The function should probably be a function of player A’s alignment with player B; for example, player A might always cooperate and player B might always defect. Then it seems reasonable to consider whether A is aligned with B (in some sense), while B is not aligned with A (they pursue their own payoff without regard for A’s payoff).
That seems to be confused reasoning. “Cooperate” and “defect” are labels we apply to a 2x2 matrix sometimes, and applying those labels changes the payouts. If I get $1 or $5 for picking “A” and $0 or $3 for picking “B” depending on a coin flip that leads me to a different choice than if A is labeled “defect” and B is labeled “cooperate” and the payout depends on another person, because I get psychic/reputational rewards for cooperating/defecting (which one is better depends on my peer group, but whichever is better the story equity is much higher than $5, so my choice is dominated by that, and the actual payout matrix is: pick S: 1000 util or 1001 util. Pick T: 2 util or 2 util.
None of which negates the original question of mapping the 8! possible arrangements of relative payouts in a 2x2 matrix game to some sort of linear scale.
That seems to be confused reasoning. “Cooperate” and “defect” are labels we apply to a 2x2 matrix sometimes, and applying those labels changes the payouts.
Not sure I follow your main point, but I was talking about actual PD, which I’ve now clarified in the original post. See also my post on What counts as defection?.
The function should probably be a function of player A’s alignment with player B; for example, player A might always cooperate and player B might always defect. Then it seems reasonable to consider whether A is aligned with B (in some sense), while B is not aligned with A (they pursue their own payoff without regard for A’s payoff).
That seems to be confused reasoning. “Cooperate” and “defect” are labels we apply to a 2x2 matrix sometimes, and applying those labels changes the payouts. If I get $1 or $5 for picking “A” and $0 or $3 for picking “B” depending on a coin flip that leads me to a different choice than if A is labeled “defect” and B is labeled “cooperate” and the payout depends on another person, because I get psychic/reputational rewards for cooperating/defecting (which one is better depends on my peer group, but whichever is better the story equity is much higher than $5, so my choice is dominated by that, and the actual payout matrix is: pick S: 1000 util or 1001 util. Pick T: 2 util or 2 util.
None of which negates the original question of mapping the 8! possible arrangements of relative payouts in a 2x2 matrix game to some sort of linear scale.
Not sure I follow your main point, but I was talking about actual PD, which I’ve now clarified in the original post. See also my post on What counts as defection?.