How Can We Quantify Player Alignment in 2×2 Normal-Form Games?

Question

This was originally posted as a question on LessWrong.

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 2×2 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.)

Thoughts:

• Assume the alignment function has range $[0,1]$ or $[-1,1]$.

  • 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.

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.

I consider this problem solved by Vanessa Kosoy

Consider any finite two-player game in normal form (each player can have any finite number of strategies, we can also easily generalize to certain classes of infinite games). Let $S_A$ be the set of pure strategies of player $A$ and $S_B$ the set of pure strategies of player $B$. Let $u_A: S_A \times S_B \rightarrow \mathbb{R}$ be the utility function of player $A$. Let $(\alpha, \beta) \in \Delta S_A \times \Delta S_B$ be a particular (mixed) outcome. Then the alignment of player $B$ with player $A$ in this outcome is defined to be:

$$a_{B / A}(\alpha, \beta)≝\frac{E_{\alpha \times \beta}\left[u_A\right]-\min _{\beta^{\prime} \in S_B} E_{\alpha \times \beta^{\prime}}\left[u_A\right]}{\max _{\beta^{\prime} \in S_B} E_{\alpha \times \beta^{\prime}}\left[u_A\right]-\min _{\beta^{\prime} \in S_B} E_{\alpha \times \beta^{\prime}}\left[u_A\right]} \in[0,1]$$

Ofc so far it doesn’t depend on $u_B$ at all. However, we can make it depend on $u_B$ if we use $u_B$ to impose assumptions on $(\alpha, \beta)$, such as:

• $\beta$ is a $u_B$-best response to $\alpha$ or
• $(\alpha, \beta)$ is a Nash equilibrium (or other solution concept)

Caveat: If we go with the Nash equilibrium option, $a_{B / A}$ can become “systematically” ill-defined (consider e.g. the Nash equilibrium of matching pennies). To avoid this, we can switch to the extensive-form game where $B$ chooses their strategy after seeing $A$’s strategy.

☙❧