Purely the information that Player One behaves irrationally doesn’t give Player Two any more information on A’s behaviour than the fact that it is not rational. So other than knowing Player One didn’t use the strategy “play A with 100% probability”, Player Two doesn’t know anything about Player One’s behaviour. What can Player Two do on that basis? They can assume that any deviations from rational choice are small, which brings us to the trembling hand solution. Or they can use a different model.
Which model of player One’s behaviour is the “correct” one to assume in this situation is not at all clear. Perhaps Player One can be modelled as a RNG? Then pick Y. Perhaps One always models its opponents as RNG’s (which is, given no information about their irrationality, irrational)? Then pick X (since One is indifferent between A and C in this case). Just as reversed stupidity is not intelligence, [not intelligence] doesn’t tell you anything about the kind of stupidity, unless given more information.
Most Game Theory Problems in general can be said to have no solution if one of the players behaves irrationally. But that’s not a problem for game theory because the Rational Choice assumption (perhaps allowing small deviations) is perfectly fine in the real world, and really the only sane way one can solve game-theoretic problems, narrowing down the space of possible behaviours tremendously!
And by the way, what use has a payoff matrix if players don’t do their best acting on that information?
But that’s not a problem for game theory because the Rational Choice assumption (perhaps allowing small deviations) is perfectly fine in the real world
Unless you have more specific information about the problem in question, it’s the best concept to consider. At least in the limit of large stacks, long pondering times, and decisions jointly made by large organizations, the assumption holds. Although thinking about it, I’d really like to see game theory for predictably irrational agents, suffering from exactly those biases untrained humans fall prey to.
Let’s consider the top headlines of the moment: the Russian separatists in the Ukraine shot down a passenger jet and the IDF invaded Gaza. Both situations (the separatist movement and the Middle Eastern conflict) could be modeled in the game theory framework. Would you be comfortable applying the “Rational Choice assumption” to these situations?
I would attribute the shooting of the passenger jet to incompetence; The IDF invading Gaza yet again certainly makes sense from their perspective.
Considering the widespread false information in both cases, I’d argue that by and large, the agents (mostly the larger ones like Russia and Israel, less so the separatists and the palestine fighters) act rationally on the information they have. Take a look at Russia, neither actively fighting the separatists nor openly supporting them. I could argue that this is the best strategy for territorial expansion, avoiding a UN mission while strengthening the separatists. Spreading false information does its part.
I don’t know enough about the palestine fighters and the information they act on to evaluate whether or not their behaviour makes sense.
I only consider instrumental rationality here, not epistemic rationality.
As I wrote above, in the limit of large stacks, long pondering times, and decisions jointly made by large organizations, people do actually behave rationally. As an example: Bidding for oil drilling rights can be modelled as auctions with incomplete and imperfect information. Naïve bidding strategies fall prey to the winner’s curse. Game theory can model these situations as Bayesian games and compute the emerging Bayesian Nash Equilibria.
in the limit of large stacks, long pondering times, and decisions jointly made by large organizations, people do actually behave rationally.
I still don’t think so. To be a bit more precise, certainly people behave rationally sometimes and I will agree that things like long deliberations or joint decisions (given sufficient diversity of the deciding group) tend to increase the rationality. But I don’t think that even in the limit assuming rationality is a “safe” or a “fine” assumption.
Example: international politics. Another example: organized religions.
I also think that in analyzing this issue there is the danger of constructing rational narratives post-factum via the claim of revealed preferences. Let’s say entity A decides to do B. It’s very tempting to say “Aha! It would be rational for A to decide to do B if A really wants X, therefore A wants X and behaves rationally”. And certainly, that happens like that on a regular basis. However what also happens is that A really wants Y and decides to do B on non-rational grounds or just makes a mistake. In this case our analysis of A’s rationality is false, but it’s hard for us to detect that without knowing whether A really wants X or Y.
“They can assume that any deviations from rational choice are small,”
Not playing A might be rational depending on Player Two’s beliefs.
“And by the way, what use has a payoff matrix if players don’t do their best acting on that information?” The game’s uncertainty and sequential moves mean you can’t use a standard payoff matrix.
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.
Purely the information that Player One behaves irrationally doesn’t give Player Two any more information on A’s behaviour than the fact that it is not rational. So other than knowing Player One didn’t use the strategy “play A with 100% probability”, Player Two doesn’t know anything about Player One’s behaviour. What can Player Two do on that basis? They can assume that any deviations from rational choice are small, which brings us to the trembling hand solution. Or they can use a different model.
Which model of player One’s behaviour is the “correct” one to assume in this situation is not at all clear. Perhaps Player One can be modelled as a RNG? Then pick Y. Perhaps One always models its opponents as RNG’s (which is, given no information about their irrationality, irrational)? Then pick X (since One is indifferent between A and C in this case). Just as reversed stupidity is not intelligence, [not intelligence] doesn’t tell you anything about the kind of stupidity, unless given more information.
Most Game Theory Problems in general can be said to have no solution if one of the players behaves irrationally. But that’s not a problem for game theory because the Rational Choice assumption (perhaps allowing small deviations) is perfectly fine in the real world, and really the only sane way one can solve game-theoretic problems, narrowing down the space of possible behaviours tremendously!
And by the way, what use has a payoff matrix if players don’t do their best acting on that information?
Not in the real world I’m familiar with.
Unless you have more specific information about the problem in question, it’s the best concept to consider. At least in the limit of large stacks, long pondering times, and decisions jointly made by large organizations, the assumption holds. Although thinking about it, I’d really like to see game theory for predictably irrational agents, suffering from exactly those biases untrained humans fall prey to.
I am not convinced about that at all.
Let’s consider the top headlines of the moment: the Russian separatists in the Ukraine shot down a passenger jet and the IDF invaded Gaza. Both situations (the separatist movement and the Middle Eastern conflict) could be modeled in the game theory framework. Would you be comfortable applying the “Rational Choice assumption” to these situations?
I would attribute the shooting of the passenger jet to incompetence; The IDF invading Gaza yet again certainly makes sense from their perspective.
Considering the widespread false information in both cases, I’d argue that by and large, the agents (mostly the larger ones like Russia and Israel, less so the separatists and the palestine fighters) act rationally on the information they have. Take a look at Russia, neither actively fighting the separatists nor openly supporting them. I could argue that this is the best strategy for territorial expansion, avoiding a UN mission while strengthening the separatists. Spreading false information does its part.
I don’t know enough about the palestine fighters and the information they act on to evaluate whether or not their behaviour makes sense.
I only consider instrumental rationality here, not epistemic rationality.
That may well be so, but this is a rather different claim than the “Rational Choice assumption”.
We know quite well that people are not rational. Why would you model them as rational agents in game theory?
As I wrote above, in the limit of large stacks, long pondering times, and decisions jointly made by large organizations, people do actually behave rationally. As an example: Bidding for oil drilling rights can be modelled as auctions with incomplete and imperfect information. Naïve bidding strategies fall prey to the winner’s curse. Game theory can model these situations as Bayesian games and compute the emerging Bayesian Nash Equilibria.
Guess what? The companies actually bid the way game theory predicts!
I still don’t think so. To be a bit more precise, certainly people behave rationally sometimes and I will agree that things like long deliberations or joint decisions (given sufficient diversity of the deciding group) tend to increase the rationality. But I don’t think that even in the limit assuming rationality is a “safe” or a “fine” assumption.
Example: international politics. Another example: organized religions.
I also think that in analyzing this issue there is the danger of constructing rational narratives post-factum via the claim of revealed preferences. Let’s say entity A decides to do B. It’s very tempting to say “Aha! It would be rational for A to decide to do B if A really wants X, therefore A wants X and behaves rationally”. And certainly, that happens like that on a regular basis. However what also happens is that A really wants Y and decides to do B on non-rational grounds or just makes a mistake. In this case our analysis of A’s rationality is false, but it’s hard for us to detect that without knowing whether A really wants X or Y.
“They can assume that any deviations from rational choice are small,”
Not playing A might be rational depending on Player Two’s beliefs.
“And by the way, what use has a payoff matrix if players don’t do their best acting on that information?” The game’s uncertainty and sequential moves mean you can’t use a standard payoff matrix.
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.