I like how this proposal makes explicit the player strategies, and how they are incorporated into the calculation. I also think that the edge case where the agents actions have no effect on the result
I think that this proposal making alignment symmetric might be undesirable. Taking the prisoner’s dilemma as an example, if s = always cooperate and r = always defect, then I would say s is perfectly aligned with r, and r is not at all aligned with s.
The result of 0 alignment for the Nash equilibrium of PD seems correct.
I think this should be the alignment matrix for pure-strategy, single-shot PD:
a=[1,11,00,10,0]
Here the first of each ordered pair represents A’s alignment with B. (assuming we use the [0,1] interval)
I think in this case the alignments are simple, because A can choose to either maximize or to minimize B’s utility.
I believe the upper right-hand corner of a shouldn’t be 1; even if both players are acting in each other’s best interest, they are not acting in their own best interest. And alignment is about having both at the same time. The configuration of Prisoner’s dilemma makes it impossible to make both players maximally satisfied at the same time, so I believe it cannot have maximal alignment for any strategy.
Anyhow, your concept of alignment might involve altruism only, which is fair enough. In that case, Vanessa Kosoy has a similar proposal to mine, but not working with sums, which probably does exactly what you are looking for.
Getting alignment in the upper right-hand corner in the Prisoner’s dilemma matrix to be 1 may be possible if we redefine u(A,B) to u(A,B)=maxu,vuT(A+B)v, the best attainable payoff sum. But then zero-sum games will have maximal instead of minimal alignment! (This is one reason why I defined u(A,B)=maxu,vuTAv+maxu,vuTBv.)
(Btw, the coefficient isn’t symmetric; it’s only symmetric for symmetric games. No alignment coefficient depending on the strategies can be symmetric, as the vectors can have different lengths.)
I like how this proposal makes explicit the player strategies, and how they are incorporated into the calculation. I also think that the edge case where the agents actions have no effect on the result
I think that this proposal making alignment symmetric might be undesirable. Taking the prisoner’s dilemma as an example, if s = always cooperate and r = always defect, then I would say s is perfectly aligned with r, and r is not at all aligned with s.
The result of 0 alignment for the Nash equilibrium of PD seems correct.
I think this should be the alignment matrix for pure-strategy, single-shot PD:
a=[1,11,00,10,0]Here the first of each ordered pair represents A’s alignment with B. (assuming we use the [0,1] interval)
I think in this case the alignments are simple, because A can choose to either maximize or to minimize B’s utility.
I believe the upper right-hand corner of a shouldn’t be 1; even if both players are acting in each other’s best interest, they are not acting in their own best interest. And alignment is about having both at the same time. The configuration of Prisoner’s dilemma makes it impossible to make both players maximally satisfied at the same time, so I believe it cannot have maximal alignment for any strategy.
Anyhow, your concept of alignment might involve altruism only, which is fair enough. In that case, Vanessa Kosoy has a similar proposal to mine, but not working with sums, which probably does exactly what you are looking for.
Getting alignment in the upper right-hand corner in the Prisoner’s dilemma matrix to be 1 may be possible if we redefine u(A,B) to u(A,B)=maxu,vuT(A+B)v, the best attainable payoff sum. But then zero-sum games will have maximal instead of minimal alignment! (This is one reason why I defined u(A,B)=maxu,vuTAv+maxu,vuTBv.)
(Btw, the coefficient isn’t symmetric; it’s only symmetric for symmetric games. No alignment coefficient depending on the strategies can be symmetric, as the vectors can have different lengths.)