Bitcoin vs. Bitcoin Cash: Coexistence or Downfall of Bitcoin Cash?

02/28/2019 ∙ by Yujin Kwon, et al. ∙ KAIST 수리과학과 0

In Aug. 2017, Bitcoin was split into the original Bitcoin (BTC) and Bitcoin Cash (BCH). Since then, miners have had a choice between BTC and BCH mining because they have compatible proof-of-work algorithms. Therefore, they can freely choose which coin to mine for higher profit, where the profitability depends on both the coin price and mining difficulty. Some miners can immediately switch the coin to mine only when mining difficulty changes because the difficulty changes are more predictable than that for the coin price, and we call this behavior fickle mining. In this paper, we study the effects of fickle mining by modeling a game between two coins. To do this, we consider both fickle miners and some factions (e.g., BITMAIN for BCH mining) that stick to mining one coin to maintain that chain. In this model, we show that fickle mining leads to a Nash equilibrium in which only a faction sticking to its coin mining remains as a loyal miner to the less valued coin (e.g., BCH), where loyal miners refer to those who conduct mining even after coin mining difficulty increases. This situation would cause severe centralization, weakening the security of the coin system. To determine which equilibrium the competing coin systems (e.g., BTC vs. BCH) are moving toward, we traced the historical changes of mining power for BTC and BCH. In addition, we analyze the recent "hash war" between Bitcoin ABC and SV, which confirms our theoretical analysis. Finally, we note that our results can be applied to any competing cryptocurrency systems in which the same hardware (e.g., ASICs or GPUs) can be used for mining. Therefore, our study brings new and important angles in competitive coin markets: a coin can intentionally weaken the security and decentralization level of the other rival coin when mining hardware is shared between them, allowing for automatic mining.

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I Introduction

Bitcoin [nakamoto2008bitcoin] is the most popular cryptocurrency based on a distributed and public digital ledger called blockchain. Nodes in the Bitcoin network store the blockchain, where transactions are recorded in a unit of a block, and the blockchain is extended by generating new blocks. The process of generating new blocks is referred to as mining, and nodes conducting mining activities are referred to as miners. To successfully mine, miners should find a solution called the proof-of-work (PoW) [pow]. In Bitcoin, miners are required to solve a cryptographic puzzle finding a hash value to satisfy specific conditions such as a certain number of leading zeroes. To solve a puzzle, miners spend their computational power, and the miner who finds the solution obtains 12.5 coins and the transaction fees in the new block as a reward. In addition, Bitcoin has an average block interval of 10 minutes by adjusting the mining difficulty (i.e., the difficulty of the puzzles).

As Bitcoin has gained popularity, the transaction scalability issue has risen, and several solutions have been proposed to address the issue. However, there were also several conflicts over these solutions. As a result, in Aug. 2017, the Bitcoin system was split into the original Bitcoin (BTC) and Bitcoin Cash (BCH) [bch, split]. The key idea of BCH is to increase a maximum block size to process more transactions than BTC. However, even with different block size limits, they have compatible proof-of-work mechanisms with each other. Therefore, miners can freely alternate between BTC and BCH mining to boost their profits [stability]. The mining profitability changes when the mining difficulty and coin price change, but some miners may be concerned only with the change in former because it is relatively easier to predict the former than the latter. More precisely, rational miners can decide which cryptocurrency is better to mine depending on the coin mining difficulty — BCH mining would be conducted by the miner only if the BCH mining difficulty is low compared to the BTC mining difficulty; otherwise, the miner does BTC mining rather than BCH mining. We call this miner’s behavior “fickle mining” in this paper. Note that the fickle miner may change the coin to mine at a specific time period whenever the coin mining difficulty changes. Thus, fickle mining leads to instability of mining power, which may eventually cause unstable coin prices [stability].

Game model and analysis. In this study, we aim to analyze the economics of fickle mining rigorously, which can later be extended to show how one coin can lead to a lack of loyal miners for other less valued coins. Here, a loyal miner represents one who conducts mining the less valued coin even after the coin mining difficulty increases. To study the economics of fickle mining, we propose a game theoretical framework of players who can conduct fickle mining between two coins (e.g., BTC and BCH). Moreover, our game model reflects coin factions that stick to mining their own coins, as they are interested in only the maintenance of their systems rather than the payoffs. Then we analyze Nash equilibria and dynamics in the game; two types of equilibria exist: the stable coexistence of two coins and the lack of loyal miners for the less valued coin. More specifically, in the latter case, only some factions (e.g., BITMAIN for BCH mining) remain as loyal miners for the less valued coin, and this fact can eventually make the coin system severely centralized, weakening its security. We describe the game model in Section IV and analyze the game in Section V.

Data analysis for BTC vs. BCH. Next, as a case study, we analyzed the mining power changes in BTC and BCH to see if our theoretical analysis matches with actual mining power changes. In this paper, we refer to the Bitcoin system as a coin system consisting of BTC and BCH. We examine the mining power history in the Bitcoin system from the release date of BCH until Dec. 2018 to 1) analyze which equilibrium its state has been moving to and 2) evaluate our theoretical analysis empirically. Our analysis results show that until the BCH mining difficulty adjustment algorithm changed (on Nov. 13, 2017), the Bitcoin state reached a lack of loyal miners for BCH. Therefore, BCH periodically became severely centralized before the update of the BCH protocol. For example, we observe a period when only five miners exist, of which two miners possess about 70 % power. However, since Nov. 13, 2017, the Bitcoin state has been close to coexistence because the change in the BCH mining difficulty adjustment algorithm with a shorter difficulty adjustment time interval (i.e., every block) has affected the game as an external factor.

Nevertheless, we explain that the state would still get closer to a lack of BCH loyal miners if automatic mining, in which miners automatically choose the most profitable coin to mine, is popularly used. Note that the main difference between fickle mining and automatic mining is that fickle miners immediately change their coin only when the mining difficulty changes while automatic miners can immediately change their coin when not only the mining difficulty but also the coin price changes. As a result, at the time of writing (Dec. 2018), if 5% of the total mining power of the Bitcoin system involves automatic mining, the current loyal miners for BCH would leave, weakening its security.

Data analysis for Bitcoin ABC vs. SV. As another case study in our game model, we also analyze the changes in the hash rate distributions of Bitcoin ABC and Bitcoin SV, before and after the recent “hash war” between those two coins. The analysis results of these case studies are presented in Section VI and VII.

Generalization. Moreover, we remark that our analysis can be generalized to any circumstance wherein two coins have compatible PoW mechanisms with each other. We believe that the generalized results bring new important angles in competitive coin markets; a coin can attempt to steal loyal miners from other rivalry coins that have compatible PoW mechanisms. In Section VIII, a risk of automatic mining and the way to intentionally reduce the number of loyal miners for other coins are described. Then, in Section IX, we discuss countermeasures and environmental factors that may make the actual coin states deviate from our game analysis.

In summary, our main contributions are as follows:

  1. To analyze the economics of fickle mining, we first model a game between two coins, considering some coin factions that stick to mining their own coin.

  2. We analyze Nash equilibria and dynamics in the game and find two types of equilibria: 1) stable coexistence of two coins and 2) a lack of loyal miners to the less valued coin. Then, we apply this game to the Bitcoin system.

  3. To determine if real-world miners’ behaviors follow our model, we investigate the mining power history in the Bitcoin system. Then we show that the state reached the lack of BCH loyal miners until Nov. 13, 2017, and we confirm that this fact periodically led the BCH system to be centralized and insecure. Moreover, for generalization, we also analyze the recent “hash war” situation between Bitcoin ABC and Bitcoin SV according to our game model.

  4. We introduce a risk of automatic mining and predict that the current BCH loyal miners would leave when 5% of the total mining power in BTC and BCH involves automatic mining.

  5. Finally, our game is generalized to any mining-compatible coins (e.g. Ethereum vs. Ethereum Classic). Therefore, our study brings a threat that one coin can intentionally steal loyal miners from other less valued coin.

Ii Preliminary

Ii-a Cryptocurrency

Many cryptocurrencies such as Bitcoin, Ethereum, and Litecoin adopt the PoW mechanism as a consensus algorithm. In the PoW mechanism, when a node solves a cryptographic puzzle, the node can generate and propagate a valid block. Then other nodes append the generated block to the existing blockchain. The puzzle is to find an inverse image of a hash function satisfying the certain condition, and thus the node should spend computational power to solve the cryptographic puzzle. The process of generating a block is called mining, and nodes participating in mining are called miners. In systems, the mining difficulty is adjusted to maintain the average time of generating one block. In particular, Bitcoin mining difficulty is adjusted to keep the average period of generating one block at 10 minutes. In addition, to incentivize mining, whenever a miner finds a valid block, the miner earns the reward for one block in compensation for the computational power spent. For example, currently, miners earn the block reward of 12.5 coins in the Bitcoin system when they find one block.

Many people have become involved in mining because of the incentive for mining, and specialized hardware for efficient mining such as application-specific integrated circuits (ASICs) has appeared. Based on the above reasons, the vast computational power is used for mining, and mining difficulty has increased significantly. Therefore, it should take a solo miner

, who mines alone, a significantly long time to find a valid block, and this causes solo miners to wait for a long time to earn block rewards. To reduce not only node costs and but also the variance of their rewards,

mining pools where miners gather together for mining have been organized. Most pools are composed of workers and a manager. The manager gives puzzles to workers, and they solve the puzzles. If a worker solves a given puzzle, the block reward is distributed to the workers in the pool.

In the past years, there have been many attacks on and problems with cryptocurrency systems, and these attacks or problems have even caused cryptocurrency systems to split. For example, because Bitcoin has become a popular cryptocurrency, the system needs to provide high transaction throughput. To address the scalability issue, several solutions such as Segregated Witness [segwit] and unlimited block size have been proposed. Because of the debate on the proposed solutions, Bitcoin was eventually split into BTC and BCH in early Aug. 2017. Even though BCH chose to increase the block size limit in order to allow more transactions per block, the mining protocol of BCH was designed to be compatible with that of BTC. Therefore, miners can conduct both BTC and BCH mining with one hardware device.

Ii-B Fickle mining

Before Nov. 13, 2017, BCH adjusted the mining difficulty every 2016 block to ensure that the average time period for generating a block is 10 minutes, like in the case of BTC. In doing so, if the time required for generating past 2016 blocks is longer than two weeks, the mining difficulty decreases, and miners can generate subsequent blocks more easily. In addition, BCH added a new difficulty adjustment algorithm called emergency difficulty adjustment (EDA) [eda] to decrease the mining difficulty without waiting for 2016 blocks to be generated when it is significantly difficult to find a valid block.

Because BTC and BCH have a PoW mechanism compatible with each other, miners can freely switch between them depending on the mining difficulty and the coin price. However, because the change in coin price is hard to predict, some miners immediately change their coin only when mining difficulty changes, where we call this behavior fickle mining. Concretely, the fickle miners first conduct BTC mining, observing the changes in the mining difficulties of BTC and BCH. Then, if the BCH mining difficulty is low, they immediately shift to BCH mining. When the BCH mining difficulty increases again thanks to its difficulty adjustment algorithm, fickle miners immediately shift to BTC mining. Fickle mining can boost profits of miners; however, this behavior might cause instability of both BTC and BCH.

This mining behavior was easily observed in Bitcoin when we monitored the mining power in pools. We collected mining power history data over the course of a week from two popular pools: ViaBTC [viabtc] and BTC.com [btc.com]. These two pools support both BTC and BCH mining; miners in the pools can choose either BTC or BCH mining by just clicking one button. Figure 1 represents the mining power data of ViaBTC and BTC.com for a week. In the figure, the grey regions show movements of mining power from BTC to BCH mining.

Figure 1: Mining power history of ViaBTC and BTC.com (Sep. 29, 2017 Oct. 6, 2017). The grey regions represent movements of mining power from BTC to BCH.
Figure 2: Mining power history of ViaBTC (Dec. 5, 2017 Dec. 8, 2017). Grey regions represent movements of mining power from BTC to BCH. Note that we only displayed the mining power history of ViaBTC because BTC.com did not evidently execute fickle mining for this period.

As fickle mining causes a sudden increase in mining power as shown in the grey zones of Figure 1, many blocks were generated quite quickly in the BCH system. For example, in the BCH system, 2016 blocks were generated within only three days in each grey zone. This caused the blockchain of BCH to be thousands of blocks ahead of BTC, and the halving time of the block reward in BCH was brought forward. To address this issue, BCH performed another hard fork on Nov. 13, 2017 [hardfork]. Currently, BCH adjusts the difficulty for each block based on the previous 144 blocks as a moving window [newdaa]. To determine if it is possible that miners conduct fickle mining even after the hard fork of Nov. 13, 2017, we investigated the BCH mining power data of ViaBTC for four days (Dec. 5, 2017 Dec. 8, 2017). Figure 2 represents the BCH mining power data of ViaBTC during this time period; as is evident from the figure, some miners still conduct fickle mining. Because the BCH mining difficulty is more quickly adjusted than before the hard fork of BCH, fickle miners should switch their mining power more quickly than before the hard fork. Indeed, fickle mining can occur in any mining difficulty adjustment algorithm.

Iii Related work

In this section, we review previous studies related to mining in PoW systems. Kroll et al. considered the Bitcoin mining process as a game among multiple players [kroll2013economics] and showed that a miner possessing 51% mining power can be motivated to disrupt the Bitcoin system. Several works [johnson2014game, laszka2015bitcoin] modeled and analyzed a game between two pools that can launch denial of service attacks against each other. Eyal and Sirer introduced the selfish mining strategy, where a malicious miner successfully mines blocks but does not immediately broadcast the blocks; instead, the attacker temporarily withholds the block [eyal2014majority]. Many researchers have intensively studied ways to optimize and extend selfish mining [sapirshtein2016optimal, nayak2016stubborn, gervais2016security, zhang2017necessity]. Bonneau introduced bribery attacks as a way for an attacker to increase her mining power [B16a]. Lewenberg et al. considered a mechanism of sharing rewards among pool miners as a cooperative game [lewenberg2015bitcoin]. In 2015, Eyal modeled a game between two pools that execute block withholding (BWH) attacks [eyal2015miner]. As a concurrent work, Luu et al. [luu2015power] modeled a power splitting game to find an optimized strategy for a BWH attacker. Kwon et al. [kwon2017selfish] proposed a new attack called a fork after withholding (FAW) attack against pools [kwon2017selfish]. Also, several works [carlsten2016instability, tsabary2018gap] analyzed a transaction-fee regime in PoW systems, where miners receive incentives for mining as transaction fees. Moreover, because many cryptocurrencies are competing with each other, there can be another incentive to execute 51% attacks. Considering this fact, Bonneau revisited the 51% attack with some basic analysis [bonneau2018hostile].

Recently, Ma et al. [ma2018market] considered a mining game of multiple miners and concluded that openness of the Bitcoin system causes the need for vast mining power. Another study [prat2018equilibrium] examined the relation between the Bitcoin/USD exchange rate and Bitcoin mining power. They first proposed an industry equilibrium model to forecast the mining power depending on the Bitcoin/USD exchange rate. Then, they showed that the real mining power data and simulated mining power according to their model are similar. Our study focuses on the relation between two coins that have compatible PoW mechanisms with each other and the miners’ behavior between two coins. Furthermore, our model can be used to forecast the ratio of mining power between two coins. To the best of our knowledge, this is the first to study the effects of fickle mining.

Iv Model

In this section, we formally model a game to represent fickle mining between two coins.

Iv-a Notation and assumptions

We consider two coins, and , which have compatible PoW mechanisms with each other. In this case, a miner with a hardware device can alternately conduct mining of and ; that is, he can conduct fickle mining between them. Meanwhile, a -faction can stick to -mining rather than fickle mining or -mining to maintain its own coin, and the set of -factions sticking to -mining is denoted by . For example, in the case where BCH is , BITMAIN [bitmain], one of the main supporters of BCH, may belong to . We aim to formalize a game considering the fickle mining and .

The proposed game consists of many players (i.e., miners), where the set of all players is denoted by Player chooses one of three strategies, : Fickle mining (), -only mining (), and -only mining (). The payoff function of player is denoted by which we will formally define later as well as fickle mining. We also define three sets , , and , indicating a set of players who conduct fickle mining, -only mining, and -only mining, respectively. Note that is a subset of because players in always choose strategy . The sum of mining powers in and is regarded as 1; mining power of a coin is expressed as a ratio to the total mining power. The mining power possessed by player is denoted by and the total computational power possessed by is denoted by We also define as the maximum of Moreover, because our game analysis result would depend on the computational power possessed by players, we use the notation to refer to the game, where

indicates a vector of computational power possessed by players except for

(i.e., ). Lastly, we denote the total mining power of , , and as (i.e., ), (i.e., ), and (i.e., ), respectively. Observe that and Namely, represents the full status of mining powers where is not less than .

For the analysis of the game, we assume the following:

Assumption 1.

A miner conducts either only or -mining (not both) at each time instance; for example, an ASIC miner cannot execute both BTC and BCH mining simultaneously. However, their choices can be time-varying; that is, miners can change their coin to mine.

Assumption 2.

The price of 1 is equal to that of . We assume that without loss of generality. In addition, rewards for mining a block in both coins are 1 and 1 , respectively.

Assumption 3.

In both and systems, mining difficulties are adjusted to maintain the average period of generating a block as the same specific time period, which we denote by 1 time and regard as a time unit; for example, 1 = 10 minutes in the Bitcoin system. Furthermore, we consider a generalized model in which mining difficulties of and are adjusted in proportion to the mining power for the previous time window, and we consider a normalized difficulty. Thus, if mining power has been engaged in coin mining, the mining difficulty would be More precisely, in our model, the coin mining difficulty decreases and increases again, considering the generation time of a specific number of blocks since the last update of coin mining difficulty. In particular, for the mining difficulty of we denote the number of considered blocks when the -mining difficulty decreases and increases as and , respectively.111In Section VI, we will show that our results can be applied to the coin system regardless of the mining difficulty adjustment algorithm of . Note that and cannot be zero. In the case of BTC and Litecoin, and are 2016.

As described previously, a fickle miner may change the preferred coin when the coin mining difficulty changes. Here we define fickle mining formally.

Definition IV.1 (Fickle mining).

Let and denote the and -mining difficulties, respectively. If or when or is updated, fickle miners () decide to conduct -mining until or is adjusted again. Otherwise, they conduct -mining.

We also emphasize that if is 0, no miner engages in fickle mining, and mining powers of and are stably maintained. On the other hand, if is , only -factions would conduct -mining after an increase in the mining difficulty of In other words, in this case, only the factions remain as loyal miners for Therefore, if the number of such factions () is small, the state would be a lack of loyal miners. Note that loyal miners refer to players who continue to conduct -mining even after an increase in -mining. In particular, if all -factions stop -mining for higher payoff (i.e., ), is 0, and no player conducts -mining after an increase in the mining difficulty of Note that the -mining difficulty cannot decrease in this case because cannot be zero. Therefore, the case indicates the complete downfall of while only survives.

Parameters used in this paper are summarized in Table I. The last parameter in the table will be introduced later.

Figure 3: Changes in the mining power of and , and mining difficulty of .

The set of -factions sticking to
mining to maintain their own coin

The set of all players

Player ’s strategy

Player ’s payoff

, ,

Fickle, -only, -only mining

, ,

The set of players with , ,

Computational power of player

Computational power possessed by

The maximum of

The vector of computational power
possessed by players in

The game of players and with
computational power and

The total computational power
fraction of , ,

The relative price of to

The time unit representing the average
period of generating one block

The number of considered past blocks when the
mining difficulty of decreases or increases

The mining difficulty of ,

The set of all Nash equilibrium in
Table I: List of parameters.

Illustration of fickle mining. Figure 3 illustrates a stream of mining power in and , as well as the mining difficulty of over time, caused by the strategies of players.
- Time : At the beginning, and mining powers are used for and -mining, respectively.
- Time : The mining difficulty of decreases because it is relatively difficult to find PoWs with

mining power. At the moment,

shifts from to , and each of and mining powers is used for and -mining, respectively.
- Time : Because the mining difficulty of is again adjusted (increases) after blocks are found in the system since the last adjustment of the mining difficulty of , the mining difficulty of would increase after time since it takes to find one valid block on average. Then, shifts again from to and conducts -mining until the mining difficulty of decreases.
- Time : Until when the mining difficulty of decreases after blocks are found in the system, would conduct -mining (for time).
- This process is continually repeated.

Iv-B Payoff function

Next, we describe payoff functions for our game model. All payoffs are expressed as a unit of and are calculated as a profit density, which is defined as an average earned reward for time divided by the player’s mining power. In other words, if player earns a reward for 1  time on average, the payoff would be Player ’s payoff function is expressed as follows:

(1)

where indicates other players’ strategies. Here, it suffices to define in the range , and respectively; for example, would be defined when (i.e, a fickle miner exists, and ).

First, we define the payoff for a player in . As shown in Figure 3, conducts -mining for   time. Therefore, a player in earns the profit per 1  time on average for   time. After that, conducts -mining for time during which a player in earns the following profit per 1 time on average:

(2)

The above formulation is due to the fact that mining powers and engage in -mining for   and times, respectively, and thus, the second factor in the right-hand side of (2) represents an inverse number of the mining difficulty of . Consequently, the payoff of a player in can be expressed as

where

Next, we provide payoffs and as follows:

where we observe that a player in earns the profit per 1 for time and profit per 1 for time, on average.

V Game analysis

In this section, we analyze Nash equilibria and dynamics in game

V-a Equilibrium in game

Characterization of equilibria. Before finding Nash equilibria of we define a pure Nash equilibrium.

Definition V.1 (Pure Nash equilibrium).

A strategy vector is a Nash equilibrium if

At an equilibrium, all rational players would not change their strategy, that is, and are not updated. We map a strategy vector to state and denote by the set of all Nash equilibria in We first determine the dynamics of player with small through Lemma V.1 to establish the characterization of

Lemma V.1.

There is such that, any player possessing does not change its strategy at state if and only if

where is a decreasing function of which input is and output ranges between 0 and Parameters and are defined in Assumption 2 and 3.

Note that is for a small value of while is 0 for a large value of The above lemma implies that, considering miners with small computational power, if a Nash equilibrium exists, only would remain as loyal miners to in the equilibrium. This is because would continually change when is greater than From Lemma V.1, we can characterize the set as stated in Theorem V.2. We present the proof of Lemma V.1 and Theorem V.2 in Appendix A.

Theorem V.2.

There is such that, when the set is as follows.

where

and range between 0 and 1.

As described above, Theorem V.2 shows that, in a game where players except for possess small computational power, there exist only Nash equilibria where the -factions sticking to -mining are loyal miners for . In the case where is small, we can certainly see that the overall health of the system would be weakened in terms of scalability, decentralization, and security, which will be discussed in more detail in Section VII-A. Indeed, even if is large, the case where is equal to would make the system significantly centralized because only a few players possessing large power are loyal miners to (this example is presented in Section VII-B). In particular, if is empty, no miner exists in the system in all Nash equilibria. Remark that this case indicates the complete downfall of As a result, Theorem V.2 implies that fickle mining can be dangerous.

When players possess infinitesimal mining power. Under the game , it is not easy to analyze movement of state (this movement will be used for data analysis in Section VII

) due to a large degree of freedom in

. Thus, we further assume that players except for (i.e., ) possess infinitesimal computational power (i.e., ). We show that this assumption is reasonable by analyzing the real-world dataset in the Bitcoin system (see Section VI). We again study the equilibria of in this case.

Theorem V.3.

When players except for possess infinitesimal mining power, the set is as follows.

(3)

Here, and are defined in Section V-B.

We present the proof of Theorem V.3 in Appendix B. Comparing with Theorem V.2, the state also becomes another Nash equilibrium when the computational power possessed by players (except for ) is infinitesimal. Note that this state indicates the stable coexistence of and Indeed, when is closer to 0, the difference among payoffs of players in , , and would also be closer to 0 at the state . Therefore, under the assumption that players possess infinitesimal power, payoffs of players in , , and are the same at the state while the mining difficulties of and are maintained as and , respectively. Meanwhile, at the remaining equilibria except for the state , only the -factions conduct -mining after the -mining difficulty increases. In particular, if no -faction sticking to -mining exists, loyal mining power to is 0 in the Nash equilibria. Note that, in this case, and would continuously conduct -mining, because the mining difficulty of has not decreased after the previous increase in difficulty. These players would not also change their strategy because the mining difficulty of increases to a significantly high value due to the heavy occurrence of fickle mining.

Example. Considering the case we give an example where and the initial mining difficulty of is 0.4. The state is not a Nash equilibrium according to Theorem V.3. Because fickle miners continuously conduct the -mining, the mining difficulty of is maintained as 1, and players in and earn the payoff of 1. If a player moves into , the player would earn for a while in the beginning. However, because the mining difficulty of decreases after finds several blocks, the player who moves to would eventually earn consistently. Note that the time duration in which the mining difficulty of is close to 0 is negligible compared to the time duration in which the mining difficulty of is 0.2. Therefore, the payoff of is and rational players tend to move to due to the higher payoff. This means that the state is not a Nash equilibrium.

V-B Dynamics in game

In this section, we analyze dynamics in the game and study how a state can reach an equilibrium.

Figure 4: Horizontal and vertical axes give the values of and , respectively, and -coordinates of vertices in zones are marked. At the vertex of and , is a solution of equation for . All points in , , and move in directions , , and , respectively.
Figure 5: Yellow points and line represent equilibria for each case.

Best response dynamics. In game , point reaches either of the two types of Nash equilibria: the stable coexistence of two coins and the lack of loyal miners to Figure 4 represents dynamics in game , where horizontal and vertical axes are and values, respectively. A line, , represents

(4)

On the line, the payoffs of (i.e., ) and (i.e., ) are the same. In addition, the line does not intersect with the line and has an intersection with the line for , where is a solution of equation for . The equation has only one solution , and it is between 0 and . Another line, , represents

(5)

and the payoffs of (i.e., ) and (i.e., ) are the same on the line. The line does not intersect with the line for and has an intersection with the line . Moreover, it is most profitable among the three strategies to continually conduct -mining () in a zone above . We let this zone be . In the zone below , it is most profitable to continually conduct -mining (), and the zone is denoted as . In the zone between and , fickle mining () is the most profitable, and this zone is denoted as . Note that the range of zones changes if the coin price changes because boundaries are functions of

The moving direction of point is expressed as a red arrow in Figure 4. For ease of reading, we express directions in which values and increase () or decrease () as . For example, indicates the direction in which both values, and , increase. In , is the most profitable strategy, and thus every point in moves in the direction . In , because is the most profitable strategy, every point moves in the direction . Finally, in , as is the most profitable strategy, every point in moves in the direction . Figure 4 shows the directions in the three zones (, , and ).

2D-Illustration of movement towards equilibria. To determine which equilibrium can be reached within each zone, we represent all Nash equilibria in game depending on a value of as yellow points and line in Figure 5. In the figure, the red dash lines represent for each case. As described in Section V-A, there are two types of equilibrium points: 1) a lack of loyal miners and 2) stable coexistence of two coins. The equilibrium point representing a lack of loyal miners would be located on a red dash line , and we can see that all cases have this equilibrium. For Cases 1, 2, and 3, the second type of equilibrium (i.e., ) representing stable coexistence of two coins is also found. A point moves in the direction depending on its zone. In the meantime, if the point meets the line then the point moves toward an equilibrium located on the line as shown in Figure 5. In particular, the value of in the equilibrium on the red dash line representing Case 3 is denoted by , where the equilibrium is the intersection point between and the red dash line. Note, a point in would not meet a red dash line because the point in moves in the direction and can always be above the red dash line. Therefore, such points in are likely to reach the stable coexistence of and However, some points (near to ) in can also move into when more miners of than that of revise their strategies, and then it is possible to reach the equilibrium, representing a lack of loyal miners to .

Vi Application to Bitcoin System

In this section, we apply our game model to Bitcoin as a case study. Specifically, we consider game when players possess sufficiently small mining power. To see if this assumption is reasonable, we investigate the mining power distribution in the Bitcoin system, referring to the power distribution provided by Slush [slush]. The distribution is depicted in Figure 6 where the -axis represents the range of the relative computational power and the -axis represents the number of miners possessing computational power in the corresponding range. The figure shows that 1) most miners possess sufficiently small mining power, and 2) even the maximum computational power is less than Note that BITMAIN’s is about as of Dec. 2018. Moreover, even though mining pools currently possess large computational power, the miners in pools can individually decide which coin to mine. We also recognize the distribution of computational power is significantly biased toward a few miners, as shown in Figure 6. However, this fact does not imply that is large. Referring to the data provided by Slush, is only about 0.05, where this value is equivalent to that for the case where all miners possess computational power.222We calculated this assuming that other pools have the computational power distribution similar to Slush. Therefore, most miners (and most mining power) would follow dynamics of game . As a result, we can apply game to the practical systems.

Figure 6: The computational power distribution in Slush.

Now, we describe how game is applied to the Bitcoin system. As described in Section II, Bitcoin was split into BTC and BCH in Aug. 2017. Thus, we can map BTC and BCH to and , respectively. For the mining difficulty adjustment algorithm of BCH, we should consider two types of BCH mining difficulty adjustment algorithms: those that BCH have before and after Nov. 13, 2017. This is because the mining difficulty adjustment algorithm of BCH changed through a hard fork of BCH (on Nov. 13, 2017).

Before Nov. 13, 2017. First, we consider the mining difficulty adjustment algorithm of BCH before Nov. 13, 2017. In this algorithm, not only the mining difficulty is adjusted for every 2016 block, but also EDA can occur as described in Section II. Note that EDA occurs if the mining is significantly difficult in comparison with the current mining power, i.e., EDA is used only for decreasing the BCH mining difficulty. Therefore, the value of is 2016 because the BCH mining difficulty can increase after 2016 blocks are found. Meanwhile, when the BCH mining difficulty decreases, the value of varies depending on and , ranging between 6 and 2016. Thus, we can consider the expected number of blocks found until the mining difficulty decreases (i.e, the mean of denoted by ) instead of , and as a function of and would continuously vary from 6 to 2016. If is 0, is 2016 because EDA does not occur, and if is 0, is 6.

As a result, the Bitcoin system before Nov. 13, 2017 can be where substitutes for This game has also Nash equilibria and dynamics as shown in Figure 4 because is a continuous function of and

After Nov. 13, 2017. Next, we consider the Bitcoin system after Nov. 13, 2017. In this case, the BCH mining difficulty adjustment algorithm is different from that assumed in our game because the mining difficulty is adjusted for every block by considering the generation time of the past 144 blocks as a moving time window. Despite that, game can be applied to this system. Indeed, in general, our results for game would appear in the Bitcoin system regardless of the BCH mining difficulty adjustment algorithm, shown below.

Theorem VI.1.

Consider the game when Then when the mining difficulty of is adjusted every block or in a short time period, the set is (V.3) presented in Theorem V.3. In addition, under this mining difficulty adjustment algorithm of has dynamics such as in Figure 4.

Because the current BCH mining difficulty is adjusted every block, Theorem VI.1 implies that results for game is also applied to the current Bitcoin system even though the BCH mining difficulty adjustment algorithm changed. The proof of Theorem VI.1 is presented in Appendix C.

(a)
(b)
(c)
(d)
Figure 7: The data for the Bitcoin system from early Aug. 2017 to Dec. 2018 is represented. Figure 7a, 7b, and 7c represent (a) relative mining power of BCH to the total mining power, (b) the ratio between mining difficulties of BCH and BTC, (c) the ratio between prices of BCH and BTC, and BCH mining profitability. Figure 7d shows the data for mining power, price, and mining difficulty of BCH. In the gray zones, fickle miners conduct BCH mining. The data from Dec. 2017 to Nov. 2018 are omitted because they are similar to the data for Dec. 2018. Each point represents a data captured every hour.
(a) Figure 7-(1)
(b) Figure 7-(2)
(c) Figure 7-(3)
(d) Figure 7-(4)
(e) Figure 7-(5)
(f) Figure 7-(6)
(g) Figure 7-(7)
(h) Figure 7-(8)
(i) Figure 7-(9)
Figure 17: Points and movements of Figure 7. Figure (a)a (i)i correspond to parts (1)(9) in Figure 7. Red arrows represent movement in agreement with our model, whereas black arrows represent movement deviating from our model. Each upper right square presents enlarged points and directions.

Vii Data analysis

Vii-a BTC vs. BCH

We analyze the mining power data in the Bitcoin system to identify to which equilibrium the state has been moving. Moreover, through this data analysis, we can find out empirically how much our theoretical model agrees with practical results. For data analysis of the Bitcoin system, we collected the mining power data of BTC and BCH from the release date of BCH (Aug. 1, 2017) until the time of writing (Dec. 10, 2018) from CoinWarz [coinwarz]. Figure 7a represents the mining power history of BCH, where the mining power is expressed as a fraction of the total power in BTC and BCH, i.e.,

In addition, we represent the data history of a ratio between difficulties of BCH and BTC (i.e., ) and a relative price of BCH to that for BTC (i.e., ) in Figure 7b and 7c, respectively. The price of BCH is depicted as a yellow line in Figure 7c (see the left -axis). Moreover, Figure 7c represents the relative BCH mining profitability () to the BTC mining profitability as a purple line, and the black dashed line represents (see the right -axis for the two lines). For this profitability, to increase reliability of data, we collected the daily BCH profitability from CoinDance [coindance], and thus a purple point is a data captured every day. Note that is less than in the case where the purple line is above the black dashed line. Figure 7d simultaneously shows all data histories (except for the BCH mining profitability) presented in Figure 7a7c. In Figure 7, the data from Dec. 2017 to Nov. 2018 are omitted because they are similar to the data for Dec. 2018. Figure (a)a(i)i correspond to parts (1)(9) of Figure 7, respectively, where the area of three zones has changed because the relative price of BCH to that for BTC has fluctuated quite frequently.

As another case study, we examine the mining power data of Bitcoin ABC and Bitcoin SV from Nov. 1, 2018 to Dec. 20, 2018 to analyze a special situation where suddenly increases due to the “hash war” caused by a hard fork in the BCH system. We describe this in Section VII-B.

Methodology. We first describe how to determine and of each state. According to the definition of fickle mining (Definition IV.1), fickle miners would conduct BCH mining from when changes to a value less than to when changes to a value greater than This is because is always less than and greater than (see Figure 7d). Therefore, Figure 7a represents the value of during the period. We indicate the fickle mining periods in gray before the hard fork of BCH (Nov. 13, 2017) in Figure 7. Figure 7d shows that changes to a value less than and greater than at the start and end of these periods, respectively. As a result, in Figure 7a, we can find out the value of for the gray colored periods and the value of for non-colored periods. Here, we can see that the mining power of BCH has fluctuated considerably when the ratio of the BCH mining difficulty to the BTC mining difficulty () changes to a value less than . Moreover, when the coin mining difficulties do not change while BCH mining is more profitable than BTC mining, large peaks (i.e., a sudden increase) do not appear. This fact is confirmed, referring to the purple line in non-colored zones (e.g., part (3) in Figure 7c). As a result, we can consider that those fluctuations occur due to fickle miners between BTC and BCH.

If a miner switches the coin to mine without changes in the coin mining difficulty, this implies that the miner’s strategy changes (e.g., from to ). From the method described above, we can determine the mining power used for fickle mining and the mining power used for BCH-only mining. The points and directions are marked roughly in Figure 17. The red arrow represents movement in agreement with our analysis, whereas the black arrow represents movement deviating from our analysis.

Next, we explain Figure 17 by matching it with each part of Figure 7.

The beginning of the game. In Figure 7-(1), the status point is initially in , and then it moves to as shown in Figure (a)a, as the BCH mining power decreases.

Towards the lack of BCH loyal miners. In Figure 7a-(2), two peaks occur when the BCH mining difficulty decreases to values less than and these peaks appear in the gray colored periods. Therefore, we can know that these peaks occur due to fickle miners. The first peak indicates that more and more miners started fickle mining (i.e., increase in ). This is because the upflow of the first peak is less steep than that for other peaks, and the downflow of the first peak is steeper than the upflow of the first peak, indicating that increases from near 0 up to near 0.4. Furthermore, one can see that increased at the beginning of Figure 7a-(2). Remark that Figure 7a shows the value of in a non-colored zone. In addition, the BCH mining power in the valley between two peaks of Figure 7a-(2) is greater than the mining power at the end of Figure 7a-(1). This fact shows again that increased at the beginning of Figure 7a-(2). After that, because the end of Figure 7a-(2) is less than the valley between the two peaks of Figure 7a-(2), we can know that decreased while increased in Figure 7a-(2). Figure (b)b represents these movements described above.

In the beginning of Figure 7a-(3), slightly increases, and it does not correspond with our model; we regard this as a momentary phenomenon because of a decrease in the BCH mining difficulty. Figure 7b shows that the BCH mining difficulty decreased at the beginning of the part (3). However, even though the BCH mining difficulty decreased, peaks due to fickle mining do not appear because