Optimal Correlated Equilibria in General-Sum Extensive-Form Games: Fixed-Parameter Algorithms, Hardness, and Two-Sided Column-Generation

TL;DR

This work tackles the computational challenge of optimal correlated equilibria (NFCCE, EFCCE, EFCE) in general-sum extensive-form games by introducing a mediator-augmented framework that captures correlation while preserving hardness. It provides fixed-parameter, information-complexity-based bounds and two complementary algorithms—the correlation DAG LP and a two-sided column-generation method—that enable practical computation across large game instances and multiple players. The authors prove separation results showing fundamental differences in complexity among the three notions, and demonstrate state-of-the-art empirical performance on a suite of benchmarks, including two new games. The methods enable efficient optimization of correlation plans in settings with imperfect recall, public actions/chance, and triangle-free structures, offering scalable tools for strategic coordination via mediators. Together, these contributions advance both the theory and practice of computing optimal correlated equilibria in complex extensive-form games, with potential impact on automated negotiation, multi-agent coordination, and mechanism design.

Abstract

We study the problem of finding optimal correlated equilibria of various sorts in extensive-form games: normal-form coarse correlated equilibrium (NFCCE), extensive-form coarse correlated equilibrium (EFCCE), and extensive-form correlated equilibrium (EFCE). We make two primary contributions. First, we introduce a new algorithm for computing optimal equilibria in all three notions. Its runtime depends exponentially only on a parameter related to the information structure of the game. We also prove a fundamental complexity gap: while our size bounds for NFCCE are similar to those achieved in the case of team games by Zhang et al., this is impossible to achieve for the other two concepts under standard complexity assumptions. Second, we propose a two-sided column generation approach for use when the runtime or memory usage of the previous algorithm is prohibitive. Our algorithm improves upon the one-sided approach of Farina et al. by means of a new decomposition of correlated strategies which allows players to re-optimize their sequence-form strategies with respect to correlation plans which were previously added to the support. Experiments show that our techniques outperform the prior state of the art for computing optimal general-sum correlated equilibria.

Figures (12)

• Figure 1: An example game, between two players $\color{p1color}\blacktriangle$ (P1) and $\color{p2color}\blacktriangledown$ (P2). The root node is a chance node, at which chance moves uniformly at random. Dotted lines connect nodes in the same information set. Bold lowercase letters are the names of nodes. We will refer to infosets by naming all the nodes within them; for example, b and de are infosets. At terminal nodes, the utility of $\color{p1color}\blacktriangle$ is listed below the name of the node. $\color{p2color}\blacktriangledown$ has utility zero at every terminal node, and in this game the only role of $\color{p2color}\blacktriangledown$ is to incentivize $\color{p1color}\blacktriangle$ to act in a certain way.
• Figure 2: Comparison of different notions of correlation in extensive-form games.
• Figure 3: Correspondence between notions in the mediator-augmented game, and notions in the original game.
• Figure 4: Augmented games $\Gamma^c$ for NFCCE (top), EFCCE (center), and EFCE (bottom), where $\Gamma$ is the example game in \ref{['fig:example-game']}. The obedient strategies $\boldsymbol o_1, \boldsymbol o_2$ are given by the thick colored lines below $\color{p1color}\blacktriangle$ and $\color{p2color}\blacktriangledown$'s decision points. Red circles denote decision points of the mediator. Augmented histories are labeled as $h^\tau$, where $h$ is the true node and $\tau$ is the trigger. If no superscript is present, there was no trigger. For cleanliness, $\tau$ is abbreviated in all three diagrams. For NFCCE, $\tau$ is the player $i$ who deviated---for example, $\mathsf{p^2}$ means terminal node p was reached, but $\color{p2color}\blacktriangledown$ deviated. For EFCCE, $\tau$ is the node at which the player deviated---for example, $\mathsf{p^d}$ means terminal node p was reached, but ${\color{p2color}\blacktriangledown}\xspace$ deviated at node d. For EFCE, $\tau$ is the node at which the player deviated, followed by the recommendation (${ }$ or ${ }$) given to the player at that node---for example, $\mathsf{q^{h{ }}}$ means terminal node q was reached but $\color{p1color}\blacktriangle$ deviated after being recommended to play ${ }$ at h.
• Figure 5: Two examples of two-player extensive-form game trees with no chance moves and large information complexity $k$. In both examples, $k$ can be increased arbitrarily by increasing the branching factor of the root node. The left example would be easily reparable with a tighter definition of information complexity (that takes into account the fact that only one of the infosets in the second layer is reachable in any pure strategy profile), but the right example is not so easily reparable, and examples such as these are the reason that the proof of \ref{['th:pubchance']} is more involved than one may initially expect.

Theorems & Definitions (30)

• Definition 2.1
• Example 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Remark 2.6
• Definition 3.1
• Definition 3.2
• Definition 3.3: Stengel08:Extensive
• Definition 4.1