So everybody just wrote down all 0s, right?

Keltham did, yes.  +50 * 10 + -50 * 2 = +400 points.

Keltham is very good at this game!  Everyone should clearly use his numbers, which say that a coin comes up Abrogail twice in two spins roughly 0 out of 100 times.

(It took Meritxell two rounds to catch on. +350.)

"...okay, I don't know how to fix that. Bigger penalties for being wrong? But then you lose - it being better to participate than not -"

"Actually, we're not going to end up with that particular property, necessarily?  When you're just - guessing things, trying to know things, it's not like something comes in and takes your real-life gold pieces when you do the thing that loses you game points.  Like, the game can just say, every time you try to guess, that means you might turn out to be wrong and lose points.  But that's fine, because just guessing and just being wrong doesn't hurt your bank storage unless you actually made a bet."

"Who, besides Asmodia, would like to now try stating a fragment of Law that scoring rules ought to obey?"

"It ought to be a better idea to give your true guess," says Meritxell.

"Can you state that more precisely?"

"Say it actually has a one in four chance of happening. Guessing one in four ought to score more points than guessing anything else."

"So if double-Abrogail has a 1 in 4 chance of happening, then whoever guesses 25 on that round should be rewarded the most, regardless of whether double-Abrogail actually happened or not?"

"....not on that round. But in the long run."

"So, if we're just playing one round of something, there's no way it could have any kind of Lawful scoring rule."

"I think it'd be something like - escalating gains from being right when you pick an extreme number and from being wrong when you pick an extreme number, so it's only worth being that extreme if you're sure - and there's going to be some rule this implies but I don't know it, that balances it exactly right -"

"When you pick ninety-nine, the penalty for being wrong has to cancel out ninety-nine of being right, so it's only worth doing it if that's exactly how sure you are."

"So, considering all the numbers P we could guess between 0 and 100, if the truth is that something happens F out of 100 times, and doesn't happen 100 minus F out of 100 times, we want..."

This room's wall does now function as a whiteboard for people who, like Keltham, can cast Prestidigitation.

argmax P of F*Yes(P) + (100 - F)*No(P) = F

"...which is to say that it seems like a Lawful scoring rule must surely have this property: for every F between 0 and 100, the answer P that maximizes the sum of F of the yes-value you get from P, plus 100 minus F of the no-value you get from P, is F."

"In other words, if something happens 25 out of 100 times, then out of every possible answer between '0' and '100', '25' should do best, when it comes to adding 25 yes-values of the answer to 75 no-values of the answer."

"Putting somebody into a situation like this is what makes their answer mean, 'How often do you actually think this happens?'  There's a lot of ways to put people in weird situations where their answer could mean something else instead, because the most rewarding answer they could give isn't the one that matches reality!  Civilization tries to avoid weird situations like that, so that our words and more importantly numbers go on meaning things."

Carissa's not back to full health yet. She can feel it, even aside from the waking up panicked. But -

- it's like she can feel the thing she's reaching for, just barely out of her reach -

The most rewarding answer is the one that matches reality. That simple, and Hell - might genuinely not have stumbled on it - no, surely they have, at the highest reaches -

Everyone in this classroom except for her is going to end up executed.  And then she'll also get executed, but she's not going to Hell.

Go master dath ilani thought.  Right.  Sevar had better actually have the pull to avoid them all getting tortured for heresy which, to be fair, it kind of seems she might.

...so on the one hand she is pretty sure this is not how Asmodeanism works, and, on the other hand, it seems pretty persuasive that this is how Lawfulness works, and, on the original hand, she is very sure that Asmodeus is Lawful.  Is she allowed to just think that she'll ask Aspexia Rugatonn about this later or is that cheating?

Pilar's curse knows the answer to this one, actually.  Is it time for Pilar to hear?

Pilar is absolutely not taking theological advice on this subject from Cayden Cailean, especially before Pilar knows the actually correct answer the Grand High Priestess will give.

No problem!  Chaotic Good tries not to force answers on people that they don't want!

(It's how Asmodeanism can work for someone if they'd turn down Elysium because they actually wanted to go to Hell.)

Meritxell isn't sure there's a contradiction here. 

There might be, but - you're not supposed to reason about everything the same way. Obviously you're not supposed to use numbers-reasoning to consider matters in which the Church has instructed you; it's for matters where you can't otherwise figure out what you're supposed to believe.

At least she hopes that's it.

Keltham gives those words some time to settle; he's kind of guessing that nobody in Golarion has ever before considered the notion of incentive conditions under which talking becomes communication.

"So now, of course, we have the problem of finding any scoring rule which has this lovely and desirable property."

"Well, and we'd also like it to have the property that, when something happens, the higher the number you assign to it, the higher your score; or, when it doesn't happen, lower numbers get higher scores."

"So long as we're making up a wishlist, we'd like the method to still work if somebody says '17.3 out of 100'."

"We can take advantage of a symmetry which is that, if something happens 47 out of 100 times, that means it doesn't happen 53 out of 100 times.  So the yes-score of 47 should equal the no-score of 53.  Or to put it another way, the no-score of 30 is just the yes-score of 70; we don't need separate yes-score and no-score rules."

"And somebody named an important final condition earlier, does anybody happen to remember it?"

"If a prediction breaks down into two separate parts, the points you get for the whole prediction being right should be the points you get for both parts being right," says Gregoria, who said it originally.

"Hm hm, what sort of scoring rule could have that property?"