Bitcoin Mining: A Game Theoretic Analysis

Bitcoin is a decentralized cryptocurrency payment system, working without a single administrator or a third party bank. A bitcoin is created by miners, using complex mathematical “proof of work” procedure by computing hashes. For each successful attempt, miners get rewards in terms of bitcoin and transaction fees. Miners participate in mining to get this reward as income. Mining of cryptocurrency such as bitcoin becomes a common interest among the miners as the bitcoin market value is very high. As a side effect of this mining process, a lot of electricity is used in it. Electricity is a semi-renewable resource — depending on the type of resource used for its production. Nevertheless, electricity plays an essential role in the bitcoin mining process since the whole mining process is based on it. The electricity consumed by the mining system is directly proportional to the computational power of that system. Moreover, powerful computers that specially designed for bitcoin mining process, consumes much more electricity than the regular computers. From the fact that at each time only one miner will be rewarded (the one who will win the mining game by first creating and updating the blocks), while the remaining miners’ effort, as well as electricity used for mining at that time, will be wasted. Therefore, optimizing the consumption of electricity is one of the essential and most challenging problems nowadays. One of the possible solutions to the problem mentioned above is to mine strategically rather than mining unreasonably. The strategy could be a plan of action designed to achieve a long-term goal, either Cooperative— where miners can benefit by cooperating and binding agreements or Non-Cooperative– where miners do not make binding agreements and compete against each other. In this paper, we create a game theoretic model in continuous time. We consider a dynamic game model of bitcoin market, where miners or players use mining systems to mine or producing the bitcoin by investing the electricity to the mining system. They sell their product — mined bitcoin in common market where the price of bitcoin is fixed and decide by the linear demand function— price of a product is linear in market demand of product (in economics it is called the linear inverse demand function). The game is played an infinite number of times. We propose two different types of game theory solutions to the game model: Social optimum: (Cooperative) when the miners altogether maximize their total profit and Nash equilibrium: (Non-Cooperative) when each miner behaves selfishly and individually wants to maximize his/her total profit. Note that in our game theory model, a player represents a single “miner” or a “mining pool” who is responsible for creating a block in the blockchain. Our work here found that the electricity depleted very fast for the Nash equilibrium even if it is sustainable for the Social optimum. Our result is quite intuitive to the common belief that mining in cooperation will give the higher payoff or profit to each miner than mining individually. Finally, to make the electricity consumption at equilibrium, we also propose a linear tax system which is of Pigovian type in order to enforce social optimality and refrain from over-consumption of electricity in our dynamic game model.