  In 2015, Eyal proposed the first game-theoretical model for analyzing the
equilibrium of blockchain pooling: when the blockchain pools are abstracted as
a non-cooperative game, two pools can reach a Nash equilibrium with a
closed-form formula; Moreover, an arbitrary number of pools still exhibit an
equilibrium as long as the pools have an equal number of miners. Nevertheless,
whether an equilibrium exists for three or more pools of distinct sizes remains
an open problem. To this end, this paper studies the equilibrium in a
blockchain of arbitrary pools. First, we show that the equilibrium among $q$
identical pools, coinciding the result demonstrated by Eyal through game
theory, can be constructed using a topological approach. Second, if the pools
are of different size, we show that (i) if the blockchain's pools exhibit two
distinct sizes, an equilibrium can be reached, and (ii) if the blockchain has
at least three distinct pool sizes, there does not exist an equilibrium.
