The Evolutionary Equilibrium of Block Withholding Attack

1.1 Related Work

When a miner plays the BWH attack in a mining pool, it only submits PPoW and discards FPoW, which makes the manager be convinced that this attacker is indeed trying to mine for the pool. Because the manager mistakes the attacker as an honest miner, it also allocate revenue among the attacker and other pool miners. The BWH attack was first proposed by [8]. In 2014, a mining pool called Eligius suffered the BWH attack and lost 300 BTC^[9]. Since then, different kinds of research have paid attention to this attack.

Eyal^[6] first applied a pool game between two mining pools to analyze the Nash equilibrium of the BWH attack. In the case of [6], two pools shall make decisions whether or not attack, which is similar to the famous Prisoners’ Dilemma and thus is called Miner’s Dilemma. In order to prevent the pools from being trapped in a miner’s dilemma and to optimize the mining model, [3] proposed a subclass Zero Determinant (ZD) strategy, by which a miner could control another miner’s payoff and increased the social revenue. [10] modeled a computational power splitting game and showed that the attacker can gain profits in the long-run and may not be so for a short time, implying that the existing pool reward sharing protocols in Bitcoin are insecure when the miners launch the BWH attack.

By deepening of study on the BWH attacks, more researchers pay attention to the mitigation strategies of the BWH attack. [2] believed that most of the countermeasures to mitigate the BWH attack change the mining algorithm, which lowers the practical adaptability. So, the authors suggested three necessary conditions for the BWH countermeasure: No loss, compatibility, and fairness. The incentive compatibility of the reward allocation mechanisms of mining pool was introduced by [11], which can encourage the miners to submit blocks immediately and guarantee the mining pool’s revenue. A concept called “special reward” was proposed by [12], granting additional incentive to the miner who submits a valid block to the pool and the BWH attacker would never receive the special reward. In this scheme, the revenue that the attacker gains is less than his expectation and thus could make a mining pool repulse the BWH attackers. [13] presented two schemes to counter the BWH attack, the one applies the cryptographic commitment schemes and the other is an alternative implementation by using hash function, both making it impossible for miners to distinguish between full proof of work and partial proof of work. A generalized model was constructed by [14] to analyze the equilibrium of the BWH attack, in which the authors found that increasing the asymmetry of information by information conceal mechanisms could decrease the negative influence of the BWH attack on the pool.

In decision-making research, evolutionary game theory is a common tool for constructing a model and analyzing the choice of strategies^[15]. Classic game theory assumes that all players have perfect rationality, while evolutionary game theory is only based on the bounded rationality^[16]. That is, the choice of each player’s equilibrium strategy is the result of the continuous learning and adjustment, rather than a one-time choice. The basic solution concept of the evolutionary game theory is evolutionary stable strategy (ESS)^[17], and it is used to describe the stable state of the evolution process. Recently, a few kinds of literatures study on the blockchain by using evolutionary game theory. [18] applied the evolutionary game to describe the dynamic mining-pool selection process in a PoW-based blockchain network, and they provided the theoretical analysis of the evolutionary stability under the two-pool condition. [19] modeled the process of mining as a two-stage game model in order to characterize the decision that the pool whether to open or not and to launch the BWH attack or not in the PoW-based blockchain network. They applied the evolutionary game theory and analyzed evolutionary stability of the strategy selection. This method could overcome the shortcoming of the NE which only describes the local optimization of the pool strategies selection. In [20], the authors investigated the evolutionary mining game with miner’s dilemma under the BWH attack and studied the population changes with the time between participated pools through the evolutionary stability. They also analyzed the mining pool dynamics affected by malicious infiltrators and the feasibility of autonomous migration among individual miners.

Few work studies the BWH attacks by the evolutionary game from the perspective of mitigating the attacks. [21] constructed a symmetrical evolutionary game model to analyze the expected benefits of the strategy selection of two miners in a mining pool, where the computational power of the two miners are the same. The authors explored how the pool administrator could mitigate the BWH attack under different supervision and punishment mechanisms. In this paper, an asymmetric evolutionary game model, more general than the symmetrical one, is constructed, in which there are two miners in a mining pool and they have different amounts of computational power. Our objects are to explore the influence of the main factors on the miners’ strategy selections, and to make suggestions to mitigate the BWH attack to promot the cooperation between miners.

1.2 Paper Organization

In this paper, we analyze the BWH attack in a pool by establishing an evolutionary game model. Motivated by [14], we construct the payoff functions under different strategies in Section 2. By establishing the replicator dynamic equations, different evolutionary stable strategies under different conditions are derived in Section 2. In Section 3, we analyze how different factors influence the miners’ strategic choice by a series of simulations. Last section provides several valuable suggestions and concludes this paper.

2.1 Basic Evolutionary Game Model

Generally, there are many miners in a Bitcoin mining pool. Since this work only concentrates on two strategies: One is honestly mining (C) and the other is the BWH attack (A), let us assume that there are two participants: Miner 1 and miner 2 to simplify our discussion, like [3, 22], each of whom has aforementioned two strategies. Therefore, we construct an evolutionary game model, in which there are four strategy profiles: (C,C), (C,A), (A,C) and (A,A). Different strategy profile would bring different payoffs to each miner. During the evolutionary game, the miners keep learning to adjust their low-income strategies and to imitate the strategy choice of the miner, who has a higher income, until the strategy profile of two miners reaches a stable state. To establish the evolutionary game model between two miners formally, following assumptions and parameters are necessary to be introduced in advance, which are similar to those in [14].

1. Two miners own different amounts of computational power. Let miner 1 and miner 2 have a₁ and a₂ units of computational power, respectively. W.l.o.g, we assume that a₂ = λa₁, where λ ∈ [0, 1].

2. The reward the mining pool obtains per unit computational power per unit time is denoted by R.

3. When the miner honestly mines and sends the full proof of work (FPoW) to the manager instantaneously, the cost of computational power per unit time to mine is C₁ (C₁> 0). If the miner employs the BWH attack to only submit the partial proof of work (PPoW), the mining cost per unit time it consumes is C₂ (0 ≤ C₂< C₁).

4. If both of the two miners honestly mine, then the probability to dig up the legal block will increase, which leads to the improvement of the expected profit. Thus we assume that the miners’ cooperation with each other would enlarge γ multiples of reward (γ > 1).

5. To encourage the miners to cooperate, the pool manager will draw an additional reward from the reward R to offer to the one who submits the FPoW. We assume the additional reward per unit computational power per unit time to be δR, where δ ∈ (0, 1).

If the two miners adopt the strategy of cooperation at the same time, then the revenue per unit computational power per unit time of the entire mining pool increases to γR. The payoff of each miner is the difference between the reward which is proportional to its computational power and the cost to mine honestly. Therefore, under the strategy profile of (C,C), miner 1 has its payoff of a₁γR−C₁ and miner 2 has its payoff of a₂γR−C₁. If miner 1 mines honestly and miner 2 employs the BWH attack, that is the strategy profile is (C,A), then the actual useful computational power to dig up a legal block is a₁ and thus the total revenue per time of the mining pool is a₁R. Under this situation, the pool manager first provides the additional reward to miner 1 to encourage its honest behavior, and then allocate the rest of reward a₁(1−δ)R to miner 1 and miner 2 proportional to their computational powers, respectively. So the payoff of miner 1 is a₁δR + a12(1−δ)R−C1.Miner 2 could obtain a partial of reward as a free rider, even though it just consumes a smaller cost. Thus its payoff is a₁a₂(1 − δ)R − C₂. For the strategy profile (A,C), the opposite symmetry case happens and then the payoffs of miner 1 and miner 2 are a₁a₂(1 −δ)R−C₂ and a₂δR+a22(1−δ)R−C1,respectively. For the last case of (A,A), as both of the miners attack, they cannot dig up a legal block, and thus no reward can be obtained. It follows that the payoff of each miner is −C₂.

Based on the above analysis and the aforementioned assumptions, the corresponding payoff matrix is shown in Table 1.

2.2 The Solutions of the Evolutionary Game

In the established evolutionary game model, the two miners have two strategies and different payoffs. Let x and y, 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, be the probabilities of miner 1 and miner 2 to play the strategy of cooperation, respectively. Therefore, the possibilities to employ the BWH attack of miner 1 and miner 2 are 1 − x and 1 − y, respectively.

Denote the payoffs of miner 1 when it takes the strategy of cooperation and adopts the BWH attack by U₁₁ and U₁₂, respectively. According to the payoff matrix in Table 1, we can obtain U₁₁ and U₁₂ as follows:

Hence, the average expected payoff of miner 1 is

Similarly, let U₂₁ and U₂₂ be the payoffs of miner 2 when it mines honestly and employs the BWH attack, respectively. From the payoff matrix in Table 1, U₂₁ and U₂₂ are as follows:

So, the average expected payoff of miner 2 is

In an evolutionary game model, each participant keeps learning and then adjusts its lower-payoff strategy to the higher-payoff strategy. Such a learning approach results in a greatly significant growth rate of a strategy in a replicator system, which reflects the evolutionary direction. By [23], the growth rate of a strategy selected by a participant is just equal to the difference between the payoff of this strategy and its average expected payoff. Because a₂ = λa₁, we have the replicator dynamic equations of miner 1 and miner 2 are as follows:

Then, Equations (1) and (2) combine the following replicator dynamic system:

By the stability theorem of the differential equations, all the solutions satisfying F(x, y) = dx dt=0 and G(x,y)=dy dt=0are the equilibrium points of the replicator dynamic system. It is not hard to see there are five fixed equilibrium points of this system: (0, 0), (1, 0), (0, 1), (1, 1) and (x^∗, y^∗), where

if x^∗ ∈ [0, 1] and y^∗ ∈ [0, 1].

From the results in [23], we know the stability condition at the fixed equilibrium points can be achieved by the application of Jacobian matrix. Therefore, the Jacobian matrix of the replicator dynamic system (3) is:

where

The determinant (Det) and the trace (Tr) of J are as follows.

As different fixed equilibrium points bring different determinants and traces of the Jacobian matrix, we then list all determinants and traces in Table 2.

where

2.3 Equilibrium Analysis for the Evolutionary Game

Based on the results in [23], given a point (x, y), if the determinant DetJ(x, y) > 0 and the trace TrJ(x, y) < 0, then this point is an evolutionary stable strategy. Thus according to Table 2.2, following five situations are necessary to be discussed.

Situation 1 When a1δR+(1−δ)a12R−C1+C2>0 and λa1γR−(1−δ)λa12R−C1+C2>0,then DetJ(1, 1) > 0 and TrJ(1, 1) < 0, implying point (1, 1) is the unique evolutionary stable strategy. The main reason is as follows. From the condition of λa1γR−(1−δ)λa12R−C1+C2>0,it is easy to deduce that a₁γR−C₁> (1−δ)Ra₁a₂−C₂, since a₂ = λa₁ and λ ∈ [0, 1]. It means that when miner 2 cooperates, then payoff of miner 1 when it cooperates is higher than the one when it attacks. On the other hand, from the condition of a1δR+(1−δ)a12R−C1+C2>0,we know a1δR+(1−δ)a12R−C1>−C2,implying that when miner 2 attacks, then payoff of miner 1 when it cooperates is also higher than the one when it attacks. Based on the analysis from two aspects, we can conclude that no matter whatever strategy miner 2 adopts, miner 1 always receives a higher benefit when it honestly mines. Therefore, miner 1 must be a cooperator. At this time, once miner 1 chooses cooperation, miner 2 adopts cooperation too, because λa1γR−(1−δ)λa12R−C1+C2>0.Thus under this situation, the system is eventually evolved to adopt cooperation strategy by both miners and then (1, 1) is the evolutionary stable strategy.

Situation 2 When a1δR+(1−δ)a12R−C1+C2<0 and λa1γR−(1−δ)λa12R−C1+C2>0,we have DetJ(0, 0) > 0, TrJ(0, 0) < 0 and DetJ(1, 1) > 0, TrJ(1, 1) < 0. It means that (0, 0) and (1, 1) are both the evolutionary stable strategies of the system. Because of the condition of a1δR+(1−δ)a12R−C1+C2<0,we can see that when miner 2 attacks, then playing the BWH attack is a better choice for miner 1, since it can obtain more payoff than the one when it cooperates. In addition, due to the condition that λa1γR−(1−δ)λa12R−C1+C2>0,it is not hard to observe that if miner 1 cooperates, then miner 2 selects the cooperation too. Thus the rational miners will choose to honestly mine if the cooperation can make them obtain higher payoffs. On the contrary, both of the participants choose to attack, if their payoffs are higher than the ones when they cooperate. So under this situation, the overall evolutionary results of the system are uncertain.

Situation 3 When a1δR+(1−δ)a12R−C1+C2<0 and λa1γR−(1−δ)λa12R−C1+C2<0,only solution (0, 0) satisfies DetJ(0, 0) > 0 and TrJ(0, 0) < 0, meaning that (0, 0) is the unique stable point of the system. Based on the condition of a1δR+(1−δ)a12R−C1+C2<0we have λa1δR+λ2(1−δ)a12R−C1<−C2(λ∈[0,1]),showing that miner 2 can obtain higher payoff by attacking if miner 1 launches the BWH attack. From the condition of λa₁γR − (1 −δ)λa12R−C1+C2<0,it is easy to deduce that λa1γR−C1<(1−δ)λa12R−C2.It follows that miner 2 also can obtain higher payoff by attacking when miner 1 cooperates. Hence, miner 2 always attacks whatever the strategy miner 1 chooses. On the other hand, once miner 2 attacks, miner 1 selects to attack as a1δR+(1−δ)a12R−C1<−C2.Therefore, under this situation, the system is eventually evolved to adopt attack strategy by both miners.

Situation 4 When a1δR+(1−δ)a12R−C1+C2>0 and λa1γR−(1−δ)λa12R−C1+C2<0,only the solution (1, 0) satisfies DetJ(1, 0) > 0 and TrJz(1, 0) < 0, meaning (1, 0) is the unique evolutionary stable strategy of the dynamic system. Since a1δR+(1−δ)a12R−C1+C2>0,when miner 2 attacks, miner 1 can obtain more payoffs by cooperating. At the same time, when miner 1 mines honestly, the payoff of miner 2 to attack is higher than the one when it cooperates, due to λa1γR−(1−δ)λa12R−C1+C2<0.Thus under this situation, the dynamic system is eventually evolved to (1, 0), that is adopting cooperation strategy by miner 1 and launching the BWH attack by miner 2.

Situation 5 When a1λδR+(1−δ)λ2a12R−C1+C2>0 and a1γR−(1−δ)λa12R−C1+C2<0then (0, 1) and (1, 0) are the stable points of the system. According to the conditions in this situation, we know that when miner 1 attacks, miner 2 gains more from honest mining, and when miner 1 honestly mines, miner 2 can gain a higher return by attacking. That is, when one side cooperates, the other side will choose to attack, and thus both sides will adopt different strategies. Under this situation, the dynamic system is eventually evolved to the state, in which one side adopts cooperation strategy and the other adopts attack strategy.

3.1 Simulations for Different Situations

To help the readers to intuitively understand the conclusions of the evolutionary game model with the BWH attack, we first simulate the dynamic evolutionary process of two miners’ strategy selection under different situations demonstrated in Subsection 2.3 by using Matlab.

1. Setting the parameters in the evolutionary game as: δ = 0.2, = 0.6, = = 0.4, a1δR+(1−δ)a12R−C1+C2>0 and a₁a₂λa₁γ = 2, R = 10, C₁ = 4, C₂ = 1. Under this setting, it followsλa1γR−(1−δ)λa12R−C1+C2>0,which is Situation 1. The dynamic evolutionary processes are shown in Figure 1(a), and the evolutionary stable strategy between the two miners is (1, 1).

2. Setting the parameters in the evolutionary game as: δ = 0.2, a₁ = 0.5, a₂ = λa₁ = 0.4, γ=2,R=10,C1=5,C2=1. Then a1δR+(1−δ)a12R−C1+C2<0,λa1γR−(1−δ)λa12R−C1+C2>0,which is Situation 2. The dynamic evolutionary processes are shown in Figure 1(b). Obviously, (0, 0) and (1, 1) are the two evolutionary stable strategies of the system.

3. Setting the parameters in the evolutionary game as: δ = 0.2, a₁ = 0.6, a₂ = λa₁ = 0.3,γ=2,R=10,C1=7,C2=0.5. It follows a1δR+(1−δ)a12R−C1+C2<0 and λa1γR−(1−δ)λa12R−C1+C2<0,which is Situation 3. The dynamic evolutionary processes are shown in Figure 1(c), and then the system converges to the evolutionary stable strategy (0, 0).

4. Setting the parameters in the evolutionary game as: δ = 0.3, a₁ = 0.8, a₂ = λa₁ = 0.6, γ=1.2,R=10,C1=7,C2=1.5. So a1δR+(1−δ)a12R−C1+C2>0 and λa1γR−(1−δ)λa12R−C1+C2<0,which is Situation 4. Then the dynamic evolutionary processes between two parties are shown in Figure 1(d), and the evolutionary stable strategy is (1, 0).

5. Setting the parameters in the evolutionary game as: δ = 0.2, a₁ = 1, a₂ = λa₁ = 0.9, γ = 1.2, R = 10, C₁ = 7, C₂ = 1. Thus we have a1λδR+(1−δ)λ2a12R−C1+C2>0and a1γR−(1−δ)λa12R−C1+C2<0.It just is Situation 4. Under this situation, the dynamic evolutionary processes between two parties are shown in Figure 1(e), and the the evolutionary stable strategies in the dynamic system are (1, 0) and (0, 1).

Figure 1

Dynamic evolution process of two miners

3.2 Simulations for the Influence of Parameters

In the mining process, the main objective is to explore the cooperation tendency between two miners, that is the evolutionary stable strategy (1, 1) is the ideal stable state which we expect for. In this subsection, we will discuss the influence of the parameters on both sides’ strategic choice through analyzing the enlarging multiple γ of rewards, the percentage δ of additional rewards, and the ratio λ of the computational powers of two miners.

1) The influence of δ on the evolutionary behavior of miners.

In order to observe the impact of the additional rewards on the evolutionary behavior, we fix other parameters, and let δ take 0.1, 0.3, 0.5 and 0.7 respectively. Furthermore, the initial probability x, y takes 0.2, 0.5 and 0.8, respectively. Figure 2(a) and Figure 2(b) demonstrate that the additional reward mechanism can encourage miners to mine honestly. With the percentage δ of additional rewards increasing, the evolutionary speed of miner 1 and miner 2 toward cooperation strategy increases too. On the one hand, with a continuous increase of the additional rewards, the rate the miner evolves to the cooperation strategy gradually increases. On the other hand, after the additional rewards increased to a certain amount, the marginal effect would reduce. As a result, a certain additional reward mechanism can motivate miners to cooperation.

Figure 2

Impact of the percentage of additional rewards on the system evolution

2) The influence of λ on the evolutionary behavior of miners.

We fix the other parameters, and let λ take 16,36,46 and 56,respectively, to analyze the impact of the ratio λ of the computational power on the system evolution. The initial probability x and y take 0.2, 0.5 and 0.8, respectively. In Figure 3(a), we can observe that when the initial probability is low, the difference of the computational power between two miners has a great impact on the convergence speed of the miner 1’s strategy. With the initial probability increasing, the difference of computational power has less influence on the convergence speed of miner 1’s strategy. On the contrary, as shown in Figure 3(b), the changes of λ always have a significant impact on miner 2’s strategy choice. When λ takes 16,namely, the computational power of miner 2 is very small, it will adopts the BWH attack. With λ increasing, the miner 2 will turn from the BWH attack to honest mining. The higher the miners’ computational power, the faster the miners’ strategy converges to cooperation. Because of the higher computational power of miner 1, the rate it evolves to cooperation faster than that of miner 2. As a result, the higher computational power of the miner, the greater the probability to adopt a cooperation strategy.

Figure 3

Impact of the ratio of the computational power on the system evolution

3) The influence of γ on the evolutionary behavior of miners.

In order to observe the impact of the enlarging multiples of rewards on the evolutionary behavior, the other parameters are fixed, and γ takes 1.5, 2, 2.5 and 3 respectively. The initial probability x and y take 0.2, 0.5 and 0.8, respectively. With the gradual increase of the enlarging multiples of reward after the cooperation, the miners’ strategy will converge to cooperate quickly. More precisely, from Figure 4, we can see that the enlarging multiples of rewards has a significant effect on miner 2. When γ takes 1.5, it takes a long time for miner 2 to evolve to be a cooperator. With γ increasing, miner 2 will turn to adopt a cooperation strategy regardless of the initial strategy selection. Comparing Figure 4(a) with Figure 4(b), the increased revenue caused by the cooperation strategy has an incentive effect on both parties, but the effect on lower computational power, such as miner 2, is more obvious.

Figure 4

Impact of the enlarging multiples on the system evolution