, “will burn or freeze together with the earthly globe.” Back to cards, or golf, or belief, or whatever passes your embarkment most easily.

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1. Happily, that unnatural state of affairs was put to rights in 1635, when Cardinal Richelieu, prime minister to Louis XIII, jumped into yet another religious dispute, the Thirty Years’ War, which killed an estimated eight million people; one of the most devastating wars in Europe. Normalcy had been restored. ↩︎
2. This is known as “the problem of points.” ↩︎
3. At the cost of some mental effort. Luca Pacioli worked it all out in his 1494 textbook Summa de arithmetica, geometrica, proportioni et proportionalità. ↩︎
4. Alias Chevalier de Méré. ↩︎
5. Blaise Pascal, Pascal’s Pensées (Urbana, Illinois: 2006, Project
   Gutenberg), https://www.gutenberg.org/files/18269/18269-h/18269-h.htm ↩︎
6. If you must know, the solution goes like this: Say A and B have each put $10 in the pot, and the game is interrupted when A needs 1 more win and B needs 2 more wins. The game could end in either the next round (A wins) or the two subsequent rounds (B wins both). The probability of A winning the next round is 1/2, and the probability of B winning two in a row is (1/2) × (1/2) = 1/4. Therefore, the fair division would be in proportion to these probabilities: A should receive 2/3 of the pot ($20), and B should receive 1/3 ($10). ↩︎
7. Such as expected value. Look it up yourself. ↩︎
8. Pascal, op. cit. ↩︎
9. Alan Hájek, “Pascal’s Wager”, The Stanford Encyclopedia of Philosophy (Winter 2022 Ed.), Edward N. Zalta & Uri Nodelman (eds.), https://plato.stanford.edu/archives/win2022/entries/pascal-wager/. ↩︎
10. Pascal, op. cit. ↩︎
11. Idem. ↩︎
12. Idem. ↩︎

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