“…Let (I, K I , D) denotes a scenario that there are I prosumers, each has K I resources and the required load adjustment for prosumer i is D i . Then, the scenario (I , K I , D ) is an equal partition of (I, K I , D) when there exists an Z ∈ Z + , such that I = ZI and K I = ZK I , the resources one prosumer possesses and required load adjustment is distributed equally to Z prosumers and satisfies, ∀i, ∀z 1 , z 2 ∈ {1, ..., Z} (19) where p * corresponds to the NE under scenario (I, K I , D). The proof can be found in Appendix G. It shows that the system under I = 1 is the most efficient; otherwise, introducing competition by spreading resources benefits social welfare.…”