Comment by Jonas Moss (jonas-moss) on Open problem: how can we quantify player alignment in 2x2 normal-form games? · 2021-06-17T07:47:53.497Z · LW · GW

Are you sure zero-sum games are maximally misaligned? Consider the joint payoff matrix 

$$P=\left[\begin{array}{cc}(1,1)& (1,1)\\ (1,1)& (1,1)\end{array}\right]\sim \left[\begin{array}{cc}(0,0)& (0,0)\\ (0,0)& (0,0)\end{array}\right]$$

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 $s^{T}Ar$.) Then they are aligned at $s,r$ if the sum of expected payoffs  $s^{T}Ar+s^{T}Br$ 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 $1$s), 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

$$Q=\left[\begin{array}{cc}(1,-1)& (-1,1)\\ (-1,1)& (1,-1)\end{array}\right],$$

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 $0$...)