Spurious Equilibrium in the Graph Model for Conflict Resolution under Catastrophic Risk

In many strategic conflicts, certain states represent catastrophic outcomes that must be avoided at all costs.When equilibrium analysis implicitly relies on threats or sanctions that traverse such catastrophic states, the resulting stability predictions may be operationally meaningless.This paper introduces the concept of spurious equilibrium to formalize this phenomenon in the Graph Model for Conflict Resolution (GMCR).A state exhibits spurious stability if its equilibrium status depends essentially on paths through catastrophic states.We provide formal characterizations of spurious equilibria for Nash stability, General Metarationality (GMR), Symmetric Metarationality (SMR), and Sequential Stability (SEQ).Each stability concept admits spurious equilibria under different structural conditions.The characterization theorems clarify when classical equilibrium predictions can be trusted in high-risk environments and provide diagnostic criteria for identifying misleading stability conclusions.Illustrative examples using Prisoner's Dilemma and Chicken Game demonstrate the practical implications of the framework.