Team Belief DAG: Generalizing the Sequence Form to Team Games for Fast Computation of Correlated Team Max-Min Equilibria via Regret Minimization

TL;DR

This paper tackles computing equilibria in adversarial team games where team members cannot privately communicate, highlighting the inadequacy of the traditional sequence-form representation. It introduces the Team Belief DAG (TB-DAG), a convex, perfect-information representation that enables regret-minimization methods (CFR and its variants) to operate efficiently on the team decision problem, with strong parameterized-size bounds. The authors establish foundational complexity results, showing distinct levels of hardness for TMECor and TME in team-vs-team settings, and provide a rigorous TB-DAG construction that is often exponentially smaller and faster to build than prior representations. Empirical results across benchmark games demonstrate state-of-the-art performance, with significant speedups over LP-based and IP-based methods, validating TB-DAG as a practical and scalable tool for computing correlated team max-min equilibria via regret minimization.

Abstract

A classic result in the theory of extensive-form games asserts that the set of strategies available to any perfect-recall player is strategically equivalent to a low-dimensional convex polytope, called the sequence-form polytope. Online convex optimization tools operating on this polytope are the current state-of-the-art for computing several notions of equilibria in games, and have been crucial in landmark applications of computational game theory. However, when optimizing over the joint strategy space of a team of players, one cannot use the sequence form to obtain a strategically-equivalent convex description of the strategy set of the team. In this paper, we provide new complexity results on the computation of optimal strategies for teams, and propose a new representation, coined team belief DAG (TB-DAG), that describes team strategies as a convex set. The TB-DAG enjoys state-of-the-art parameterized complexity bounds, while at the same time enjoying the advantages of efficient regret minimization techniques. We show that TB-DAG can be exponentially smaller and can be computed exponentially faster than all other known representations, and that the converse is never true. Experimentally, we show that the TB-DAG, when paired with learning techniques, yields state of the art on a wide variety of benchmark team games.

Figures (5)

• Figure 1: An example extensive-form game tree with two players (a), and its decision problems for $\color{p1color}\blacktriangle$ (b), $\color{p2color}\blacktriangledown$ (c), and the team consisting of both $\color{p1color}\blacktriangle$ and $\color{p2color}\blacktriangledown$ (d). Dotted lines connect nodes in the same infoset. Note that $\color{p1color}\blacktriangle$ has perfect information, $\color{p2color}\blacktriangledown$ has perfect recall, and the team has neither. In the decision problems, black nodes are active and white inactive.
• Figure 2: (a) Team decision problem from \ref{['fi:example-game']} (nodes named for ease of reference), its connectivity graph (b), and its TB-DAG (c). We remark that per our definition of the construction procedure of TB-DAG, the root is always a decision node; when the root of the original problem is an observation node, this creates a trivial layer in the decision tree. (Of course, it does not affect the complexity guarantees, and in fact the layer might be removed as a postprocessing step---we do this in the experiments; see also \ref{['se:optimizations']}, point 2).
• Figure 3: A team decision problem showing that public-state-based approaches do not subsume inflation.
• Figure 4: A pictoral representation of the proof of \ref{['pr:inflate1']}. Since $h$ and $h'$ can be played simultaneously but $u$ and $u'$ cannot, there must be an infoset like the red dotted one connecting a child of $h$ to a child of $h'$. Therefore, inflation cannot break existing edges between played nodes.
• Figure 5: The counterexample for \ref{['pr:inflate2']}, for $C=6$. All solid-colored nodes ($\color{p1color}\blacktriangle$, $\color{p2color}\blacktriangledown$, and $\blacksquare$) are active (we split them into three different symbols and types so that we can discuss each one separately).

Theorems & Definitions (40)

• Definition 2.1
• Theorem 2.2: Romanovskii62:ReductionKoller94:Fast
• Definition 2.3
• Definition 2.4
• Theorem 2.5: Zinkevich07:Regret
• Definition 4.1
• Theorem 4.2
• Theorem 4.3
• Definition 6.1
• Proposition 6.2