Xie Dan, CEO of Xinmai Microelectronics: How to Achieve Pareto Optimality in Mathematical Game Theory of Mining Machines (Brain Burning Depth)

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Author: Dan Xie core clock Microelectronics CEO

The mining of cryptocurrency is an emerging industry. Its short industry chain and technical foothold make the pricing of mining machines a very wonderful game: if the price is high, the mining machine business cannot sell it; if the price is low, the mining farm earns 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 power consumption. Here, let’s take a simple case to illustrate: assuming that regardless of the currency price rise and fall, the daily output of a certain mining coin is 1 million yuan, the current total mining power of A is 10T, and the power consumption per T is 50,000 yuan. Assuming that A has no rivals, then A produces 1 million, and the electricity cost is 500,000, and thus has a profit of 500,000.

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

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

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Of course, in a real situation, A will reduce its own production capacity, so how much? This is easy to calculate. Assuming that the computing power of A is x, 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 of A, we can use calculus to find the derivation, the result of the solution is

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

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

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

According to game theory, this means that the Nash equilibrium has been reached. Changes in A and B alone will be detrimental to them. This is a stable solution.

Case 1, the first level, under the Nash balance, A’s income has increased from 250,000 to 457,200; B’s income has increased from 0 to 86,000.

In the above case, there is only one variable parameter, which is computing power. We assume that another technical optimization variable is introduced: lowering the voltage to reduce the power consumption of computing power. There is the possibility of voltage drop on some mining machines, that is, reducing the computing power, which can reduce the power consumption. Let us first assume that the simplest voltage reduction model is that computing power is the square of power consumption.

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

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Under this 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 is also better than the original 457,200

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

Case 2, the second layer, through technical improvements, realized A revenue of 550,000, and B realized revenue of 137,500.

Nash equilibrium is a non-cooperative game theory, in fact, there is a cooperative game theory.

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

Case 3, in this case, AB mines 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 computing power), thus B The income is 25* 0.95/1.25 – 2.5*2.5 = 125,500, and A income is 55 + 1 = 560,000.

On top of Case 3, there is a better cooperation model:

Case 4: A’s mining machine has a lease 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.

In case 1, B still has an income of 86,000, and A’s income is 100-25-8.6 = 66.4, which is a 45% increase in income.

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

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

From a small mining machine case, we can see the economic embodiment of game theory: Nash Equilibrium-“Pareto Optimization of Technology -” Commercial Division and Cooperation.

In a more realistic business environment, currency prices are constantly fluctuating. In addition to large mine owners such as AB, there are generally small mine owners of C, D, and E. Some mines have advantages in electricity charges (equivalent to low power consumption). ), some mines have operational technology advantages (the firmware can reduce voltage), some mines can get computing power faster (market advantage), and some mines have capital advantages. However, there is a significant difference between the non-cooperative Nash equilibrium and the optimal cooperative Pareto optimum. For the entire industry to achieve Pareto optimality, a credible and neutral mining alliance is needed.

Xie Dan’s old articles: Rewarding old articles: The former technical director of Bitmain reveals the secret of the first generation of S9

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