# Diversification across mining pools: optimal mining strategies under PoW

## Abstract

mining is a central operation of all proof-of-work ( PoW ) -based cryptocurrencies. The huge majority of miners today participate in “ mining pools ” alternatively of “ solo mine ” in order to lower risk and achieve a more steady income. however, this advance of participation in mining pools negatively affects the decentralization levels of most cryptocurrencies. In this solve, we look into mining pools from the point of opinion of a miner : We present an analytic mannequin and implement a computational tool that allows miners to optimally distribute their computational exponent over multiple pools and PoW cryptocurrencies ( i.e. build a mine portfolio ), taking into report their gamble aversion levels. Our joyride allows miners to maximize their risk-adjusted earnings by diversifying across multiple mining pools. Our underlying techniques are drawn from both the areas of fiscal economy and calculator skill since we use computer science-based approaches ( i.e. optimization techniques ) to experimentally prove how parties ( and in particular miners ) interact with cryptocurrencies in a room of increasing their Sharpe proportion. To showcase our model, we run an experiment in Bitcoin historic data and demonstrate that a miner diversifying over multiple pools, as instructed by our model/tool, receives a higher overall Sharpe ratio ( i.e. median excess reinforce over its standard deviation/volatility ) .

## Introduction

The majority of cryptocurrencies use some character of proof-of-work ( PoW ) -based consensus mechanism to decree and finalize transactions stored in the blockchain. At any given time, a set of users all over the global ( called miners or maintainers ) competes in solving a PoW puzzle that will allow them to post the adjacent block in the blockchain and at the same time claim the “ Coinbase ” reward and any relevant transaction fees. In the early years of cryptocurrencies solo mine was the average, and a miner using his own hardware would attempt to solve the PoW puzzle himself, earning the reward. however, as the exchange rate of cryptocurrencies increased, the PoW competition became boisterous, specialized hardware was manufactured just for the aim of mining particular types of PoW ( e.g. Bitcoin or Ethereum mining ASICs [ 1 ] ), and finally users formed coalitions for better chances of solving the perplex. These coalitions known as “ mine pools, ” where miners are all continuously trying to mine a block with the “ pool coach ” being the reward recipient role, enabled participating users to reduce their mining risks. 1 After the establishment of mine pools, it became about impossible for “ solo ” miners to compete on the mining game, even if they were using specialized hardware, or else they could risk not to earn any rewards at all during the hardware ’ second life.

The survival of a mine pool is not a superficial undertaking. A large number of pools exist each offer different advantage distribution methods and earning fees ( as we further discus in the “ Reward methods in mine pools ” section ). At the same time different pools control a different ratio of the overall hashish rate consumed by a cryptocurrency and larger pools ( in terms of hash pace ) extend lower risk, as they typically offer more patronize payouts to the miners. But how can a miner make an optimum decision about which mine pools to participate in and for which cryptocurrencies at any given time considering the assortment of potential options ?

### Our contributions

We present an analytic tool that allows risk-averse miners to optimally create a mine portfolio that maximizes their risk-adjusted rewards. We characterize miners by their entire computational resources ( i.e. hash power ) and their gamble antipathy level, and mine pools by their total computational world power ( i.e. hashish pace ) and the reward mechanism they offer. We model the hash rate allotment as an optimization problem that aims to maximize the miner ’ south expected utility. In the “ Active Miner ’ second Problem ” section, we provide three different versions of our model. The foremost one, inspired by [ 3 ], concerns a miner who wishes to mine on a single cryptocurrency, while aiming to diversify among any act of mining pools ( including the solo mining choice ). The second version captures miners who diversify across different cryptocurrencies that use the like PoW mining algorithm. In our one-third interpretation, we model miners who besides wish to diversify across cryptocurrencies with different PoW mine algorithm. Our model technique is based on standard utility maximization, and extends the Markowitz advanced portfolio theory [ 4 ] to multiple mine pools, preferably than multiple assets. In the “ execution of Our Model ” section, we present an execution of our model. We develop a Python tool that uses the stiffen optimization by analogue approximation ( COBYLA ) [ 5 ] method to automate the pool distribution for an active miner. A miner can use our tool by providing as input its own mining ability ( for any PoW character ) angstrom well as his risk aversion rate and any act of pools he wishes to take into history when computing the optimum distribution of his mining power. As expected, we observe that for “ reasonable ” values of risk aversion level, the miner would generally allocate more of his resources to pools offering large hash world power combined with little fees, without however neglecting early pools that are not as “ lucrative. ” finally, to illustrate the utility of our joyride, we run an experiment on Bitcoin historical data ( see the “ Evaluation and Simulated Results ” incision ). We start by considering a Bitcoin miner who starts mining passively on a unmarried choose pool for 4 months and computes his earnings on a day by day footing. then, we consider a miner with the same hash power who using our joyride, would “ actively ” diversify every 3 days over three Bitcoin pools and reallocate his hash world power accordingly. In our experiment we observe that the “ active ” miner improves his reward over risk ratio ( Sharpe ratio [ 6 ] ) by 260 % compared with the “ passive ” miner. In our experiments, for the “ passive ” miner we picked a big and reputable pool ( SlushPool ) while for the “ active ” miner we added one more pool of equivalent size and fee structure ( ViaBTC ), arsenic well as a lower fee smaller consortium ( DPool ) for a better illustration. note that there are respective degrees of exemption in our experiments ( i.e. time periods, typeset of pools selected and so forth ). frankincense, we include replications with different parameters ( time menstruation, pools, hazard aversion, miner ’ randomness baron, and frequency of diversification ) to show how each one of them can affect the end result.

### Related work

There exist a few prevail tools that on inputting a miner ’ south hashing ability indicate which cryptocurrency is presently most profitable. For example, tools like MultiPoolMiner [ 7 ], SmartMine [ 8 ], or MinerGate [ 9 ] startle by benchmarking the CPU/GPU of the miner, ( which we consider an orthogonal service to the third version of our model ) and then suggest a cryptocurrency that would offer the best wages at that time. To make that decision, they look into respective cryptocurrencies ’ parameters ( i.e. block fourth dimension, reward etc. ) and the current difficulty ( their exact model is ill-defined ). Which pools the miner will use toward mining the indicate cryptocurrency is either hard coded by the joyride, or chosen by the miner. Our method differs from such tools in respective aspects. Most importantly, in our model we take the gamble antipathy rate of the miner into report, which is an crucial gene when making fiscal decisions. furthermore, we focus on allocating a miner ’ mho might over multiple pools, as opposed to barely different cryptocurrencies, which can benefit the decentralization within a cryptocurrency. Miner ’ south hazard antipathy was taken into report by Fisch et aluminum. [ 10 ] who provided a top-down ( from pool ’ s sharpen of view ) analysis, i.e. focused on optimum pool operation strategies toward maximizing the pool ’ mho expected utility. Cong et aluminum. [ 3 ] besides took hazard aversion into report in their model ; however, the stress of their solve was different from ours. In especial, they focus on the interaction between miners and pools. They first base demonstrate the significant risk-diversification profit offered by mine pools for individual miners, highlighting risk sharing as a natural centralize coerce. then, they demonstrate that the risk-sharing benefit within a bombastic pool could be alternatively obtained through miner diversification across multiple small pools. last, they present an chemical equilibrium model where multiple pool managers compete in fees to attract customer miners. In our work, we focus on the miners ’ side : we develop tools to help miners diversify among different pools and cryptocurrencies to maximize their risk-adjusted earnings. Because we emphasize on hash rate allocations across multiple mining pools, either within the same or over different cryptocurrencies, our analysis distinguishes from contemporaneous works such as [ 11 ], who present an economic model of hash power allotment over different cryptocurrencies sharing the lapp PoW algorithm in a Markowitz manner. This position besides sets us apart from [ 12 ] who study mining across different currencies in a strategic manner, without accounting for pool mine. finally, some holocene works examined the case where miners change the mine pool they mine with, in order to optimize their rewards from a network performance telescope ( communications delay ). Lewenberg et alabama. [ 13 ] showed how network delays incentivize miners to switch among pools in order to optimize their payoffs due to the nonlinearity introduced. Liu et alabama. [ 14 ] discussed how to dynamically select a mine pool taking into history the pool ’ s calculation office ( hashish rate ) and the network ’ second generation delay. Network operation is an authoritative aspect when diversifying across pools, and we view [ 13, 14 ] as complementary color to our employment.

## Mining Background

As of today, the majority of blockchain-based cryptocurrencies use PoW for maintaining their daybook. Miners listen the network for ( one ) pending transactions and ( two ) new blocks of transactions to be posted on the daybook. The function of a miner is to select a subset of pending transactions, assemble them to a raw block and perform computational work toward finding a random time being radius that will make the blocking valid and allow the miner to append it on the blockchain and frankincense win the reinforce. The brute-forcing action of finding a desirable time being gas constant such that together with the rest of the blockage contents b satisfies the property H ( b||r ) < T for some target value T is called “ mining. ” For Bitcoin, this translates to finding a desirable hash pre-image using the double SHA256 hash function. note that, although solving PoW puzzles was initially done using ordinary CPUs, the increasing prices of cryptocurrencies have resulted in a “ hardware race ” to develop the most effective mining hardware using application-specific incorporate circuits ( ASICs ), designed to perform SHA256 hashing operations in many orders of magnitude faster than CPUs or GPUs.

### Mining pools

mine is a random process, in most cryptocurrencies the computational world power devoted to solve the PoW perplex is very high, which implies a eminent division on the miner ’ s wages. In Bitcoin, flush for a miner using state of the art ASICs, there is a estimable probability that he never gets a block mined during the hardware ’ sulfur life. This led to the formation of mining pools, where coalitions of miners are all continuously trying to mine a barricade, with the “ pool coach ” being the honor recipient. If any of the participating miners finds a solution to the PoW puzzle, the consortium coach receives the block reward R and distributes to the participants, while possibly keeping a small cut ( or fee degree fahrenheit ). The block reward R distribution is based on how much workplace these miners performed. A method acting for the consortium coach to measure how much effort each miner has put into the pool is by keeping a record of shares, which are “ near solutions ” to the PoW puzzle ( or “ near-valid ” blocks ), satisfying the property Ts < H ( b||r ) < T where Ts is the “ share difficulty. ” There are several methods to distribute the advantage R to the miners, which we analyze late. ( In Table 1 : we provide a referece for note used throughout our influence ).

### Reward methods in mining pools

different pools offer slenderly different reinforce methods ( or a mix of them ), with the most popular being : pay per parcel ( PPS ), proportional, and pay per stopping point N shares ( PPLNS ). In the PPS reward method, the miners are not immediately paid when a block is found ; however, each block honor is deposited into a “ central pool fund ” or “ bucket. ” The miners are paid proportionately to the shares submitted throughout their engagement in the pool, regardless of if and when the pool has found a solution to the PoW puzzle. PPS is broadly considered to offer a firm, about guarantee income, independent of the pool ’ second “ luck ” finding a block. In the proportional wages method, whenever a pool solves the perplex for a fresh jam, the fresh obstruct reward is distributed to the pool ’ s miners proportionately to the number of shares each miner has submitted to the pool for that particular stop. This method acting however was found to be vulnerable to the “ pool-hopping ” attack, where the miners could exploit the pond ’ second expected earnings, division and maturity prison term and “ hop aside ” to another consortium or solo mining when the pool ’ south attractiveness is abject [ 15 ]. The PPLNS method acting was implemented to counter this attack, where the miners ’ reward is distributed according to the “ holocene ” number of render shares, thus invalidating shares submitted early on. As shown in Appendix 3, the PPLNS method in mine pools is the most popular honor method nowadays. In addition, many other PPLNS variants are presently being used by mining pools, e.g the RBPPS ( round-based yield per share ) method acting where the pool only pays the reinforce after the stuff finally gets confirmed by the network ( frankincense excluding deprecated blocks ). We refer the reader to [ 16 ] and [ 15 ] for exhaustive and complete analysis of pool advantage methods. In our set, we largely consider pools that offer a PPLNS wages method ( or its variants ). We by and large do not consider pools that use the PPS method, as the miners ’ expected earnings do not depend on the variability of finding blocks.

#### Mining pools offering multiple reward methods

As mentioned earlier, some mining pools offer multiple reward systems ( i.e. the Coinotron consortium [ 17 ] offers both PPS and PPLNS ). We study these types of pools individually, as some miners might opt for different fee contracts within the same pool. Let a mine pool thousand with total hash pace Λm, offering both PPLNS and PPS reward systems to choose from, where z is the share of pool ’ s hashing office paid using a PPLNS fee contract, and ( 1 − z ) is the percentage paid using a PPS fee abridge. besides let λm be a miner ’ mho hashing power allocated to pool m and | $R_ { { \lambda _ { megabyte } } }$ | the miner ’ s wages when a forget is finally mined by the pool bringing a total reward R. The pool director should make sure to keep paying its PPS miners at a steadily rate, compensating for any pool “ luck ” fluctuations in any given time period when trying to find a block. To achieve this, the director needs to maintain a “ bucket ” containing an adequate come of coins, and keep replenishing it with | $R_ { \mathsf { PPS } }$ | ( i.e. reward of PPS ) each prison term a block is “ mined ” by the pond with blockage honor R, to keep paying PPS miners during periods of bad “ luck. ” Consequently, when the pool jointly “ mines ” a block with reward R, the pool director can select one of the follow three miner payment strategies, which besides determine the accurate PPLNS miners ’ reward .

• Strategy 1: The pool coach splits R into | $R_ { \mathsf { PPS } } = ( 1-z ) R_ { }$ | and | $R_ { \mathsf { PPLNS } } = zR_ { }$ |⁠. By this scheme, | $R_ { \lambda _ { m } } = \frac { \lambda _ { megabyte } } { z\Lambda _ { megabyte } } zR_ { } = R_ { } \frac { \lambda _ { thousand } } { \Lambda _ { megabyte } }$ |⁠. In other words, the miner having contributed a hash rate of λm will get a reward based on the percentage of the hashish rate he contributed with deference to the sum hash rate of PPLNS miners, multiplied by | $R_ { \mathsf { PPLNS } }$ | ( i.e. reward of PPLNS ) .
• Strategy 2: The consortium director pays the PPLNS miners based on the total hashish rate of the pool Λm, and then allocates the end of the rewards to the PPS bucket. By this strategy, | $R_ { \lambda _ { megabyte } } = R_ { } \frac { \lambda _ { m } } { \Lambda _ { megabyte } }$ |⁠, which effectively results in the same nonrecreational total as in Strategy 1 .
• Strategy 3: First, replenish the PPS “ bucket ” based on the total measure | $\tilde { R }$ | that was paid away to the PPS miners since the last block was found, and then pay PPLNS based on what is left of the total reward. Following this strategy, | $R_ { \mathsf { PPLNS } } = R_ { \ } – \tilde { R_ { } }$ | and miner ’ s reward is | $R_ { \lambda _ { molarity } } = R_ { \mathsf { PPLNS } } \frac { \lambda _ { m } } { z\Lambda _ { megabyte } } = ( R_ { } – \tilde { R_ { } } ) \frac { \lambda _ { meter } } { z\Lambda _ { megabyte } }$ |⁠. effectively, the pool coach by this scheme transfers some of his hazard to the PPLNS miners .

To our cognition, no mining pool that offers both PPLNS and PPS reward systems specifies which scheme it follows. Using public data to prove which scheme a mining consortium follows is a nontrivial serve. We assume that pools offering both PPS and PPLNS reward mechanisms follow either Strategy 1 or 2, which are the most intuitive and produce the like end leave for the miners.

## Active Miner’s Problem

We study the problem faced by a miner, who given a set of | $\mathsf { C }$ | different cryptocurrencies and M mine pools for each currentness, where each pool thousand has a sum hashish pace Λm, coke and fee frequency modulation, hundred, maximizes his expected utility. In the rest of the section, we first characterize the miner ’ s payoff, then specify his optimization problem, and last derive a amenable version under constant-absolute-risk-aversion ( CARA ) utilities for subsequently numeral analysis. We besides discuss how miners with more general utilities may make use of our model.

### Miner’s payoff

| $\lambda _ { \mathsf { A } }$ | and is mining on | $\mathsf { C }$ | different cryptocurrencies, each with total hash rate Λc. The miner can distribute | $\lambda _ { \mathsf { A } }$ | among different cryptocurrencies and different mining pools that offer different fee structures (while possibly keeping a portion of his power for zero-fee solo mining). Therefore the miner’s payoff | $\mathsf { P }$ | is given by the following equation.  \begin { eqnarray * } \tilde { \mathsf { P } } & = & \sum _ { c\in \mathsf { C } } \Bigg ( \sum _ { m=1 } ^ { M_c } \underbrace { \frac { \lambda _ { megabyte, c } } { \lambda _ { megabyte, coke } + \Lambda _ { megabyte, c } } } _ { \text { within-pool hashrate share } } \\ & & \times\ ; \underbrace { ( 1-f_ { thousand, c } ) ( R_c/D_ { deoxycytidine monophosphate } +\mathsf { texas } _ { carbon } \mathsf { texas } _ { thousand, c } ) \tilde { N } _ { pool, meter, c } } _ { \text { Pool ‘s total reinforce to miners } } \\ & & +\ ; \underbrace { ( R_c/D_ { degree centigrade } +\mathsf { texas } _ { vitamin c } \mathsf { texas } _ { thousand, c } ) \tilde { N } _ { solo, c } } _ { \text { solo reward } } \\ & & +\ ; \underbrace { ( R_c/D_ { hundred } +\mathsf { texas } _ { c } \mathsf { texas } _ { megabyte, c } ) \frac { \lambda _ { \mathsf { PPS }, hundred } ( 1-f_ { \mathsf { PPS }, speed of light } ) } { \Lambda _c } } _ { \text { PPS pools advantage } } \Bigg ) \end { eqnarray * }  ( 1 ) A miner owns mining hardware with PoW hashing powerand is mining ondifferent cryptocurrencies, each with entire hashish rate Λ. The miner can distributeamong different cryptocurrencies and different mining pools that offer different tip structures ( while possibly keeping a parcel of his world power for zero-fee solo mine ). Therefore the miner ’ second payoffis given by the keep up equality. The above wages for each cryptocurrency c includes the follow terms : ( one ) the slant sum of each pool molarity ’ s wages, with the weights being the miner ’ s hash rate share in the pool, where λm, c denotes the mining office allocated to pool m in cryptocurrency degree centigrade, Λm, hundred and Rc denote the full hash rate of pool meter and the block reward of cryptocurrency c, respectively, | $\mathsf { texas } _ { c }$ | denotes the average transaction fee for cryptocurrency c observed during a late period of time, | $\mathsf { texas } _ { thousand, c } = 1$ | if the pool pays transaction fees to the miner, else we set | $\mathsf { texas } _ { molarity, cytosine } = 0$ |⁠, fermium, c the pond fee share ( which is subtracted from the miner ’ s bribe ), and | $\tilde { N } _ { consortium, m, speed of light } \sim { Poisson } ( \frac { \lambda _ { thousand, speed of light } +\Lambda _ { molarity, c } } { \Lambda _c } )$ | denotes the ( random ) number of blocks pool m for cryptocurrency cytosine mines. bill as each cryptocurrency speed of light has its own block wages Rc 2 and its own average block time Dc, we consider the reward over time proportion | $\frac { R_ { c } } { D_ { c } }$ |⁠, as it effectively normalizes Rc over different cryptocurrencies. ( two ) The reinforce from solo mine, where | $\tilde { N } _ { solo } \sim Poisson ( \frac { \lambda _ { solo } } { \Lambda } )$ | denotes the ( random ) total of blocks a solo miner mines. ( three ) The honor from a PPS pool in cryptocurrency cytosine, where Λm, c denotes the total hash rate of that cryptocurrency.

### Miner’s allocation problem

), a miner with a utility function u( · ) and initial wealth W0 would solve the following problem. Choose vector {λm,c} to maximize

(2)

where | $\tilde { \mathsf { P } }$ | is given by Equation ( ), subject to  \begin { eqnarray * } \sum _ { coke } \sum _ { m=1 } ^ { M } \lambda _ { megabyte, deoxycytidine monophosphate } + \lambda _ { PPS, vitamin c } + \lambda _ { solo, degree centigrade } \le \lambda _ { \mathsf { A } } \ \text { and } \ \lambda _ { meter, c } \ge 0, \forall m\in M. \end { eqnarray * }  Given the expression of miner ’ s bribe in Equation ( 1 ), a miner with a utility function u ( · ) and initial wealth Wwould solve the following problem. Choose vector { λ } to maximizewhereis given by Equation ( 1 ), subject to We can promote generalize to allow a miner to distribute his power across | $\mathsf { C }$ | unlike cryptocurrencies, and across | $\mathsf { A }$ | different PoW algorithm, which of naturally assumes that the miner owns CPU/GPU mine hardware, since ASICs are restricted to specific PoW algorithm. We discuss this case in more detail in Appendix 1.

### Analytical forms under CARA utility

For general utility functions, Equation ( 2 ) does not have an analytic expression, rendering foster analysis difficult. One exception is with CARA utilities, where Equation ( 2 ) can be obtained as the moment-generating routine of the weighted total of freelancer Poisson random variables.

(3)

where the CARA parameter ρ quantifies how risk averse a miner is (e.g. ρ = 0 means that the miner is risk neutral). Then notice that for a Poisson distributed variable x with parameter λ, its moment generating function E[ewx] for any parameter w is given by | $e^ { \lambda ( e^w-1 ) }$ |⁠. Therefore, if we plug Equation ( ) into Equation ( specifically, suppose the miner has CARA utility u ( · ) aswhere the CARA parameter ρ quantifies how risk averse a miner is ( e.g. ρ = 0 means that the miner is risk neutral ). then notice that for a Poisson distributed variable star ten with argument λ, its consequence generating function E [ e ] for any parameter west is given by. therefore, if we plug Equation ( 1 ) into Equation ( 3 ), the miner ’ s objective function can be expressed as certainty equivalents in analytic forms. The most cosmopolitan one will be given in Equation ( 6 ) late. Before presenting the most general case, we besides highlight a few special cases that will be far implemented in the “ implementation of Our Model ” department .

#### Mining on a single cryptocurrency without PPS pools

, and the miner looks for a vector | ${ \lbrace \lambda _ { thousand } \rbrace } _ { m=1 } ^ { M }$ | to maximize it. This vector expresses an allocation of the miner’s mining power | $\lambda _ { \mathsf { A } }$ | over M different pools, where λm denotes the mining power allocated to pool m  \begin { eqnarray * } & & \sum _ { m=1 } ^ { M } ( \lambda _ { m } + \Lambda _ { m } ) ( 1 – e^ { -\rho R_ { } ( 1-f_ { m } ) \frac { \lambda _ { meter } } { \lambda _ { meter } + \Lambda _ { molarity } } } ) \\ & & \quad+\ ; \left ( \lambda _ { \mathsf { A } } – \sum _ { m=1 } ^ { M } \lambda _ { megabyte } \right ) ( 1-e^ { -\rho R_ { } } ) \end { eqnarray * }  ( 4 )  \begin { eqnarray * } \sum _ { m=1 } ^ { M } \lambda _ { thousand } \le \lambda _ { \mathsf { A } } \ \text { and } \ \lambda _ { m } \ge 0, \forall m\in M. \end { eqnarray * }  If a miner alone mines in one crytocurrency and alone allocates hashish rates to mine pools offering a PPLNS reward method, the miner ‘s certainty equivalent is given by Equation 4, and the miner looks for a vectorto maximize it. 3 This vector expresses an allotment of the miner ‘s mine powerover M different pools, where λdenotes the mine power allocated to pool munder constraints By solving this optimization problem, we are given the optimum distribution of the miner ’ south sum hash ability | $\lambda _ { \mathsf { A } }$ | to M pools. note that the second term of Equation ( 4 ) expresses the leftover hash power for the miner to mine “ solo ”. If the miner is risk neutral ( i.e. ρ = 0 ), solo mining ( which has a zero fee ) is the optimum solution.

#### Allowing selection of PPS pools

) restricts the miners to choose between pools that only offer the PPLNS reward method. We now allow miners to choose mining pools also offering PPS reward systems in our model. In this case, a rational miner will choose to add only the PPS pool that offers the smaller fee, and disregard pools with higher PPS fees. Equation ( ) is then transformed as follows (changes denoted in blue color):  \begin { eqnarray * } & & \sum _ { m=1 } ^ { M } ( \lambda _ { meter } + \Lambda _ { m } ) ( 1 – e^ { -\rho R_ { } ( 1 – f_ { m } ) \frac { \lambda _ { thousand } } { \lambda _ { thousand } + \Lambda _ { thousand } } } ) + \left ( \lambda _ { \mathsf { A } } { \color { aristocratic } – \lambda _ { \mathsf { PPS } } } – \sum _ { m=1 } ^ { M } \lambda _ { megabyte } \right ) \\ & & \quad\times\ ; ( 1-e^ { -\rho R_ { } } ) { \color { blue } + \lambda _ { \mathsf { PPS } } ( 1 – f_ { \mathsf { PPS } } ) \rho R_ { } } \end { eqnarray * }  ( 5 ) Equation ( 4 ) restricts the miners to choose between pools that only offer the PPLNS honor method. We nowadays allow miners to choose mine pools besides offering PPS wages systems in our exemplar. In this case, a rational miner will choose to add alone the PPS pond that offers the smaller fee, and ignore pools with higher PPS fees. Equation ( 4 ) is then transformed as follows ( changes denoted in blue color ) :  \begin { eqnarray * } \sum _ { m=1 } ^ { M } \lambda _ { molarity } { \color { blue } +\lambda _ { \mathsf { PPS } } } \le \lambda _ { \mathsf { A } }, \ \ \lambda _ { megabyte } \ge 0, \forall m\in M, \ \text { and } \ { \color { blue } \lambda _ { \mathsf { PPS } } \ge 0 }. \end { eqnarray * }  under constraints

#### Mining across multiple cryptocurrencies

| $\lambda _ { \mathsf { A } }$ | and wants to maximize his “risk-sharing benefit” value by mining over | $\mathsf { C }$ | different cryptocurrencies and M pools in total, provided that each cryptocurrency | $c\in \mathsf { C }$ | uses the same PoW mining algorithm α. The allocation of the miner’s hashing power | $\lambda _ { \mathsf { A } }$ | will now be | ${ \lbrace \lambda _ { thousand, vitamin c } \rbrace } _ { m=1 } ^ { M }, c\in \mathsf { C }$ | and the first constraint in Equation  takes the form  \begin { eqnarray * } \sum _ { c\in \mathsf { C } } \sum _ { m=1 } ^ { M } \lambda _ { molarity, coke } \le \lambda _ { \mathsf { A } }. \end { eqnarray * }  We now consider a miner who owns mining hardware with PoW hashish powerand wants to maximize his “ risk-sharing benefit ” value by mining overdifferent cryptocurrencies and M pools in entire, provided that each cryptocurrencyuses the lapp PoW mining algorithm α. The allocation of the miner ’ mho hashing powerwill nowadays beand the first constraint in Equation 4 takes the formm allocated to pool m that mines cryptocurrency c should be normalized to each cryptocurrency’s total hash rate Λc. We also now include the transaction fees into the block reward. So, in this (more general) case, Equation  takes the following form, under the new constraints outlined earlier:

(6)

where λ0,c denotes solo mining for cryptocurrency c, and the first constraint of Equation  is more precisely expressed as  \begin { eqnarray * } \sum _ { c\in \mathsf { C } } \left ( \sum _ { m=1 } ^ { M } \lambda _ { megabyte, coulomb } + \lambda _ { 0, c } \right ) \le \lambda _ { \mathsf { A } }. \end { eqnarray * }  In accession, a miner ’ south hash rate λallocated to pool megabyte that mines cryptocurrency c should be normalized to each cryptocurrency ’ s total hash pace Λ. We besides now include the transaction fees into the blockage reward. indeed, in this ( more general ) character, Equation 4 takes the follow imprint, under the new constraints outlined earlier : where λdenotes alone mine for cryptocurrency hundred, and the first constraint of Equation 4 is more precisely expressed as The above constraint shows that the miner could choose to diversify his solo mining ( which was initially expressed by the second term in Equation 4 ) over multiple currencies as well, in a similar fashion as the miner would do by diversifying across multiple mine pools.

#### A discussion over more general utility functions

From a theoretical position, the miner ’ sulfur problem in the “ Miner ‘s allotment trouble ” section is well specified for any increasing and concave utility function u ( · ). Our option of CARA utilities here is entirely for computational efficiency in later numeral solutions in the “ implementation of Our Model ” section. otherwise, without an analytic formula for the miner ’ s aim function, the objective has to be calculated by first simulating random variables and then taking numeric consolidation. furthermore, such procedures need to be repeated every meter we iterate through a campaigner allotment, dramatically increasing calculation complexity. That said, we are aware that a miner ’ mho utility may not inevitably conform to constant absolute risk aversion. For example, a miner may have a constant-relative-risk-aversion ( CRRA ) utility. We argue that such miners can however find our tool utilitarian for finding approximate solutions to their problems. specifically, if a miner has initial wealth W0 and a CRRA utility with parameter η, i.e. | $uracil ( W_0+\mathsf { P } ) = \frac { ( W_0+\mathsf { P } ) ^ { 1-\eta } -1 } { 1-\eta }$ |⁠, then since | $E [ \mathsf { P } ]$ | is typically little relative to W0, one can approximate the miner ’ mho “ imposter ” absolute gamble antipathy as | $\rho = W_0^ { -1 } \eta$ | and plug it into Equation ( 6 ). One can then efficiently solve the problem in the “ Miner ‘s allotment problem ” section with the “ imposter ” objective, and the result allocations would give an approximate solution to the miner ’ s original problem. Over time, as the miner ’ s wealth gradually changes to some W1 ( so that the utility becomes | $\frac { ( W_1+\mathsf { P } ) ^ { 1-\eta } -1 } { 1-\eta }$ |⁠ ), the miner can recalibrate their “ pseudo ” absolute risk antipathy as | $\rho = W_1^ { -1 } \eta$ | and readjust their hash rate allotment. however, such readjustment lone needs to be sporadically, and the numeral solution is computationally efficient.

## Implementation of Our Model

In this section, we present an execution of our mining resources allotment mechanism. We developed a Python creature that automates the decision for an active miner, who owns either commercial off-the-rack ( COTS ) hardware ( e.g. CPUs/GPUs ) or application-specific integrated circuit hardware ( ASICs ). 4 Our cock covers all the cases discussed in the “ Active Miner ’ s Problem ” section. In Appendix 2, we provide details for choosing the right optimization method acting of our joyride. Tool description and instantiation assumptions. The basic individual cryptocurrency interpretation of our tool, given as inputting the miner ’ s hashing power | $\lambda _ { \mathsf { A } }$ |⁠, coin ’ sulfur change rate E, chosen pool data | $[ \Lambda _ { i }, f_ { i } ] _ { i=1 } ^ { M }$ | ( pool sum hashish rate and fee, respectively ) and risk aversion ρ, outputs the optimum distribution of the miner ‘s hashish power over these pools plus a “ solo-mining ” remainder. Some instantiations of our tool are outlined in the “ Single cryptocurrency ” section as examples for a typical respect range of ρ. Our instrument can besides provide the optimum distribution for the “ multicryptocurrency, unmarried PoW algorithm ” ( see the “ Analytical forms under CARA utility ” section ) and “ multicryptocurrency, multi-PoW algorithm ” ( Appendix 1 ) cases, and we besides outline an extend instantiation in the “ Multiple cryptocurrencies ” incision. The results of our tool can be well applied for mining in big scale, where a miner can allocate a share of his hardware to mine on a specific pool. The march of applying the results on a individual mine hardware piece is not fiddling, as to our cognition, no ASIC or GPU miner application exists that enables the drug user to allocate his mining baron over many pools by a specific share, even if theoretically it ’ mho technically feasible. The majority of ASICs utilize a fork of the cgminer tool, which initially offered a “ multipool strategy ” choice for the miner, but was belated deprecated as it was no longer compatible with the modern class mining protocol [ 18 ]. Multipool mine in a round-robin fashion is not effective for the miner adenine well, as this would result in a decrease in his overall reward, given the nature of reward schemes that prevent pool “ hopping. ” We encourage ASIC manufacturers and mining lotion developers to enable user-specified multipool mine in future releases, for the benefit of the miners and the whole community. We assume that the miner possesses an average wealth of | $\mathcal { W }$ | = $100k, while having typical values for the constant relative hazard aversion CRRA metric between 1 and 10 [ 19 ]. Given that CARA = CRRA|$ / \mathcal { W } $|⁠, we take as typical values for CARA ρ between 10−5 and 10−4, which we by and large assume throughout the rest of this newspaper. note that changing our assumption for our miner ’ s wealth is equivalent to changing the typical value stove for ρ accordingly ( we include extra evaluation analysis for a broader range of ρ values ). ### Single cryptocurrency #### Evaluation I We instantiate the first experiment of our creature by using the come parameters : a miner with total hashish world power |$ \lambda _ { \mathsf { A } } $| = 40 hashes/s, wishing to mine on a single cryptocurrency with engine block honor R =$ 50 000, having picked four mine pools with parameters shown in board 2. These values do not correspond to “ real ” mine pools ( or an exist cryptocurrency ), but are representatives for different classes of pools in terms of relative size, as larger real-world pools blame higher fees ( but have less fluctuations on the miner ’ s income ), while smaller pools have lower ( or evening zero ) fees to attract new miners to them. The results depicted in Fig. 1 express that our model produces the expect choices for rational miners. For smaller values of ρ the miner is will to “ risk ” more, and would dedicate much of his hash might to the little Pool 4, but for larger values of ρ the miner would diversify among larger pools for a steady income. Another significant observation is that for ρ > 6 · 10−5 the miner would allocate some of his office at both Pools 1 and 2 to diversify his risk ( which are the “ largest ” pools, having the same 2 % tip ), although he would show a strong preference for Pool 1 that is 10 times larger than Pool 2. note that for simplicity, we do not take any transaction fees kept by pools into report ; however, as shown in the previous section, our evaluation would produce equivalent results.

figure 3 :Open in new tabDownload slide Single currency with PPS pool and large values of ρ. figure 3 :Open in new tabDownload slide Single currentness with PPS pool and bombastic values of ρ .

Table 1:

 Total number of pools M Hash rate and fee of pool m Λm, fm Transaction fee | $\mathsf { texas } _ { }$ | Miner’s hashing power allocated to pool m (andcryptocurrency c) λm(λm,c) Miner’s total hashing power (for mining algorithm α) | $\lambda _ { \mathsf { A } } ( \lambda _ { \alpha _ { } }$ |⁠) Constant absolute risk aversion (CARA) ρ Cryptocurrency c total hash rate Λc Block time and block reward of cryptocurrency c Dc, Rc Total number of cryptocurrencies | $\mathsf { C }$ | Total number of PoW mining algorithms | $\mathsf { A }$ | (Average) network difficulty T Cryptocurrency exchange rate Ec Number of blocks found on a day d Bd Diversification interval (days) t Sharpe ratio, total accumulated reward/payoff | $\mathsf { S }, \mathsf { P }$ |
 Total number of pools M Hash rate and fee of pool m Λm, fm Transaction fee | $\mathsf { texas } _ { }$ | Miner’s hashing power allocated to pool m (andcryptocurrency c) λm(λm,c) Miner’s total hashing power (for mining algorithm α) | $\lambda _ { \mathsf { A } } ( \lambda _ { \alpha _ { } }$ |⁠) Constant absolute risk aversion (CARA) ρ Cryptocurrency c total hash rate Λc Block time and block reward of cryptocurrency c Dc, Rc Total number of cryptocurrencies | $\mathsf { C }$ | Total number of PoW mining algorithms | $\mathsf { A }$ | (Average) network difficulty T Cryptocurrency exchange rate Ec Number of blocks found on a day d Bd Diversification interval (days) t Sharpe ratio, total accumulated reward/payoff | $\mathsf { S }, \mathsf { P }$ |

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Table 1:

 Total number of pools M Hash rate and fee of pool m Λm, fm Transaction fee | $\mathsf { texas } _ { }$ | Miner’s hashing power allocated to pool m (andcryptocurrency c) λm(λm,c) Miner’s total hashing power (for mining algorithm α) | $\lambda _ { \mathsf { A } } ( \lambda _ { \alpha _ { } }$ |⁠) Constant absolute risk aversion (CARA) ρ Cryptocurrency c total hash rate Λc Block time and block reward of cryptocurrency c Dc, Rc Total number of cryptocurrencies | $\mathsf { C }$ | Total number of PoW mining algorithms | $\mathsf { A }$ | (Average) network difficulty T Cryptocurrency exchange rate Ec Number of blocks found on a day d Bd Diversification interval (days) t Sharpe ratio, total accumulated reward/payoff | $\mathsf { S }, \mathsf { P }$ |
 Total number of pools M Hash rate and fee of pool m Λm, fm Transaction fee | $\mathsf { texas } _ { }$ | Miner’s hashing power allocated to pool m (andcryptocurrency c) λm(λm,c) Miner’s total hashing power (for mining algorithm α) | $\lambda _ { \mathsf { A } } ( \lambda _ { \alpha _ { } }$ |⁠) Constant absolute risk aversion (CARA) ρ Cryptocurrency c total hash rate Λc Block time and block reward of cryptocurrency c Dc, Rc Total number of cryptocurrencies | $\mathsf { C }$ | Total number of PoW mining algorithms | $\mathsf { A }$ | (Average) network difficulty T Cryptocurrency exchange rate Ec Number of blocks found on a day d Bd Diversification interval (days) t Sharpe ratio, total accumulated reward/payoff | $\mathsf { S }, \mathsf { P }$ |

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Table 2:

 Pool 1 Λ1 106 hashes/s f1 2% Pool 2 Λ2 105 hashes/s f2 2% Pool 3 Λ3 104 hashes/s f3 1% Pool 4 Λ4 103 hashes/s f4 0%
 Pool 1 Λ1 106 hashes/s f1 2% Pool 2 Λ2 105 hashes/s f2 2% Pool 3 Λ3 104 hashes/s f3 1% Pool 4 Λ4 103 hashes/s f4 0%

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Table 2:

 Pool 1 Λ1 106 hashes/s f1 2% Pool 2 Λ2 105 hashes/s f2 2% Pool 3 Λ3 104 hashes/s f3 1% Pool 4 Λ4 103 hashes/s f4 0%
 Pool 1 Λ1 106 hashes/s f1 2% Pool 2 Λ2 105 hashes/s f2 2% Pool 3 Λ3 104 hashes/s f3 1% Pool 4 Λ4 103 hashes/s f4 0%

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We then pick some actual Bitcoin pools : SlushPool, ViaBTC, and KanoPool. These pools, as indicated by their parameters shown in table 3 ( values as of September 2019 ), are representatives of the options available to a miner, as they cover a wide range of pond hash exponent Λm and pool tip fermium. The above pools use either PPLNS or Score ( form of proportional ) reward methods. We do not include a PPS pool in this exemplar, although taken into score in Equation 5, as the results turned out to be identical for the typical value image of ρ. Using our parameters, a PPS pool would participate in the diversification only for large values of ρ that are not within the distinctive range ( we show such an exercise by and by in this section ). For the early parameters, we consider a large-scale miner who owns total mining world power of | $\lambda _ { \mathsf { A } }$ | = 3000 TH/s ( roughly about 100 units of Antminer S15 ASICs ), and the Bitcoin sum block reward R = $129 502. 5 The pool parameters are shown in mesa 3 and the resulting diversification graph in Fig. 2, where we observe a alike convention to the former “ representative ” pools exemplar ( i.e. a miner ’ mho predilection for larger pools and steadier income as ρ increases ). Table 3:  SlushPool Λ1 7380 PH/s f1 2% ViaBTC Λ2 6210 PH/s f2 2% KanoPool Λ3 194 PH/s f3 0.9%  SlushPool Λ1 7380 PH/s f1 2% ViaBTC Λ2 6210 PH/s f2 2% KanoPool Λ3 194 PH/s f3 0.9% Open in new tab Table 3:  SlushPool Λ1 7380 PH/s f1 2% ViaBTC Λ2 6210 PH/s f2 2% KanoPool Λ3 194 PH/s f3 0.9%  SlushPool Λ1 7380 PH/s f1 2% ViaBTC Λ2 6210 PH/s f2 2% KanoPool Λ3 194 PH/s f3 0.9% Open in new tab #### Diversifying on a PPS pool In the previous evaluation we showed that a PPS pool would not participate in the diversification using parameters for actual pools shown in postpone 3 and distinctive values for ρ. In Fig. 3, we show how a PPS pool would affect a miner ’ randomness diversification for nontypical big values of ρ, using the parameters in table 4. This would be applicable only for a miner who is very risk antipathetic, as he would show a stronger preference to the sweetheart income a PPS pool provides, as the value of ρ increases. Table 4:  SlushPool Λ1 7380 PH/s f1 2% ViaBTC Λ2 6210 PH/s f2 2% KanoPool Λ3 194 PH/s f3 0.9% PPS pool f4 4%  SlushPool Λ1 7380 PH/s f1 2% ViaBTC Λ2 6210 PH/s f2 2% KanoPool Λ3 194 PH/s f3 0.9% PPS pool f4 4% Open in new tab Table 4:  SlushPool Λ1 7380 PH/s f1 2% ViaBTC Λ2 6210 PH/s f2 2% KanoPool Λ3 194 PH/s f3 0.9% PPS pool f4 4%  SlushPool Λ1 7380 PH/s f1 2% ViaBTC Λ2 6210 PH/s f2 2% KanoPool Λ3 194 PH/s f3 0.9% PPS pool f4 4% Open in new tab #### Small-scale miners An concern observation is in regards of smaller scale miners and pools with higher fees. For exemplify, an active miner with 10 ASICs alternatively of 100, when following our method acting, would allocate all his hash might to the smaller pool ( KanoPool ) with the lowest tip. In Fig. 4, we show a minor ( or “ home plate ” ) Bitcoin miner as an case ( ⁠|$ \lambda _ { \mathsf { A } } $| = 125 TH/s ). Given his relatively small hash baron, he would only choose the lowest tip pool to mine ( KanoPool ), without allocating any resources to larger pools with higher fees, except for the upper values of ρ. basically, it is shown that hazard antipathy has less effect on small-scale miners. ### Multiple cryptocurrencies We now consider a miner who diversifies over unlike cryptocurrencies. For simplicity, we just switch 6 the currency in the 2nd pool ( ViaBTC ) from Bitcoin to Bitcoin Cash. The pool parameters are shown in table 5 : |$ \Lambda _ { \textsf { BTC } } $| = 91.26 EH/s, |$ \Lambda _ { \textsf { BCH } } $| = 2.5 EH/s and |$ R_ { \textsf { BCH } } $| =$ 3967.5 The resulting graph in Fig. 5 shows the diversification of his computational world power for assorted values of ρ. We observe that in this exemplify, for modest values of ρ his optimum strategy would be to keep most of his resources for zero-fee Bitcoin solo mining. however, for increasing values of ρ he would diversify his world power to larger pools, allocating some of his baron to the SlushPool, even though it has the same fee and the pond might not be deoxyadenosine monophosphate profitable as the Bitcoin Cash pool.

Table 5:

 SlushPool Λ1 7380 PH/s f1 2% ViaBTC Λ2 157 PH/s (Bitcoin Cash) f2 2% KanoPool Λ3 194 PH/s f3 0.9%
 SlushPool Λ1 7380 PH/s f1 2% ViaBTC Λ2 157 PH/s (Bitcoin Cash) f2 2% KanoPool Λ3 194 PH/s f3 0.9%

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Table 5:

 SlushPool Λ1 7380 PH/s f1 2% ViaBTC Λ2 157 PH/s (Bitcoin Cash) f2 2% KanoPool Λ3 194 PH/s f3 0.9%
 SlushPool Λ1 7380 PH/s f1 2% ViaBTC Λ2 157 PH/s (Bitcoin Cash) f2 2% KanoPool Λ3 194 PH/s f3 0.9%

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#### Impact of exchange rates

We noted that our results are highly sensible evening to very small changes to any of the parameters, such as the switch over rates. For case, the lapp miner having chosen the same pools, based on the historical datum, would allocate all of his exponent to the Bitcoin Cash pond the previous day, while after a few days he would transfer all of his baron to the Bitcoin pools. In Fig 6a and 6b, we show how humble daily fluctuations in the exchange rate between two same-PoW cryptocurrencies can affect the miner ’ s diversification for these cryptocurrencies. While in our previous instantiation the exchange rate was BTC/BCH = 0.028, a little increase in favor of Bitcoin Cash ’ s measure wholly eliminates the bearing of Bitcoin pools from the diversification, leaving only the Bitcoin Cash pool and Bitcoin Cash solo mining for the miner as his options. On the other hand, a little increase in privilege of Bitcoin ’ s value eliminates the presence of the Bitcoin Cash consortium, and the miner would merely diversify among the Bitcoin pools.

## Evaluation and Simulated Results

To showcase the advantage ( and risks ) of diversifying over multiple pools, we present an evaluation of our exemplar using Bitcoin data extracted from Smartbit Block Explorer API [ 20 ], as shown in table 6. first, we consider a Bitcoin miner owning some hashing power | $\lambda _ { \mathsf { A } }$ | = 1200 TH/s, who would start mining passively on a single chosen pool on 1 February 2018 for Δ = 4 months. then, we consider a Bitcoin miner having the same hashing world power | $\lambda _ { \mathsf { A } }$ |⁠, who using our instrument over the same time menstruation, would “ actively ” diversify every thyroxine = 3 days over M = 3 Bitcoin pools of his choose and reallocate his hashish baron consequently. In postpone 7, we outline the mining pools chosen for our evaluation along with their respective fees and early pretense parameters. We choose these pools as representatives of unlike pool sizes and fees, and to show how such pools influence the active miner ’ randomness diversification over time. As discussed in the “ implementation of Our Model ” section, we pick the mean respect for ρ = 0.00005. note that as we mentioned in the “ Introduction ” section, all of the above parameters constitute respective degrees of exemption in our experiments. In the subsequent sections, we show how each one of them can affect the end solution derived from our independent evaluation.

Table 6:

 Days ∈ Δ d Exchange rate Ed Network difficulty Td Participating pools md Total number of blocks Bd Number of blocks found by pool m Bm,d
 Days ∈ Δ d Exchange rate Ed Network difficulty Td Participating pools md Total number of blocks Bd Number of blocks found by pool m Bm,d

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Table 6:

 Days ∈ Δ d Exchange rate Ed Network difficulty Td Participating pools md Total number of blocks Bd Number of blocks found by pool m Bm,d
 Days ∈ Δ d Exchange rate Ed Network difficulty Td Participating pools md Total number of blocks Bd Number of blocks found by pool m Bm,d

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Table 7:

 Miners’ hash power | $\lambda _ { \mathsf { A } }$ | 1200 TH/s Diversification interval t 3 days Fee | $f_ { \textsf { SlushPool } }$ | 2% Fee | $f_ { \textsf { ViaBTC } }$ | 2% Fee | $f_ { \textsf { DPOOL } }$ | 1% CARA ρ 0.00005
 Miners’ hash power | $\lambda _ { \mathsf { A } }$ | 1200 TH/s Diversification interval t 3 days Fee | $f_ { \textsf { SlushPool } }$ | 2% Fee | $f_ { \textsf { ViaBTC } }$ | 2% Fee | $f_ { \textsf { DPOOL } }$ | 1% CARA ρ 0.00005

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Table 7:

 Miners’ hash power | $\lambda _ { \mathsf { A } }$ | 1200 TH/s Diversification interval t 3 days Fee | $f_ { \textsf { SlushPool } }$ | 2% Fee | $f_ { \textsf { ViaBTC } }$ | 2% Fee | $f_ { \textsf { DPOOL } }$ | 1% CARA ρ 0.00005
 Miners’ hash power | $\lambda _ { \mathsf { A } }$ | 1200 TH/s Diversification interval t 3 days Fee | $f_ { \textsf { SlushPool } }$ | 2% Fee | $f_ { \textsf { ViaBTC } }$ | 2% Fee | $f_ { \textsf { DPOOL } }$ | 1% CARA ρ 0.00005

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We retroactively compute the earnings on a casual basis for both miners. To take both overall advantage and miner ’ second income variance into account, we utilize the Sharpe proportion | $\mathsf { S } = \frac { \mathsf { P } – \mathsf { P } _ { \textsf { PPS } } } { \sigma _ { \Delta } }$ | as our chief comparison metric function, where | $\mathsf { P }$ | is the sum accumulate reinforce of the miner during the prison term period Δ, | $\mathsf { P } _ { \textsf { PPS } }$ | is the estimate miner ’ s reward during time time period Δ using a PPS pool 7 and σΔ stands for the standard deviation of miner ’ s wages over the time menstruation Δ. Remark. hera, we should note that while the CARA utility function is used to capture preferences over different wealth levels, one could argue that the miner should be amortizing his costs and tax income over a time period, since the average bribe per block would have little variation and a diversification over multiple pools is not necessary. however, the union ( or the rebate rate weighted kernel ) of many Poisson variables linearly scales both the mean and division of each single Poisson variable, so our analysis, which apparently looks like “ to evaluate the honor from a single block, ” is indeed equivalent to capturing preferences over different wealth levels ( this trap is a common “ fallacy of large numbers ” [ 21, 22 ] ).

### Evaluation assumptions

We assume that both miners choose pools that do not use the PPS reward system and that all pool ’ mho fees and reward schemes remain ceaseless over time. In addition, the miners are assumed to have constant hashish might ( i.e. do not use any of their rewards to buy more mine hardware ), and convert their rewards into USD on a day by day footing. For our analysis, we derive the daily net hash rate Λd from the daily network difficulty Td using the approximation | $\Lambda _ { five hundred } = \frac { 2^ { 32 } } { 600 } T_ { five hundred }$ |⁠. We besides approximate the casual pool hash power by | $\Lambda _ { thousand, five hundred } = \frac { B_ { molarity, five hundred } } { B_ { five hundred } }$ | where Bm, d the count of blocks found by pool thousand and Bd the total actual number of blocks found on each day d. The smaller the mine pool however, the less precise this approximation becomes ( i.e. a small pool might be “ doomed ” and would not find a block for several back-to-back days, while on some day it might become “ golden ” and find several blocks in a single sidereal day ), so we employ averaging techniques over a prison term window of 14 days ( which we believe is a fair period ) to improve our approximation accuracy for computing Λm, d. however a miner in the “ real earth ” would better use the self-reporting pool hash rates ( based on put in shares ) for a more accurate solution ( to our cognition, such historical data is not available on any engine block internet explorer ). besides as noted before, we do not take any transaction fees kept by pools into report for simplicity purposes. however, a miner can use the methodology discussed in the “ Analytical forms under CARA utility ” section to compute the average transaction fees over a holocene menstruation of prison term from on-line blockchain explorers and make a projection for fees in the future, then add | $\mathsf { texas } _ { megabyte, c }$ | and | $\mathsf { texas } _ { c }$ | to Equation ( 6 ) accordingly. finally, we show the earnings in USD rather of cryptocurrency ( Bitcoins ) in order to take the switch over rate into report, which is an significant parameter for the active miner ( used to calculate the obstruct reward R ).

### Main evaluation results

In Fig. 7, we show the earnings over time for both of the miners for comparison ( in Fig. 7 we besides show how the diversification changes over clock in light colors ). We observe a slenderly increased division for the active miner ( blue tune spikes ), since he chose to include a third pool ( DPool ) with smaller entire hash power Λ. however his sum accumulate honor would be | $\mathsf { P } _ { \textsf { A } }$ | = $101 221 for the active miner, compared with |$ \mathsf { P } _ { \textsf { P } } $| =$ 97 101, which finally leads to a Sharpe ratio | $\mathsf { S } _ { \mathsf { A } }$ | = 0.156 compared with | $\mathsf { S } _ { \mathsf { P } }$ | = 0.060. note that since we assumed both miners ’ hash power | $\lambda _ { \mathsf { A } }$ | remains ceaseless throughout this period, we observe a general decline blueprint in their day by day honor, because of the increasing difficulty T immediately affecting | $\frac { \lambda _ { \mathsf { A } } } { \Lambda _ { } }$ |⁠. We besides include an equivalent analysis using the lapp pools over a 1-year period in the next evaluation, where such a decline can be observed more distinctly. last, an important parameter to consider is the block fourth dimension D [ 23 ]. We used Bitcoin for our pretense, where D is relatively high ( roughly 10 min ) ; this resulted in high reward variance, specially when using smaller pools as shown in Figs 7 and 8. If we used a cryptocurrency with more frequent blocks for our evaluation ( e.g. Ethereum ), then the resultant role would be more predictable with lower observed overall variance values.
calculate 7 :Open in new tabDownload slide active miner on SlushPool ( up ) and on three pools ( down ). Hash might distribution over 4-month data. design 7 :Open in new tabDownload slide active miner on SlushPool ( up ) and on three pools ( down ). Hash exponent distribution over 4-month data .
figure 8 :Open in new tabDownload slide active miner on SlushPool ( up ) and on three pools ( down ). Hash power distribution over 1-year data. figure 8 :Open in new tabDownload slide active miner on SlushPool ( up ) and on three pools ( down ). Hash office distribution over 1-year data .

### Analysis for a large Δ

In Fig. 8, we repeat the pretense discussed in the “ Evaluation and Simulated Results ” section over the period of Δ = 1 year ( 1 January 2018–31 December 2018 ). The decline of miner rewards ascribable to the increasing difficulty can be observed more distinctly. In such a shell, a miner would most likely reinvest his earnings on mine hardware to keep | $\frac { \lambda _ { \mathsf { A } } } { \Lambda _ { } }$ | equally steady as possible. We besides observe high division around January 2018, generated by the eminent volatility in the Bitcoin/USD exchange rate | $E_ { \mathsf { BTC } }$ |⁠. The results for this evaluation are | $\mathsf { P } _ { \textsf { A } }$ | = $233 293 and |$ \mathsf { S } _ { \mathsf { A } } $| = 0.064 five |$ \mathsf { P } _ { \textsf { P } } $| =$ 224 293 and | $\mathsf { S } _ { \mathsf { P } }$ | = 0.023, which are consistent with the results derived from the 4-month simulation.

### Analysis for a different Δ and different pools

We now replace the third small pool ( DPool ) from our nonpayment set of pools with a larger one ( AntPool ) and set our period from ( 1 January 2018–30 June 2018 ) for Δ = 6 months, frankincense diversifying over the three largest PPLNS pool during that time period. Again, we observe an improvement in our metrics, | $\mathsf { P } _ { \textsf { A } }$ | = $172 719 and |$ \mathsf { S } _ { \mathsf { A } } $| = 0.041 volt |$ \mathsf { P } _ { \textsf { P } } $| =$ 172 092 and | $\mathsf { S } _ { \mathsf { P } }$ | = 0.032. Detailed analysis is shown in Fig. 9.
digit 9 :Open in new tabDownload slide active miner on SlushPool ( up ) and on three pools ( down ). Hash might distribution over 6-month data and different pool set. design 9 :Open in new tabDownload slide active agent miner on SlushPool ( up ) and on three pools ( down ). Hash power distribution over 6-month data and different pool set .

### Appendix 2: Choosing the Right Optimization Method

In order to find the best potential allotment of the miner ’ mho hash power we tried a few optimization methods. First, we applied the consecutive least squares programming algorithm, which uses the Han–Powell quasi-newton method with a BFGS update of the B-matrix and an L1-test serve for the steplength algorithm [ 30 ]. second, we implemented a modification of Newton ’ s method acting that solves the Lagrange system of equations for the active constraints. To our surprise, both mentioned gradient-based optimization methods experienced difficulties in obtaining accurate solutions to the optimization problem for very belittled values of ρ, about 10−5. A possible explanation to that phenomenon is that the instances with a wide range of hashes/s from a few to quintillion 1018 make the gradients calculated with significant computational errors. The presence of exponential functions sensitive to the scale of their argument is a conducive factor for the loss of the accuracy for the prevail gradients under the finite preciseness calculator arithmetical. even though those methods could be used for the cases with larger values of ρ, we abandoned them. The most successful algorithm for the optimization problem was the COBYLA [ 5 ]. COBYLA was developed for solving nonlinear constrained optimization problems via a sequence of linear programming subproblems, each solved on an update simplex. COBYLA is a thoroughly match for our optimization problem for the following reasons. First, our feasible set is a simplex, so the vertices of the feasible determined shape a good initial analogue estimate. Second, our trouble is low dimensional ( no more than a twelve of variables ). The first gear dimensionality of the problem results in a relatively little number of simplexes that need to be constructed before the solution is found. finally, COBYLA is a gradient-free algorithm. consequently, to update iterates, COBYLA does not rely on the gradient obtained locally in one compass point, which may not be accurate for this trouble. rather, in its research, COBYLA relies on the slope of a linear n-dimensional approximation calculated out of readily available nitrogen + 1 feasible points. We believe that all these factors in concert contribute to the efficiency of the algorithm for finding the best possible allocation of the miner ’ second hash power. consequently, our cock use COBYLA optimization solving method acting.

### Appendix 3: Mining Pools for Major Cryptocurrencies

In Table C1, we summarize a list of mining pools for major cryptocurrencies and the reward type each offers, as of 24 September 2019. We have already discussed PPS, PPLNS and proportional in the “ Reward methods in mine pools ” section. In the pursue mesa, we besides come across some variants of the standard reward types. In especial, FPPS ( full give per share ) is alike to PPS but payments take average transaction fees into score, sexual conquest is based on the proportional reward method weighed by time the share was submitted, and exponential is a PPLNS discrepancy with exponential decay of share values. We manually collected the data using the respective mining pool websites and the cryptocurrencies ’ engine block explorers.

Table C1:

Pool name . Coin . Reward type . Hash power .
BTC.com  BTC  FPPS  13.88 EH/s
BCH  FPPS  211.00 PH/s
AntPool  BTC  PPLNS, PPS  10.07 EH/s
BCH  PPLNS, PPS  160 PH/s
ETH  PPLNS, PPS  497 GH/s
LTC  PPLNS, PPS  29.9 TH/s
ETC  PPLNS, PPS  76.3 GH/s
ZEC  PPLNS, PPS  802 MSol/s
DASH  PPLNS, PPS  949 TH/S
SIA  PPLNS, PPS  206 TH/s
ViaBTC  BTC  PPLNS, PPS  6.78 EH/s
BCH  PPLNS, PPS  155.90 PH/s
ETH  PPLNS, PPS  219.79 GH/s
LTC  PPLNS, PPS  24.76 TH/s
ETC  PPLNS, PPS  13.51 GH/s
ZEC  PPLNS, PPS  230.77 MSol/s
DASH  PPLNS, PPS  16.81 TH/s
Miningpoolhub  ETH  PPLNS  8.22 TH/s
LTC  PPLNS  257.02 GH/s
ETC  PPLNS  1.67 TH/s
ZEC  PPLNS  49.68 MH/s
DASH  PPLNS  1.65 TH/s
XMR  PPLNS  4.62 MH/s
DGB  PPLNS  777.1 TH/s
Bitcoin.com  BTC  PPS  253.24 PH/s
BCH  PPS  215.34 PH/s
Nanopool  ETH  PPLNS  22.28 TH/s
ETC  PPLNS  1.96 TH/s
ZEC  PPLNS  40.74 MSol/s
GRIN  PPLNS  39.9 Kgp/s
XMR  PPLNS  53.228 MH/s
Litecoinpool.org  LTC  PPS  25.98 TH/s
Slush  BTC  Score  5.36 EH/s
Ethermine  ETH  PPLNS  40.7 TH/s
ETC  PPLNS  3.8 TH/s
ZEC  PPLNS  442.6 MSol/s
f2pool  BTC  PPS  13.83 EH/s
ETH  PPS  22.07 TH/s
LTC  PPS  48.65 TH/s
ETC  PPS  180.40 GH/s
ZEC  PPS  945.76 MSol/s
DASH  PPS  462.33 TH/s
SIA  PPS  11.61 TH/s
XMR  PPS  35.02 MH/s
Multipool  BTC  Exponential  0.36 PH/s
BCH  PPLNS  0.081 PH/s
LTC  PPLNS  14.45 GH/s
DGB  Proportional  0.3 PH/s
MinerGate  BTG  PPLNS, PPS  4 Sol/s
ETH  PPLNS  3.7 GH/s
ETC  PPLNS  1.4 GH/s
ZEC  PPLNS  7 KSol/s
XMR  PPLNS, PPS  3.4 MH/s
BCN  PPLNS, PPS  414.5 MH/s
Suprnova  BTG  Proportional  159 KSol/s
ZEC  Proportional  4.29 KSol/s
DGB  Proportional  3.1 TH/s
DASH  Proportional  192.46 TH/s
Coinotron  ETH  PPLNS, RBPPS  806.6 GH/s
ETC  PPLNS, RBPPS  630.7 GH/s
ZEC  PPLNS, PPS  1.3 MSol/s
BTG  PPLNS, PPS  13.9 KH/s
DASH  PPLNS, PPS  65.5 TH/s
Pool name . Coin . Reward type . Hash power .
BTC.com  BTC  FPPS  13.88 EH/s
BCH  FPPS  211.00 PH/s
AntPool  BTC  PPLNS, PPS  10.07 EH/s
BCH  PPLNS, PPS  160 PH/s
ETH  PPLNS, PPS  497 GH/s
LTC  PPLNS, PPS  29.9 TH/s
ETC  PPLNS, PPS  76.3 GH/s
ZEC  PPLNS, PPS  802 MSol/s
DASH  PPLNS, PPS  949 TH/S
SIA  PPLNS, PPS  206 TH/s
ViaBTC  BTC  PPLNS, PPS  6.78 EH/s
BCH  PPLNS, PPS  155.90 PH/s
ETH  PPLNS, PPS  219.79 GH/s
LTC  PPLNS, PPS  24.76 TH/s
ETC  PPLNS, PPS  13.51 GH/s
ZEC  PPLNS, PPS  230.77 MSol/s
DASH  PPLNS, PPS  16.81 TH/s
Miningpoolhub  ETH  PPLNS  8.22 TH/s
LTC  PPLNS  257.02 GH/s
ETC  PPLNS  1.67 TH/s
ZEC  PPLNS  49.68 MH/s
DASH  PPLNS  1.65 TH/s
XMR  PPLNS  4.62 MH/s
DGB  PPLNS  777.1 TH/s
Bitcoin.com  BTC  PPS  253.24 PH/s
BCH  PPS  215.34 PH/s
Nanopool  ETH  PPLNS  22.28 TH/s
ETC  PPLNS  1.96 TH/s
ZEC  PPLNS  40.74 MSol/s
GRIN  PPLNS  39.9 Kgp/s
XMR  PPLNS  53.228 MH/s
Litecoinpool.org  LTC  PPS  25.98 TH/s
Slush  BTC  Score  5.36 EH/s
Ethermine  ETH  PPLNS  40.7 TH/s
ETC  PPLNS  3.8 TH/s
ZEC  PPLNS  442.6 MSol/s
f2pool  BTC  PPS  13.83 EH/s
ETH  PPS  22.07 TH/s
LTC  PPS  48.65 TH/s
ETC  PPS  180.40 GH/s
ZEC  PPS  945.76 MSol/s
DASH  PPS  462.33 TH/s
SIA  PPS  11.61 TH/s
XMR  PPS  35.02 MH/s
Multipool  BTC  Exponential  0.36 PH/s
BCH  PPLNS  0.081 PH/s
LTC  PPLNS  14.45 GH/s
DGB  Proportional  0.3 PH/s
MinerGate  BTG  PPLNS, PPS  4 Sol/s
ETH  PPLNS  3.7 GH/s
ETC  PPLNS  1.4 GH/s
ZEC  PPLNS  7 KSol/s
XMR  PPLNS, PPS  3.4 MH/s
BCN  PPLNS, PPS  414.5 MH/s
Suprnova  BTG  Proportional  159 KSol/s
ZEC  Proportional  4.29 KSol/s
DGB  Proportional  3.1 TH/s
DASH  Proportional  192.46 TH/s
Coinotron  ETH  PPLNS, RBPPS  806.6 GH/s
ETC  PPLNS, RBPPS  630.7 GH/s
ZEC  PPLNS, PPS  1.3 MSol/s
BTG  PPLNS, PPS  13.9 KH/s
DASH  PPLNS, PPS  65.5 TH/s

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Table C1:

Pool name . Coin . Reward type . Hash power .
BTC.com  BTC  FPPS  13.88 EH/s
BCH  FPPS  211.00 PH/s
AntPool  BTC  PPLNS, PPS  10.07 EH/s
BCH  PPLNS, PPS  160 PH/s
ETH  PPLNS, PPS  497 GH/s
LTC  PPLNS, PPS  29.9 TH/s
ETC  PPLNS, PPS  76.3 GH/s
ZEC  PPLNS, PPS  802 MSol/s
DASH  PPLNS, PPS  949 TH/S
SIA  PPLNS, PPS  206 TH/s
ViaBTC  BTC  PPLNS, PPS  6.78 EH/s
BCH  PPLNS, PPS  155.90 PH/s
ETH  PPLNS, PPS  219.79 GH/s
LTC  PPLNS, PPS  24.76 TH/s
ETC  PPLNS, PPS  13.51 GH/s
ZEC  PPLNS, PPS  230.77 MSol/s
DASH  PPLNS, PPS  16.81 TH/s
Miningpoolhub  ETH  PPLNS  8.22 TH/s
LTC  PPLNS  257.02 GH/s
ETC  PPLNS  1.67 TH/s
ZEC  PPLNS  49.68 MH/s
DASH  PPLNS  1.65 TH/s
XMR  PPLNS  4.62 MH/s
DGB  PPLNS  777.1 TH/s
Bitcoin.com  BTC  PPS  253.24 PH/s
BCH  PPS  215.34 PH/s
Nanopool  ETH  PPLNS  22.28 TH/s
ETC  PPLNS  1.96 TH/s
ZEC  PPLNS  40.74 MSol/s
GRIN  PPLNS  39.9 Kgp/s
XMR  PPLNS  53.228 MH/s
Litecoinpool.org  LTC  PPS  25.98 TH/s
Slush  BTC  Score  5.36 EH/s
Ethermine  ETH  PPLNS  40.7 TH/s
ETC  PPLNS  3.8 TH/s
ZEC  PPLNS  442.6 MSol/s
f2pool  BTC  PPS  13.83 EH/s
ETH  PPS  22.07 TH/s
LTC  PPS  48.65 TH/s
ETC  PPS  180.40 GH/s
ZEC  PPS  945.76 MSol/s
DASH  PPS  462.33 TH/s
SIA  PPS  11.61 TH/s
XMR  PPS  35.02 MH/s
Multipool  BTC  Exponential  0.36 PH/s
BCH  PPLNS  0.081 PH/s
LTC  PPLNS  14.45 GH/s
DGB  Proportional  0.3 PH/s
MinerGate  BTG  PPLNS, PPS  4 Sol/s
ETH  PPLNS  3.7 GH/s
ETC  PPLNS  1.4 GH/s
ZEC  PPLNS  7 KSol/s
XMR  PPLNS, PPS  3.4 MH/s
BCN  PPLNS, PPS  414.5 MH/s
Suprnova  BTG  Proportional  159 KSol/s
ZEC  Proportional  4.29 KSol/s
DGB  Proportional  3.1 TH/s
DASH  Proportional  192.46 TH/s
Coinotron  ETH  PPLNS, RBPPS  806.6 GH/s
ETC  PPLNS, RBPPS  630.7 GH/s
ZEC  PPLNS, PPS  1.3 MSol/s
BTG  PPLNS, PPS  13.9 KH/s
DASH  PPLNS, PPS  65.5 TH/s
Pool name . Coin . Reward type . Hash power .
BTC.com  BTC  FPPS  13.88 EH/s
BCH  FPPS  211.00 PH/s
AntPool  BTC  PPLNS, PPS  10.07 EH/s
BCH  PPLNS, PPS  160 PH/s
ETH  PPLNS, PPS  497 GH/s
LTC  PPLNS, PPS  29.9 TH/s
ETC  PPLNS, PPS  76.3 GH/s
ZEC  PPLNS, PPS  802 MSol/s
DASH  PPLNS, PPS  949 TH/S
SIA  PPLNS, PPS  206 TH/s
ViaBTC  BTC  PPLNS, PPS  6.78 EH/s
BCH  PPLNS, PPS  155.90 PH/s
ETH  PPLNS, PPS  219.79 GH/s
LTC  PPLNS, PPS  24.76 TH/s
ETC  PPLNS, PPS  13.51 GH/s
ZEC  PPLNS, PPS  230.77 MSol/s
DASH  PPLNS, PPS  16.81 TH/s
Miningpoolhub  ETH  PPLNS  8.22 TH/s
LTC  PPLNS  257.02 GH/s
ETC  PPLNS  1.67 TH/s
ZEC  PPLNS  49.68 MH/s
DASH  PPLNS  1.65 TH/s
XMR  PPLNS  4.62 MH/s
DGB  PPLNS  777.1 TH/s
Bitcoin.com  BTC  PPS  253.24 PH/s
BCH  PPS  215.34 PH/s
Nanopool  ETH  PPLNS  22.28 TH/s
ETC  PPLNS  1.96 TH/s
ZEC  PPLNS  40.74 MSol/s
GRIN  PPLNS  39.9 Kgp/s
XMR  PPLNS  53.228 MH/s
Litecoinpool.org  LTC  PPS  25.98 TH/s
Slush  BTC  Score  5.36 EH/s
Ethermine  ETH  PPLNS  40.7 TH/s
ETC  PPLNS  3.8 TH/s
ZEC  PPLNS  442.6 MSol/s
f2pool  BTC  PPS  13.83 EH/s
ETH  PPS  22.07 TH/s
LTC  PPS  48.65 TH/s
ETC  PPS  180.40 GH/s
ZEC  PPS  945.76 MSol/s
DASH  PPS  462.33 TH/s
SIA  PPS  11.61 TH/s
XMR  PPS  35.02 MH/s
Multipool  BTC  Exponential  0.36 PH/s
BCH  PPLNS  0.081 PH/s
LTC  PPLNS  14.45 GH/s
DGB  Proportional  0.3 PH/s
MinerGate  BTG  PPLNS, PPS  4 Sol/s
ETH  PPLNS  3.7 GH/s
ETC  PPLNS  1.4 GH/s
ZEC  PPLNS  7 KSol/s
XMR  PPLNS, PPS  3.4 MH/s
BCN  PPLNS, PPS  414.5 MH/s
Suprnova  BTG  Proportional  159 KSol/s
ZEC  Proportional  4.29 KSol/s
DGB  Proportional  3.1 TH/s
DASH  Proportional  192.46 TH/s
Coinotron  ETH  PPLNS, RBPPS  806.6 GH/s
ETC  PPLNS, RBPPS  630.7 GH/s
ZEC  PPLNS, PPS  1.3 MSol/s
BTG  PPLNS, PPS  13.9 KH/s
DASH  PPLNS, PPS  65.5 TH/s

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