Abstract: Decentralized blockchain is a distributed ledger that may be updated for conducting a sequence of timed transactions representing assets or ownership etc., which is used as a platform for exchanging goods and services. This distributed ledger is maintained by a network of nodes obeying a set of rules, called a consensus protocol, to help them resolve inconsistencies over their local copies of the blockchain. The consensus protocol designer needs to specify: (a) a mapping to an ownership of a substantial resource to allow nodes to invest strategically in this resource to compete for the right to build new blocks in the blockchain, and (b) a reasonable reward function for their efforts. In this way, the equilibrium of the associated stochastic differential game can be implemented to select nodes in proportion to this specified resource and the dishonest nodes will be penalised by loss of resources, thereby encouraging nodes to behave honestly. Thus, the consensus is reached and the underlying security of the blockchain is governed by this equilibrium. Though the induced associated game is complex, we can simplify the analysis using an approximation methodology, known as mean-field games. Without loss of generality, we choose the bitcoin blockchain mechanism as an example to demonstrate the existence and uniqueness of such mean-field games equilibrium.

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