Open-loop Pareto-Nash equilibria in multi-objective interval differential games

TL;DR

This paper introduces the concept of open-loop Pareto-Nash equilibria in multi-objective interval differential games, establishing existence conditions and applying the theory to a linear quadratic example.

Contribution

It defines and proves existence of open-loop Pareto-Nash equilibria in multi-objective interval differential games, extending game theory to interval-valued payoffs.

Findings

Established existence theorems for equilibria.

Derived necessary and sufficient conditions using Hamilton functions.

Applied theory to a two-player linear quadratic game.

Abstract

The paper explores n-player multi-objective interval differential games, where the terminal payoff function and integral payoff function of players are both interval-vector-valued functions. Firstly, by leveraging the partial order relationship among interval vectors, we establish the concept of (weighted) open-loop Pareto-Nash equilibrium for multi-objective interval differential games and derive two theorems regarding the existence of such equilibria. Secondly, necessary conditions for open-loop Pareto-Nash equilibria in n-player interval differential games are derived through constructing Hamilton functions in an interval form and applying the Pontryagin maximum principle. Subsequently, sufficient conditions for their existence are provided by defining a maximization Hamilton function and utilizing its concavity. Finally, a two-player linear quadratic interval differential game is discussed along with a specific calculation method to determine its open-loop Pareto-Nash equilibrium.