Available via license: CC BY 4.0

Content may be subject to copyright.

Modeling and Simulation of the Economics of Mining

in the Bitcoin Market

Luisanna Cocco,1∗Michele Marchesi,1

1Department of electrical and electronic engineering, University of Cagliari,

Piazza D’Armi, 09123 Cagliari, Italy.

∗E-mail: {luisanna.cocco, michele}@diee.unica.it

Abstract

In January 3, 2009, Satoshi Nakamoto gave rise to the ”Bitcoin Block Chain” creating

the ﬁrst block of the chain hashing on his computers central processing unit (CPU). Since

then, the hash calculations to mine Bitcoin have been getting more and more complex,

and consequently the mining hardware evolved to adapt to this increasing difﬁculty. Three

generations of mining hardware have followed the CPU’s generation. They are GPU’s,

FPGA’s and ASIC’s generations.

This work presents an agent based artiﬁcial market model of the Bitcoin mining process

and of the Bitcoin transactions. The goal of this work is to model the economy of the mining

process, starting from GPU’s generation, the ﬁrst with economic signiﬁcance.

The model reproduces some ”stylized facts” found in real time price series and some

core aspects of the mining business. In particular, the computational experiments performed

are able to reproduce the unit root property, the fat tail phenomenon and the volatility

clustering of Bitcoin price series. In addition, under proper assumptions, they are able to

reproduce the price peak at the end of November 2013, its next fall in April 2014, the

generation of Bitcoins, the hashing capability, the power consumption, and the mining

hardware and electrical energy expenses of the Bitcoin network.

keywords: Artiﬁcial Financial Market, Bitcoin, Heterogeneous Agents, Market Simu-

lation

1 Introduction

Bitcoin is a digital currency alternative to the legal ones, as any other crypto currency. Nowa-

days, Bitcoin is the most popular cryptocurrency. It was created by a cryptologist known as

”Satoshi Nakamoto”, whose real identity is still unknown [40]. Like other cryptocurrencies,

1

arXiv:1605.01354v1 [q-fin.TR] 4 May 2016

Bitcoin uses cryptographic techniques and, thanks to an open source system, anyone is allowed

to inspect and even modify the source code of the Bitcoin software.

The Bitcoin network is a peer-to-peer network that monitors and manages both the genera-

tion of new Bitcoins and the consistency veriﬁcation of transactions in Bitcoins. This network

is composed of a high number of computers connected to each other through the Internet. They

perform complex cryptographic procedures which generate new Bitcoins (mining) and manage

the Bitcoin transactions register, verifying their correctness and truthfulness.

Mining is the process which allows to ﬁnd the so called ”proof of work” that validates a

set of transactions and adds them to the massive and transparent ledger of every past Bitcoin

transaction known as the ”Blockchain”. The generation of Bitcoins is the reward for the val-

idation process of the transactions. The Blockchain was generated starting since January 3,

2009 by the inventor of the Bitcoin system himself, Satoshi Nakamoto. The ﬁrst block is called

”genesis block” and contains a single transaction, which generates 50-Bitcoin for the beneﬁt of

the creator of the block. The whole system is set up to yield just 21 million Bitcoins by 2040,

and over time the process of mining will become less and less proﬁtable. The main source of

remuneration for the miners in the future will be the fees on transactions, and not the mining

process itself.

In this work, we propose an agent-based model with the aim to study and analyse the mining

process and the Bitcoin market starting from September 1, 2010, the approximate date when

miners started to buy mining hardware to mine Bitcoins. The proposed model simulates the

mining process and the Bitcoin transactions, by implementing a mechanism for the formation

of the Bitcoin price, and speciﬁc behaviors for each typology of trader. We try to reproduce

the generation of Bitcoins, the main stylized facts present in the real Bitcoin market and the

economy of the mining process. The model described is built on a previous work of the authors

[15], which modeled the Bitcoin market under a purely ﬁnancial perspective. In this work, we

fully consider also the economics of mining.

The paper is organized as follows. In Section Related Work we discuss other works related

to this paper, in Section Mining Process we describe brieﬂy the mining process and we give

an overview on the mining hardware and on its evolution over time. In Section The Model we

present the proposed model in detail. Section Simulation Results presents the values given to

several parameters of the model and reports the results of the simulations, including an analysis

of Bitcoin real prices, and a robustness analysis. The conclusions of the paper are reported in

Section Conclusions. Finally, appendix deals with the calibration to some parameters of the

model.

2

2 Related Work

The study and analysis of the cryptocurrency market is a relatively new ﬁeld. In the last years,

several papers appeared on this topic given its potential interest and the many issues related to

it (see for instance the works [3, 5, 7, 24, 28, 29, 39, 53]).

However, very few works were made to model the cryptocurrencies market. We can cite

the works by Luther [37], who studied why some cryptocurrencies failed to gain widespread

acceptance using a simple agent model; by Bornholdt et al. [6], who proposed a model based

on Moran process to study the cryptocurrencies able to emerge; by Garcia et al. [27], who

studied the role of social interactions in the creation of price bubbles; by Kristoufek [31] who

analysed the main drivers of the Bitcoin price; and by Kaminsky et al. [30] who related the

Bitcoin market with its sentiment analysis on social networks.

In this paper we propose a complex agent-based model in order to reproduce the economy of

the mining process and the main stylized facts of the Bitcoin price series. Our model is inspired

by business, economic and ﬁnancial agent-based models that depict how organizations, or in

general the economy of a country, create, deliver, and capture value.

As regards the business models, Amini et al. [2] presented a agent-based model with the

aim to analyze the impact of alternative production and sales policies on the diffusion of a new

product; Cocco et al. [12, 13, 14] proposed agent-based models to simulate the software market

and analyze the business processes and policies adopted by proprietary software ﬁrms and Open

Source software ﬁrms; Li et al. [35] researched the dominant players behavior in supply chains

and the relationship between the selling prices and purchasing prices in supply chains by using a

multi-agent simulation model; Rohitratana et al. [51] studied the pricing schemes of the market

of the Software as-a-Service and on the market of the proprietary or traditional software; ﬁnally,

Xiaoming et al. [56] studied how a ﬁrm maximizes its proﬁt by determining the production and

sales policies for a new product during the lifetime of the product.

Concerning economic models, in [11] the authors presented one of the most signiﬁcant

agent-based model developed to date in order to study the European economy. In particular,

they show how monetary policies, i.e, credit money supplied by commercial banks as loans

to ﬁrms, inﬂuence the economy of a country. In [21, 23] agent based keynesian models are

presented in order to investigate the properties of macro economic dynamics and the impact

of public polices on supply, demand and the fundamentals of the economy, and to study the

interactions between income distribution and monetary and ﬁscal policies.

As regards artiﬁcial ﬁnancial market models, they reproduce the real functioning of markets,

trying to explain the main stylised facts observed in ﬁnancial markets, such as the fat-tailed dis-

tribution of returns, the volatility clustering, and the unit-root property. For a review, see works

[8] and [9]. Raberto et al. [47] and Cincotti et al. [10] proposed the Genoa Artiﬁcial Stock Mar-

3

ket (GASM) an agent-based artiﬁcial ﬁnancial market characterized by actual tracking of status

and wealth of each agent, and by a realistic trading and price clearing mechanisms. GASM is

able to reproduce some of the main stylised facts observed in real ﬁnancial markets.

This paper is built on GASM, adding speciﬁc features and a mix of zero-intelligence and

trend-following traders with the aim to model the Bitcoin exchange market and the economics

of mining.

3 Mining Process

Today, every few minutes thousands of people send and receive Bitcoins through the peer-to-

peer electronic cash system created by Satoshi Nakamoto. All transactions are public and stored

in a distributed database called Blockchain which is used to conﬁrm transactions and prevent

the double-spending problem.

People who conﬁrm transactions of Bitcoins and store them in the Blockchain are called

”miners”. As soon as new transactions are notiﬁed to the network, miners check their validity

and authenticity and collect them in a block. Then, they take the information contained in the

block of the transactions, which include a variable number called ”nonce” and run the SHA-

256 hashing algorithm on this block, turning the initial information into a sequence of 256 bits,

known as Hash [18].

There is no way of knowing how this sequence will look before calculating it, and the

introduction of a minor change in the initial data causes a drastic change in the resulting Hash.

The miners cannot change the data containing the information of transactions, but can

change the ”nonce” number used to create a different hash. The goal is to ﬁnd a Hash hav-

ing a given number of leading zero bits. This number can be varied to change the difﬁculty

of the problem. The ﬁrst miner who creates a proper Hash with success (he ﬁnds the ”proof-

of-work”), gets a reward in Bitcoins, and the successful Hash is stored with the block of the

validated transactions in the Blockchain.

In a nutshell,

”Bitcoin miners make money when they ﬁnd a 32-bit value which, when hashed together

with the data from other transactions with a standard hash function gives a hash with a

certain number of 60 or more zeros. This is an extremely rare event”, [18].

The steps to run the network are the followings:

” New transactions are broadcast to all nodes; each node collects new transactions into a

block; each node works on ﬁnding a difﬁcult proof-of-work for its block; when a node

ﬁnds a proof-of-work, it broadcasts the block to all nodes; nodes accept the block only if

all transactions in it are valid and not already spent; nodes express their acceptance of the

4

block by working on creating the next block in the chain, using the hash of the accepted

block as the previous hash”, [40].

Producing a single hash is computationally very easy, consequently in order to regulate the

generation of Bitcoins, over time the Bitcoin protocol makes this task more and more difﬁcult.

The proof-of-work is implemented by incrementing the nonce in the block until a value

is found that gives the block’s hash with the required leading zero bits. If the hash does not

match the required format, a new nonce is generated and the Hash calculation starts again [40].

Countless attempts may be necessary before ﬁnding a nonce able to generate a correct Hash.

The computational complexity of the process necessary to ﬁnd the proof-of-work is adjusted

over time in such a way that the number of blocks found each day is more or less constant (ap-

proximately 2016 blocks in two weeks, one every 10 minutes). In the beginning, each generated

block corresponded to the creation of 50 Bitcoins, this number being halved each four years,

after 210,000 blocks additions. So, the miners have a reward equal to 50 Bitcoins if the created

blocks belong to the ﬁrst 210,000 blocks of the Blockchain, 25 Bitcoins if the created blocks

range from the 210,001th to the 420,000th block in the Blockchain, 12.5 Bitcoins if the created

blocks range from the 420,001th to the 630,000th block in the Blockchain, and so on.

Over time, mining Bitcoin is getting more and more complex, due to the increasing number

of miners, and the increasing power of their hardware. We have witnessed the succession of

four generations of hardware, i.e. CPU’s, GPU’s, FPGA’s and ASIC’s generation, each of them

characterized by a speciﬁc hash rate (measured in H/sec) and power consumption. With time,

the power and the price of the mining hardware has been steadly increasing, though the price

of H/sec has been decreasing. To face the increasing costs, miners are pooling together to share

resources.

3.1 The evolution of the mining hardware

In January 3, 2009, Satoshi Nakamoto created the ﬁrst block of the Blockchain, called ”Genesis

Block”, hashing on the central processing unit (CPU) of his computer. Like him, the early min-

ers mined Bitcoin running the software on their personal computers. The CPU’s era represents

the ﬁrst phase of the mining process, the other eras being GPU’s, FPGA’s and ASIC’s eras (see

web site https://tradeblock.com/blog/

the-evolution-of-mining/).

Each era announces the use of a speciﬁc typology of mining hardware. In the second era,

started about on September 2010, boards based on graphics processing unit (GPU) running in

parallel entered the market, giving rise to the GPU era.

About in December 2011, the FPGA’s era started and hardware based on ﬁeld programmable

gate array cards (FPGA) speciﬁcally designed to mine Bitcoins was available in the market.

5

Finally, in 2013 fully customized application speciﬁc integrated circuit (ASIC) appeared, sub-

stantially increasing the hashing capability of the Bitcoin network and marking the beginning

of the fourth era.

Over time, the different mining hardware available was characterized by an increasing hash

rate, a decreasing power consumption per hash, and increasing costs. For example, NVIDIA

Quadro NVS 3100M, 16 cores, belonging to the GPU generation, has a hash rate equal to 3.6

MH/s and a power consumption equal to 14 W [16]; ModMiner Quad, belonging to the FPGA

generation, has a hash rate equal to 800 MH/s and power consumption equal to 40 W [16];

Monarch(300), belonging to the ASIC generation, has a hash rate equal to 300 GH/s and power

consumption equal to 175 W (see web site https://tradeblock.com/mining/.

3.2 Modeling the Mining Hardware Performances

The goal of our work is to model the economy of the mining process, so we neglected the ﬁrst

era, when Bitcoins had no monetary value, and miners used the power available on their PCs,

at almost no cost. We simulated only the remaining three generations of mining hardware.

We gathered information about the products that entered the market in each era to model

these three generations of hardware, in particular with the aim to compute:

•the average hash rate per US$ spent on hardware, R(t), expressed in H

sec∗$;

•the average power consumption per H/sec,P(t), expressed in W

H/sec .

The average hash rate and the average power consumption were computed averaging the

real market data at speciﬁc times and constructing two ﬁtting curves.

To calculate the hash rate and the power consumption of the mining hardware of the GPU

era, that we estimate ranging from September 1st, 2010 to September 29th, 2011, we computed

an average for Rand Ptaking into account some representative products in the market during

that period, neglecting the costs of the motherboard.

In that era, motherboards with more than one Peripheral Component Interconnect express

(PCIe), started to enter the market allowing to install, by using adapters, multiple video cards in

only one system and to mine criptocurrency, thanks to the power of the GPUs. In Table 1, we

describe the features of some GPUs in the market in that period. The data reported are taken

from the web site http://coinpolice.com/gpu/.

As regards the FPGA and ASIC eras, starting about on September 2011 and on December

2013, respectively, we tracked the history of the mining hardware by following the introduction

into the market of Butterﬂy Labs company’s products. We extracted the data illustrated in

Table 2 from the history of the web site http://www.butterﬂylabs.com/ through the web site

web.archive.org.. For hardware in the market in 2014 and 2015 we referred to the Bitmain

6

Table 1: GPU Mining Hardware.

Date Product Hash Rate GH/$ Consumption W/GH

23/09/2009

Radeon 5830 0.001475 593.22

Radeon 5850 0.0015 398.94

Radeon 5870 0.0015 467.66

Radeon 5970 0.0023 392

22/10/2010

Radeon 6870 0.0015 503.33

Radeon 6950 0.002 500

Radeon 6990 0.0018 328.95

Technologies Ltd company, and in particular, to the mining hardware called AntMiner (see web

site https://bitmaintech.com and Table 2).

Starting from the mining products in each period (see Tables 1 and 2), we ﬁtted a ”best hash

rate per $” and a ”best power consumption function” (see Table 3). We call the ﬁtting curves

R(t)and P(t), respectively.

We used a general exponential model to ﬁt the curve of the hash rate, R(t)obtained by using

eq. 1:

R(t) = a∗e(b∗t)(1)

where a= 8.635 ∗104and b= 0.006318.

The ﬁtting curve of the power consumption P(t)is also a general exponential model:

P(t) = a∗e(b∗t)(2)

where a= 4.649 ∗10−7and b=−0.004055.

Fig. 1 (a) and (b) show in logaritmic scale the ﬁtting curves and how the hash rate increases

over time, whereas power consumtpion decreases.

4 The Model

We used the Blockchain.info, a web site which displays detailed information about all trans-

actions and Bitcoin blocks, providing graphs and statistics on different data, for extracting the

empirical data used in this work. This web site provides several graphs and statistical analysis

of data about Bitcoins. In particular, we can observe the time trend of the Bitcoin price in the

market, the total number of Bitcoins, the total hash rate of the Bitcoin network and the total

number of Bitcoin transactions.

7

Table 2: Butterﬂy Labs Mining Hardware: FPGA Hardware from 09/29/2011 to 12/17/2012, ASIC

Hardware from 12/17/2012 to December 2013 and AntMiner Hardware for 2014 and 2015.

Date Product Price $ Hash Rate GH/s Hash Rate GH

sec∗$Power Consumption W

GH/sec

09/29/2011- 12/2/2011 The Single 699 1 0.0014 19.8

12/2/2011- 12/28/2011 The Single 699 1 0.0014 19.8

Rig Box 24980 50.4 0.0021 49

12/28/2011- 05/1/2012 The Single 599 0.832 0.0014 96.15

Rig Box 24980 50.4 0.0021 49

05/1/2012- 12/17/2012 The Single 599 0.832 0.0014 96.15

Mini Rig 15295 25.2 0.0016 49

12/17/2012- 04/10/2013 BitForce Jalapeno 149 4.5 0.0302 1

BitForce Little Single SC 649 30 0.0462 1

BitForce Single SC 1299 60 0.0462 1

BitForce Mini Rig SC 29899 1500 0.0502 1

04/10/2013- 05/31/2013 Bitcoin Miner 274 5 0.0182 6

Bitcoin Miner 1249 25 0.02 6

Bitcoin Miner 2499 50 0.02 6

05/31/2013- 10/15/2013 Bitcoin Miner 274 5 0.0182 6

Bitcoin Miner 1249 25 0.02 6

Bitcoin Miner 2499 50 0.02 6

Bitcoin Miner 22484 500 0.0222 6

10/15/2013- 12/10/2013 Bitcoin Miner 274 5 0.0182 6

Bitcoin Miner 2499 50 0.02 6

Bitcoin Miner 22484 500 0.0222 6

Bitcoin Minin Card 2800 300 0.1071 0.6

Bitcoin Minin Card 4680 600 0.1282 0.6

12/10/2013- 01/22/2014 AntminerS1 734.18 180 0.245 2

01/22/2014- 07/4/2014 AntminerS2 1715 1000 0.583 1.1

07/4/2014- 10/23/2014 AntminerS4-B2 1250 2000 1.6 0.69

10/23/2014- 03/25/2015 AntminerS5-B5 419 1155 2.756 0.51

03/25/2015-30/09/2015 AntminerS7-B8 454 4730 10.42 0.27

Table 3: Average of Hash Rate and of Power Consumption over time.

Date ⇒Simulation Step Average of Hash Rate GH

sec∗$Average of power Consumption W

GH/sec

September 1, 2010 ⇒1 0.0017 454.87

September 29, 2011 ⇒394 0.0014 19.8

December 2,2011 ⇒458 0.00175 34.4

December 28,2011 ⇒484 0.0017 72.575

May 1, 2012 ⇒608 0.0029 72.575

December 17, 2012 ⇒835 0.03565 1

April 10, 2013 ⇒953 0.0194 6

May 31, 2013 ⇒1004 0.0201 6

October 15, 2013 ⇒1141 0.1351 3.84

December 10, 2013 ⇒1197 0.0595 3.84

January 22, 2014 ⇒1240 0.245 2

July 4, 2014 ⇒1403 0.583 1.1

October 23, 2014 ⇒1484 1.6 0.69

March 25, 2015 ⇒1667 2.756 0.51

September 30, 2015 ⇒1856 10.42 0.27

8

(a) (b)

Figure 1: (a) Fitting curve of R(t) and (b) ﬁtting curve of P(t).

The proposed model presents an agent-based artiﬁcial cryptocurrency market in which

agents mine, buy or sell Bitcoins.

We modeled the Bitcoin market starting from September 1st, 2010, because one of our goals

is to study the economy of the mining process. It was only around this date that miners started

to buy mining hardware to mine Bitcoins, denoting a business interest in mining. Previously,

they typically just used the power available on their personal computers.

The features of the model are:

•there are various kinds of agents active on the BTC market: Miners, Random traders and

Chartists;

•the trading mechanism is based on a realistic order book that keeps sorted lists of buy and

sell orders, and matches them allowing to fulﬁll compatible orders and to set the price;

•agents have typically limited ﬁnancial resources, initially distributed following a power

law;

•the number of agents engaged in trading at each moment is a fraction of the total number

of agents;

•a number of new traders, endowed only with cash, enter the market; they represent people

who decided to start trading or mining Bitcoins;

•miners belong to mining pools. This means that at each time tthey always have a pos-

itive probability to mine at least a fraction of Bitcoin. Indeed, since 2010 miners have

been pooling together to share resources in order to be able avoiding effort duplication to

9

optimally mine Bitcoins. A consequence of this fact is that gains are smoothly distributed

among miners.

On July 18th, 2010,

”ArtForz establishes an OpenGL GPU hash farm and generates his ﬁrst Bitcoin block”

and on September 18th, 2010,

”Bitcoin Pooled Mining (operated by slush), a method by which several users work

collectively to mine Bitcoins and share in the beneﬁts, mines its ﬁrst block”,

(news from the web site http://historyofBitcoin.org/).

Since then, the difﬁculty of the problem of mining increased exponentially, and nowaday it

would be almost unthinkable to mine without participating to a pool.

In the next subsections we describe in detail the model simulating the mining, the Bitcoin

market and the related mechanism of Bitcoin price formation.

4.1 The Agents

Agents, or traders are divided into three populations: Miners, Random traders and Chartists.

Every i-th trader enters the market at a given time step, tE

i. Such a trader can be either

a Miner, a Random trader or a Chartist. All traders present in the market at the initial time

tE

i= 0 holds an amount ci(0) of ﬁat currency (cash, in US dollars) and an amount bi(0) of

crypto currency (Bitcoins), where iis the trader’s index. They represent the persons present in

the market, mining and trading Bitcoins, before the period considered in the simulation. Each

i-th trader entering the market at tE

i>0holds only an amount ci(tE

i)of ﬁat currency (cash, in

dollars). These traders represent people interested in entering the market, investing their money

in it.

The wealth distribution of traders follows a Zipf law [34]. The set of all traders entering

the market at time tE

i>0are generated before the beginning of the simulation with a Pareto

distribution of ﬁat cash, and then are randomly extracted from the set, when a given number

of them must enter the market at a given time step. Also, the wealth distribution in crypto

cash of the traders in the market at initial time follows a Zipf law. Indeed, the wealth share in

the world of Bitcoin is even more unevenly distributed than in the world at large (see web

site http://www.cryptocoinsnews.com/owns-Bitcoins-infographic-wealth-distribution/). More

details on the trader wealth endowment are illustrated in Appendix.

10

Miners Miners are in the Bitcoin market aiming to generate wealth by gaining Bitcoins. At

the initial time, the simulated Bitcoin network is calibrated respecting the Satoshi original idea

of Bitcoin network where each node participates equally to the process of check and validation

of the transactions and mining. We assumed that miners in the market at initial time (tE

i= 0)

own a personal PC such as Core i5 2600K, and hence they are initially endowed with a hashing

capability ri(0) equal to 0.0173GH/sec, that implies a power consumption equal to 75W [16].

Core i5 is a brand name of a series of fourth-generation x64 microprocessor developed by Intel

and brought to market in October 2009.

Miners entering the market at time tE

i>0acquire mining hardware, and hence a hashing

capability ri(t), which implies a speciﬁc electricity cost ei(t), investing a fraction γ1,i(t)of their

ﬁat cash ci(t).

In addition, over time all miners can improve their hashing capability by buying new mining

hardware investing both their ﬁat and crypto cash. Consequently, the total hashing capability of

i−th trader at time t,ri(t)expressed in [H/sec], and the total electricity cost ei(t)expressed

in $ per day, associated to her mining hardware units, are deﬁned respectively as:

ri(t) =

t

X

s=tE

i

ri,u(t)(3)

and

ei(t) =

t

X

s=tE

i

∗P(s)∗ri,u(s)∗24 (4)

where:

ri,u(t=tE

i>0) = γ1,i(t)ci(t)R(t)(5)

ri,u(t > tE

i)=[γ1,i(t)ci(t) + γi(t)bi(t)p(t)]R(t)(6)

•R(t)and P(t)are, respectively, the hash rate which can be bought with one US$, ex-

pressed in H

sec∗$, and the power consumption, expressed in W

H/sec . At each time t, their

values are given by using the ﬁtting curves described in subsection Modeling the Mining

Hardware Performances;

•ri,u(t)is the hashing capability of the hardware units ubought at time tby i−th miner;

•γi= 0 and γ1,i = 0 if no hardware is bought by i−th trader at time t. When a trader

decides to buy new hardware, γ1,i represents the percentage of miner’s cash devoted to

buy it. It is equal to a random variable characterized by a lognormal distribution with

11

average 0.15 and standard deviation 0.15. γirepresents the percentage of miner’s Bitcoins

to be sold for buying the new hardware. It is equal to a random variable characterized

by a lognormal distribution with average 0.175 and standard deviation 0.075. The term

γ1,i(t)ci(t) +γi(t)bi(t)p(t)expresses the amount of personal wealth that the miner wishes

to devote to buy new mining hardware, meaning that on average the miner will devote

35% of her cash and 17.5% of her bitcoins to this purpose. If γi>1or γ1,i >1, they are

set equal to one;

•is the ﬁat price per Watt and per hour. It is assumed equal to 1.4∗10−4$, consid-

ering the cost of 1 KWh equal to 0.14$, that we assumed to be constant throughout

the simulation. This electricity price is computed making an average of the electricity

prices in the countries in which the Bitcoin nodes distribution is higher; see web sites

https://getaddr.bitnodes.io and http ://en.wikipedia.org/wiki/Electricity pricing.

The decision to buy or not new hardware is taken by every miner from time to time, on

average every two months (60 days). If i−th miner decides whether to buy new hardware

and/or to divest the old hardware units at time t, the next time, tI−D

i(t), she will decide again is

given by eq. 7:

tI−D

i(t) = t+int(60 + N(µid, σid)) (7)

where int rounds to the nearest integer and N(µid, σid)is a normal distribution with average

µid = 0 and standard deviation σid = 6.tI−D

i(t)is updated each time the miner takes her

decision.

Miners active in the simulation since the beginning will take their ﬁrst decision within 60

days, at random times uniformly distributed. Miners entering the simulation at time t > 1will

immediately take this decision.

In deeper detail, at time t=tI−D

i(t), every miner can decide to buy new hardware units, if

her ﬁat cash is positive, and/or to divest the old hardware units. If trader’s cash is zero, she issues

a sell market order to get the cash to support her electricity expenses, ci,a(t) = γi(t)bi(t)p(t).

Each i−th miner belongs to a pool, and consequently at each time tshe always has a

probability higher than 0 to mine at least some sub-units of Bitcoin. This probability is inversely

proportional to the hashing capability of the whole network. Knowing the number of blocks

discovered per day, and consequently knowing the number of new Bitcoins Bto be mined per

day, the number of Bitcoins bimined by i−th miner per day can be deﬁned as follows:

bi(t) = ri(t)

rT ot(t)B(t)(8)

where:

12

•rT ot(t)is the hashing capability of the whole population of miners Nmat time tdeﬁned

as the sum of the hashing capabilities of all miners at time t,PNm

iri(t);

•the ratio ri(t)

rT ot(t)deﬁnes the relative hash rate of i−th miner at time t.

Note that, as already described in section Mining Process, the parameter Bdecreases over

time. At ﬁrst, each generated block corresponds to the creation of 50 Bitcoins, but after four

years, such number is halved. So, until November 27, 2012, 100,800 Bitcoins were mined in

14 days (7200 Bitcoins per day), and then 50,400 Bitcoins in 14 days (3600 per day).

The decision of a miner to buy and/or divests a hardware unit udepends on the Bitcoins

potentially obtained mining with the new hardware. A miner buys new hardware units if the

daily cost, given by the expense in electricity, ei,u (t), associated to these units is smaller than

the gain expected in Bitcoin. Hence, before buying new hardware units the following constraint

has to be evaluated:

ei,u(t)< bi,u(t)p(t)(9)

where:

•bi,u are the Bitcoins potentially mined by unit uat time t:bi,u(t) = ri,u (t)

rT ot(t)B(t)

•p(t)is the Bitcoin price at time t.

Only if this constraint is respected the miner can buy new hardware. In this case, she issues

a market order acquiring an amount of ﬁat cash ci,a(t) = γi(t)bi(t)p(t)in the next time steps.

She invests 50% of this amount to buy new hardware and keeps the remaining 50% as cash, to

pay the electricity bill for her hardware.

If the constraint in eq. 9 is not respected, the miner anyway issues a market order equal to

ci,a(t) = γi(t)bi(t)p(t)

2to support her electricity expenses.

A miner divests her old hardware units if the expense in electricity associated to that units

is 20% higher than the gain expected in Bitcoins using that hardware, at the current price.

Therefore, the following constraint has to be respected for each value of k, with kgoing from 0

to current time t:

ei,u(k)≤1.2ri,u(k)

rT ot(t)B(t)p(t)(10)

The model also includes a mechanism that enables 10% of miners to invest and/or divest

their hardware also at a time t6=tI−D

i(t). This mechanism is triggered when the price relative

variation, in a time window τMequal to 15 days, is positive and is higher than a threshold

T hMequal to 0.016. This because, in the real market, the investments of miners grow when the

13

proﬁtability of mining activity increases. So, increasing the interest of miners in buying new

hardware in these periods is a plausible assumption.

Random Traders Random Traders represent persons who enter the crypto-currency market

for various reasons, but not for speculative purposes. They issue orders for reasons linked to

their needs, for instance they invest in Bitcoins to diversify their portfolio, or they disinvest to

satisfy a need for cash. They issue orders in a random way, compatibly with their available

resources. In particular, buy and sell orders are always issued with the same probability. The

speciﬁcs of their behavior is described in section Buy and Sell Orders.

Chartists Chartists represent speculators, aimed to gain by placing orders in the Bitcoin mar-

ket. They speculates that, if prices are rising, they will keep rising, and if prices are falling,

they will keep falling. In particular, i−th chartist issues a buy order when the price relative

variation in a time window τC

i, is higher than a threshold T hC= 0.01, and issues a sell order

if this variation is lower than T hC.τC

iis speciﬁc for each chartist, and is characterized by a

normal distribution with average equal to 20 and standard deviation equal to 1. Chartists usu-

ally issue buy orders when the price is increasing and sell orders when the price is decreasing.

However, 10% of Chartists decide, instead, to adopt a contrary strategy, and place a sell order

instead of a buy order, or vice-versa. This contrarian behavior is common in ﬁnancial markets,

and is typically modeled also in market models [49]. Note that a Chartist will issue an order

only when the price variation is above a given threshold. So, in practice, the extent of Chartist

activity varies over time. In general the modelled Chartists’ behavior is key to produce large

price variations, and to the reproduction of the basic statistical proprieties of the real returns.

All Random traders and Chartists entering the market at t=tE>0, issue a buy order to

acquire their initial Bitcoins. Over time, at time t>tEonly a fraction of Random traders and

Chartists is active, and hence enabled to issue orders. Active traders can issue only one order

per time step, which can be a sell order or a buy order.

Orders already placed but not yet satisﬁed or withdrawn are accounted for when determining

the amount of Bitcoins a trader can buy or sell. Details on the percentage of active traders, the

number of the traders in the market and on the probability of each trader to belong to a speciﬁc

traders’ population are described in Appendix.

4.2 Buy and Sell Orders

The Bitcoin market is modelled as a steady inﬂow of buy and sell orders, placed by the traders

as described in [15]. Both buy and sell orders are expressed in Bitcoins, that is, they refer to a

given amount of Bitcoins to buy or sell. In deeper detail, all orders have the following features:

14

•amount, expressed in $ for buy order and in Bitcoins for sell order: the latter amount is a

real number, because Bitcoins can be bought and sold in fractions as small as a ”Satoshi”;

•residual amount (Bitcoins or $): used when an order is only partially satisﬁed by previous

transactions;

•limit price (see below), which in turn can be a real number;

•time when the order was issued;

•expiration time: if the order is not (fully) satisﬁed, it is removed from the book at this

time.

The amount of each buy order depends on the amount of cash, ci(t), owned by i-th trader at

time t, less the cash already committed to other pending buy orders still in the book. Let us call

cb

ithe available cash. The number of Bitcoins to buy, bais given by eq. 11

ba=cb

iβ

p(t)(11)

where p(t)is the current price and βis a random variable drawn from a lognormal distribu-

tion with average and standard deviation equal to 0.25 and 0.2, respectively for Random traders

and equal to 0.4and 0.2, respectively for Chartists. In the unlikely case that β > 1,βis set

equal to 1.

Similarly, the amount of each sell order depends on the number of Bitcoins, bi(t)owned by

i-th trader at time t, less the Bitcoins already committed to other pending sell orders still in the

book, overall called bs

i. The number of Bitcoins to sell, sais given by

sa=bs

iβ(12)

where βis a lognormal random variable as above. Short selling is not allowed.

The limit price models the price to which a trader desire to conclude his/her transaction.

An order can also be issued with no limit (market order), meaning that its originator wishes

to perform the trade at the best price she can ﬁnd. In this case, the limit price is set to zero.

The probability of placing a market order, Plim, is set at the beginning of the simulation and is

equal to 1 for Miners, to 0.2 for Random Traders and to 0.7 for Chartists. This because, unlike

Random Traders, if Miners and Chartists issue orders, then they wish to perform the trade at

the best available price, the formers because they need cash, the latters to be able to gain by

following the price trend.

Let us suppose that i-th trader issues a limit order to buy ab

i(t)Bitcoins at time t. Each buy

order can be executed if the trading price is lower than, or equal to, its buy limit price bi. In the

15

case of a sell order of as

i(t)Bitcoins, it can be executed if the trading price is higher than, or

equal to, its sell limit price si. As said above, if the limit prices bi= 0 or si= 0, then the orders

can be always executed, provided there is a pending complementary order.

The buy and sell limit prices, biand si, are given respectively by the following equations:

bi(t) = p(t)∗Ni(µ, σi)(13)

si(t) = p(t)

Ni(µ, σi)(14)

where

•p(t)is the current Bitcoin price;

•Ni(µ, σc

i)is a random draw from a Gaussian distribution with average µ'1and standard

deviation σi1.

The limit prices have a random component, modelling the different perception of Bitcoin

value, that is the fact that what traders ”feel” is the right price to buy or to sell is not constant,

and may vary for each single order. In the case of buy orders, we stipulate that a trader wishing

to buy must offer a price that is, on average, slightly higher than the market price.

The value of σiis proportional to the ”volatility” σ(Ti)of the price p(t)through the equation

σi=Kσ(Ti), where Kis a constant and σ(Ti)is the standard deviation of price absolute

returns, calculated in the time window Ti.σiis constrained between a minimum value σmin and

a maximum value σmax (this is an approach similar to that of [47]). For buy orders µ= 1.05,

K= 2.5,σmin = 0.01 and σmax = 0.003.

In the case of sell orders, the reasoning is dual. For symmetry, the limit price is divided by

a random draw from the same Gaussian distribution Ni(µ, σc

i).

An expiration time is associated to each order. For Random Traders, the value of the expira-

tion time is equal to the current time plus a number of days (time steps) drawn from a lognormal

distribution with average and standard deviation equal to 3 and 1 days, respectively. In this way,

most orders will expire within 4 days since they were posted. Chartists, who act in a more

dynamic way to follow the market trend, post orders whose expiration time is at the end of

the same trading day. Miners issue market orders, so the value of the expiration time is set to

inﬁnite.

4.3 Price Clearing Mechanism

We implement the price clearing mechanism by using an Order Book similar to that presented

in [50].

16

At every time step, the order book holds the list of all the orders received and still to be

executed. Buy orders are sorted in descending order with respect to the limit price bi. Orders

with the same limit price are sorted in ascending order with respect to the order issue time. Sell

orders are sorted in ascending order with respect to the limit price sj. Orders with the same

limit price are sorted in ascending order with respect to the order issue time.

At each simulation step, various new orders are inserted into the respective lists. As soon as

a new order enters the book, the ﬁrst buy order and the ﬁrst sell order of the lists are inspected

to verify if they match. If they match, a transaction occurs. The order with the smaller residual

amount is fully executed, whereas the order with larger amount is only partially executed, and

remains in the head of the list, with its residual amount reduced by the amount of the matching

order. Clearly, if both orders have the same residual amount, they are both fully executed.

After the transaction, the next pair of orders at the head of the lists are checked for matching.

If they match, they are executed, and so on until they do not match anymore. Hence, before the

book can accept new orders, all the matching orders are satisﬁed.

A sell order of index jmatches a buy order of index i, and vice versa, only if sj≤bi, or if

one of the two limit prices, or both, are equal to zero.

As regards the price, pT, to which the transaction is performed, the price formation mecha-

nism follows the rules described below. Here, p(t)denotes the current price:

•when one of the two orders has limit price equal to zero:

–if bi>0, then pT=min(bi, p(t)),

–if sj>0, then pT=max(sj, p(t)),

•when both orders have limit price equal to zero, pT=p(t);

•when both orders have limit price higher than zero, pT=bi+sj

2.

5 Simulation Results

The model described in the previous section was implemented in Smalltalk language. Before the

simulation, it had to be calibrated in order to reproduce the real stylized facts and the mining

process in the Bitcoin market in the period between September 1st, 2010 and September 30,

2015. The simulation period was thus set to 1856 steps, a simulation step corresponding to one

day. We included also weekends and holidays, because the Bitcoin market is, by its very nature,

accessible and working everyday.

We set the initial value of several key parameters of the model by using data recovered from

the Blockchain Web site. The main assumption we made is to size the artiﬁcial market at about

17

Figure 2: Price of Bitcoins in US$.

1/100 of the real market, to be able to manage the computational load of the simulation. Table

4 shows the parameter values and their computation assumptions in detail.

In Appendix other details about the calibration of the model are shown. Speciﬁcally, the

calibration of the trader wealth endowment, the number of active traders, the total number of

traders in the market and the probability of a trader to belong to a speciﬁc traders’ population

are described in detail.

The model was run to study the main features which characterize the Bitcoin market and the

traders who operate in it. In order to assess the robustness of our model and the validity of our

statistical analysis, we repeated 100 simulations with the same initial conditions, but different

seeds of the random number generator. The results of all simulations were consistent, as shown

in the followings.

5.1 Bitcoin prices in the real and simulated market

We started studying the real Bitcoin price series between September 1st, 2010 and September

30, 2015, shown in Fig. 2. The ﬁgure shows an initial period in which the price trend is

relatively constant, until about 950th day. Then, a period of volatility follows between 950th and

1150th day, followed by a period of strong volatility, until the end of the considered interval.

18

Table 4: Values of simulation parameters and the assumptions behind them.

Param. Initial Value Description and discussion

Nt(0) 160 Number of initial traders. Obtained dividing by

100 the number of traders on September 1st,

2010 estimated through the ﬁtting curve shown

in eq. 15 (see Appendix).

Nt(T)39,649 Total number of traders at the end of the sim-

ulation. Obtained dividing by 100 the num-

ber of traders on September 30, 2015 estimated

through the ﬁtting curve shown in eq. 15.

B72 or 36 Bitcoins mined per day. Obtained dividing

by 100 the Bitcoins which are mined every

day. They are 72 until 853th simulation step

(November 27th, 2012), and 36 from 853th

simulation step onwards.

p(0) 0.0649 $ Initial price. The average price as of September

2010.

BT(0) 23,274 $ Total initial crypto cash. Obtained dividing by

100 the number of Bitcoins on September 1st,

2010 and keeping just 60% of this value, be-

cause we assume that 40% of Bitcoins are not

available for the trade.

q200,000 $ Constant used in Zipf’s law ( q

i0.6), used to assign

the initial cash for traders entering at t > 1.

cs

120,587 $ Initial cash of the richest trader entering the

simulation at t= 1.

bs

14,117 $ Initial Bitcoin cash of the richest trader entering

the simulation at t= 1.

19

Figure 3: Bitcoin simulated Price in one simulation run.

The Bitcoin price started to fall at the beginning fo 2014, and is continuing on its downward

slope until September 2015.

It is well known that the price series encountered in ﬁnancial markets typically exhibit some

statistical features, also known as ”stylized facts” [42, 38]. Among these, the three uni-variate

properties which appear to be the most important and pervasive of price series, are (i) the unit-

root property, (ii) the fat tail phenomenon, and (iii) the Volatility Clustering. We examined daily

Bitcoin prices and found that also these prices exhibit these properties as discussed in detail in

[15].

As regards the prices in the simulated market, we report in Fig. 15 the Bitcoin price in one

typical simulation run. It is possible to observe that, as in the case of the real price, at ﬁrst

the price keeps its value constant, but then, after about 1000 simulation steps, contrary to what

happens in the reality, it grows and continues on its upward slope until the end of the simulation

period.

Figs. 16 (a) and (b) report the average and the standard deviation of the simulated price,

taken on all 100 simulations. Note that the average value of prices steadily increases with time,

in contrast with what happens in reality. Fig. 16 (b) shows that the price variations in different

simulation runs increase with time, as the number of traders, transactions and the total wealth

in the market are increasing.

In the proposed model, the upward trend of the price depends on an intrinsic mechanism

– in fact, the average price tends to the ratio of total available cash to total available Bitcoins.

Since new traders bring in more cash than new mined Bitcoins, the price tends to increase.

In reality, Bitcoin price is also heavily affected by exogenous factors. For instance, in the

past the price strongly reacted to reports such as those regarding Bitcoin ban in China, or the

MtGox exchange going bust. Moreover, the total capitalization of the Bitcoin market is of the

20

(a) (b)

Figure 4: (a) Average Price and (b) standard deviation computed on the 100 Monte Carlo simulations

performed.

order of just some billions of US$, so if a large hedge fund decided to invest in Bitcoins, or if

large amounts of Bitcoins disappeared because of theft, fraud or mismanagement, the effect on

price would be potentially quite large. All these exogenous events, that can trigger strong and

unexpected price variations, obviously cannot be part of our base model.

In section Other Results, we shall describe the results obtained when some random traders

adopt speculative behaviors, in addition to the speculative behaviour that characterizes Chartists.

Simulating this behavior allows to reproduce the Bitcoin price peak in December 2013 and its

subsequent fall.

Despite inability to reproduce the decreasing trend of the price, the model presented in

Section The Model, is able to reproduce quite well all statistical properties of real Bitcoin prices

and returns. The stylized facts, robustly replicated by the proposed model, are the same of a

previous work of Cocco et al. [15], and do not depend on the addition of the miners to the

model.

5.2 Traders’ Statistics

Figs. 5 - 7 show the average and the standard deviation of the crypto and ﬁat cash, and of the

total wealth, A(t), of trader populations, across all 100 simulations. These simulations were

carried with miners buying new hardware using an average percentage of 15% of their wealth,

that demonstrated to be optimal.

Figure 7(a) highlights how Miners represent the richest population of traders in the market

in the beginning of the simulation. However, from about 1400th step onwards, Random traders

become the richest population in the market. This is mainly due to the higher number of Random

21

traders with respect to Miners. Note also that the standard deviation of the total wealth is much

more variable than the former two ﬁgures. This is due to the fact that the wealth is obtained

by multiplying the number of Bitcoins by their price, which is very volatile among the various

simulations, as shown in Fig. 16(b).

(a) (b)

Figure 5: (a) Average and (b) standard deviation of the Bitcoin amount for all trader populations during

the simulation period across all Monte Carlo simulations.

(a) (b)

Figure 6: (a) Average and (b) standard deviation of the cash amount for all trader populations during the

simulation period across all Monte Carlo simulations.

Fig. 8, shows the average of the total wealth per capita for all trader populations, across all

100 Monte Carlo simulations. Miners are clearly the winners about from the 380th simulation

step onwards, thanks to their ability to mine new Bitcoins. Speciﬁcally, thanks to the percentage

22

(a) (b)

Figure 7: (a) Average and (b) standard deviation of the total wealth for all trader populations during the

simulation period across all Monte Carlo simulations.

of cash that Miners devot to buy new mining hardware, Miners are able to acquire a wealth per-

capite that ranges about between $1,000 at the beginning of the simulation and $14,000 at the

end. This is due to the optimal percentage of cash devoted to buy new hardware, that is drawn

from a lognormal distribution γwith both average and standard deviation set to 0.15, as already

mentioned in The Agents.

Figure 8: Average across all Monte Carlo simulations of the total wealth per capita, for all trader popu-

lations.

We varied the average percentage of their wealth that Miners devote for buying new hard-

ware, γ, to verify how this variation can impact on Miners’ success. Remember that the actual

percentage for a given Miner is drawn from a log-normal distribution, so these percentages are

fairly different among Miners.

23

Figures 9 (a) and (b) show the total wealth per capita for Miners, for increasing values

of the average of γ. It is apparent that Miners’ gains are inversely proportional to γ, so the

general strategy of devoting more money to buy hardware is not successful for Miners. This

is because if all Miners devote an increasing amount of money to buy new mining hardware,

the overall hashing power of the network increases, and each single Miner does not obtain the

expected advantage of having more hash power, whereas the money spent on hardware and

energy increases. The wealth per-capite ranges between about $1,000 at the beginning of the

simulation and $8,000 at the end, for γ= 0.25 (see ﬁg. 9 (a)) and about between $1,000 and

$6,000 for γ= 0.35 (see ﬁg. 9 (b)).

(a) (b)

Figure 9: Average across all Monte Carlo simulations of the total wealth average per capita for all trader

populations (a) for γ= 0.25 and (b) for γ= 0.35.

Having found that Miners’ wealth decreases when too much of it is used to buy new hard-

ware, we studied if increasing money spent in mining hardware is a successful strategy for

single Miners, when most other Miners do not follow it. Fig. 10 (a) shows the ratio of initial

Miners’ total wealth computed at the end and at the beginning of a single simulation, Afm

i(T)

Afm

i(0) ,

versus their actual value of γi, that is their propension to spend money to buy mining hardware.

The average < γ >= 0.15 in this simulation. The correlation coefﬁcients is equal to -0.14, so

it looks that there is no meaningful correlation between mining success and the propension to

invest in hardware. In Fig. 10 we can see that two of the three most successful Miners, able

to increase their wealth of about 100 and 45 times, have a very low value of γi, (less than 0.1),

whereas the third one, who was able to increase his wealth forty times, has a high propension

to invest (γi'0.62).

On the contrary, we found that the total wealth, Afm

i(T), of the miners at the end of the

simulation is correlated with their hashing capability rfm

i(T), being the correlation coefﬁcient

24

equal to 0.788, as shown in Fig. 10 (b). This result is not unexpected because wealthy Miner

can buy more hardware, that in turn helps them to increase their mined Bitcoins.

(a) (b)

Figure 10: Scatterplots of (a) the increase in wealth of single Miners versus their average wealth percent-

age used to buy mining hardware, and (b) the total wealth of Miners versus their hashing power at the

end of the simulation.

Figures 11 - 14 show some signiﬁcant quantities related to the Miner’s population.

(a) (b)

Figure 11: (a) Comparison between real hashing capability and average of the simulated hashing capa-

bility across all Monte Carlo simulations in log scale, and (b) average and standard deviation of the total

expenses in electricity across all Monte Carlo simulations in log scale.

Fig. 11(a) shows the average hashing capability of the whole network in the simulated

market across all Monte Carlo simulations and the hashing capability in the real market, being

25

both these quantities expressed in log scale. Note that the simulated hashing capability should

be about 100 times lower than the real one, due to the reduced dimension of the simulated

market with respect to the real one. The simulated hash rate does not follow the upward trend

of the Bitcoin price at about 1200th time step that is due to an exogenous cause (the step price

increase at the end of 2013), that is obviously not present in our simulations. However, in Fig.

11(a) the simulated hashing capability is actually about two orders of magnitude lower than the

real one, as it should be.

In general, Bitcoin mining hardware become obsolete from a few months to one year after

you purchase them. ”Serious” miners usually buy new equipment every month, re-investing

their proﬁts into new mining equipment, if they want that their Bitcoin mining operation to

run long term (see web site http://coinbrief.net/proﬁtable-bitcoin-mining-farm/. In our model,

miners divest their mining equipment about every ten months.

Figure 12 (a) shows the average and standard deviation of the power consumption across

all Monte Carlo simulations. Figure 12 (b) shows an estimated minimum and maximum power

consumption of the Bitcoin mining network, together with the average of the power consump-

tion of Fig. 12 (a), in logarithmic scale. The estimated theoretical minimum power consumption

is obtained by multiplying the actual hash rate of the network at time t(as shown in Fig. 11(a))

with the power consumption P(t)given in eq. 2. This would mean that the entire hashing

capability of miners is obtained with the most recent hardware. The estimated theoretical maxi-

mum power consumption is obtained by multiplying the actual hash rate of the network with the

power consumption P(t−360), referring to one year before. This would mean that the entire

hashing capability of miners is obtained with hardware one year old, and thus less efﬁcient. The

estimated obsolescence of mining hardware is between six months and one year, so the period

of one year should give a reliable maximum value for power consumption.

The simulation results, averaged on 100 simulations, show a much more regular trend,

steadily increasing with time – which is natural due to the absence of external perturbations

on the model. However, the power consumption value is of the same order of magnitude of the

”real” case. Note that the simulated consumption shown in Fig. 12 (b) is multiplied by 100, that

is the scaling factor of our simulations, that have 1/100th of the real number of Bitcoin traders

and miners.

Fig. 12 (b) also shows a white circle, at time step corresponding to April 2013, with a value

of 38.8 MW. This value has been taken by Courtois et al, who in work [18] write:

In April 2013 it was estimated that Bitcoin miners already used about 982 Megawatt hours

every day. At that time the hash rate was about 60 Tera Hash/s. (Refer to article by Gimein

Mark ”Virtual Bitcoin Mining Is a Real-World Environmental Disaster”, 13 April 2013

published on web site www.Bloomberg.com.).

In fact, the hash rate quoted is correct, but the consumption value looks overestimated of

26

(a) (b)

Figure 12: (a) Average and standard deviation of the power consumption across all Monte Carlo simula-

tions, and (b) Estimated minimum and maximum power consumption of the real Bitcoin Mining Network

(solid lines), and average of the power consumption across all Monte Carlo simulations, multiplied by

100, the scaling factor of our simulations (dashed line). For the meaning of the circles, see text.

one order of magnitude, even with respect to our maximum power consumption limit. We

believe this is due to the fact that the authors still referred to FPGA consumption rates, not fully

appreciating how quickly the ASIC adoption had spread among the miners.

As of 2015, the combined electricity consumption was estimated equal to 1.46 Tera Wh

per year, that corresponds to about 167 MW (see article ”The magic of mining”, 13 January

2015 published on web site www.economist.com.). This value is reported in Fig. 12 (b) as a

black circle. This time, the value is slightly underestimated, being at the lower bound of power

consumption estimate, and is practically coincident with the average value of our simulations.

Figures 13 (a) and (b) show an estimate of the expenses incurred every six days in electricity

(a) and in hardware (b) for the new hardware bought each day in the real and simulated market.

Note that also the values of the simulated expenses are average values across all Monte

Carlo simulations.

These expenses were computed assuming that the new hardware bought each day in the real

(simulated) market, and hence the additional hashing capability acquired each day, is equal to

the difference between the real (simulated) hash rate in tand the real (simulated) hash rate in

t−1.

For both these expenses, contrary to what happens to the respective real quantities, the

simulated quantities do not follow the upward trend of the price due to the constant investment

rate in mining hardware.

Figure 14 (a) and (b) show the average and standard deviation, across all Monte simulations,

27

(a) (b)

Figure 13: Real expenses and expenses average in electricity across all Monte Carlo simulations (a) and

real expenses and expenses average in hardware across all Monte Carlo simulations every six days (b).

of the expenses incurred every six days in electricity and in new hardware respectively, showing

the level of the variation across the simulations.

(a) (b)

Figure 14: Average and standard deviation of the expenses in electricity (a) and of the expenses in new

hardware across all Monte simulations.

Remembering that our model sizes the artiﬁcial market at about 1/100 of the real market

and that the number of traders, their cash and their trading probabilities are rough estimates of

the real ones, the simulated market outputs can be considered reasonably close to the real one.

28

5.3 Other Results

It is known that Bitcoin price is driven by speculation, government regulation and investors

behavior, and its volatility depends also on Bitcoin acceptance and usage. In 2012 and 2013

prices had a wild ride, until they reached a peak of $1,150 in December 2013. In 2014 Bitcoin

price fell following the shutdown of historical Mt. Gox exchange site and reports regarding

Bitcoin ban in China.

Trying to reproduce this market trend, we introduced in the model a particular speculative

behaviour by some traders. The speculative mechanism implemented stems from a report, called

the ”Willy Report”, published by an anonymous researcher, which alleges suspicious trading

activity at Mt. Gox. ”The Willy Report: proof of massive fraudulent trading activity at Mt.

Gox, and how it has affected the price of Bitcoin”, was posted on May 25, 2014 in web site

https://willyreport.wordpress.com/.

The anonymous researcher claims to have noted a suspicious bot behavior on Mt. Gox, that

spread its trading activity over many accounts, and how this fraudulent massive trading activity

impacted on the price, causing bubble and crash.

In the report the researcher writes:

”Somewhere in December 2013, a number of traders including myself began noticing sus-

picious bot behavior on Mt. Gox. Basically, a random number between 10 and 20 bitcoin

would be bought every 510 minutes, nonstop, for at least a month on end until the end of

January. The bot was dubbed Willy . . . its trading activity was spread over many accounts.

. . .Their trading activity went back all the way to September 27th.. . .

In total, a staggering about $112 million was spent to buy close to 270,000 BTC - the

bulk of which was bought in November. So if you were wondering how Bitcoin suddenly

appreciated in value by a factor of 10 within the span of one month, well, this may be why.

Not Chinese investors, not the Silkroad bust - these events may have contributed, but they

may not have been main reason. . . .

. . . there was another timetraveller account with an ID of 698630 - and this account, after

being active for close to 8 months, became completely inactive just 7 hours before the ﬁrst

Willy account became active! So it is a reasonable assumption that these accounts were

controlled by the same entity. . . . There were several peculiar things about Markus. First,

its fees paid were always 0 (unlike Willy, who paid fees as usual). Second, its ﬁat spent

when buying coins was all over the place, with seemingly completely random prices paid

per bitcoin.. . .

. . .Since there are no logs past November 2013, the following arguments are largely based

on personal speculation, and that of other traders. . .

on January 26th, Willy suddenly became inactive – and with it, the price retraced back to

a more reasonable spread with the other exchanges. Shortly after – on February 3rd to be

29

Figure 15: Price of the Bitcoin in the simulated market.

precise – it seemed as if Willy had begun to run in reverse, although with a slightly altered

pattern: it seemed to sell around 100 BTC every two hours. . . . There’s some additional

evidence on the chart that a dump bot may have been at play. At several points in time,

starting from Feb. 18th, it seemed that some bot was programmed to sell down to various

ﬁxed price levels.. . .

At this point, I guess the straightforward conclusion would be that this is how the coins

were stolen: a hacker gained access to the system or database, was able to assign him-

self accounts with any amount of USD at will, and just started buying and withdrawing

away.. . . ” .

According to what just mentioned, we modeled a similar behaviour. We assumed that, until

the end of January 2014, 40% of Random traders entering the market were Mt. Gox accounts.

The Mt. Gox accounts have a behaviour equal to that of Random traders described in Paragraph

Random Traders until July 2012. Then, from August 2012 and until the end of January 2014,

they issue only buy orders. Next, from February 2014 they issue only sell orders. Their trading

probability is set equal to 0.1 in every period.

Fig. 15 shows the Bitcoin price in one typical simulation run, under these conditions. At

ﬁrst, the price keeps its value constant, then at about 700 simulation steps, it grows as happens

in reality. The price maintains its value high for about 500 simulation steps, then its value falls

down, but after a short delay it continues on its upward slope until the end of the simulation,

due to the intrinsec mechanism of our model already previous described.

The MtGox accounts’ behaviour has a key rule in the reproduction of the price that has a

trend more similar to the real one (shown in Fig. 2) than that described in section Bitcoin prices

in the real and simulated market.

Figs. 16 (a) and (b) report the average and the standard deviation of the simulated price

30

across all Monte Carlo simulations showing the consistency of the results.

(a) (b)

Figure 16: (a) Average Price and (b) standard deviation computed on the 100 Monte Carlo simulations

performed.

All the analysis described in the previous sections was performed also for the model includ-

ing ”Mt.Gox” accounts, producing results consistent with those studied in the previous sections.

6 Conclusions

In this work, we propose an heterogenous agent model of the Bitcoin market with the aim

to study and analyse the mining process and the Bitcoin market starting from September 1st,

2010, the approximate date when miners started to buy mining hardware to mine Bitcoins, for

ﬁve years.

The proposed model simulates the mining process and the Bitcoin transactions, by imple-

menting a mechanism for the formation of the Bitcoin price, and speciﬁc behaviors for each

typology of trader. It includes different trading strategies, an initial distribution of wealth fol-

lowing Pareto law, a realistic trading and price clearing mechanism based on an order book, the

increase with time of the total number of Bitcoins due to mining, and the arrival of new traders

interested in Bitcoins.

The model was simulated and its main outputs were analysed and compared to respective

real quantities with the aim to demonstrate that an artiﬁcial ﬁnancial market model can repro-

duce the stylized facts of the Bitcoin ﬁnancial market.

The main result of the model is the fact that some key stylized facts of Bitcoin real price

series and of Bitcoin market are very well reproduced. Speciﬁcally, the model reproduces quite

well the unit root property of the price series, the fat tail phenomenon, the volatility clustering

31

of the price returns, the price peak in November 2013, its next fall in April 2014, the generation

of Bitcoins, the hashing capability, the power consumption, and the hardware and electricity

expenses incurred by Miners.

The proposed model is fairly complex. It is intrinsically stochastic and of course it in-

cludes endogenous mechanisms affecting the market dynamics. The Zipf distribution of traders’

wealth, that impacts to the size of the orders and the ”herding” effect of Chartists, when a price

trend is established, play a key role in the distribution of the price returns, and hence in the

reproduction of the fat tail phenomenon. The Chartist behavior and also the variability of the

spread of limit prices as a function of past price volatility contribute to the volatility cluster-

ing. The threshold of activation of Chartists based on price relative variation, and the past price

volatility used to determine the spread of limit prices impact on the unit-root property of the

price series. The percentage of each trader’s population, the choice of a trader that trades at

a given time step, and the type of trading (buy or sell), as well as the setting of the quantity

to trade, impact on the price trend. The setting of the amount of cash to devote to buy new

hardware impacts on the wealth and hashing capability of Miners, and consequently on their

hardware and electricity expenses.

Future research will be devoted to study in deeper detail the mechanisms impacting on the

model dynamics. In particular, we will perform a comprehensive analysis of the sensitivity

of the model to the various parameters, and will add traders with more sophisticated trading

strategies, to assess their proﬁtability in the simulated market. In addition, since the calibration

of our model is based on very few speciﬁc real data, and on many assumptions aiming to derive

the needed data from indirect real data, we plan to perform a deeper analysis of the Block Chain,

and to gather ﬁnancial data from the existing exchanges, in order to extract speciﬁc information

needed for a better calibration of our model.

A Appendix

B Trader Wealth Endowment

The distributions of cash and Bitcoins, for traders in the market at initial time, follow a power-

law with exponent αset equal to 1, a value yielding the distribution known as Zipf’s law [41].

This is the same assumption made in other papers [47, 49]. We assumed that 40% of the total ini-

tial crypto cash is not involved in the trading activity, given the initial Miners hoarded a fraction

of their Bitcoins (see web site http://www.coindesk.com/mit-report-bitcoin-more-likely-spent-

hoarded/.).

To create the Zipf’s distribution we used the ranking property of the power-law [54]. If

the total number of traders is Ntand the number of Bitcoins owned by them, bi, follows a

32

Pareto law with exponent α= 1, it is well-known from Harmonic-series theory that the total

number of Bitcoins BT=b1ln(Nt) + γ, where γis the Euler-Mascheroni constant and b1is the

number of Bitcoins owned by the richest trader. The number of Bitcoins owned by i-th trader

is consequently: b1

i. We set the cash of each initial trader equal to ﬁve times the value of their

crypto cash.

A similar approach was followed to set the wealth of traders who enter the simulation at

t > 0, but in this case the traders are only endowed with cash. In this case, we had no speciﬁc

data to calibrate the wealth of these traders. We stipulated that the cash, cs

1, of the richest trader

is about ﬁve times the cash owned by the richest initial trader, and that the exponent of the

Pareto law is in this case α= 0.6. Overall, we performed various simulations varying these

parameters, with no signiﬁcant impact on the results.

The set of ”new” traders are generated before the simulation starts. When new traders are

needed to enter the simulation, they are chosen randomly in this set.

C Active Traders

As mentioned in subsection The agents, only a given percentage of traders is active in the mar-

ket, and hence enabled to issue orders. To compute this percentage we made some assumptions

starting from the work [52] from which we extracted Table 5, which shows the distribution of the

transactions number per entity (Entity means the common owner of multiple Bitcoin addresses)

and per address on a period between January 3rd, 2009 and May 13th 2012.

Table 5: The distribution of the number of transactions per entity and per address.

Number of Number of Number of

Transactions (n) entities addresses

1≤n < 2557,783 495,773

2≤n < 41,615,899 2,197,836

4≤n < 10 222,433 780,433

10 ≤n < 100 55,875 228,275

100 ≤n < 1,000 8,464 26,789

1,000 ≤n < 5,000 287 1,032

5,000 ≤n < 10,000 35 51

10,000 ≤n < 100,000 32 24

100,000 ≤n < 500,000 7 3

n≥500,000 1 2

By analyzing Table 5, we can observed that 97.37% of all entities had fewer than 10 trans-

actions each, 2.27% of all entities had a number of transactions ranging from 10 to 100, 0.34 %

33

of all entities had a number of transactions ranging from 100 to 1,000, and the remaining 0.02

% had a number of transactions higher than 1,000.

According to the insights coming from this work, we neglected the entities having fewer than

10 transactions each and the entities having more than 1,000 transactions each. This was done

hypothesizing that the formers, typically involving a small number of Bitcoins, refer to entities

who made transactions by chance ”only to use” this new coin, whereas the latters, involving

a very high of transactions, are probably not linked to single traders, but are the addresses of

exchange sites, or of retailers accepting Bitcoins.

We considered the remaining entities, that is the entities with a number of transactions in

the range from 10 to 1,000. They issue orders with a period ranging from about one day to

about 122 days, because these data are computed on 1227 days. As a result, the daily trading

probability of an entity ranges from 0.008 to 1.

We set the values of trading probabilities for Random traders and Chartists in this range.

Speciﬁcally, we assumed that, Random Traders are active with a probability pt

R= 0.1, whereas

the Chartists are active with a probability pt

C= 0.5. This because the interest of Chartists in

purchasing or selling Bitcoins is higher than that of Random traders. Random traders issue

orders to satisfy their needs, whereas Chartists issue orders for speculative reasons, study care-

fully the price variation over time and are readier to place orders. Note that active Chartists

actually place orders only if the price variation is above a given threshold.

D Number of Traders

One of the most attractive property of Bitcoin is to provide quasi-anonymous transactions, so

knowing the number of traders in the real market is very difﬁcult. The Bitcoin addresses used

for the transactions are known, but a user can have, and typically has, more than one address.

At the moment of writing (last quarter of 2015) we had three ﬁgures:

1. the massive and transparent ledger of every Bitcoin transaction was generated starting

since January 3, 2009 presumably by the inventor of the Bitcoin system itself, Nakamoto

and so in January 2009 there was only one person owning Bitcoins;

2. ”according to rough estimates, 280,000 people owned Bitcoins at the end of 2013”

(http://www.whoishostingthis.com/blog/2014/03/03/who-owns-all-the-bitcoins/);

3. on April 22nd, 2014 the total number of holders was estimated equal to 1.0 million

(https://Bitcointalk.org/index.php?topic=316297.0).

In addition to these ﬁgures, we have other data related to the period between January 2009

and September 2010. These data were extracted from an analysis on the daily number of down-

loads of the ofﬁcial Bitcoin software client from the SourceForge platform (http://sourceforge.net/projects/bitcoin).

34

In the period just mentioned, the Bitcoin network began to spread and Bitcoin had no mon-

etary value. So, for this period we assumed the number of downloads of the ofﬁcial Bitcoin

software client equal to the number of traders in the market. We made the assumption that a

person who downloads the Bitcoin software is mainly interested to use it to mine Bitcoins. So,

we extracted two ﬁgures from this data: the total number of downloads on May 1st, 2010 equal

to 2,769, and the total number of downloads on September 1st, 2010, equal to 30,589, and we

set the number of downloads equal to the number of traders.

So, we computed the number of traders in the market ﬁtting the curve NTthrough the ﬁve

ﬁgures available:

1. 1 people owned Bitcoins on January 2009. He was Satoshi Nakamoto;

2. 2769 people had downloaded Bitcoin mining software on May 2010;

3. 30589 people had downloaded Bitcoin mining software on September 2010;

4. 280,000 people owned Bitcoins at the end of 2013;

5. 1000000 people owned Bitcoins on April 2014.

The ﬁtting curve of the number of traders NTis deﬁned by using a general exponential

model:

NT(t) = a∗e(b∗(608+t)) (15)

where a= 2624,b= 0.002971.

Fig. 17 (a) show the ﬁtting curve and how the number of traders increases over time.

E Probability of a trader to belong to a speciﬁc traders’ pop-

ulation

At ﬁrst, users in the Bitcoin network were mainly miners. Subsequently, when the Bitcoin

network started to grow and Bitcoins started to acquire a monetary value, new users entered the

network. Most of these users did not mine, but simply traded Bitcoins. They are represented in

the model as Random traders and Chartists.

Since the percentages of the different trader populations is not known, to compute the prob-

ability of a trader to be a Miner we performed an analysis of the Bitcoin Blockchain. We

analysed the Blockchain until May 1st, 2010 and then until September 1st, 2010. Each block

in the Blockchain contains a list of validated transactions. Each transaction has in input all

35

(a) (b)

(c)

Figure 17: Fitting curve (a) of NT(x), (b) of the probability of a trader to be a Miner, and (c) number of

real Bitcoin transactions.

the addresses containing the amount of Bitcoins to transfer, and in output all the addresses that

receive the Bitcoins. Users can use multiple addresses.

We assumed, as in [27], that the input addresses that send Bitcoins to the same output

address must belong to an unique owner, since to proceed with the transaction it is necessary to

know the private keys of all input addresses.

In addition, we considered the input addresses that transfer more than 20 Bitcoins owned

by the same owner of the corresponding outputs. In fact, in the period from May 1st, 2010,

to September 1st, 2010 Bitcoins had no monetary value, and so it is acceptable to consider

that Miners exchanged only small amounts of Bitcoins to test the operation of the network. Of

course, this last assumption can be valid only for the period under study.

With these assumptions, we found 46,005 unique addresses on May 1st, 2010 and 82,294

36

unique addresses on September 1st, 2010. With further analysis, we found that on May 1st,

2010, out of 46,005 addresses, 43,389 were addresses that mined, and on September 1st, 2010,

out of 82,294 addresses, 55,974 were addresses that mined, so we computed the probability of

an user to be a miner. This probability is equal to 0.94% on May 1st, 2010 and equal to 0.68%

on September 1st, 2010.

Using these two ﬁgures we computed a ﬁtting curve of the probability of an user to be a

miner pM. Again, it is deﬁned by using a general exponential model:

pM(t) = a∗e(b∗t)(16)

where a= 0.9425,b=−0.002654.

Fig. 17 (b) shows the ﬁtting curve and how this probability decreases over time. We see that

the probability of a trader to be a Miner decreases over time, going about from 0.38 to less than

0.01. Of course, deﬁning this probability using a ﬁtting curve computed from only two points

is questionable. In the followings, we made some considerations to validate the adoption of this

curve.

At ﬁrst, the number of Bitcoin transactions was low because in the market there were mainly

miners. Over time, as Bitcoin was acquiring monetary value, the number of users interested in

exchanging Bitcoins increased. So, while the percentage of Miners in the market was decreas-

ing, also due to the increasing difﬁcult to mine Bitcoins, the percentage of Random traders pR

and Chartists pCgreatly increased, according to the growth of the number of transactions, which

slowly rose until a peak on May 2012 and then at the end of the 2013 (see ﬁgure 17 (c)). .

We assumed that the random traders to chartist ratio is 7/3, meaning that 30% of traders

who are not miners are speculators, whereas the remaining 70% are non-speculative investors.

These ﬁgures are consistent with recent data obtained for the foreign exchange market [19].

The probabilities of non-miners to be a random trader or a chartist, pRand pCrespectively, are

deﬁned as a function of pM, respectively as:

pR= 0.7(1 −pM)(17)

and

pC= 0.3(1 −pM)(18)

With these probabilities, we have at the end of the simulations a number of Miners equal to

about 1000, corresponding to 100,000 miners in the real world. This is in agreement with what

an Australian bitcoin miner, Andrew Geyl, estimated (see web site http://bravenewcoin.com/news/number-

of-bitcoin-miners-far-higher-than-popular-estimates/).

37

References

[1] Alfarano S, Lux T, Wagner F. Time variation of higher moments in a ﬁnancial market

with heterogeneous agents: An analytical approach. Journal of Economic Dynamics &

Control. 2008; 32: 101–136.

[2] Amini M, Wakolbinger T, Racer M, and Nejad M.G. Alternative supply chain produc-

tionsales policies for new product diffusion: An agent-based modeling and simulation

approach .European Journal of Operational Research. 2012; 216: 301–311.

[3] Androulaki E, Karame G, Roeschlin M, Scherer T, and Capkun S. Evaluating User Pri-

vacy in Bitcoin. Proceedings of the Financial Cryptography and Data Security Confer-

ence (FC). 2013.

[4] Arthur W. B, Holland J. H, LeBaron B, Palmer R, and Tayler P. Asset pricing under en-

dogenous expectations in an artiﬁcial stock market. The Economy as an Evolving Com-

plex System II, ser. SFI Studies in the Sciences of Complexity, Arthur W.B, Durlauf S.N,

and Lane D.A. (Eds). Addison Wesley Longman. 1997;15–44.

[5] Bergstra J. A, and Leeuw dK. Questions related to Bitcoin and other Informational

Money. CoRR. 213; 1305.5956.

[6] Bornholdt S, and Sneppen K. Do Bitcoins make the world go round? On the dynamics

of competing crypto-currencies. CoRR. 2014; 1403.6378.

[7] Brezo F, and Bringas P. G. Issues and Risks Associated with Cryptocurrencies such as

Bitcoin. The Second International Conference on Social Eco-Informatics. 2012.

[8] Chakraborti A, Toke I.M, Patriarca M, and Abergel F. Econophysics review: II. Agent-

based models. Quantitative Finance. 2011; 11(7): 1013-1041.

[9] Chen S.H, Chang C. L, and Du Y. R. Agent-Based Economic Models and Econometrics.

The Knowledge Engineering Review. 2012; 27: 187–219.

[10] Cincotti S, Focardi S, Marchesi M, and Raberto M. Who wins? Study of long-run trader

survival in an artiﬁcial stock market. Physica A: Statistical Mechanics and its Applica-

tions, Elsevier. 2003; 324(1): 227-233.

[11] Cincotti S, Raberto M, and TeglioA. The EURACE macroeconomic model and simulator.

16th IeA World Congress. 2011.

38

[12] Cocco L, Concas G, Marchesi M, and Destefanis G. Agent-Based Modelling and Simu-

lation of the Software Market, Including Open Source Vendor. 2013; Journal of Informa-

tion Technology Managemen; 24:1.

[13] Cocco L, Mannaro K, Concas G, Marchesi M. Study of the Competition between Propri-

etary Software Firms and Free/ Libre Open Source Software using a Simulation model.

Software Business Lecture Notes in Business Information Processing, Springer. 2011;

80: 56–69.

[14] Cocco L, Concas G, Marchesi M. Simulation of the On Demand Software Market, In-

cluding Vendors Offering Source Code Availability. International Journal of Modelling

and Simulation. 2014; 34(3).

[15] Cocco L, Concas G, Marchesi M. Using an Artiﬁcial Financial Market for studying a

Cryptocurrency Market. Journal of Economic Interaction and Coordination, Springer.

2015; DOI:10.1007/s11403-015-0168-2.

[16] Courtois N. T. Crypto Currencies And Bitcoin. http :

//blog.bettercrypto.com/?pageid= 20. 2014.

[17] Courtois N.T, and Bahack L. On Subversive Miner Strategies and Block Withholding

Attack in Bitcoin Digital Currency. CoRR. 2014; 1402.1718.

[18] Courtois N. T, Grajek M, and NaikR. The Unreasonable Fundamental Incertitudes Be-

hind Bitcoin Mining. CoRR. 2014; 1310.7935v3.

[19] Dick C. D, and MenkhoffL. Exchange rate expectations of chartists and fundamentalists.

Journal of Economic Dynamics & Contro2013; 37: 1362–1383.

[20] Dosi G, Fagiolo G, and A. Roventini. Animal Spirits, Lumpy Investment and Endogenous

Business Cycles. 2006.

[21] Dosi G, Fagiolo G, and A. Roventini. Schumpeter meeting Keynes: A policy-friendly

model of endogenous growth and business cycles. Journal of Economic Dynamics &

Control. 2010; 34: 1748–1757.

[22] Dosi G, Fagiolo G, and A. Roventini. The microfoundations of business cycles: an evo-

lutionary, multi-agent model. J. Evol Econ DOI 10.1007/s00191-008-0094-8, Springer.

2008.

39

[23] Dosi G, Fagiolo G, Napoletano M, and Roventini A. Income distribution, credit and ﬁscal

policies in an agent-based keynesian model, Journal of economic Dynamics & control 37.

2013; 1598–1625.

[24] Eyal I, and Sirer E. Majority is not Enough: Bitcoin Mining is Vulnerable. CoRR. 2013;

1311.0243.

[25] Feng L, Podobnik B. L, Preis T, and StanleyH.E. Linking agent-based models and

stochastic models of ﬁnancial markets. PNAS 109:22. 2012; 8388–8393.

[26] Tzamaloukas A, and Garcia Luna AcevesJ. J. Channel-Hopping Multiple Access. 2000;

I-CA2301. Department of Computer Science, University of California. Berkeley, CA.

[27] Garcia D, Tessone C.J, Mavrodiev P, and Perony N. Feedback cycles between socio-

economic signals in the Bitcoin economy The digital traces of bubbles: feedback cycles

between socio-economic signals in the Bitcoin economy. CoRR. 2014; 1408.1494.

[28] Hanley B.P. The False Premises and Promises of Bitcoin. CoRR. 2013; 1312.2048.

[29] Hout M. C. V, and Bingham T. Responsible vendors, intelligent consumers: Silk Road,

the online revolution in drug trading. International Journal of Drug Policy, Elsevier. 2014;

25(4): 183-189.

[30] Kaminsky J, and GloorP.A. Nowcasting the Bitcoin Market with Twitter Signals. CoRR.

2014; 1406.7577.

[31] Kristoufek L. What are the main drivers of the Bitcoin price? Evidence from wavelet

coherence analysis. CoRR. 2014; 1406.0268.

[32] LeBaron B. Agent-based computational ﬁnance, Handbook of Computational Eco-

nomics. Agent-based Computational Economics. Elsevier. 2006; 2: 1187-1233.

[33] Lengnick M. Agent-based macroeconomics - a baseline model. Economics working pa-

per / Christian-Albrechts-Universitt Kiel, Department of economics, No. 2011,04, Pro-

vided in Cooperation with: Christian-Albrechts-University of Kiel, Department of eco-

nomics.

[34] Levy M, and Solomon S. New evidence in the power-law distribution of wealth. Physica

A.1997; 242(12):9094.

[35] Li J, Sheng Z, and Liu H. Multi-agent simulation for the dominant players behavior in

supply chains. Simulation Modelling Practice and Theory. 2010; 18: 850–859.

40

[36] Liua X, Gregorc S, Yang J. The effects of behavioral and structural assumptions in artiﬁ-

cial stock market. Physica A. 2008; 387: 25352546.

[37] Luther W. Cryptocurrencies, Network Effects, and Switching Costs. Mercatus Center

Working Paper No. 13-17.2013.

[38] Lux T, Marchesi M. Volatility Clustering in Financial Markets: a Microsimulation of

Interacting Agents. International Journal of Theoretical and Applied Finance. 2000; 3(4):

675-702.

[39] Moore T. The promise and perils of digital currencies, International Journal of Critical In-

frastructure Protection. Agent-based Computational Economics. 2013; 6(3-4):147-149.

[40] Nakamoto S. Bitcoin: A Peer-to-Peer Electronic Cash System. www.Bitcoin.org. 2009.

[41] Newman M.E.J. Power laws, Pareto distributions and Zipf’s law, Contemporary Physics.

2005; 46(5): 323-351.

[42] Pagan A. The econometrics of ﬁnancial markets. J. Empirical Finance. 1996; 3: 15-102.

[43] Palmer R, Arthur W. B, Holland J. H, LeBaron B, and Tayler P. Artiﬁcial economic life:

a simple model of a stock market. Physica D. 1994; 75: 264–274.

[44] Ponta L, Pastore S, and Cincotti S. Information-based multi-assets artiﬁcial stock mar-

ket with heterogeneous agents. Nonlinear Analysis: Real World Applications. 2011; 12:

1235–1242.

[45] Ponta L, Scalas E, Raberto M. , and Cincotti S. Statistical Analysis and Agent-Based

Microstructure Modeling of High-Frequency Financial Trading. IEEE Journal of selected

topics in signal processing. 2012; 6(4): 381-387.

[46] Raberto M, and Cincotti S, and Focardi S, and Marchesi M. Agent-based simulation of a

ﬁnancial market. Physica A. 2001; 299(1): 319-327.

[47] Raberto M, Cincotti S, Focardi S, and Marchesi M. Agent-based simulation of a ﬁnancial

market. Physica A. 2001; 299(1): 319–327.

[48] Raberto M, Cincotti S, Focardi S, and Marchesi M.Traders’ long-run wealth in an arti-

ﬁcial ﬁnancial market. Society for Computational Economics, Computing in Economics

and Finance. 2002; 301.

[49] Raberto M, Cincotti S, Focardi S, and Marchesi M. Traders’ Long-Run Wealth in an

Artiﬁcial Financial Market. Computational Economics, Kluwer. 2003; 22(3): 255–272.

41

[50] Raberto M, and Cincotti S, and Dose C, and Focardi S, and Marchesi M. Price formation

in an artiﬁcial market: limit order book versus matching of supply and demand. Nonlinear

Dynamics and Heterogenous Interacting Agents, Springer-Verlag. Berlin 2005.

[51] Rohitratana J, and Altmann J. Impact of pricing schemes on a market for Softwareas-a-

Service and perpetual software. 2012.

[52] Ron D, and Shamir A. Quantitative Analysis of the Full Bitcoin Transaction Graph. Fi-

nancial Cryptography and Data Security Volume 7859 of the series Lecture Notes in

Computer Science. 2013; 6–24.

[53] Singh P, and Chandavarkar B. R, and Arora S, and Agrawal N. Performance Comparison

of Executing Fast Transactions in Bitcoin Network Using Veriﬁable Code Execution.

Second International Conference on Advanced Computing, Networking and Security.

2013.

[54] Takayasu, H. Fractals in the Physical Sciences. John Wiley and Sons, New York.1990.

[55] Westerhoff F, and Franke R. Converse trading strategies, intrinsic noise and the stylized

facts of ﬁnancial markets. Quantitative Finance. 2012; 12(3): 425–436.

[56] Xiaoming Y, Cao P, Zhang M, and Liu K. The optimal production and sales policy for a

new product with negative word-of-mouth. Journal of Industrial and Management Opti-

mization 7. 2011; 117–137.

[57] O’Dwyer Karl J, and David Malone. Bitcoin Mining and its Energy Footprint. Irish Sig-

nals and Systems Conference 2014 and 2014 China-Ireland International Conference on

Information and Communications Technologies. 2014.

42