The Framework of Consensus Equilibria for Mining-Pool Games and Related Stability of Gap Games Behaviors in Blockchain Ecosystems

The goal of this paper is to establish the general framework of consensus equilibria for Mining-Pool Games in Blockchain Ecosystems, and in particular to explain the stable in the sense for the existence of consensus equilibria related to mining gap game's behaviors by using one new concept called “consensus games (CG)” 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 phenomenons on the study of mining- pool games with possible impacts due to miners' gap behaviors with scenarios embedded in 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