# 2947: Pascal's Wager Triangle

 Pascal's Wager Triangle Title text: In contrast to Pascal's Wager Triangle, Pascal's Triangle Wager argues that maybe God wants you to draw a triangle of numbers where each one is the sum of the two numbers above it, so you probably should, just in case.

## Explanation

The comic is a conflation of Pascal's Wager and Pascal's Triangle. It's structured as a layout that emulates Pascal's triangle, an infinite triangle of numbers where the top number is 1 and each value below is the sum of the adjacent number(s) above it. The second row has two 1s (each the sum of the single 1 above), and the third row has a 1 (the sum of a single 1 in the second row), a 2 (the sum of both 1s above it), and another 1, and so on. It plays important roles in binomial expansion, probability theory, and other areas of math. While Blaise Pascal did not invent the triangle, it is named after him (an example of Stigler's law of eponymy).

 … seed1 … … … + 11 1 + …1 … … … + 11 1 + 12 1 + …1 … … … + 11 1 + 23 2 + 13 1 + …1 … … … + 11 1 + 34 3 + 36 3 + 14 1 + …1 … ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

Pascal's Wager is a philosophical argument proposed by the same Pascal. Essentially it says that if God exists, both the rewards for believing in God and the punishment for nonbelief are infinite; if not, the cost of belief and benefit for nonbelief are negligible. Therefore, if there is a finite possibility that God exists, however small, one should believe in God. One problem with that is that there may be more than one God to believe in, even if only one truly exists. Which one of all the possible Gods should you choose to believe in could be problematic, if the real God insists that you only believe in Him and punishes you for believing in any other gods (even if you somehow also believed in Him). A further problem is that committing to any particular belief in a deity is not a totally zero cost option, and thus affects your life in many needless ways if you subscribe to any particular practice of religion not actually required by any extant god(s), even if any of them exist in the first place.

The comic merges the two Pascalian concepts: each Cueball is wagering his proof of a god or gods to the Cueballs below him, thereby creating Cueballs that believe in the sum of the number of gods of the Cueballs above them. In the second row, the two Cueballs each believe in one god, as intended by the original Cueball. However, in the third row, the Cueball in the middle interprets the two proofs offered to him as proving the existence of two gods. Theoretically, this expansion would continue for all integers as the triangle grows, giving rise to a belief in escalating numbers of gods going down and towards the middle of the triangle. This is clearly not the intent of the first Cueball, who simply offered a proof of his one god, but he has no control over the situation below him.

It is unclear why the Cueballs behave in this fashion, instead of treating all the proofs as proving the existence of the same god. Perhaps each one rewords their arguments for god(s) sufficiently to make them sound different than other gods. This is not without precedent. Scholars of comparative mythology believe that the religion of Proto-Indo-European peoples splintered into many disparate religions of Europe and West Asia; for instance, Dyeus phter (sky father) became Zeus in Greece and Jupiter in Italy.

This comic may be referencing a common counterargument to Pascal's Wager — that it works equally well for any hypothetical god which offers eternal paradise for one action and eternal damnation otherwise. This can even include hypothetical gods with contradictory criteria for entrance into paradise. In this case, the Cueballs apparently chose to believe in all the deities they've heard of in order to cover their bases.

The title text suggests that everyone should draw a proper Pascal's Triangle, since there is a possibility that God wants you to do so, and if they do then the benefits of pleasing God or the costs of displeasing God could be high, whereas if they have no such desire then there is minimal cost to drawing one anyway. The failing of this logic is that God may have a positive preference for you not to draw a Pascal's Triangle (though at least according to the Catholic Church this is unlikely, as Pascal himself is on the way to beatification.)

Pascal's Wager was previously mentioned in the title text of 525: I Know You're Listening.

## Transcript

[Cueballs, each holding some document, are shown in a triangular arrangement, with arrows pointing from upper to lower Cueballs:]
[At the top, row 1 has a Cueball, unnamed but described below as "C1", holding a piece of paper with a crossed-square shape possibly resembling a Punnet Square diagram and with a speech-bubble]
C1: Hey, you two below me! Here's a proof that you should believe in my god!
[Two arrows lead diagonally down-left and down-right from C1 to the second row, having two similarly drawn Cueballs (differing only by slight changes in pose) described here as "C2" and "C3"]
C2 & C3: I'm convinced! Hey, you two below me! Here's a proof that you should believe in my god!
[Two arrows lead down from each of Row 2's Cueballs to three similar Cueballs on Row 3, "C4", "C5" and "C6", the central Cueball being the target of arrows from both of the predecessors, and the speech-bubbles partly obscuring the predecessor Cueballs and the lines of the arrows]
C4 & C6: I'm convinced! Hey, you two below me! Here's a proof that you should believe in my god!
C5: Ok, I believe you both! Hey, you two below me! Here's a proof that you should believe in my two gods!
[Two further arrows lead down from each Row 3 Cueball to a total of four Row 4 Cueballs, all but the edge ones having two incoming arrows, "C7" to "C10"]
C7: I'm convinced! Hey, you two below me! Here's a proof that you should believe in my god!
C8 & C9: Ok, I believe you both! Hey, you two below me! Here's a proof that you should believe in my three gods!
[C10 has no visible bubble, as there is no room for one in-frame, and is itself also slightly obscured by C9's bubble, but would logically be considered to have a "believe in my (singular) god" monologue]
[Continuing the pattern, Row 5 has "C11" to "C15" Cueballs (with only a small amount of leg visible above the lower edge of the comic frame), each is led to by diagonal arrow(s) originating from the Row4 Cueballs]
[C11 and C15 have no visible speech bubbles, due to being even more edge-adjacent and C15 even reaching off the edge of the frame with his 'punnet paper', but can be assumed to have a "believe in my god" statement]
C12 & C14: Ok, I believe you both! Hey, you two below me! Here's a proof that you should believe in my four gods!
C13: Ok, I believe you both! Hey, you two below me! Here's a proof that you should believe in my six gods!
[No further arrows or Cueballs can be visible, and no further speech-bubbles obscure Row 5, but without any reason to believe they aren't just off frame]
[Caption below the panel:]
Pascal's Wager Triangle

# Discussion

Apparently, if two people are writing a first draft at the same time, the wiki appends one to the other. Welp. GreatWyrmGold (talk) 02:29, 18 June 2024 (UTC)

If you add another one it puts it beside the second, and you have Pascal's explanation.172.70.85.102 08:33, 18 June 2024 (UTC)

Some religions (such as my own) prohibit polytheism, so that's an added wrench in the works. --141.101.98.119 06:56, 18 June 2024 (UTC)

I think MOST religions are quite strict about the idea that you are supposed to CHOOSE single religion, preferably the one in question, and not trying to cover all bases by believing in multiple ones. Which is the answer to Pascal's Wager: choosing wrong God is likely to result in worse punishment than choosing none, so better NOT believe. -- Hkmaly (talk) 09:44, 18 June 2024 (UTC)
Well, it might be considered a bad "memetic trait" to have "as well as believing in our thing, you can believe in anything else". (Much as a number of holy books include the instruction that you should not change anything when copying the book, and this would clearly be a "dominant meme" as soon as someone spontaneously thinks to add it to a previously mutable version of the text.) Although there's syncratic religions which are effectively a case of "horizontal meme transfer", philoso-evolutionarily.
But polythesitic pantheons aren't unusual within a (structurally singular) religion. The three branches of abriamic religion ("There is no god but Jehovah", "There is no god but God (sic)", there is no god but Allah", at least unless you start going into the prevalence of trinitarianism and "praying to individual saints") and I think Sikhism is rather good at "everything is but an aspect of the one..." (even when it comes to considering other religions' own ideas, and thus nominally folding ). But multi-deity (and zero-deity) religions/beliefs/etc are quite widespread, so an exclusivity of "my god" might be considered rare. (Though, numbers-wise, Christianity (in all its flavours) and Islam (ditto) add up to just over half of the world's population, perhaps being generous with 'habitual followers' rather than just the most devout. So it would be fairly accurate to say that most people are living under a (theoretically) quite strict monotheistic situation. Not that they all agree with each other quite what 'the singly god' is (even within themselves, ask a Protestent what they think about what a Catholic thinks, as the famous joke about this goes; and that's definitely not solely a Christian trait), although that's not what we're counting. 172.71.242.54 21:05, 18 June 2024 (UTC)
This reminds me of a novel (I think it was Arthur C. Clarke's 3001: The Final Odyssey) where in the distant future, all religions ended up merging into two - one where there is at most one god and one where there is at least one god. Shamino (talk) 14:15, 18 June 2024 (UTC)
A surprisingly large number of religions actually don't care if you additionally are a member of another religion. Happens all the time in eastern Asia.162.158.212.173 21:05, 18 June 2024 (UTC)
If you take a wider perspective, Abrahamic religions are the exception to the rule. In European history, lots of polytheists listened to Christian missionaries and decided to add this Jesus guy to their pantheon (until Christianity drowned the old religion in a well). Before that, Romans and Greeks and basically every other civilization whose religious history can be deduced from available evidence exchanged gods with neighboring cultures or trading partners as readily as they exchanged useful vocabulary or sword designs or whatever. Outside Europe, these patterns hold, up until some pale jerks from across the sea start threatening people who worship gods they don't like. (And sometimes even after that.) The exceptions to the rule are exceptionally popular and influential (especially in English-speaking parts of the world, natch), but that doesn't make them the Real Rule that all other religions deviate from. GreatWyrmGold (talk) 04:16, 25 June 2024 (UTC)

What if there is a God, but they don't want you to believe in them?172.70.86.35 11:48, 18 June 2024 (UTC)

Or a God, but they have impostor syndrome? P?sych??otic?pot??at???o (talk) 12:34, 18 June 2024 (UTC)
Or a God, and he just wants to screw over us all (not in the Yivo sense, mind you)? (Under this condition, all three outcomes are possible: he wants us to believe, he wants to hide, he couldn't care less) 198.41.242.174 08:05, 19 June 2024 (UTC)

How did the middle Cueball on line 3 not realize that both Gods he was being asked to believe in are the same God (since the ones on line 2 both got it from the same Cueball on line 1)? Is there also a game of Telephone going on? Barmar (talk) 14:29, 18 June 2024 (UTC)

Tbf there's no clarification by the Line 2 guys that they're talking about the same God. It makes sense that he'd assume they're different. If two different religious people told you at the same time "My God is real" and you'd never really been exposed to religion or atheism, you would assume they were talking about two. Forgive me if I made any unfair assumptions here. P?sych??otic?pot??at???o (talk) 15:34, 18 June 2024 (UTC)
Look at the schism in Islam and the huge number of divisions in Christianity. That it may be the same god and same basic religion doesn't mean that it isn't presented entirely differently. For example (what I'm familiar with), Roman Catholic versus Church of England. Similar, but also completely different. 141.101.69.118 19:03, 19 June 2024 (UTC)
Presumably each person interprets the diety slightly differently mouse

>>then there is minimal cost to drawing one anyway<< I argue that drawing a complete pascal's triangle will take infinite time and infinite resources. Which is slightly above "minimal cost" 108.162.221.61 05:34, 19 June 2024 (UTC)

The use of the words "complete" and "infinite" while referring to the same thing is funny to me. I don't know why, though. P?sych??otic?pot??at???o (talk) 13:19, 20 June 2024 (UTC)

I very nearly inserted the following 'argument' paragraph in the Explanation, but decided that it perhaps was a bit of a 'belief' of my own.

```(A more realistic approach could be to just outwardly conform to the social expectations of the society to which you belong. With or without any actual core belief, which cannot be tested in any realistic manner during your lifetime, maintaining a performative cohesion with your neighbours and acquaintences would still benefit you in avoiding standing out from the commonly accepted ideology. This may still count as a 'belief' (desirable or undesirable) should this stance ever come to be judged by any deity, with your ultimate fate probably being mirrored by others that you know who are in your exact same ideological position and merely outwardly perpetuting the 'accepted' demonstrations of faith. Assuming that the society's religion is not a self-destructive one, you would have as good a lifetime as circumstances would generally allow, with the possibility of an afterlife that is at least not uniquely bad for you; unlike the smart-arse who presumes to know what the single best theological stance is and then has to rationalise their attempt to cynically play the odds upon Judgement Day, against a deity who fully understands their base motivations and may even take far less kindly towards a Wagerer than with someone who never ever tried to 'believe' but at least was generally and secularly philanthropic in a way that the deity might be pleased by.)
```

I stand by the logic, as far as it can ever be taken (without knowing anything useful about the actual mind of God/gods it has to be tested against) but it got a bit long as a 'minor philosophical interlude'. And if the Ultimate Arbiter does not like Bill-And-Ted-ism (i.e. "be excellent to each other") and getting along with your community, instead prefers more outstandingly misanthropic worldly behaviours (because the only 'heavenly' afterlife is more Valhalla than Asphodel Meadows; anyone who even mildly disappoints goes to the Hades/Tartarus equivalent) without actually clueing us in on this particular Ineffable Plan, then the fate of many (who don't make earthly lives more miserable for the rest of us) is pretty much doomed from the start and even the most perfect Pascalian Wagerer probably never managed to stand on the right side of the philosophical barrier either. This whole argument is not even a novel philosophy, of course, but it's mine. 172.69.195.56 16:36, 21 June 2024 (UTC)

Doesn't the exact wording of the alt text not technically call for a classic Pascal's Wager, but rather an inverted triangle where there is only one number at the bottom, two in the row above the bottom, et cetera? (Since there have to be two numbers above every number: the traditional exception for the top row and the numbers at the sides isn't present.) Given such a triangle, wouldn't the only possible solutions be all-0, all-positive-infinity, or all-negative-infinity, which certain readings of "number" might restrict to the all-0 solution? WingedCat (talk) 00:20, 22 June 2024 (UTC)

PROBLEM: Currently, the transcript includes a recreation of the triangle, nicknaming the Cueballs C1 through 15, laid out in a triangle. This seems too visual for the transcript. The thing is, it seems like a major goal of the transcript is for blind people, who have a reader program read them the transcript, and they follow XKCD solely in this way. Wouldn't a reader program render this as "C1 C2 C3..." etc? Giving no indication as to the layout? As the fix feels rather clunky and I'm not 100% sure this is an issue, I felt I should mention it first before enacting change. NiceGuy1 (talk) 04:07, 23 June 2024 (UTC)

Extended the Transcript to perhaps help cater for this issue. Maybe too wordy. Maybe a mistake to mention what isn't there (except in explaining the full spirit of the diagram). But hopefully more mentally re-constructable by anyone with a visual imagination. 172.70.86.9 19:25, 23 June 2024 (UTC)