Xie Dan: How to achieve Pareto optimality in the mathematical game theory of mining machines

Cryptocurrency mining is an emerging industry. Its short industrial chain and technology-based characteristics make mining machine pricing a very wonderful game: if the price is high, the mining machine manufacturer cannot sell it; if the price is low, the mine will make money. most of the money. In fact, the operation of the mining machine itself in the mine also needs the support of mathematical game theory.

The main mathematical models of mining focus on a few: currency price, computing power, and computing power consumption. Here, we take the simplest case to illustrate: Assuming that the daily output of a certain mining coin is 1 million yuan without considering the fluctuation of the currency price, A’s current total mining power is 10T, and the power consumption per T is 50,000 Yuan. Assuming that A has no opponent, then A produces 1 million, and the electricity cost is 500,000, so there is a profit of 500,000.

At this time, a new mining farm B appeared, and the power consumption per T of its mining machine was only half of that of A, that is, 25,000 yuan per T. B also has 10T computing power in hand. After B joins, the distribution of the mining market is a huge change.

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Now the whole mine has a total of 20T computing power, and the daily output is still 1 million. B is a latecomer and leads in power consumption. The daily output of B 10T is 500,000, and the cost is 250,000, so there is a daily profit of 250,000. But A is only 500,000 output, and the cost is 500,000, so the profit is 0.

Of course, in the real situation, A will reduce its own production capacity, so how much will it be reduced? This is easy to calculate. A assumes that the computing power is x, and the computing power of the entire network is 10+X, so A’s income is 100*X/(10+x), and its cost is 5X, so the profit is 100X/(10+X ) - 5X. For the optimal solution of A, we can perform calculus derivation, and the result of the solution is

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X=10*(√2 -1) = 4.1 T. Of course, the more direct way is to pull an excel sheet.

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At this time, B's income is 100* 10 /14.14 -2.5*10 = 45.72.

Moreover, after the derivation of the B income calculus, it is one-way, that is, 10T is the maximum income.

According to game theory, this is the Nash equilibrium, where A and B change independently, which is a stable solution.

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Case 1, the first layer, under Nash equilibrium, A's income ranges from 250,000 to 457,200; B's income ranges from 0 to 86,000.

In the above case, there is only one parameter that can be changed, which is computing power. We assume that another technical optimization variable is introduced: voltage reduction reduces computing power consumption. There is a possibility of reducing the voltage on some mining machines, that is, reducing the computing power, which can reduce the power consumption of computing power. Let's first assume the simplest voltage reduction model, that is, computing power is the square relationship of power consumption.

Therefore, another option for the above A is to reduce the voltage of 10T, for example, from 1v to 0.7v. At this time, the computing power consumption can basically be reduced by half, and the power consumption per 25,000/T can be reached, but the machine computing power Then it drops from 10T to 1/4 and becomes 2.5T.

Under such an assumption, the best points of A and B are both fully loaded, so A’s income:

2.5*100 /12.5 - 2.5*2.5 = 13.75 is better than the original 86,000.

B's income is:

10 *100 /12.5 -25 = 55   also better than the original 457,200

Voltage reduction is a technology, which means that we have achieved Pareto optimization through technical improvement.

Case 2, the second layer, achieved a profit of 550,000 for A and 137,500 for B through technical improvement.

Nash equilibrium is a non-cooperative game theory, but there is actually a cooperative game theory.

In the above two cases, both are non-cooperative games. If A has the voltage reduction technology, he can earn 137,500 yuan, but if he does not have the voltage reduction technology, he can only gain 86,000 yuan, which is a profit difference of 60%.

Case 3, in this case, the AB mine can carry out a cooperation model, which is mostly an authorization model. If AB has a good relationship and can cooperate, then B will transfer the voltage reduction technology to A for a fee (for example, charge 5% of the computing power), so that B The income is 25* 0.95/1.25  - 2.5*2.5 = 127,500 , and the income of A is 55 + 1 = 560,000.

On top of case 3, there is a better cooperation mode:

Case 4: A's mining machine has a rental or sale mode, B directly buys A's mining machine, and then turns off A's machine, so as to maintain the total computing power of 10T unchanged.

In Case 1, B’s income is still 86,000, while A’s income is 100-25-8.6 = 66.4, which is an increase of 45%.

In case 2, B is still 137,500, and A's income is 100-25 -13.75 = 61.25, an increase of 11.4%

This is the third layer, through business cooperation, to achieve higher returns.

From a small case of a mining machine, we can see the economics of game theory: Nash Equilibrium-"Pareto Optimization of Technology-"Commercial Division of Labor and Cooperation.

In a more real business environment, currency prices are always fluctuating. In addition to large mine owners like AB, there are generally small mine owners of C, D, and E. Some mines have an advantage in electricity costs (equivalent to low power consumption) ), some mines have advantages in operating technology (capable of lowering voltage by firmware), some mines can get computing power faster (market advantage), and some mines have capital advantages. However, there is a significant difference between the uncooperative Nash equilibrium and the optimal cooperative Pareto optimum. In order for the entire industry to achieve Pareto optimality, a mining alliance with credibility and neutrality is needed.

Author: Xinmai Microelectronics CEO Xie Dan

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