I might get around to editing the post at some point, but unless/until I do, a note:
Anti-coordination: ∙>∙>W>Z (two games)
In this game, the goal is to play a different move than your playmate. If X=Y then there’s no reason to prefer one move over another, but if they’re not equal there’ll be some maneuvering around who gets which reward. If you’re not happy with the outcome, then changing the move you play will harm your playmate more than it harms you. The Nash equilibria are when you play different moves, and these are socially optimal.
Correction: if you’re not happy with the outcome, changing your move will harm your playmate more than it harms you if Y>X, and harm you more than them if X>Y.
Also, Battle of the Sexes is an instance of this game if you relabel the moves. If you label them “do the thing you/they prefer”, you get X>Y>W≥Z. (X is “we both do my thing”, Y is “we both do their thing”, W is “I do my thing and they do theirs” and Z is “I do their thing and they do mine”. W=Z if going alone to either is equally bad.) If you label them “do the thing they/you prefer”, you get Y>X>W=Z. You won’t get Y>X>W>Z. (That would imply payoffs like: I most want to do “my thing” together. Failing that I want to do “their thing” together. But if we do go separately, I’d rather do “their thing” than “my thing”. Which wouldn’t be a BoS.)
I might get around to editing the post at some point, but unless/until I do, a note:
Correction: if you’re not happy with the outcome, changing your move will harm your playmate more than it harms you if Y>X, and harm you more than them if X>Y.
Also, Battle of the Sexes is an instance of this game if you relabel the moves. If you label them “do the thing you/they prefer”, you get X>Y>W≥Z. (X is “we both do my thing”, Y is “we both do their thing”, W is “I do my thing and they do theirs” and Z is “I do their thing and they do mine”. W=Z if going alone to either is equally bad.) If you label them “do the thing they/you prefer”, you get Y>X>W=Z. You won’t get Y>X>W>Z. (That would imply payoffs like: I most want to do “my thing” together. Failing that I want to do “their thing” together. But if we do go separately, I’d rather do “their thing” than “my thing”. Which wouldn’t be a BoS.)