I have put the preferred state for each player in bold. I think by your rule this works out to 50% aligned. However, the Nash equilibrium is both players choosing the 1⁄1 result, which seems perfectly aligned (intuitively).
1⁄0.50⁄0
0⁄00.5/1
In this game, all preferred states are shared, yet there is a Nash equilibrium where each player plays the move that can get them 1 point 2⁄3 of the time, and the other move 1⁄3 of the time. I think it would be incorrect to call this 100% aligned.
(These examples were not obvious to me, and tracking them down helped me appreciate the question more. Thank you.)
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!
1/1 0/0
0⁄0 0.8/-1
I have put the preferred state for each player in bold. I think by your rule this works out to 50% aligned. However, the Nash equilibrium is both players choosing the 1⁄1 result, which seems perfectly aligned (intuitively).
1⁄0.5 0⁄0
0⁄0 0.5/1
In this game, all preferred states are shared, yet there is a Nash equilibrium where each player plays the move that can get them 1 point 2⁄3 of the time, and the other move 1⁄3 of the time. I think it would be incorrect to call this 100% aligned.
(These examples were not obvious to me, and tracking them down helped me appreciate the question more. Thank you.)
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!