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Playing by the rules November 15, 2014

Posted by Ezra Resnick in Game theory, Logic.
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chess-cube“Want to play a game?”

“Sure! But first we need to agree on the rules.”

“Of course. I propose that we take turns proposing rules.”

“Agreed, and since you just proposed the first rule, I guess I get to propose the next one.”

“Wait a minute: I didn’t propose a rule for the game itself — I merely proposed a rule for how we ought to go about proposing the game rules.”

“Apologies; yours was indeed a meta-rule. In that case, let me propose a meta-rule of my own: Any disagreement about a proposed game rule will be decided by flipping a coin.”

“I’m not sure I agree with that.”

“Well, we haven’t yet agreed on a method for resolving disagreements about meta-rules. Do you have a suggestion?”

“How about we take turns: one of us gets to decide the first disagreement, the other decides the next disagreement, and so on.”

“OK, then: following your meta-meta-rule, I now get to decide our meta-rule disagreement about how to resolve disagreements about game rules.”

“Hold on: Who said you get to decide the first meta-rule disagreement?”

“Well, I let you determine the meta-meta-rule on how to decide meta-rule disagreements, so now it’s my turn.”

“Nice try, but we never agreed on how to resolve disagreements about meta-meta-rules. You can’t just make unilateral assumptions.”

“Well, how come you got to propose the first meta-rule to begin with? If you get to propose the first meta-rule then I should get to decide the first meta-rule disagreement.”

“Then I get to propose the first game rule.”

“Agreed.”

“First, I propose the following meta-rule: If the first rule proposal is challenged and loses the coin flip, the challenger must propose the following as his next rule: ‘The winner is whoever proposed playing the game.'”

“I don’t agree to that!”

“Noted, but according to our meta-meta-rule, it’s my turn to decide in case of disagreement on a meta-rule. And now for my first proposed game rule: The winner is whoever proposed playing the game.”

“Even if I disagree I still lose. Nicely played.”

“Thanks! That was fun.”

“Indeed. But maybe we should play a different game next time?”

“Sure! As long as we can agree on the rules…”

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