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Enhanced Equilibria-Solving via Private Information Pre-Branch Structure in Adversarial Team Games

Chen Qiu, Haobo Fu, Kai Li, Weixin Huang, Jiajia Zhang, Xuan Wang

TL;DR

The paper tackles TMECor computation in ex ante coordinated ATGs, where existing action-based transformations (e.g., TPICA) incur exponential growth and poor scalability. It introduces a private-information–driven transformation, MPTA, built on a private information pre-branch (PIPB) that replaces team members with a coordinator while dummy players encode teammates’ private information. The authors prove equilibrium equivalence between TMECor in the original game and NE in the transformed game and demonstrate dramatic empirical speedups (up to 694.44×) across Kuhn, Leduc, and Goofspiel testbeds, including large-scale and dynamically changing-private-information scenarios. This approach expands the class of solvable ATGs and offers substantial practical benefits for automated coordination in adversarial settings.

Abstract

In ex ante coordinated adversarial team games (ATGs), a team competes against an adversary, and the team members are only allowed to coordinate their strategies before the game starts. The team-maxmin equilibrium with correlation (TMECor) is a suitable solution concept for ATGs. One class of TMECor-solving methods transforms the problem into solving NE in two-player zero-sum games, leveraging well-established tools for the latter. However, existing methods are fundamentally action-based, resulting in poor generalizability and low solving efficiency due to the exponential growth in the size of the transformed game. To address the above issues, we propose an efficient game transformation method based on private information, where all team members are represented by a single coordinator. We designed a structure called private information pre-branch, which makes decisions considering all possible private information from teammates. We prove that the size of the game transformed by our method is exponentially reduced compared to the current state-of-the-art. Moreover, we demonstrate equilibria equivalence. Experimentally, our method achieves a significant speedup of 182.89$\times$ to 694.44$\times$ in scenarios where the current state-of-the-art method can work, such as small-scale Kuhn poker and Leduc poker. Furthermore, our method is applicable to larger games and those with dynamically changing private information, such as Goofspiel.

Enhanced Equilibria-Solving via Private Information Pre-Branch Structure in Adversarial Team Games

TL;DR

The paper tackles TMECor computation in ex ante coordinated ATGs, where existing action-based transformations (e.g., TPICA) incur exponential growth and poor scalability. It introduces a private-information–driven transformation, MPTA, built on a private information pre-branch (PIPB) that replaces team members with a coordinator while dummy players encode teammates’ private information. The authors prove equilibrium equivalence between TMECor in the original game and NE in the transformed game and demonstrate dramatic empirical speedups (up to 694.44×) across Kuhn, Leduc, and Goofspiel testbeds, including large-scale and dynamically changing-private-information scenarios. This approach expands the class of solvable ATGs and offers substantial practical benefits for automated coordination in adversarial settings.

Abstract

In ex ante coordinated adversarial team games (ATGs), a team competes against an adversary, and the team members are only allowed to coordinate their strategies before the game starts. The team-maxmin equilibrium with correlation (TMECor) is a suitable solution concept for ATGs. One class of TMECor-solving methods transforms the problem into solving NE in two-player zero-sum games, leveraging well-established tools for the latter. However, existing methods are fundamentally action-based, resulting in poor generalizability and low solving efficiency due to the exponential growth in the size of the transformed game. To address the above issues, we propose an efficient game transformation method based on private information, where all team members are represented by a single coordinator. We designed a structure called private information pre-branch, which makes decisions considering all possible private information from teammates. We prove that the size of the game transformed by our method is exponentially reduced compared to the current state-of-the-art. Moreover, we demonstrate equilibria equivalence. Experimentally, our method achieves a significant speedup of 182.89 to 694.44 in scenarios where the current state-of-the-art method can work, such as small-scale Kuhn poker and Leduc poker. Furthermore, our method is applicable to larger games and those with dynamically changing private information, such as Goofspiel.
Paper Structure (28 sections, 8 theorems, 13 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 28 sections, 8 theorems, 13 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Extensive-Form Games and Nash Equilibrium
  5. Adversarial Team Games and Team-Maxmin Equilibrium with Correlation
  6. Team-Public-Information Representation for Extensive-Form Games
  7. Method
  8. The Structure of Private Information Pre-Branch
  9. Multi-Player Transformation Algorithm
  10. Equilibrium Equivalence
  11. Experimental Evaluation
  12. Experimental Setting
  13. Experimental Results
  14. Execution Efficiency.
  15. Solving Efficiency.
  16. ...and 13 more sections

Key Result

Theorem 1

Given an ATG $G$ with visibility that satisfies the public-turn-taking property, and its transformed game $G^{\prime}=\emph{MPTA}(G)$. The size of any episode in $G^{\prime}$ is $\mathcal{O}((\frac{(\lvert \Omega\rvert-1)!}{(\lvert \Omega\rvert-\lvert \mathcal{T}\rvert)!}\lvert A\rvert)^{\lvert \mat

Figures (4)

  • Figure 1: Example of game transformation. "…" indicates omitted branches. The nodes of a player with the same number are in the same infoset. Left: Original ATG omitting the opponent. Right: Result of transforming the game on the left using MPTA.
  • Figure 2: Comparison of runtime within the same number of iterations. All schemes except for 12K6 have been iterated for 20,000 rounds, as the TPICA is too time-consuming to run more rounds.
  • Figure 3: Comparison of exploitability in the same running time. All experiments except 21G run for 100,000 seconds. TPICA fails to work due to out-of-memory in 14K6 and 14L33 and cannot run on Goofspiel due to changes in private information.
  • Figure 4: Example of game transformation. "…" indicates omitted branches. The nodes of a player with the same number are in the same infoset. Left: Original ATG omitting the opponent. Right: Result of transforming the game on the left using TPICA.

Theorems & Definitions (13)

  • Definition 1
  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Lemma 2
  • proof
  • Theorem 5
  • ...and 3 more