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.
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.