One of the things one might hope to get from an equilibrium concept in a game is that it might plausibly describe or predict behavior in the real world. That’s one reason negative results suggesting that certain equilibria can’t be computed efficiently are so interesting—they call those equilibrium concepts into question. Of course, there’s much more to it than that, but the existence of efficient algorithms (or, even better, simple, distributed dynamics) for computing equilibria is a big selling point for a solution concept.

In that sense, correlated equilibria (CE) and coarse correlated equilibria (CCE) are much more promising than Nash equilibria (NE). Even in games with many players, there exist a number of natural dynamics that quickly converge to these solution concepts. In particular, these dynamics induce efficient computation of approximate CE and CCE in multiplayer games; by contrast, computation of approximate Nash equilibria is computationally hard in multiplayer games.

If you aren’t so familiar with CEs and CCEs, you can think of them as very similar to Nash equilibria, in that they also consist of a distribution over players’ action profiles at which no player can benefit by unilateral deviation, and hence represent stable choices of distributions over player actions. The big difference is that, whereas a Nash equilibrium is defined to be a product of independent distributions over actions (one for each player); correlated and coarse correlated equilibria are general (joint) probability distributions—CEs and CCEs allow players to correlate their choices of actions. For the purposes of this blog post, let’s not worry too much about the distinction between CEs and CCEs. Notice that CE and CCE are more general than NE—every NE is a CE (CCE), but the converse is not true. So it makes sense that one might be able to compute a CE efficiently, even in settings where computing a NE is hard.

Beyond computation of equilibria, another significant thread of research in algorithmic game theory has been the study of the *quality* of equilibria, often as measured by the social welfare of the equilibrium or its ratio to the social welfare of the socially optimal outcome (c.f. the extensive literature on the price of anarchy (PoA)). Given that we know it is possible to efficiently compute CE and CCE, it is natural to ask *how good* are the equilibria we can efficiently compute? For example, do existing efficient dynamics find the best such equilibria, or at least ones that approximately optimize the social welfare? Since the gap between the worst and the best equilibria (CE or CCE), in terms of social welfare, can be large in natural games, it is interesting to understand if there exist efficient dynamics or algorithms that avoid—at least to some extent—the bad outcomes.

In their notable work, Papadimitriou and Roughgarden [PR] show that determining a socially optimal CE is NP-hard, in a number of succinct multiplayer games*. (What’s socially optimal? The equilibrium that maximizes the total (or average) welfare of the players.) This result intuitively follows from the fact that determining an action profile with maximum welfare—i.e., solving the problem of welfare optimization even without equilibrium constraints—is NP-hard in general. The hardness result of [PR] leaves open the question of computing *near-optimal* CE/CCE, i.e., whether there exist efficient algorithms that compute CE/CCE with welfare at least, say, *alpha* times the optimal, for a nontrivial approximation ratio *alpha* < 1.

In recent work with Sid Barman, a postdoc at Caltech, we consider exactly this question. We establish that, unless P=NP, there does not exist any efficient algorithm that computes a CCE with welfare better than the *worst possible* CCE, in succinct multiplayer games*. We also establish similar hardness results for computing equilibria under the egalitarian objective or Pareto-optimality.

Analogous hardness results hold for CE. A classical interpretation of a CE is in terms of a mediator who has access to the players’ payoff functions and who draws outcomes from a correlated equilibrium’s joint distribution over player actions and privately recommends the corresponding actions to each player. The equilibrium conditions ensure that no player can benefit in expectation by unilaterally deviating from the recommended actions. Therefore, the problem we study here is exactly the computational complexity of the problem that a mediator faces if she wishes to maximize social welfare.

We also extend the hardness result to approximate CE and CCE. Therefore, while one can efficiently compute an approximate CE/CCE in succinct multiplayer games, one cannot provide any nontrivial welfare guarantees for the resulting equilibrium (unless P=NP).

In addition, we show that this hardness result also holds specifically for potential games (generally considered to be a very tractable class of games), and persists even in settings where the gap between the best and worst equilibrium is large.

We also have some positive results, but I think the negative ones are the most interesting—this says that those appealing dynamics that compute CE/CCE can’t be guaranteed to be finding anything other than the worst equilibrium. To me, that’s some surprising bad news. One silver lining is that this provides new motivation for studying the price of anarchy (the quality of the worst equilibrium) for CE/CCE, since generally that’s the best thing we can hope to compute.

*Technical Aside (succinct games): In general multiplayer games the size of the normal form representation, $N$, is exponentially large in the number of players; one can compute a CE/CCE that optimizes a linear objective by solving a linear program of size polynomial in $N$, and hence the computational complexity of equilibrium computation is not interesting for general games. However, most games of interest—such as graphical games, polymatrix games, congestion games, local effect games, network design games, anonymous games, and scheduling games—admit a succinct representation (wherein the above-mentioned linear program can be exponentially large in the size of the representation), and hence it is such succinctly representable games that we (and previous works) study.

[PR] PAPADIMITRIOU, C. H. AND ROUGHGARDEN, T. 2008. Computing correlated equilibria in multi-player games. Journal of the ACM (JACM) 55, 3, 14.