It’s like an instance of the problem involves not two, but four agents that should coordinate: a possible winner/loser pair, and a corresponding impossible pair. The impossible pair has a bizarre property that they know themselves to be impossible, like self-defeating theories PA+NOT(Con(PA)) (except that we’re talking about agent-provability and not provability), which doesn’t make them unable to reason. These four agents could form a coordinated decision, where the coordinated decision problem is obtained by throwing away the knowledge that’s not common between these four agents, in particular the digit of pi and winner/loser identity. After the decision is made, they plug back their particular information.
It’s like an instance of the problem involves not two, but four agents that should coordinate: a possible winner/loser pair, and a corresponding impossible pair. The impossible pair has a bizarre property that they know themselves to be impossible, like self-defeating theories PA+NOT(Con(PA)) (except that we’re talking about agent-provability and not provability), which doesn’t make them unable to reason. These four agents could form a coordinated decision, where the coordinated decision problem is obtained by throwing away the knowledge that’s not common between these four agents, in particular the digit of pi and winner/loser identity. After the decision is made, they plug back their particular information.