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Nash Equilibrium in Games on Graphs with Incomplete Preferences

Abhishek N. Kulkarni, Jie Fu, Ufuk Topcu

TL;DR

The paper addresses computing Nash equilibria in two-player deterministic turn-based games on graphs with incomplete preferences over temporal goals. It introduces an automata-theoretic product that lifts each player's PrefLTL$_f$-based preferences to a state preorder on $V$, connecting equilibria to sure winning and Pareto efficiency across fully aligned, completely opposite, and partially aligned regimes. Key contributions include a complete characterization of NE under each alignment, a proof of existence, and an efficient, linear-scaling method to compute maximal sure winning strategies; the framework is illustrated via drone-delivery experiments demonstrating the impact of alignment and cooperation attitudes on equilibrium outcomes. The work advances our understanding of rational play with incomplete preferences and has practical implications for mechanism design and multi-agent planning in robotics and AI, where cooperation and opposition coexist under temporal objectives.

Abstract

Games with incomplete preferences are an important model for studying rational decision-making in scenarios where players face incomplete information about their preferences and must contend with incomparable outcomes. We study the problem of computing Nash equilibrium in a subclass of two-player games played on graphs where each player seeks to maximally satisfy their (possibly incomplete) preferences over a set of temporal goals. We characterize the Nash equilibrium and prove its existence in scenarios where player preferences are fully aligned, partially aligned, and completely opposite, in terms of the well-known solution concepts of sure winning and Pareto efficiency. When preferences are partially aligned, we derive conditions under which a player needs cooperation and demonstrate that the Nash equilibria depend not only on the preference alignment but also on whether the players need cooperation to achieve a better outcome and whether they are willing to cooperate.We illustrate the theoretical results by solving a mechanism design problem for a drone delivery scenario.

Nash Equilibrium in Games on Graphs with Incomplete Preferences

TL;DR

The paper addresses computing Nash equilibria in two-player deterministic turn-based games on graphs with incomplete preferences over temporal goals. It introduces an automata-theoretic product that lifts each player's PrefLTL-based preferences to a state preorder on , connecting equilibria to sure winning and Pareto efficiency across fully aligned, completely opposite, and partially aligned regimes. Key contributions include a complete characterization of NE under each alignment, a proof of existence, and an efficient, linear-scaling method to compute maximal sure winning strategies; the framework is illustrated via drone-delivery experiments demonstrating the impact of alignment and cooperation attitudes on equilibrium outcomes. The work advances our understanding of rational play with incomplete preferences and has practical implications for mechanism design and multi-agent planning in robotics and AI, where cooperation and opposition coexist under temporal objectives.

Abstract

Games with incomplete preferences are an important model for studying rational decision-making in scenarios where players face incomplete information about their preferences and must contend with incomparable outcomes. We study the problem of computing Nash equilibrium in a subclass of two-player games played on graphs where each player seeks to maximally satisfy their (possibly incomplete) preferences over a set of temporal goals. We characterize the Nash equilibrium and prove its existence in scenarios where player preferences are fully aligned, partially aligned, and completely opposite, in terms of the well-known solution concepts of sure winning and Pareto efficiency. When preferences are partially aligned, we derive conditions under which a player needs cooperation and demonstrate that the Nash equilibria depend not only on the preference alignment but also on whether the players need cooperation to achieve a better outcome and whether they are willing to cooperate.We illustrate the theoretical results by solving a mechanism design problem for a drone delivery scenario.
Paper Structure (9 sections, 15 theorems, 8 equations, 6 figures)

This paper contains 9 sections, 15 theorems, 8 equations, 6 figures.

Table of Contents

  1. Main Results
  2. Fully Aligned Preferences
  3. Completely Opposite Preferences
  4. Partial Aligned Preferences
  5. Experiment
  6. Aligned Preferences
  7. Completely Opposite Preferences
  8. Partially Aligned Preferences
  9. Conclusion

Key Result

Proposition 1

Given any finite paths $\rho, \rho'$ in $G$, let $\varrho, \varrho'$ be their traces in $H$. Then, for $i = 1, 2$, $L(\rho) \succeq_i L(\rho')$ if and only if $\mathsf{Last}(\varrho) \succeq_{\mathcal{E}_i} \mathsf{Last}(\varrho')$.

Figures (6)

  • Figure 1: Two drone delivery environments.
  • Figure 2: Preference automaton for PrefLTL$_f$ formula $\psi = (\varphi_4 \triangleright \varphi_1) \mathop{\mathrm{\&}}\limits (\varphi_4 \triangleright \varphi_2) \mathop{\mathrm{\&}}\limits (\varphi_4 \triangleright \varphi_3)$.
  • Figure 3: The rank of maximal reachable state in scenario 1.
  • Figure 4: The smallest rank achievable by drone A by following a maximal sure winning strategy in scenario 2.
  • Figure 5: Preference automaton for PrefLTL$_f$ formula $\psi_1 = (\varphi_1 \triangleright \varphi_2) \mathop{\mathrm{\&}}\limits (\varphi_1 \triangleright \varphi_3) \mathop{\mathrm{\&}}\limits (\varphi_4 \triangleright \varphi_2) \mathop{\mathrm{\&}}\limits (\varphi_4 \triangleright \varphi_3)$.
  • ...and 1 more figures

Theorems & Definitions (34)

  • Definition 8
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • Proposition 2
  • proof
  • Definition 9
  • Proposition 3
  • ...and 24 more