I explained it poorly in my comment above. The mind-reading analogy is useful, but it’s just an analogy. Otherwise the solution would be “Use your amazing psionic powers to level both enemy cities without leaving your room”.
If I had to extend the analogy, it might be something like this: we take a pair of strategies and run two checks on it. The first check is “If your opponent’s choice was fixed, and you alone had mind-reading powers, would you change your choice, knowing your opponent’s?”. The second check, performed in a different reality unbeknownst to you, is “If your choice was fixed, and your opponent alone had mind-reading powers, would she change her choice, knowing yours?” If the answer to both checks is “no”, then you’re at Nash equilibrium. You don’t get to use your mind-reading powers for two-way communication.
You can do something like what you described—if you and your girlfriend realize you’re playing the game above and both share the same payoff matrix, then (go home, go home) is the obvious Schelling point because it’s a just plain better option, and if you have good models of each others’ minds you can get there. But both that and (stay, stay) are Nash equilibria.
I explained it poorly in my comment above. The mind-reading analogy is useful, but it’s just an analogy. Otherwise the solution would be “Use your amazing psionic powers to level both enemy cities without leaving your room”.
If I had to extend the analogy, it might be something like this: we take a pair of strategies and run two checks on it. The first check is “If your opponent’s choice was fixed, and you alone had mind-reading powers, would you change your choice, knowing your opponent’s?”. The second check, performed in a different reality unbeknownst to you, is “If your choice was fixed, and your opponent alone had mind-reading powers, would she change her choice, knowing yours?” If the answer to both checks is “no”, then you’re at Nash equilibrium. You don’t get to use your mind-reading powers for two-way communication.
You can do something like what you described—if you and your girlfriend realize you’re playing the game above and both share the same payoff matrix, then (go home, go home) is the obvious Schelling point because it’s a just plain better option, and if you have good models of each others’ minds you can get there. But both that and (stay, stay) are Nash equilibria.