Now I think the reasoning presented is correct in both cases, and the lesson here is for our expectations of rationality.
I agree that the reasoning is correct in both cases (or rather: could be correct, assuming some details), but the lesson I derive is that we have to be really careful about our assumptions here.
Normally, in game theory, we’re comfortable asserting that re-labeling the options doesn’t matter (and re-numbering the players also doesn’t matter). But normally we aren’t worried about anthropic uncertainty in a game.
If we suppose that players can see their numbers, as well, this can be used as a signal to break symmetry for anti-matching. Player 1 can choose option 1, and player 2 can choose option 2. (Or whatever—they just have to agree on an anti-matching policy acausally.)
Thinking physically, the question is: are the two players physically precisely the same (including environment), at least insofar as the players can tell? Then anti-matching is hard. Usually we don’t need to think about such things for game theory (since a game is a highly abstracted representation of the physical situation).
But this is one reason why correlated equilibria are, usually, a better abstraction than Nash equilibria. For example, a game of chicken is similar to anti-matching. In correlated equilibria, there is a “fair” solution to chicken: each player goes straight with 50% probability (and the other player swerves). This corresponds to the idea of a traffic light. If traffic lights were not invented, some other correlating signal from the environment might be used (particularly as we assume increasingly intelligent agents). This is a possible game-theoretic explanation for divination practices such as reading entrails.
Nash equilibria, otoh, are a better abstraction for the case where there truly is no “environment” to take complicated signals from (besides what you explicitly represent in the game). It better fits a way of thinking where models are supposed to be complete.
are the two players physically precisely the same (including environment), at least insofar as the players can tell?
In the examples I gave yes. Because thats the case where we have a guarantee of equal policy, from which people try to generalize. If we say players can see their number, then the twins in the prisoners dilemma needn’t play the same way either.
But this is one reason why correlated equilibria are, usually, a better abstraction than Nash equilibria.
The “signals” players receive for correlated equilibria are already semantic. So I’m suspicious that they are better by calling on our intuition more to be used, with the implied risks. For example I remember reading about a result to the effect that correlated equilibria are easier to learn. This is not something we would expect from your explanation of the differences: If we explicitly added something (like the signals) into the game, it would generally get more complicated.
The “signals” players receive for correlated equilibria are already semantic. So I’m suspicious that they are better by calling on our intuition more to be used, with the implied risks. For example I remember reading about a result to the effect that correlated equilibria are easier to learn. This is not something we would expect from your explanation of the differences: If we explicitly added something (like the signals) into the game, it would generally get more complicated.
It’s not something we would naively expect, but it does further speak in favor of CE, yes?
In particular, if you look at those learnability results, it turns out that the “external signal” which the agents are using to correlate their actions is the play history itself. IE, they are only using information which must be available to learning agents (granted, sufficiently forgetful learning agents might forget the history; however, I do not think the learnability results actually rely on any detailed memory of the history—the result still holds with very simple agents who only remember a few parameters, with no explicit episodic memory (unlike, eg, tit-for-tat).
I agree that the reasoning is correct in both cases (or rather: could be correct, assuming some details), but the lesson I derive is that we have to be really careful about our assumptions here.
Normally, in game theory, we’re comfortable asserting that re-labeling the options doesn’t matter (and re-numbering the players also doesn’t matter). But normally we aren’t worried about anthropic uncertainty in a game.
If we suppose that players can see their numbers, as well, this can be used as a signal to break symmetry for anti-matching. Player 1 can choose option 1, and player 2 can choose option 2. (Or whatever—they just have to agree on an anti-matching policy acausally.)
Thinking physically, the question is: are the two players physically precisely the same (including environment), at least insofar as the players can tell? Then anti-matching is hard. Usually we don’t need to think about such things for game theory (since a game is a highly abstracted representation of the physical situation).
But this is one reason why correlated equilibria are, usually, a better abstraction than Nash equilibria. For example, a game of chicken is similar to anti-matching. In correlated equilibria, there is a “fair” solution to chicken: each player goes straight with 50% probability (and the other player swerves). This corresponds to the idea of a traffic light. If traffic lights were not invented, some other correlating signal from the environment might be used (particularly as we assume increasingly intelligent agents). This is a possible game-theoretic explanation for divination practices such as reading entrails.
Nash equilibria, otoh, are a better abstraction for the case where there truly is no “environment” to take complicated signals from (besides what you explicitly represent in the game). It better fits a way of thinking where models are supposed to be complete.
In the examples I gave yes. Because thats the case where we have a guarantee of equal policy, from which people try to generalize. If we say players can see their number, then the twins in the prisoners dilemma needn’t play the same way either.
The “signals” players receive for correlated equilibria are already semantic. So I’m suspicious that they are better by calling on our intuition more to be used, with the implied risks. For example I remember reading about a result to the effect that correlated equilibria are easier to learn. This is not something we would expect from your explanation of the differences: If we explicitly added something (like the signals) into the game, it would generally get more complicated.
It’s not something we would naively expect, but it does further speak in favor of CE, yes?
In particular, if you look at those learnability results, it turns out that the “external signal” which the agents are using to correlate their actions is the play history itself. IE, they are only using information which must be available to learning agents (granted, sufficiently forgetful learning agents might forget the history; however, I do not think the learnability results actually rely on any detailed memory of the history—the result still holds with very simple agents who only remember a few parameters, with no explicit episodic memory (unlike, eg, tit-for-tat).