Are you sure zero-sum games are maximally misaligned? Consider the joint payoff matrix
P=[(1,1)(1,1)(1,1)(1,1)]∼[(0,0)(0,0)(0,0)(0,0)]
This matrix doesn’t appear minimally aligned to me; instead, it seems maximally aligned. It might be a trivial case but has to be accounted for in the analysis, as it’s simultaneously a constant sum game and a symmetric/common payoff game.
I suppose alignment should be understood in terms of payoff sums. Let s be the (random!) strategy of player 1 and r be the strategy of player 2, and A and B be their individual payoff matrices. (So that the expected payoff of player 1 is sTAr.) Then they are aligned at s,r if the sum of expected payoffs sTAr+sTBr is “large” and misaligned if it is “small”, where “large” and “small” need to be quantified, perhaps in relation to the maximal individual payoff, or perhaps something else.
For the matrix P above (with 1s), every strategy will yield the same large sum compared to the maximal individual payoff, and appears to be maximally aligned. In the case of, say
Q=[(1,−1)(−1,1)(−1,1)(1,−1)],
any strategy will yield a sum that is minimally mall (0) compared to the maximal individual payoff (1), which isn’t minimally small, and it is minimally aligned.
(Comparing the sum of payoffs to the maximal individual may be wrong though, as it’s not invariant under affine transformations. For instance, the sum of payoffs in the (0,0) representation of P is 0 and the individual payoffs are 0…)
Hmm, a very interesting case! Intuitively, I would think the function would be undefined for P. Is it really a “game” at all, when neither player has a decision that has any affect on the game?
I could see “undefined” coming naturally from a division by 0 here, where the denominator has something to do with the difference in the payouts received in some way. Indeed, you probably need some sort of division like that, to make the answer invariant under affine transformation.
It’s a game, just a trivial one. Snakes and Ladders is also a game, and its payoff matrix is similar to this one, just with a little bit of randomness involved.
My intuition says that this game not only has maximal alignment, but is the only game (up to equivalence) game with maximal alignment for any set of strategies s,r. No matter what player 1 and player 2 does, the world is as good as it could be.
The case can be compared to the R2 when the variance of the dependent variable is 0. How much of the variance in the dependent variable does the independent variable explain in this case? It’d say it’s all of it.
Are you sure zero-sum games are maximally misaligned? Consider the joint payoff matrix
P=[(1,1)(1,1)(1,1)(1,1)]∼[(0,0)(0,0)(0,0)(0,0)]This matrix doesn’t appear minimally aligned to me; instead, it seems maximally aligned. It might be a trivial case but has to be accounted for in the analysis, as it’s simultaneously a constant sum game and a symmetric/common payoff game.
Q=[(1,−1)(−1,1)(−1,1)(1,−1)],I suppose alignment should be understood in terms of payoff sums. Let s be the (random!) strategy of player 1 and r be the strategy of player 2, and A and B be their individual payoff matrices. (So that the expected payoff of player 1 is sTAr.) Then they are aligned at s,r if the sum of expected payoffs sTAr+sTBr is “large” and misaligned if it is “small”, where “large” and “small” need to be quantified, perhaps in relation to the maximal individual payoff, or perhaps something else.
For the matrix P above (with 1s), every strategy will yield the same large sum compared to the maximal individual payoff, and appears to be maximally aligned. In the case of, say
any strategy will yield a sum that is minimally mall (0) compared to the maximal individual payoff (1), which isn’t minimally small, and it is minimally aligned.
(Comparing the sum of payoffs to the maximal individual may be wrong though, as it’s not invariant under affine transformations. For instance, the sum of payoffs in the (0,0) representation of P is 0 and the individual payoffs are 0…)
Hmm, a very interesting case! Intuitively, I would think the function would be undefined for P. Is it really a “game” at all, when neither player has a decision that has any affect on the game?
I could see “undefined” coming naturally from a division by 0 here, where the denominator has something to do with the difference in the payouts received in some way. Indeed, you probably need some sort of division like that, to make the answer invariant under affine transformation.
It’s a game, just a trivial one. Snakes and Ladders is also a game, and its payoff matrix is similar to this one, just with a little bit of randomness involved.
My intuition says that this game not only has maximal alignment, but is the only game (up to equivalence) game with maximal alignment for any set of strategies s,r. No matter what player 1 and player 2 does, the world is as good as it could be.
The case can be compared to the R2 when the variance of the dependent variable is 0. How much of the variance in the dependent variable does the independent variable explain in this case? It’d say it’s all of it.