Existence of nash equilibrium in single-leader multiple-follower games with min-max interval payoffs

Abstract

In decision-making problems with hierarchical structures, leader–follower games are highly prevalent. As a core concept in game theory, the existence of Nash equilibrium is crucial. However, in reality, complex uncertainties often lead to imprecise game outcomes, and interval representations are an effective tool for capturing such uncertainties. To address the issue of imprecise payoffs in complex environments, this paper proposes the concept of min-max interval (for short, MMI) and studies the existence of Nash equilibrium in single-leader multiple-follower (for short, SLMF) games with MMI payoffs. MMI is an appropriate extension of the traditional interval-providing a more flexible tool for representing uncertain payoffs. We propose an MMI expected payoff ranking method to address the issue of players ranking MMIs. Based on this, operational rules for MMIs and concepts such as limits, continuity, and concavity of MMI-valued functions (for short, MIVFs) are defined. After extending key theorems of real-valued functions to the case of MIVFs, we combine these extended theorems with set-valued mapping theory and Kakutani’s fixed point theorem to prove the existence of Nash equilibrium in SLMF MMI-valued games. Additionally, we compare existing works to verify the innovativeness of the proposed method and provide numerical examples to demonstrate its applicability.