We establish the existence of a best proximity pair for which the best proximity set is nonempty for a finite family of multimaps whose product is either an 𝔄 𝐜 κ -multimap or a multimap T : A → 2 B such that both T and S ∘ T are closed and have the KKM property for each Kakutani multimap S : B → 2 A . As applications, we obtain existence theorems of equilibrium pairs for free n -person games as well as for free 1-person games. Our results extend and improve several well-known and recent results.