Competitive Equilibrium for Bitcoin Mining

Understanding the group effect and deriving an equilibrium

Bitcoin mining is like a race in some ways. You want to be faster than anyone else, and your chance of winning depends on your competition.

Bitcoin miners need to invest time and effort in order to obtain future cash flows. However, they do not act in isolation. Competing market participants may simultaneously pursue the discovery of a cryptocurrency via costly exploration. They may also affect the total supply and market price.

Nowadays cryptocurrency mining requires a rising power consumption. In some cases, depending on the prevailing market price, the mining cost even exceeds the reward for some miners.

The performance of a chosen action is affected by the corresponding choices of others. As a consequence, the group effect of other participants must be taken into account when deciding an optimal strategy.

Mean-Field Equilibrium

In a recent paper, we analyze the competitive equilibrium for markets with a very large number of competitors, like bitcoin mining. We consider a collection of agents that are able to expend costly effort continuously through time, and each agent is rewarded for their effort at a random time by receiving a stream of cash flows.

The time until the cash flows begin for a particular agent can be shortened, although it remains random, by expending more effort. As more participants succeed in generating income, the reward for any particular participant may diminish.

We work in a mean-field setting with a continuum of agents and search for a mean-field Nash equilibrium.

In such an equilibrium, the dynamics of population entry into the state of receiving cash flows is determined by the aggregate strategies of all agents, and under these dynamics no individual agent can improve their performance by altering their strategy.

We analyze the problem under three different specifications for the cash flow horizon: (i) finite, (ii) random (exponentially distributed), and (iii) infinite. When the horizon is exponentially distributed or infinite, we are able to find a class of equilibria in closed form parameterized by two real numbers. The specification which is used depends on the context of the problem being modeled.

Analyzing the behavior of equilibrium with respect to input parameters produces some initially counterintuitive results. For example, in some cases an increase in the cost of an agent’s effort to receive cash flows can increase the value function rather than decrease it as might be expected. This is a result of the endogenous structure of cash flows in equilibrium caused by competition.

There are many directions for future research. To facilitate research collaboration and strategic partnership between CFRM and the industry, the CFRM Quantitative Analytics Lab (QAL) is created to focus on the design and development of quantitative models, analytical tools, and algorithms to solve problems in the finance industry and beyond. For new collaborative projects on cryptocurrency, reach out to the QAL director here.

Disclaimer: this is not intended to be investment advice.


Effort Expenditure for Cash Flow in a Mean-Field Equilibrium [pdf], International Journal of Theoretical & Applied Finance, Vol. 22, №04, p.1950014, 2019

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