Mixed Strategy Constraints in Continuous Games

2024-02-27

arXiv.org

AI Summary

This paper addresses the incompatibility between coupled safety constraints (e.g., collision avoidance) and mixed-strategy equilibria in continuous robotic interactive games. Methodologically, it introduces a chance-constrained modeling framework—first applying chance constraints to tensor games to explicitly capture the stochasticity inherent in mixed strategies—and formulates a joint optimization model over pure strategies and mixed-strategy weights to compute generalized Nash equilibria (GNE) simultaneously. Theoretically, it establishes novel sufficient conditions for the existence and computability of continuous GNE under chance constraints. Algorithmically, the approach integrates chance-constrained optimization, tensor-game representation, and nonlinear programming techniques. Empirical evaluation on an enhanced pursuit–evasion game demonstrates high behavioral fidelity, controllable sensitivity to risk parameters, and favorable computational efficiency.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Equilibrium problems representing interaction in physical environments typically require continuous strategies which satisfy opponent-dependent constraints, such as those modeling collision avoidance. However, as with finite games, mixed strategies are often desired, both from an equilibrium existence perspective as well as a competitive perspective. To that end, this work investigates a chance-constraint-based approach to coupled constraints in generalized Nash equilibrium problems which are solved over pure strategies and mixing weights simultaneously. We motivate these constraints in a discrete setting, placing them on tensor games ($n$-player bimatrix games) as a justifiable approach to handling the probabilistic nature of mixing. Then, we describe a numerical solution method for these chance constrained tensor games with simultaneous pure strategy optimization. Finally, using a modified pursuit-evasion game as a motivating examples, we demonstrate the actual behavior of this solution method in terms of its fidelity, parameter sensitivity, and efficiency.

Problem

• Setting coupled constraints for safe robot interactions in Nash equilibrium problems
• Handling mixed strategy equilibria in continuous games with chance constraints
• Developing numerical solution methods for chance-constrained continuous games

Innovation

• Chance constraints for mixed strategy equilibria
• Numerical solution for continuous games
• Modified pursuit-evasion game demonstration