Mean Field Game for Equilibrium Analysis of Mining Computational Power in Blockchains

Abstract

In a blockchain network, to mine new blocks like in cryptocurrencies or secure IoT networks, each node or player specifies the amount of computational power as its strategy by compromising between the cost and expected utility. Since the strategies of all players affect the expected utility of others through the probability of success, in this article, we first formulate the mining competition among the players in a blockchain network as a noncooperative game. The existence and uniqueness of the Nash equilibrium (NE) point of the game are proven. We consider a gradient learning strategy for the players while preserving their private information as a bounded rational learning model. Furthermore, the convergence of this learning strategy to the E-NE point of the game is studied analytically using the concept of the mean field (MF) game theory. While conventional analytical tools face problems in dealing with a large number of participants, which is a key feature in many IoT networks, deploying the MF game theory facilitates analyzing the behavior of a large population of players by encapsulating the network behavior in an MF term. As the number of players becomes larger, the accuracy of the MF method becomes greater. Moreover, in the MF approach, no information exchange among the agents is needed for optimal decision making and the privacy of the players is preserved. The minimal information exchange is also a proper motivation for using the MF approach in the IoT networks.