Stability analysis and control of decision-making of miners in blockchain

Blockchain is a distributed ledger technology for recording transactions that underlies various fields such as digital currency like Bitcoin [1], data sharing [2], and computer security [3]. Blockchain-based services use cryptography to record transactions as a chain of blocks. A block consists of a block header and transaction data. The block header contains a cryptographic hash of its previous block, which makes blockchain-based services resistant to tampering. In these services, participants called miners create blocks in a distributed manner, and the longest chain of blocks is considered to be legitimate. When a miner succeeds in creating a block, he/she gets a reward called a mining reward.

Blockchain-based services approve transactions through a consensus algorithm. As a consensus algorithm, proof-of-work (PoW) is typically used. In this algorithm, the mining difficulty is set using a scalar value called a nonce in the block header. To create a block, miners must find a nonce such that the cryptographic hash value for the previous block satisfies specific conditions. The process of creating blocks is called a mining. In general, a cryptographic hash value for a block is unique according to the nonce contained in the block. Moreover, a nonce that satisfies the specific conditions cannot be calculated directly. As a result, an exhaustive search imposes a large computational cost on miners, which contributes to the resistance to tampering. Because transaction approvals depend on miner calculations (such calculations are very costly and require a lot of energy [4, 5]), the participation of many miners is needed to maintain blockchain-based services and ensure blockchain system security [6, 7]. Therefore, it is important to analyze the decision-making problem of whether miners participate in the mining according to the energy consumption and mining rewards.

Game theory is used to analyze interactions among rational decision-makers. Many studies have adopted game theory to analyze blockchain-related issues with PoW [8], such as decision-making problems in the mining considering the energy consumption [9, 10]. Evolutionary game theory has been used as a powerful mathematical tool for analyzing dynamical models of evolutionary selection [11]. Dynamical characteristics of the selection process are modeled by replicator dynamics. Control methods for the replicator dynamics have been studied in [12, 13, 14]. Evolutionary game models and replicator dynamics are also used in analyzing blockchain-related issues such as mining pool selection problems [15, 16], and attack scenarios [17].

We previously focused on a decision-making problem of whether miners participate in the mining according to the energy consumption and the mining rewards, and modeled it as a non-cooperative game. Through theoretical and numerical analysis, we showed the property of Nash equilibria [18]. However, in this study, we assumed that, once miners choose a strategy (i.e., participation in the mining or not), they do not change their strategies. Practically, the miners may decide to participate in the mining dynamically based on their current earned mining rewards. It is important to analyze such a dynamical decision-making process.

In this paper, we propose a dynamical model of the decision-making problem for miners, by applying an evolutionary game approach. We analyze the stability of its equilibrium points and show the existence of transcritical bifurcations and hysteresis phenomena with the coexistence of two asymptotically stable equilibrium points: one corresponds to the state where all miners participate in the mining and the other to the state where the number of participating miners is minimum. The former equilibrium point is preferable to maintain blockchain-based services. We propose a controller that determines the mining reward based on the number of current participating miners so as to stabilize the equilibrium point if at least one miner participates in the initial time.

The remainder of this paper is organized as follows. In Section 2, we propose an evolutionary game-based dynamical model of the decision-making process. In Section 3, we analyze the stability of its equilibrium point. In Section 4, we design a state feedback controller to let all miners participate in the mining.

We assume that miners in a blockchain network are partitioned into two sets $\mathcal{M}$ and $\mathcal{N}$, where miners in $\mathcal{M}$ always participate in the mining and those in $\mathcal{N}$ have two strategies, participating in the mining (strategy $s_{k}=1$) and not participating in the mining (strategy $s_{k}=0$), where $k\in\mathcal{N}$. Note that $\mathcal{M}\cap\mathcal{N}=\emptyset$. Denoted by $m$ and $n$ are the cardinalities of $\mathcal{M}$ and $\mathcal{N}$, respectively (we assume $m\geq 1$ and $n\geq 1$). We define $x_{0}$ and $x_{1}$ as the ratios of miners in $\mathcal{N}$ that choose strategies $0$ and $1$, respectively. Note that

Miners need to find a nonce such that the first $h$ bits of the hash of the block are all $0$. Then, $D=2^{h}$ is the difficulty parameter and $1/D$ is the probability that a miner creates a block with one hash calculation [19]. When miner $k\in\mathcal{M}\cup\mathcal{N}$ participates in the mining, he/she needs a cost $c$ per unit operating time. The average number $w_{k}=f_{k}(c)$ of hash queries calculated per unit operating time by miner $k$ depends on the cost $c$, and we assume that $f_{k}(c)$ is the same for all miners. In this paper, for simplicity, we assume $f_{k}(c)=vc\,(v>0)$ for any $k\in\mathcal{M}\cup\mathcal{N}$.

The mining of blocks can be described as a Poisson process [20, 21]. That is, the block creation time is exponentially distributed [22]. The rate $\lambda_{k}$ of the Poisson process of miner $k\in\mathcal{M}\cup\mathcal{N}$ is given by $\lambda_{k}=w_{k}/D$ [21]¹¹1 Note that a combination of independent Poisson processes is still a Poisson process. Thus, the rate of the Poisson process of all miners is written as $\sum_{i\in\mathcal{M}\cup\mathcal{N}}\lambda_{i}$. . If miner $k$ chooses $s_{k}=1$, then the rate of the Poisson process is $\lambda_{k}=s_{k}f_{k}(c)/D=s_{k}c/d$ (we define $d=D/v$, in this paper). Let $R$ be the mining reward. Based on the previous work [18], the expected reward $R_{k}$ and the expected cost $CS_{k}$ for the mining of miner $k$ are calculated as follows.

We define the utility function $u_{i}(x_{0},x_{1})$ of miners that choose the strategy $i\in\{0,\ 1\}$ as

which means that the utility of a miner who participates in the mining is the difference between the expected reward $R_{k}$ and the expected cost $CS_{k}$. Based on the principle of the evolutionary game [11], the dynamics of the ratio of miners that choose the strategy $i$ is given by

where $\bar{u}(x_{0},x_{1})=\sum_{i=0}^{1}x_{i}u_{i}(x_{0},x_{1})$ is the average utility of all miners. According to (2.4), (2.5) is rewritten as

Thus, the dynamics of the decision-making of miners is described by the above 1st-order differential equation and the reward $R$ plays an important role in the decision-making of the miners for the participation in the mining. In the following, we investigate stability and stabilization of equilibrium points of (2.6). For that purpose, the concept of a basin of attraction [23] is important. Let $\xi(t;x_{1}^{\mathrm{init}})$ be the solution of (2.6) that starts from an initial state $x_{1}^{\mathrm{init}}$ at time $t=0$. For a given asymptotically stable equilibrium point $x_{1}^{\prime}$ of (2.6), the basin of attraction is defined as the set of all initial states $x_{1}^{\mathrm{init}}$ such that $\xi(t;x_{1}^{\mathrm{init}})$ is defined for all $t\geq 0$ and $\lim_{t\to\infty}\xi(t;x_{1}^{\mathrm{init}})=x_{1}^{\prime}$.

In this section, we investigate the stability of the equilibrium point $x_{1}=0,\ 1,\ x_{1}^{*}$ of (2.6), where

When $1/(m+n)<R/d<1/m$, the equilibrium point $x_{1}^{*}$ satisfies $0<x_{1}^{*}<1$. This equilibrium point is the state where the utility $u_{1}(1-x_{1}^{*},x_{1}^{*})$ is equal to $0$, that is, the utility for the strategy $0$ is equal to that for the strategy $1$.

We investigate the local stability of the three equilibrium points. The derivative of $\varphi_{R}(x_{1})$ with respect to $x_{1}$ is

Thus, we obtain

Thus, we have their stability conditions as shown in Table 1, where S (resp. U) represents an asymptotically stable (resp. unstable) point.

Fig. 1 shows the $m-R/d$ parameter plane with $n=2$ where the meaning of each region is as follows. In the region $(A)$, both $x_{1}=1$ and $x_{1}^{*}<0$ are asymptotically stable equilibrium points, and the basin of attraction of $x_{1}=1$ is $(0,\ \infty)$, that is, every solution of (2.6) starting in $(0,1]$ converges to $1$. In the region $(B)$, both $x_{1}=0$ and $x_{1}=1$ are asymptotically stable equilibrium points, and basins of attraction of $x_{1}=0$ and $x_{1}=1$ are $(-\infty,\ x_{1}^{*})$ and $(x_{1}^{*},\ \infty)$, respectively, that is, every solution of (2.6) starting in $[0,\ x_{1}^{*})$ converges to $0$, and every solution of (2.6) starting in $(x_{1}^{*},\ 1]$ converges to $1$. In the region $(C)$, both $x_{1}=0$ and $x_{1}^{*}>0$ are asymptotically stable equilibrium points, and the basin of attraction of $x_{1}=0$ is $(-\infty,\ 1)$, that is, every solution of (2.6) starting in $[0,\ 1)$ converges to $0$.

Shown in Fig. 2 is a bifurcation diagram with respect to the bifurcation parameter $R/d$, where $m=n=2$. The solid (resp. dashed) line represents an asymptotically stable (resp. unstable) equilibrium point. Two curves of equilibrium points pass through $(x_{1},R)=(1,d/(m+n))$ (resp. $(x_{1},R)=(0,d/m)$), one given by $x_{1}=x_{1}^{*}$, the other by $x_{1}=1$ (resp. $x_{1}=0$). Both curves exist on both sides of $R=d/(m+n)$ (resp. $R=d/m$). The stability along each curve exchanges on passing through $R=d/(m+n)$ (resp. $R=d/m$). Thus, the exchange of stability (known as a transcritical bifurcation) [24] is observed when $(x_{1},R)=(0,d/m),(1,d/(m+n))$. We show in A that the vector field (2.6) satisfies the condition of the transcritical bifurcation shown in [24].

Since the values of $x_{i}\,(i=0,1)$ satisfy $0\leq x_{i}\leq 1$, we observe jump phenomena owing to these transcritical bifurcations. Moreover, $R>d/m$ needs to be satisfied so that all miners in $\mathcal{N}$ participate in the mining. It is noted that, once the miners participate in the mining, they continue to participate in the mining until the reward $R$ becomes $d/(m+n)$. Thus, a hysteresis phenomenon of the equilibrium points is observed.

Fig. 3 shows trajectories of (2.6) from an initial state $x_{1}^{\mathrm{init}}=0.1$ (blue) and $x_{1}^{\mathrm{init}}=0.9$ (red). When $d/(m+n)<R<d/m$, both $x_{1}=0$ and $x_{1}=1$ are asymptotically stable points whose basins of attraction are $(-\infty,\ x_{1}^{*})$ and $(x_{1}^{*},\ \infty)$, respectively. Thus, the number of miners who participate in the mining converges to $0$ if the initial ratio is less than $x_{1}^{*}$ because their utility is negative and they prefer non-participation.

The result in Section 3 implies that no miner in $\mathcal{N}$ participates in the mining in the steady state when the mining reward $R^{*}$ satisfies $R^{*}<d/(m+n)$. When the mining reward $R^{*}$ satisfies $d/(m+n)<R^{*}<d/m$, miners’ behaviors depend on the initial state $x_{1}^{\mathrm{init}}$, i.e., no miner in $\mathcal{N}$ participates in the mining in the steady state when $x_{1}^{\mathrm{init}}<x_{1}^{*}$. We propose a state feedback controller to adjust the reward based on the ratio $x_{1}$ so that all miners in $\mathcal{N}$ participate in the mining, i.e., let every trajectory of $x_{1}$ with its initial state in $(0,\ 1]$ converge to $1$.

We proposed a dynamical model of the decision-making of miners in the blockchain. The proposed model is described by the 1st-order differential equation. So, it is simple but its theoretical analysis gives an insight into the characteristics of the decision-making. We analyzed the stability of its equilibrium points. We showed the occurrence of the transcritical bifurcations and observed a hysteresis phenomenon. We also proposed a feedback controller and showed that it can stabilize the state where all miners participate in the mining from any non-zero initial participation ratio of the miners. Our future work is to extend our model to the case where miners’ computational performances are different from each other.

This research was supported by JST ERATO JPMJER1603.