In my experience, constant-sum games are considered to provide "maximally unaligned" incentives, and common-payoff games are considered to provide "maximally aligned" incentives. How do we quantitatively interpolate between these two extremes? That is, given an arbitrary payoff table representing a two-player normal-form game (like Prisoner's Dilemma), what extra information do we need in order to produce a real number quantifying agent alignment?
If this question is ill-posed, why is it ill-posed? And if it's not, we should probably understand how to quantify such a basic aspect of multi-agent interactions, if we want to reason about complicated multi-agent situations whose outcomes determine the value of humanity's future. (I started considering this question with Jacob Stavrianos over the last few months, while supervising his SERI project.)
- Assume the alignment function has range or .
- Constant-sum games should have minimal alignment value, and common-payoff games should have maximal alignment value.
- The function probably has to consider a strategy profile (since different parts of a normal-form game can have different incentives; see e.g. equilibrium selection).
- The function should probably be a function of player A's alignment with player B; for example, in a prisoner's dilemma, 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).
- So the function need not be symmetric over players.
- The function should be invariant to applying a separate positive affine transformation to each player's payoffs; it shouldn't matter whether you add 3 to player 1's payoffs, or multiply the payoffs by a half.
The function may or may not rely only on the players' orderings over outcome lotteries, ignoring the cardinal payoff values. I haven't thought much about this point, but it seems important.EDIT: I no longer think this point is important, but rather confused.
If I were interested in thinking about this more right now, I would:
- Do some thought experiments to pin down the intuitive concept. Consider simple games where my "alignment" concept returns a clear verdict, and use these to derive functional constraints (like symmetry in players, or the range of the function, or the extreme cases).
- See if I can get enough functional constraints to pin down a reasonable family of candidate solutions, or at least pin down the type signature.
In a sense, your proposal quantifies the extent to which B selects a best response on behalf of A, given some mixed outcome. I like this. I also think that "it doesn't necessarily depend on uB" is a feature, not a bug.
EDIT: To handle
common-constant-payoff games, we might want to define the alignment to equal 1 if the denominator is 0. In that case, the response of B can't affect A's expected utility, and so it's not possible for B to act against A's interests. So we might as well say that B is (trivially) aligned, given such a mixed outcome?