  The goal of this paper is to establish the general framework of consensus
equilibria for Mining-Pool Games in Blockchain Ecosystems, and with the
explanation for the stability of in terms of the existence of consensus
equilibria related to mining gap game's behaviors by using one new concept
called consensus games in Blockchain Ecosystems, here, the Blockchain ecosystem
mainly means the economic activities by taking into the account of three types
of different factors which are expenses, reward mechanism and mining power for
the work on blockschain by applying the key consensus called Proof of Work due
to Nakamoto in 2008.
  In order to do so, we first give an outline how the general existence of
consensus equilibria for Mining Pool Games is formulated, and then used to
explain the stable for Gap Games for Bitcoin in the sense by the existence of
consensus equilibria under the framework of Blockchain consensus, we then
establish a general existence result for consensus equilibria of general mining
gap games by using the profit functions for miners as the payoffs in game
theory. As applications, the general existence results for consensus equilibria
of Gap games are established, which not only help us to claim the existence for
the general stability for Gap games under the general framework of Blockchain
ecosystems, but also allow us to illustrate a number of different phenomenon on
the study of mining-pool games with possible impacts due to miners's gap
behaviors with scenarios embedded n Bitcoin economics. Our study on the
explanation for the stability of mining gap game for Blockchain ecosystems
shows that the concept of consensus equilibria may play a important role for
the development of fundamental theory for consensus economics.
