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James_Miller comments on A simple game that has no solution - Less Wrong Discussion
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If Player 2 gets to move he is uncertain as to what Player 1 did. He might have a different probability estimate in the game I gave than one in which strategy A did not exist, or one in which he is told what Player 1 did.
I'm not convinced that the game has any equilibrium unless you allow for trembling hands. For A,A to be an equilibrium you have to tell me what belief Player 2 would have if he got to move, or tell me that Player 1's belief about Player 2's belief can't effect the game.
In a classical game all the players move simultaneously. So to repeat, your game is: * player 1 chooses A, B or C * then, player 2 is told whether player 1 chose B or C, and in that case he chooses X or Y * payoffs are (A,-) -> (3,0); (B,X) -> (2,0); (B,Y) -> (2,2); (C,X) -> (0,1); (C,Y) -> (6,0)
The classical game equivalent is * player 1 chooses A, B or C * without being told the choice of player 1, player 2 chooses X or Y * payoffs are as before, with (A,X) -> (3,0); (A,Y) -> (3,0).
I hope you agree that the fact that player 2 gets to make a (useless) move in the case that player 1 chooses A doesn't change the fundamentals of the game.
In this classic game player 2 also has less information before making his move. In particular, player 2 is not told whether or not player 1 choose A. But this information is completely irrelevant for player 2's strategy, since if player 1 chooses A there is nothing that player 2 can do with that information.
If the players choose (A,X), then the payoff is (4,0). Changing his choice to B or C will not improve the payoff for player 1, and switching to Y doesn't improve the payoff for B. Therefore this is a Nash equilibrium. It is not stable, since player 2 can switch to Y without getting a worse payoff.
I'm not sure what you mean by "classical game" but my game is not a simultaneous move game. Many sequential move games do not have equivalent simultaneous move versions.
"I hope you agree that the fact that player 2 gets to make a (useless) move in the case that player 1 chooses A doesn't change the fundamentals of the game."
I do not agree. Consider these payoffs for the same game:
A 3,0 [And Player Two never got to move.]
B,X 2,10000
B,Y 2,2
C,X 0,1
C,Y 4,4
Now although Player 1 will never pick A, its existence is really important to the outcome by convincing Player 2 that if he moves C has been played.
Different payoffs imply a different game. But even in this different game, the simultaneous move version would be equivalent. With regards to choosing between X and Y, the existence of choice A still doesn't matter, because if player 1 chose A X and Y have the same payoff. The only difference is how much player 2 knows about what player 1 did, and therefore how much player 2 knows about the payoff he can expect. But that doesn't affect his strategy or the payoff that he gets in the end.