Xie Dan: How to achieve Pareto optimality through mathematical game theory of mining machines (deep brain-burning)

Author: Xie Dan, CEO of Xinmai Microelectronics

Cryptocurrency mining is an emerging industry. Its short industry chain and technological advantages make the pricing of mining machines a very interesting game: if the price is too high, the mining machine vendors cannot sell them; if the price is too low, the mining farms make most of the money. In fact, the operation of the mining machines in the mining farms also requires the support of mathematical game theory.

The main mathematical models of mining focus on the following: coin price, computing power, and computing power consumption. Here, we take the simplest example to illustrate: Assuming that the price fluctuations are not taken into account, the daily output of a certain mining coin is 1 million yuan, and A's current total mining computing power is 10T, and the power consumption per T is 50,000 yuan. Assuming that A has no competitors, then A's output is 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. Its mining machines consumed only half the power per ton of A, which is 25,000 yuan per ton. B also had 10 ton of computing power. After B joined, the distribution of the mining market was a huge change.

Now the total mining power of the mine is 20T, and the daily output is still 1 million. B is a latecomer, and its power consumption is leading. B's daily output of 10T is 500,000, and the cost is 250,000, so there is a profit of 250,000 per day. But A only has an output of 500,000, and the cost is also 500,000, so the profit is 0.

Of course, in reality, A will reduce its production capacity, so how much will it reduce? This is easy to calculate. Assume that A's 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. The optimal solution for A can be derived by calculus, and the solution is

X=10*(√2 -1) = 4.1 T. Of course, a more direct way is to pull an Excel spreadsheet.

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

Moreover, after taking the calculus derivative of B's ​​benefit, it is one-way, that is, 10T is the maximum benefit.

According to game theory, this is the Nash equilibrium. Any change in A or B alone will be disadvantageous to itself, which is a stable solution.

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

In the above case, there is only one variable parameter, which is the hashrate. Let's assume that we introduce another technical optimization variable: lowering the voltage reduces the power consumption of the hashrate. Some mining machines may reduce the voltage, that is, reduce the hashrate, which can reduce the power consumption of the hashrate. Let's first assume the simplest voltage reduction model, that is, the hashrate is the square of the power consumption.

Therefore, another option for A above is to lower the voltage of 10T, for example, from 1v to 0.7v. At this time, the power consumption of computing power can be basically reduced by half, reaching a power consumption of 25,000/T, but the computing power of the machine will drop from 10T to 1/4, becoming 2.5T.

Under this assumption, the optimal points of A and B are both fully loaded, so A earns:

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

B's income is:

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

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

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

Nash equilibrium is a non-cooperative game theory, but there is also 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. If he does not have the voltage reduction technology, he can only earn 86,000 yuan, which is a 60% profit difference.

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

In addition to case 3, there is a better cooperation model:

Case 4: A's mining machine is available for rent or sale. B directly purchases A's mining machine and then shuts down A's machine, thereby maintaining the total computing power of 10T unchanged.

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

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

This is the third layer, achieving higher profits through business cooperation.

From a small mining machine case, we can see the economic embodiment of game theory: Nash equilibrium -> Pareto optimization of technology -> business division of labor and cooperation.

In a more realistic business environment, the price of coins is constantly fluctuating. In addition to large mine owners like AB, there are generally small mine owners like C, D, and E. Some mines have electricity cost advantages (equivalent to low power consumption), some mines have operational technology advantages (firmware can reduce voltage), some mines can get computing power faster (market advantages), and some mines have financial advantages. However, there is a significant difference between the non-cooperative Nash equilibrium and the optimal cooperative Pareto optimality. In order for the entire industry to achieve Pareto optimality, a mining alliance with credibility and neutrality is needed.

Xie Dan's old article: Reward for old article: Bitmain's former technical director reveals the secrets of the flagship S9

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