I do not have a formal proof, but here is an outline:
Consider a message
one player can transmit to the group .If it is beneficial to transmit
, it must be detrimental to a subset , and at least neutral for .The subsets
and are in zero-sum conflict, so as an entity does not benefit from transmitting .Thus, for
to be beneficial to a player within , it must be at the expense of the other players, contradicting the definition.Also, if there are several coalition-forming messages, and one of them leads to the highest payoff whether or not it is true, it is always beneficial to transmit that message, so it is cheap talk.
Therefore, the highest-payoff coalition-forming message is either true or silence.
The third step is a little tricky. What if the coalition forming itself benefits everyone in
I appreciate your replies. I did talk explicitly with them about this before writing my comment. I learned they were committed to social manipulation, though they did agree to target me less. I like your suggestion of a handicap, and I might bring that up next time we play.
I agree that I was underestimating how fun social manipulation is. Looking back, when I play Secret Hitler online I absolutely lie all the time, just because it’s fun to sew chaos. So, I think I’m being hypocritical and annoyingly principled. Why draw the line at in-person games? [1]
I remember them repeating something like, “it’s really fun to play the social manipulation game, it’s not all about winning.” I told them (paraphrasing), “okay, but can’t you do that after following the norm for awhile, so everyone has gotten better at the game? Then you get the enjoyment of social manipulation plus the enjoyment of an interesting game!” They said they didn’t really care for that. They didn’t elaborate. My guess is their discount rate is lower, or maybe it’s more fun to manipulate people who don’t understand the game.
I did tell them that I’m probably going to just stop playing Settler’s of Catan with them. As you said, I should seek to strike a mutually beneficial deal, and I don’t think that’s actually possible here. The game is just as fun without me—maybe more fun, because the competition is gone. What selfish rational incentive is there to play a less fun game just so I can play a more fun game?
To be more charitable, the line is when with significantly less skilled players. Online Catan? Not cool. Offline SH? Not cool. Online SH? Go for it. It’s still norm breaking since players want to win, and sewing chaos hurts your team’s chances. It’s a different norm, but maybe one people care about more, and makes me a hypocrite regardless.