" - I don't know of one but if people were using coin flips for important things it'd definitely be invented."

"I wouldn't be surprised if my god has a spell for that which is incredibly difficult to mess with, but I don't know if it'd be fourth-circle or lower.  It would be on-theme with the truthspell and the pricing spell."

"Anyways, I now pose to you this devastating question: who says that a 1/2 probability of an apple is worth exactly half as much as an apple?  Maybe somebody is like, 'All this uncertainty about getting the apple makes me feel terrible; a 1/2 probability of an apple is really only worth to me a third as much as the certainty of an apple.'  And so, you can't use probabilistic trades to determine people's real trading ratios."

"Who says that the probabilities of things need to combine with their values by multiplication?  How simple, how naive!  Perhaps there is some more clever way to do things.  By what Law is it a regulation of our city, that we must do things in exactly that way, or else suffer some terrible misfortune?"

"- I mean, in this thought experiment we can't cut apples and we can't make promises. I guess you can say that we also don't like uncertainty but that'd be - you adding that - if you were perfect you just wouldn't care about uncertainty except for planning costs. Gods don't, I'm pretty sure."

"What I'm asking here is why there's a rule to combine probabilities with values by multiplication in the first place.  Why not square the probabilities and multiply the values by that, instead?"

" - there's the thing you said about how we could conspire against the Conspiracies in worlds that have Conspiracies, and I don't know if you literally mean there are uncountable worlds or if it's just a way of thinking, but - getting an apple in half of worlds is half as good as getting an apple -"

"There's countably infinite worlds, not uncountable ones - there's as many worlds as there are counting numbers, not as many as there are possible infinite sets of counting numbers.  You can only get ratios between the reality-weight of worlds if they're countable, so only those can be real.  Any time a world is divided more finely than that, only the parts that - I can't actually say this in Taldane, sorry."*

"And nice try, but we have ways to banish the 'anthropics' out of the conversation.  We can bet on a mathematical fact that neither of us know and seems to have the right random properties, like whether the remainder of 1001 divided by 17 is less than 6, say, which should give us a probability pretty close to 6/17.  I could truthspell myself and promise you that I hadn't secretly calculated the result in advance."

"There's no other worlds where that fact will be different.  So why do I need to value that bet of an apple at 6/17ths the value of an apple?"

(*) Realityfluid can be spread out over continuous distributions, but only chunks of those distributions large enough to integrate up to finite measures have people finding themselves inside.  People themselves are not so finely divisible, in regards to finding yourself to be one of them.  If you tried to make a continuity of different people in order to have an uncountable population of distinct people, sufficiently close parts of that continuity that they couldn't tell themselves apart would add realityweight from the perspective of whether you find yourself to be one of them, and so again you'd find yourself as something whose realityweight sums up to a finite fraction of everything there is.  This is why nobody ever finds themselves to be an entity with an actually-infinite number of introspectively-distinguishable distinct parts.

"Some dreadful misfortune will happen to you if you don't, involving people trading around probabilities of apples with you in an arrangement that leaves you with only a tiny chance of getting an apple."

"Or you passing up a trade series to increase your chance of getting an apple from some tiny amount to almost certainty, would be the complete spec of the dreadful misfortune."

"...and what would be an example case of an arrangement like that?  Suppose I'd trade a 1/2 probability of an apple for 1/4 the value of an apple, and a 1/3 probability of an apple for 1/9 the value of an apple.  What dread fate must now befall me?"

"I pay a whole apple to you for nine 1/3 chances to win an apple."

"And?  Just as I don't value a 1/3 chance of winning an apple at 1/3 of an apple's value, I don't value a 512-in-19683 chance of ending up paying you zero apples at around 1/38th the value of paying you zero apples.  Why, I value it at around 9/10th the value of paying you zero apples with surety!  As for all those other outcomes where I end up paying you one, two, or even nine apples, I value all of those put together with the remaining 1/10th of the weight that I put on things, when I weigh possibilities."

"If I haven't started out by already accepting that chances of things happening, ought to be weighted proportional to their probability, then when I look at all the things that might probably happen when you ask to buy four 1/2 chances of winning an apple from me, I don't have to value the 1/16th chance of paying you zero apples at 1/16th the value I put on that outcome."

"In other words, you're trying to convince me to accept the principle of weighting outcomes proportional to their probabilities, using an argument that only works on people who've already accepted that."

It's frustrating, she can feel that Keltham is doing something very bad and punishable, and it should be possible to Lawfully argue him out of it; but Pilar can't see the argument.  Talking to the Elysians was less frustrating than this.

She does not of course feel any anger at Keltham; it is clear that the fault lies within her for being unable to refute him.

"You've at least got to value two 1/2 chances of getting one apple, the same as, a 1/4 chance of getting no apples plus a 1/2 chance of getting one apple plus a 1/4 chance of getting two apples," Pilar states.  "Which means you've got to value one 1/2 chance of getting one apple the same as a 1/4 chance of getting no apples plus a 1/4 chance of getting two apples."

"Well, I accept you could mess with me if I didn't value two 1/2 chances of getting one apple, the same as 1/4 of zero plus 1/2 of one plus 1/4 of two.  But where was it said that I have to value two 1/2 chances of getting an apple, twice as much as I value one 1/2 chance of getting an apple?  We can stipulate that I value two apples twice as much as one apple, in cherries.  That was already said, but who says that packs of chances add up the same way?"

"Does it work as an argument if I say that people who think like you do won't have a lot of kids and eventually there won't be any of them left?"

"No, for it has not been stipulated that I care."

"Also we do tend, in Civilization, to regard that as an invalid argument generally.  If we look at the statistics and find that currently wizards are having fewer children, or for that matter, masochists are having fewer children, it doesn't follow that Cheliax should heritage-optimize wizards or masochists out of existence.  Maybe the thing to do instead is subsidize them so that they go on existing."

"If you do trades like that you'll go out of business. And I don't think Civilization would intervene to stop you going out of business."

"With some tiny probability I'll make a ton of money, though, if all the gambles I take pay off their maximum amounts.  Maybe I just happen to weight that tiny probability by a huge amount in my calculations, much larger than I'd weight it if I was multiplying outcome-values by probability-weights the way you think I should."

"So once again, you have not yet justified the principle of multiplying by probability, except by appealing to the principle of multiplying by probability."

"Now, if you could show that I was going to trade in a pattern that led me into strictly lower probabilities of getting an apple, or passing up strictly higher chances, that'd be another matter.  I do accept the principle that I should always want more apples and a greater probability of apples, all my other resources being equal or undiminished."

"Nethysian advisory, it's getting close to dinnertime, Keltham.  You should choose between wrapping up or going into deliberate overtime."

...he has not forgotten what is awaiting him after dinner.  Namely Yaisa.  He will not be delaying dinner today.

"All right, let's wrap up.  The general section of Law we are entering into is that which governs planning, paths through time, and its central principle is that of outcomes or destinations with consistent values, to which we navigate paths through time governed by the Law of Probability, and the value to us in this moment of a probable outcome is that outcome's value times the probability we place on it."

"From a more advanced perspective, the Law of Utility, or the Law of Probable Utility, is something that stands before the Law of Probability, even if the Law of Probability seems simpler.  The reason to think about events and reality using chances-out-of-100, instead of scales from 1 to 12, is that the chances out of 100 are what we have to plug into our plans, and not the scale from 1 to 12."

"Or at least, that's the perspective you'd take if you weren't coming at things from the angle of 'anthropics', but this, we should not do until a whole lot later."

"Cautions that I remember getting about this:  First, the same basic caution as for Probability.  If you try to think about something using numbers using this portion of Law, and the conclusions that result make no sense, and you are not already very skilled in this art, throw away the numbers and start over; do not follow the numbers off a cliff.  It has ever happened, in a case like that, that the conclusions were true and the flaw was in your own ability to make sense of them.  But in that case, the remedy is to first improve your intuitions until you can feel how the numbers make sense, not to go rush out and follow that advice before it has made sense to you."

"Second - though this part feels intuitively incredibly obvious to me, now that I'm no longer seven years old, I don't know if it actually is obvious, it wasn't when I was seven - you cannot by any amount of cleverness, reason from the mathematics of Probable Utility, to conclusions about it being Lawful or un-Lawful to value particular things.  Zon-Kuthon is Lawful Evil, he isn't making a math error by valuing endless suffering above happy people leading complicated worthwhile lives.  The Law says that, for Zon-Kuthon to get what he values, Zon-Kuthon must either behave a certain way, or else end up with pointlessly flawed plans that stumble over themselves and don't lead to the endless suffering that Zon-Kuthon prefers.  The Law says nothing of Zon-Kuthon's first preference from which his plans begin."

The students nod. That feels theologically right, not that it's exactly easy to map to a particular theological teaching.

"And it was also said to me from the very beginning:  For all the beauty of the Law as Law, and all the reasoning you might ever do about it as mathematics, the only reason to ever take that Law upon yourself, is if it is the correct Law of obtaining what you desire.  Not, necessarily, desire in a selfish sense, for this is Civilization's teaching of which we speak.  Good people desire Good ends, and this Law is their Law too."

"The meaning of the caution is rather that if you think, at some point, that this Law is telling you to do a thing, which will not lead best of all your available choices to whatever destination you seek, then most likely, vastly likely, you have made a mistake somewhere.  You may be mistaken about the Law, you might be correct in calculating what the Law must say and wrong in thinking that some other way is better, you may be correct about some derivations in mathematics but be wrong about which mathematics you should be using.  It is not likely the case that the Law is telling you a worse way and some other pattern is telling you a better one."

"But if - we are always also told - if some very clever person at some point demonstrated that the Law as taught in Civilization's lessons, did fail to be the best way of choosing in one part of reality so as to make another part of reality conform to our desire - then we should at once discard that old Law and seek another.  That, after all, was presumably how that whole branch of mathematics was invented in the first place.  In its final form, dealing with choices that are themselves mathematics, the Law of decision is a touch complicated; there must have been a time when people did not know it, and used simpler math instead.  If at that time they had thought to themselves that the Law they held was the final and ultimate principle and the definition in itself, of what should be done - and not instead thought of there being an ultimate goal to find that mathematics which best describes how choices in one place operate to constrain reality in another - they would have been unable to move on."

"But as far as dath ilan knows, this is the Law the gods use too? It's - right no matter how smart you are, or how vast your goals?" If she wants to be an archdevil she'd better go right for learning the Law gods use.

"I'd guess!  But if Civilization got a portal to Golarion and the gods said they had a different decision theory, everyone would be listening very attentively, much more so than they listened to me when I was twelve and had a better decision theory."

"What was your better decision theory when you were twelve." Meritxell says. 

"...sort of embarrassing, and complicated, and really blatantly wrong once you understand what Law is even supposed to look like normally, and it had a lot of terms in it you haven't learned yet."

"So I'm going to delay explaining it at least until everybody knows what the correct theory was supposed to be, to avoid misleading you.  That is the only reason I am delaying explaining it.  It's not at all because some part of my brain is worried that nobody in this room will want to have sex with me if you know about my early attempts at decision theory."

"dath ilan gives people weird sexual hangups," Meritxell says.