Software engineer and small time DS/ML practitioner.
Thanks for careful analysis, I must confess that my metric does not consider the stochastic strategies, and in general works better if players actions are taken consequently, not simultaneously (which is much different from the classic description).
The reasoning being that for maximal alignment each action of P1 there exist exactly one action of P2 (and vice versa) that is Nash equilibrium. In this case the game stops in stable state after single pair of actions. And maximally unaligned game will have no nash equilibrium at all, meaning the players actions-reactions will just move over the matrix in closed loop.
Overall, my solution as is seems not fitted for the classical formulation of the game :) but thanks for considering it!
So, something like “fraction of preferred states shared” ?
Describe preferred states for P1 as cells in the payoff matrix that are best for P1 for each P2 action (and preferred stated for P2 in a similar manner)
Fraction of P1 preferred states that are also preferred for P2 is measurement of alignment P1 to P2.
Fraction of shared states between players to total number of preferred states is measure of total alignment of the game.
For 2x2 game each player will have 2 preferred states (corresponding to the 2 possible action of the opponent). If 1 of them will be the same cell that will mean that each player is 50% aligned to other (1 of 2 shared) and the game in total is 33% aligned (1 of 3), This also generalize easily to NxN case and for >2 players.
And if there are K multiple cells with the same payoff to choose from for some opponent action we can give 1/K to them instead of 1.
(it would be much easier to explain with a picture and/or table, but I’m pretty new here and wasn’t able to find how to do them here yet)
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