How does it change the structure of the game? Of course, it was in normal form before, and is now in extensive form, but really, the way you set it up means it shouldn’t matter which representation we choose, since player two is getting exactly the same information.
Also, your argument about player two getting information about One’s behaviour can easily be applied to “normal” extensive form games. Regardless of whether you intended to, if your argument were correct, it would render the concept of subgame perfect equilibrium useless.
I know that credibility is normally applied as in “make credible threats”. But if I change payoffs to A: (3, 5) and add a few nodes above, then player 1′s threat to not play A (which in this case is a threat) is not credible, and (3,5) carries over to the parent node.
By the logic of the extensive form formulation, Two should simply play the equilibrium strategy for the 2x2 subgame.
Edit: Here is what the game looks like in extensive form:
The dotted ellipse indicates that 2 can’t differentiate between the two contained nodes. I don’t see how any of the players has any more or less information or any more or less choices available.
But I just showed that this is irrational as they would get less payoff in that subgame!
If that’s your attitude, then you have to abandon the concept of subgame perfect equilibrium entirely. Are you willing to do that?
I think that adding the extra node does change the structure of the game. I also think that we have different views of what credibility means.
How does it change the structure of the game? Of course, it was in normal form before, and is now in extensive form, but really, the way you set it up means it shouldn’t matter which representation we choose, since player two is getting exactly the same information.
Also, your argument about player two getting information about One’s behaviour can easily be applied to “normal” extensive form games. Regardless of whether you intended to, if your argument were correct, it would render the concept of subgame perfect equilibrium useless.
I know that credibility is normally applied as in “make credible threats”. But if I change payoffs to A: (3, 5) and add a few nodes above, then player 1′s threat to not play A (which in this case is a threat) is not credible, and (3,5) carries over to the parent node.
By the logic of the extensive form formulation, Two should simply play the equilibrium strategy for the 2x2 subgame.
Edit: Here is what the game looks like in extensive form:
The dotted ellipse indicates that 2 can’t differentiate between the two contained nodes. I don’t see how any of the players has any more or less information or any more or less choices available.
Yes this is the same game, but you can not create a subgame that has B and C but not A.