"Unifying Bargaining Notions (1/2)" by Diffractor

https://www.lesswrong.com/posts/rYDas2DDGGDRc8gGB/unifying-bargaining-notions-1-2

Crossposted from the AI Alignment Forum. May contain more technical jargon than usual.

This is a two-part sequence of posts, in the ancient LessWrong tradition of decision-theory-posting. This first part will introduce various concepts of bargaining solutions and dividing gains from trade, which the reader may or may not already be familiar with.

The upcoming part will be about how all introduced concepts from this post are secretly just different facets of the same underlying notion, as originally discovered by John Harsanyi back in 1963 and rediscovered by me from a completely different direction. The fact that the various different solution concepts in cooperative game theory are all merely special cases of a General Bargaining Solution for arbitrary games, is, as far as I can tell, not common knowledge on Less Wrong.

Bargaining Games

Let's say there's a couple with a set of available restaurant options. Neither of them wants to go without the other, and if they fail to come to an agreement, the fallback is eating a cold canned soup dinner at home, the worst of all the options. However, they have different restaurant preferences. What's the fair way to split the gains from trade?

Well, it depends on their restaurant preferences, and preferences are typically encoded with utility functions. Since both sides agree that the disagreement outcome is the worst, they might as well index that as 0 utility, and their favorite respective restaurants as 1 utility, and denominate all the other options in terms of what probability mix between a cold canned dinner and their favorite restaurant would make them indifferent. If there's something that scores 0.9 utility for both, it's probably a pretty good pick!