Not playing A might be rational depending on Player Two’s beliefs.
That, too, is the case in many games. The success in always assuming mutually rational behaviour first lies in its nice properties, like inability to be exploited, existence of equilibrium, and a certain resemblance (albeit not a perfect one) to the real world.
The game’s uncertainty and sequential moves mean you can’t use a standard payoff matrix.
Well, I mean, why specify payoffs at all if you then assume players won’t care about them? If Player One cared about their payoff and were modelling Two as a rational choice agent, they would’ve played A. So either they don’t model Two as a rational choice agent (which is stupid in its own right), or they simply don’t care about their payoff.
In any case, the fact that a game with irrational players doesn’t have a solution, at least as long as the nature of players’ irrationality is not clear, doesn’t surprise me. Dealing with madmen has never been easy.
If Player One cared about their payoff and were modelling Two as a rational choice agent, they would’ve played A
You can only prove this if you first tell me what Player two’s beliefs would be if he got to move.
In any case, the fact that a game with irrational players doesn’t have a solution, at least as long as the nature of players’ irrationality is not clear, doesn’t surprise me.
I agree, but I meant for the players in the game to be rational.
Player two could simply play the equilibrium strategy for the 2x2-subgame.
And to counter your response that it’s all one game, not two games, I can split the game by adding an extra node without changing the structure of the game. Then, we arrive at the standard subgame perfect equilibrium, only that the 2x2 subgame is in normal form, which shouldn’t really change things since we can just compute a Nash equilibrium for that.
After solving the subgame, we see that player one not playing A is not credible, and we can eliminate that branch.
I can split the game by adding an extra node without changing the structure of the game
That actually does change the structure of the game, if we assume that physical mistakes can happen with some probability, which seems realistic. (Think about playing this game in real life as player 2. If you do get to move, you must think there’s some non-zero probability that it was because Player 1 just accidentally pressed the wrong button, right?) With the extra node, you get the occasional “crap, I just accidentally pressed not-A, now I have to decide which of B or C to choose” which has no analogy in the original game where you never get to choose between B or C without A as an option.
The structure isn’t changed there, but without the extra node, there is no subgame. That extra node is necessary in order to have a subgame, because only then can Player 2 think “the probabilities I’m facing is the result of Player 1′s choice between just B and C” which allows them to solve that subgame independently of the rest of the game. Also, see this comment and its grandchild for why specifically, given possibility of accidental presses, I don’t think Player 2′s strategy in the overall game should be same as the equilibrium of the 2x2 “reduced game”. In short, in the reduced game, Player 2 has to make Player 1 indifferent between B and C, but in the overall game with accidental presses, Player 2 has to make Player 1 indifferent between A and C.
In the 2x2 reduced game, Player One’s strategy is 1⁄3 B, 2⁄3 C; Two’s strategy is 2⁄3 X, 1⁄3 Y.
In the complete game with trembling hands, Player Two’s strategy remains unchanged, as you wrote in the starter of the linked thread, invoking proper equilibrium.
Later on in the linked thread, I realized that the proper equilibrium solution doesn’t make sense. Think about it: why does Player 1 “tremble” so that C is exactly twice the probability of B? Other than pure coincidence, the only way that could happen is if some of the button presses of B and/or C are actually deliberate. Clearly Player 1 would never deliberately press B while A is still an option, so Player 1 must actually be playing a mixed strategy between A and C, while also accidentally pressing B and C with some small probability. But that implies Player 2 must be playing a mixed strategy that makes Player 1 indifferent between A and C, not between B and C.
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.
That, too, is the case in many games. The success in always assuming mutually rational behaviour first lies in its nice properties, like inability to be exploited, existence of equilibrium, and a certain resemblance (albeit not a perfect one) to the real world.
Well, I mean, why specify payoffs at all if you then assume players won’t care about them? If Player One cared about their payoff and were modelling Two as a rational choice agent, they would’ve played A. So either they don’t model Two as a rational choice agent (which is stupid in its own right), or they simply don’t care about their payoff.
In any case, the fact that a game with irrational players doesn’t have a solution, at least as long as the nature of players’ irrationality is not clear, doesn’t surprise me. Dealing with madmen has never been easy.
You can only prove this if you first tell me what Player two’s beliefs would be if he got to move.
I agree, but I meant for the players in the game to be rational.
Player two could simply play the equilibrium strategy for the 2x2-subgame.
And to counter your response that it’s all one game, not two games, I can split the game by adding an extra node without changing the structure of the game. Then, we arrive at the standard subgame perfect equilibrium, only that the 2x2 subgame is in normal form, which shouldn’t really change things since we can just compute a Nash equilibrium for that.
After solving the subgame, we see that player one not playing A is not credible, and we can eliminate that branch.
That actually does change the structure of the game, if we assume that physical mistakes can happen with some probability, which seems realistic. (Think about playing this game in real life as player 2. If you do get to move, you must think there’s some non-zero probability that it was because Player 1 just accidentally pressed the wrong button, right?) With the extra node, you get the occasional “crap, I just accidentally pressed not-A, now I have to decide which of B or C to choose” which has no analogy in the original game where you never get to choose between B or C without A as an option.
Okay, I agree. But what do you think about the extensive-form game in the image below? Is the structure changed there?
The structure isn’t changed there, but without the extra node, there is no subgame. That extra node is necessary in order to have a subgame, because only then can Player 2 think “the probabilities I’m facing is the result of Player 1′s choice between just B and C” which allows them to solve that subgame independently of the rest of the game. Also, see this comment and its grandchild for why specifically, given possibility of accidental presses, I don’t think Player 2′s strategy in the overall game should be same as the equilibrium of the 2x2 “reduced game”. In short, in the reduced game, Player 2 has to make Player 1 indifferent between B and C, but in the overall game with accidental presses, Player 2 has to make Player 1 indifferent between A and C.
In the 2x2 reduced game, Player One’s strategy is 1⁄3 B, 2⁄3 C; Two’s strategy is 2⁄3 X, 1⁄3 Y. In the complete game with trembling hands, Player Two’s strategy remains unchanged, as you wrote in the starter of the linked thread, invoking proper equilibrium.
Later on in the linked thread, I realized that the proper equilibrium solution doesn’t make sense. Think about it: why does Player 1 “tremble” so that C is exactly twice the probability of B? Other than pure coincidence, the only way that could happen is if some of the button presses of B and/or C are actually deliberate. Clearly Player 1 would never deliberately press B while A is still an option, so Player 1 must actually be playing a mixed strategy between A and C, while also accidentally pressing B and C with some small probability. But that implies Player 2 must be playing a mixed strategy that makes Player 1 indifferent between A and C, not between B and C.
But Player 1 can make this perfectly credible by actually not Playing A.
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.