Pair Equilibrium under Pair Invexity: Conditions for Mutual Feasibility

Let X ⊂ Rn be nonempty and F (x) = PNi=1 fi(x) be the aggregate potential.Classical Nash equilibrium is defined via unilateral deviations and fails to capturemutual feasibility. We introduce a new level of equilibrium, called pair equilibrium,which is defined via pair invexity with respect to a map P : X × X → Rn.F (x) − F (x∗) ≥ ⟨ξ∗, P (x, x∗)⟩, ∀x ∈ X,where ξ∗ ∈ ∂F (x∗). We prove that if F is pair invex w.r.t. P and 0 ∈ ∂F (x∗),then x∗ is a pair equilibrium (Theorem 3d.1). This new level strictly generalizesNash equilibrium and provides a sufficient condition for mutual feasibility beyondunilateral stability.