"It's a local optimum if there aren't any trades anyone can make that leave both parties to the trade better off, and it's global if there are no possible states of the jellychips that leave all three people better off."

"And can you prove that a local-not-global optimum is impossible for three players, each with one jellychip, of three different flavors?  Proving something for a simple special case is often easier than proving it for the general case, and sometimes is a good start on a general proof, if the problem hasn't been selected by some sadist... that is not what the dath ilani word 'troll' means but okay fine.  Anyways, proving it impossible for three players with three jellychips might be a start on proving it impossible in general, and in fact, there would be a lot of really interesting other proofs you could derive from that one."

Asmodia doesn't feel particularly driven to succeed in class, today, but -

"I have a chip Meritxell wants more than hers, but I don't want her chip more than mine.  Meritxell has a chip Paxti wants.  Paxti has the chip I want.  We can do a three-way trade but no two-way ones."  Sadist, she mentally completes.

"If they're continuous you can make that work, with partial jellychip trades -"

"They're not continuous!"

"Continuous doesn't help," says Asmodia.  "Meritxell wants my one chip but without Paxti she doesn't have anything I want.  Moving fractional chips around doesn't help with that.  Not unless there's continuous players, like every possible mix of Asmodia plus Meritxell plus Paxti."

Some students scribble in their notes for a little while until they are satisfied with this.

Asmodia, who is of course still pretending to be cheerful and energetic, will have enthusiastically written out the example:

Asmodia:  Has banana, prefers apple < banana < cherry
Meritxell:  Has apple, prefers cherry < apple < banana
Paxti:        Has cherry, prefers banana < cherry < apple

Asmodia wants Paxti's cherry!  But Paxti doesn't want Asmodia's banana!

All right, on to the notion of non-1-to-1 trades and quantitative indifferences.  New game, but instead of just saying that you prefer some flavors to others, you say things like, 'I'm indifferent between having 3 apple jellychips and 4 banana jellychips.'

This opens up the possibility of trading jellychips in a non-1-for-1 way.  Anybody want to try playing that game, if they're running quick simulations?

This seems like it makes it much harder to get stuck but no one has an impossibility proof yet. They do not seem to have...noticed the fairness problem? Or, they're writing down different possible trade outcomes but not with any sense that some of them are more desirable except subjectively.

Keltham quietly hands Meritxell a folded-up note telling her to try to end up with as many chips as possible for herself, in the course of suggesting mutually beneficial trades to the others.

- well all right then. 

The other students do notice this. "You recorded five to seven as the canonical one, but it could be four to eight too."

"Guess you should be the one writing it down," Meritxell says. "Paxti, seven blue for nine green?"

"Give me ten."

"You have recorded that you like green only nine percent less than blue, so I'm offering you nine."

"You have it recorded that you like blue a third more than green, so -"

"But I'm not offering you ten. Carissa, six blue for four red?"

"...is that allowed?"

"Is what allowed?"

"Are you allowed to not make trades that your utility function says you'll take. In this game."

"Well, you did it first, you turned down seven for nine. Carissa, six blue for four red, or if you make it five red, I'll throw in refusing to trade with whoever your least favorite student is."

"What if it's you?"

"Done and done, give me five red, I promise I won't trade with myself all day long. Gregoria, twelve red for....thirteen green -"

"Am I allowed to change my preference-weightings -"

"Obviously not."

"Keltham, am I?"

"Definitely no."

Gregoria hands over thirteen imaginary green. Meritxell turns around and hands eight of them to Tonia for blue. She looks supremely in her element and she's talking several miles a minute, withdrawing any trades the other girls don't agree to instantly.

...you would think they would somehow teach kids about this sort of principle before they let them have investment accounts let alone allocate years of training to wizard school.

Keltham will wait until they have ended up in a multi-agent-optimum; one of the many possible multi-agent-optima, which happens to have a lot of imaginary jellychips in the possession of Meritxell; such that, indeed, it is not possible to make all the students including her better off, by taking some of those away from her and looking for a more evenly distributed optimum.

It takes a while because Meritxell refuses so many trades but they get there eventually. 

Keltham shall now observe to them that, if Meritxell has 12 green and prefers two blue to three green, and Gregoria has 12 blue and prefers two green to three blue, then all of the trades "5 green for 7 blue", "6 green for 6 blue", and "7 green for 5 blue", are mutually beneficial, but differently divide up the gains from trade.

There's a lot of different ways for jellychips to be arranged such that they can't be moved around without making at least one player worse off.  For example, Meritxell could have all the chips, and nobody else could have any.  Then any other way of arranging the chips will make Meritxell worse off!  So that's one of the many possible global optima.

Different paths through the mutually beneficial trades will take you to different global optima.  So long as all the trades are mutually beneficial, you won't end up worse off than if you never traded, at the end; but you might end up much worse off than if you'd traded more carefully.

Keltham is a bit surprised that they didn't more quickly see the way in which this game resembled real life, since they seem pretty good at mathematical comprehension of the sort of structure that this game has in common with real life.  But that will come with having more than one day's practice with parsing up games and real life into the pure abstract structures and simple mathematical properties they have in common - with parsing up real life as a shadow of Law, that is then recognized at once when incarnated in some much simpler game.

All of what Keltham's saying makes sense to them! 

Ione wonders, in the back of her mind, if there's some way to actually go between worlds that way - by understanding real life as an instance of Law, and then sort of going through that Law to end up in a different instance of real life...

You know what, she's going to stop thinking that now.  Thinking things in the back of her mind has gotten her into enough trouble already.

Well, now that they've seen the problem of dividing gains from trade in a simpler form, re-encountering it as a more mathematical structure, have they got any new ideas about how to decide how many blue jellychips to accept for how many green jellychips?

"...it depends what the other person will accept?"

"You want to be keeping the books," Meritxell says. "Then everyone knows you'll be doing the most favorable trades you can and if they don't want to trade with you they're just out of luck. Or have some other kind of - asymmetric reason you can say 'I'm holding out for better' which they can't."

"They could, in fact, have mostly stopped trading with you, and traded with each other instead, until the game was almost ready to end.  And even then, if you'd tried to make trades too sharp, they could have just said no and offered you more even ones; and if you refused those trades, well then, the game ends without being multi-player-optimal."

"Even if you make your mutually beneficial trades very slanted in your own favor, people can't end up worse off, from trading with you, compared to if they didn't trade with anyone at all."

"They can end up worse off by trading with you, compared to if they'd traded with other people instead."

"So they walk away, and find other trade partners, if you try to capture too much of the gains from trade for yourself."

"This, too, is a lesson with a mathematical structure that appears in both this game and in real life."

"Sure, but it's costly to go around trying to find possible trade partners. In practice if you own the books you get the bulk of the gains from trade."

Then some book-owners are going to really lose out once Keltham figures out cheaper roads and bicycles; so Keltham thinks, but also meta-thinks fast enough not to say out loud.  He is not quite sure of his larger social situation, and maybe he is better off quietly not pointing out certain winners and losers just yet.

"Dath ilan has some excess wealth beyond bare living needs," and, now that Keltham thinks about it, probably a much more structured investment scene, "which a hundred thousand annoyed customers can easily use to pay the startup costs of a new company that makes whatever you make, and contracts to sell it more cheaply for the first ten years to its founding customers.  So 'I'm the only trade partner around' does play less well there."  If he emphasizes the part with the vast wealth Golarion won't have for a while, that'll maybe sound less threatening to anybody reading these reports, compared to if they realize that roads will apply the same market pressure.  "Does Cheliax have a lot of places where, say, there's only one seller of food...?"