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Dynamic Coalitions in Games on Graphs with Preferences over Temporal Goals

A. Kaan Ata Yilmaz, Abhishek Kulkarni, Ufuk Topcu

TL;DR

This work addresses the synthesis of strategies in deterministic multiplayer games on graphs where players have partial-order preferences over temporal goals, enabling dynamic coalitions. It introduces a rank-based, product-game framework to compute maximal sure winning strategies, defines a state-value function that captures each player’s worst guaranteed outcome, and provides a polynomial-time algorithm to synthesize admissible strategies for all players, with a leader proposing coalitions each round. The BlocksWorld-inspired experiments show that dynamic coalitions can outperform fixed coalitions and reveal nontrivial effects of preference alignment on cooperation and welfare. The approach unifies coalition formation with temporal-goal satisfaction under rationality constraints, offering scalable tools for multi-agent systems with temporally extended objectives.

Abstract

In multiplayer games with sequential decision-making, self-interested players form dynamic coalitions to achieve most-preferred temporal goals beyond their individual capabilities. We introduce a novel procedure to synthesize strategies that jointly determine which coalitions should form and the actions coalition members should choose to satisfy their preferences in a subclass of deterministic multiplayer games on graphs. In these games, a leader decides the coalition during each round and the players not in the coalition follow their admissible strategies. Our contributions are threefold. First, we extend the concept of admissibility to games on graphs with preferences and characterize it using maximal sure winning, a concept originally defined for adversarial two-player games with preferences. Second, we define a value function that assigns a vector to each state, identifying which player has a maximal sure winning strategy for certain subset of objectives. Finally, we present a polynomial-time algorithm to synthesize admissible strategies for all players based on this value function and prove their existence in all games within the chosen subclass. We illustrate the benefits of dynamic coalitions over fixed ones in a blocks-world domain. Interestingly, our experiment reveals that aligned preferences do not always encourage cooperation, while conflicting preferences do not always lead to adversarial behavior.

Dynamic Coalitions in Games on Graphs with Preferences over Temporal Goals

TL;DR

This work addresses the synthesis of strategies in deterministic multiplayer games on graphs where players have partial-order preferences over temporal goals, enabling dynamic coalitions. It introduces a rank-based, product-game framework to compute maximal sure winning strategies, defines a state-value function that captures each player’s worst guaranteed outcome, and provides a polynomial-time algorithm to synthesize admissible strategies for all players, with a leader proposing coalitions each round. The BlocksWorld-inspired experiments show that dynamic coalitions can outperform fixed coalitions and reveal nontrivial effects of preference alignment on cooperation and welfare. The approach unifies coalition formation with temporal-goal satisfaction under rationality constraints, offering scalable tools for multi-agent systems with temporally extended objectives.

Abstract

In multiplayer games with sequential decision-making, self-interested players form dynamic coalitions to achieve most-preferred temporal goals beyond their individual capabilities. We introduce a novel procedure to synthesize strategies that jointly determine which coalitions should form and the actions coalition members should choose to satisfy their preferences in a subclass of deterministic multiplayer games on graphs. In these games, a leader decides the coalition during each round and the players not in the coalition follow their admissible strategies. Our contributions are threefold. First, we extend the concept of admissibility to games on graphs with preferences and characterize it using maximal sure winning, a concept originally defined for adversarial two-player games with preferences. Second, we define a value function that assigns a vector to each state, identifying which player has a maximal sure winning strategy for certain subset of objectives. Finally, we present a polynomial-time algorithm to synthesize admissible strategies for all players based on this value function and prove their existence in all games within the chosen subclass. We illustrate the benefits of dynamic coalitions over fixed ones in a blocks-world domain. Interestingly, our experiment reveals that aligned preferences do not always encourage cooperation, while conflicting preferences do not always lead to adversarial behavior.

Paper Structure

This paper contains 15 sections, 6 theorems, 5 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multiplayer Games on Graphs
  4. Specifying Temporal Goals
  5. Preference Modeling
  6. Problem Formulation
  7. Main Results
  8. Maximal Sure Winning Strategy
  9. Value of a State
  10. Synthesis of Admissible Strategies
  11. Experiments
  12. Advantages of Dynamic Coalitions
  13. Preference Alignment and Coalition Formation
  14. Conclusion
  15. Omitted Proofs

Key Result

Lemma 1

Given a player $P_i$ strategy $\pi_i$, $\mathsf{MaxRank}_i(\pi_i)=k$ holds only if there exists a path ${\rho \in \mathsf{Min}\{\mathsf{Paths}(\pi_i)\}}$ such that $\mathsf{Rank}_{i}(w) = k$.

Figures (3)

  • Figure 1: Preferences over temporal objectives for arm P1, P2, and P3, where $\phi_1 = \Diamond (e_1 \land \Diamond e_2), \phi_2 = \neg e_2 \mathsf{U} e_3,$ and $\phi_3 = \Diamond (e_3 \land \Diamond e_2)$.
  • Figure 2: The first dynamic coalition formation strategy that achieves the most preferred outcome for $P_1$
  • Figure 3: The second dynamic coalition formation strategy that achieves the most preferred outcome for $P_1$

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Lemma 1
  • ...and 12 more