When Bitcoin Mining Pools Run Dry
A Game-Theoretic Analysis of the Long-Term Impact of
Attacks Between Mining Pools
Aron Laszka1 , Benjamin Johnson2 , and Jens Grossklags3
1
Institute for Software Integrated Systems, Vanderbilt University, Nashville, USA
2
CyLab, Carnegie Mellon University, Pittsburgh, USA
3
College of Information Sciences and Technology, Pennsylvania State University,
University Park, USA
Abstract. Bitcoin has established itself as the most successful cryp-
tocurrency with adoption seen in many commercial scenarios. While
most stakeholders have jointly benefited from the growing importance
of Bitcoin, conflicting interests continue to negatively impact the ecosys-
tem. In particular, incentives to derive short-term profits from attacks
on mining pools threaten the long-term viability of Bitcoin.
We develop a game-theoretic model that allows us to capture short-term
as well as long-term impacts of attacks against mining pools. Using this
model, we study the conditions under which the mining pools have no
incentives to launch attacks against each other (i.e., peaceful equilibria),
and the conditions under which one mining pool is marginalized by at-
tacks (i.e., one-sided attack equilibria). Our results provide guidelines
for ensuring that the Bitcoin ecosystem remains long-term viable and
trustworthy.
Keywords: Bitcoin, Mining, Attacks, DDoS, Game Theory
1 Introduction
Conceived in 2008, Bitcoin is a cryptocurrency system which is controlled through
an online communication protocol and facilitated in a decentralized fashion [15].
Bitcoin has experienced considerable growth in popularity and constitutes the
dominant cryptocurrency [2]. It has also increasingly found adoption as a viable
payment scheme in mainstream electronic commerce.
Despite setbacks, such as the closure of the Mt. Gox exchange, most stake-
holders of the Bitcoin ecosystem have profited from its development and expect
to benefit also in the future from the trust placed in the cryptocurrency. As such,
participants in the Bitcoin ecosystem share a common goal with the improve-
ment (or avoidance of the erosion) of trust of the currency system. Interactions in
most economic systems usually involve such common but at the same time also
conflicting interests [18]. In the Bitcoin ecosystem, such misaligned incentives
are manifested in several ways.
Most centrally, the process of mining new bitcoins is organized in the form of
a race in which the miner that solves a proof-of-work task first will be rewarded;
all other miners will leave empty-handed. Mining involves a probabilistic ele-
ment, so that not only the most powerful miner would win a particular round
of this competition with certainty. Nevertheless, individual miners have found
it beneficial to join forces in the form of mining pools. For example, averag-
ing mining proceeds across many participants makes earnings more predictable.
The specific setup of each mining pool typically differs across several dimensions
which can be tangible (e.g., related to the computing and communication infras-
tructure) or intangible (such as reputation or details of the payout schemes). We
term the sum of these factors the attractiveness of a mining pool.
However, the decentralized and quasi-anonymous nature of the Bitcoin ecosys-
tem also lowers the bar for unfair competition in the form of different attacks
which can benefit a malicious mining pool. First, attackers may abuse the re-
sources of unsuspecting computer users for mining purposes through security
compromises [8,16]. Second, attackers may attempt to redirect or siphon off
mining capabilities from a competing pool [12]. Third, attackers may diminish
the mining power of competing pools, for example, through Distributed Denial
of Service (DDoS) attacks [20], or by exploiting specific weaknesses in the im-
plementation of the procedure/software used by a particular pool.
The third dimension of unfair competition has been the focus of two recent
research contributions. Vasek et al. provided empirical evidence that, during a
two-year period, about 29% of all known mining pools had been the subject of
at least one DDoS attack, and for mining pools above 5% share of hash rate, the
likelihood of suffering an attack was 63%. They further characterized the types
of pools that are more likely to be attacked [20]. Building on these findings,
we developed a game-theoretic model in [9], which investigated the adversarial
interaction between two representative mining pools that can choose between
productive and destructive investments (i.e., computing power vs. DDoS attack
on its competitor). We found that the relative size of the mining pools is a critical
factor for the incentives to engage in attacks.
Both research studies primarily focus on the immediate impact of attacks
on mining pools, i.e., the temporary shutdown of mining power and its payoff
consequences. However, previous research on DDoS attacks in the context of
electronic commerce has shown that the future (medium or long-term) impact
of service unavailability can actually be more significant [6]. In the context of
mining pools, individual members can permanently shift to an unaffected pool,
which lowers the future prospects of the attacked pool. Even though the long-
term impact of attacks can have a strong influence on the behavior of service
providers, this phenomenon has not been studied from a theoretical perspec-
tive for DDoS attacks in general, or in the context of Bitcoin mining pools in
particular [9].
In this paper, we develop a game-theoretic model that allows us to investigate
the long-term impact of attacks against mining pools. Using this model, we study
the conditions under which the mining pools have no incentives to launch attacks
against each other (i.e., peaceful equilibria), and the conditions under which one
mining pool is marginalized by attacks (i.e., one-sided attack equilibria). Our
results provide guidelines for ensuring that the Bitcoin ecosystem remains long-
term viable and trustworthy.
The remainder of this paper is organized as follows. In Section 2, we discuss
related work relevant to the context of DDoS attacks in networked systems. We
describe our model and key assumptions in Section 3. We conduct our analysis
and present numerical results/illustrations in Sections 4 and 5, respectively. We
offer concluding remarks in Section 6.
2 Related Work
Decision-making in the context of security has been extensively studied using
various game-theoretic approaches [10,14]. Of particular interest to our work
are studies which address the incentives for adversarial behaviors. For example,
Schechter and Smith [17] build upon the literature on the economics of crime to
construct a model of attackers in the computer-security context. The authors de-
rive penalties and probabilities of enforcement that will deter a utility-optimizing
attacker, who evaluates the risks and rewards of committing an offense. Clark
and Konrad propose a game-theoretic model with one defender and one attacker
[4]. In their model, the defending player has to successfully protect multiple
nodes, while the attacker needs to compromise only a single node. Fultz and
Grossklags study the competition between multiple strategic attackers in differ-
ent interdependent decision-making scenarios [5,7].
Previous economic work has improved our understanding of DDoS attacks
and potential countermeasures. Focusing on defender behaviors, Christin et al.
investigate the incentives of a group of bounded rational agents when they face
the threat of being absorbed into a botnet, e.g., for the purpose of a DDoS attack
[3]. In contrast, Liu et al. model attackers and work towards identifying DDoS
attacker strategies in a specific case study [13]. Li et al. model the incentives of
a botnet master to maintain a zombie network for the primary purpose of rent-
ing a sufficiently large subset to a DDoS attacker [11]. The authors investigate
whether this business relationship can remain profitable if defenders can pollute
the botnet with decoy machines (which lowers the effectiveness of a DDoS at-
tack). In addition, there are several other research studies which are concerned
with the organization of effective countermeasures against DDoS [19,21].
As discussed in the introduction, our current work draws on Vasek et al. who
provided empirical evidence on the prevalence of DDoS attacks in the Bitcoin
economy [20]. They showed, for example, that the size of mining pools is related
to the probability of being targeted by an attack. Those findings motivated us to
develop a game-theoretic model of attack behaviors between two mining pools
which is, however, restricted to studying short-term effects of attacks [9].
3 Model
3.1 Overview
Our modeling framework is designed to capture two distinct effects of attacks
against mining pools. The first, and most obvious is a short-term effect on the
revenue of the attacked pool. While an attack is ongoing, the communication
of the pool is disrupted, and hence the revenue decreases. The second effect
is a longer term decrease in the size of the pool. Due to the myopic behavior
of miners, an ongoing attack may cause some miners to permanently leave the
attacked pool and mine for other pools.
In our previous paper [9], we focused on the first effect and did not take into
account the second. In this paper, we extend our analysis to incorporate both
effects using a sequential game played over an indefinite number of rounds. In
each round, the choices of players result in both short-term revenue consequences,
which affect player utilities, as well as long-term migration consequences, which
affect the relative sizes of pools in the next round.
Miner migration is an interesting feature in itself. Our modeling framework
assumes that there is some level of migration in each round, regardless of any
attacks. That is to say there is a percentage of miners who are sufficiently fluid
in their preferences that they re-evaluate their choice of pool each round. The
remaining percentage of miners in a given round will continue mining for the
same pool in the next round.
Our main focus in studying this model will be determining steady-state equi-
librium strategies. These are strategies that stabilize the long-term migration
effects, so that the players’ sizes remain the same from round to round; and that
also constitute best-response strategies for each player. Steady-state equilibria
are consistent with what we observe in the Bitcoin ecosystem, where we observe
little change in the relative sizes of pools from round to round.4
Table 1 summarizes the notations used in the model.
3.2 Players
Our game has exactly two players: a bigger mining pool B and a smaller mining
pool S. Each pool has a base level of attractiveness which we parameterize with
two constants AB , AS . We may interpret AB (for example) as the percentage of
fluid miners who will migrate to pool B in the next round.
In contrast to the attractiveness levels which are fixed for the duration of
(k)
the game, each pool also has a current size for each round. For example, sB
is the relative size of pool B in round k. Relative size is interpreted to mean
the percentage of hash power possessed by its miners, compared to the entire
Bitcoin ecosystem.
4
Steady-state equilibrium analysis has been used relatively sparingly in the security
economics literature (see, for example, [1]), while it is a frequently employed solution
concept in other areas of economics.
Table 1: Table of Notations
Symbol(s) Constraints Description
M ∈ [0, 1] Base miner migration rate
C ∈ [0, 1] Unit cost of attack
AB , AS ∈ [0, 1] and Relative attractiveness of the pool
AB + AS ∈ [0, 1]
(k) (k)
sB , sS ∈ [0, 1] and Relative size of the pool in round k
(k) (k)
sB + sS ∈ [0, 1]
(k) (k)
aB , aS ∈ [0, 1] Attack level of the pool in round k
From round to round, the sizes of pools may change; and in fact it may
happen that a bigger pool becomes a smaller pool in the next round. To be
consistent with our terminology then, the salient feature we use to distinguish
B from S is the assumption that AB ≥ AS .
3.3 Choices
In each round, players simultaneously choose an attack level in [0, 1]. In round
(k) (k)
k, pool B chooses aB , while pool S chooses aS . The attack level is intended to
be interpreted generically, independent of the specific attack form. The attack
could be distributed denial of service (DDoS), or any other form of adversarial
action that disrupts the attacked pool’s mining efforts.
3.4 Consequences
The choices of mining pools affect both the short-term utilities of players, as
well as the longer-term size of each pool as a result of miner migration.
Short-Term Consequences Let C ∈ [0, 1] be the per unit cost of an attack,
then the utility of pool B in round k can be expressed in terms of the relative
sizes of B and S via
(k) (k)
(k) sB · (1 − aS ) (k)
uB = (k) (k) (k) (k)
− C · aB . (1)
1 − sB · aS − sS · aB
(k) (k)
In the above formula, sB · (1 − aS ) is the mining power of B considering S’s
(k) (k) (k)
attack, 1 − sB · aS − sS is the mining power of the whole Bitcoin ecosystem
(k)
considering both attacks, and C · aB is the total cost of attack incurred by B.
The utility function is designed to correspond directly to the percentage of
mining revenue obtain by the pool in the given round. The relative amount of
coins being mined in round k is decreased by the two players’ attacks, which
explains the denominator; while the relative amount of coins being mined by
pool B in round k is affected proportionally to the the attack level against pool
B by pool S, which explains the numerator.
Note that by symmetry, we have the utility of S in round k as
(k) (k)
(k) sS · (1 − aB ) (k)
uS = (k) (k) (k) (k)
− C · aS . (2)
1 − sS · aB − sB · aS
Long-Term Consequences Attack strategies also have long-term consequences,
that do not affect the players’ immediate revenue, but do affect miner migration,
and hence the relative sizes of the pools in the next round.
In each round, the miners that are affected by an attack re-evaluate their
(k) (k) (k) (k)
choices and start to migrate. Formally, in each round, sB · aS (or sS · aB )
miners leave pool B (or S) due to attacks. The group of migrating miners redis-
tribute themselves in the next round among the pools in a manner proportional
to each pool’s relative attractiveness level (AB , AS , or 1 − AB − AS , for pool B,
pool S, and all other pools, respectively).
Miners may also re-evaluate their choices from time to time even when they
are not affected by an attack. Let M ∈ [0, 1] denote the level of this base mi-
(0) (0)
gration. If M = 0, then from any initial state sB , sS , the pool sizes remain
constant from round to round when there are no attacks. If M = 1, then every
miner re-evaluates her pool choice in each round. Similarly to migration due to
attacks, the group of migrating miners redistribute themselves among the pools
in a manner proportional to each pool’s relative attractiveness level.
We may thus express the relative size of pool B in round k + 1 in terms of
the relative sizes of pools in round k, together with attack levels and the base
migration rate:
(k+1) (k)
sB = sB
(k) (k) (k)
+ AB · [(1 − sB ) · M + sS aB (1 − M )] (migration into B)
(k) (k)
− sB · (1 − AB ) · [M + aS (1 − M )] (migration out of B). (3)
Analogously, the relative size of pool S in round k + 1 may be expressed as
(k+1) (k)
sS = sS
(k) (k) (k)
+ AS · [(1 − sS ) · M + sB aS (1 − M )] (migration into S)
(k) (k)
− sS · (1 − AS ) · [M + aB (1 − M )] (migration out of S). (4)
4 Model Analysis
We begin the analysis of our modeling framework by proving a uniqueness result
for each pool’s relative size in a steady-state equilibrium.
4.1 Steady-State Pool Sizes
Theorem 1. If M > 0, then for any strategy profile (aS , aB ), there exists a
unique pair of relative sizes (s∗S , s∗B ) such that
(k+1) (k+1) (k) (k)
(sS , sB ) = (sS , sB ) = (s∗S , s∗B ).
Proof. Given a strategy profile (aS , aB ), the conditions of the theorem require
(s∗S , s∗B ) to satisfy
s∗B · (1 − AB ) · [M + aS (1 − M )] (migration out of B)
= AB · [(1 − s∗B ) ·M + s∗S aB (1 − M )] (migration into B)
and
s∗S · (1 − AS ) · [M + aB (1 − M )] (migration out of S)
= AS · [(1 − s∗S ) ·M + s∗B aS (1 − M )] (migration into S).
Solving B’s migration equation for s∗B , we obtain
AB · [M + s∗S aB (1 − M )]
s∗B = .
M + aS (1 − AB )(1 − M )
This linear constraint
is satisfied by all points
(s
S , sB ) on the line segment
con-
AB M AB [M +aB (1−M )]
necting the points 0, M +aS (1−A B )(1−M )
and 1, M +aS (1−AB )(1−M ) .
Similarly, the migration equation for S may be reduced to the linear con-
straint
−AS M + s∗S [M + aB (1 − AS )(1 − M )]
s∗B = ,
AS aS (1 − M )
which
is satisfied
by
all pairs (sS , sB ) on the
line segment connecting the points
−M (1−AS )[M +aB (1−M )]
0, aS (1−M ) and 1, aS AS (1−M ) .
We now wish to show that these two segments must intersect in the interval
[0, 1]. More precisely, we claim that the second segment starts below the first
segment when sS = 0; and ends above the first segment when sS = 1.
The two relevant inequalities are:
−M AB M
< (5)
aS (1 − M ) M + aS (1 − AB )(1 − M )
and
AB [M + aB (1 − M )] (1 − AS )[M + aB (1 − M )]
< (6)
M + aS (1 − AB )(1 − M ) aS AS (1 − M )
Inequality (5) follows because the first term is negative while the second term
is positive (or zero if AB = 0). Inequality (6) follows from:
AB [M + aB (1 − M )] (1 − AS )[M + aB (1 − M )]
≤ (AS + AB ≤ 1)
M + aS (1 − AB )(1 − M ) M + aS AS (1 − M )
(1 − AS )[M + aB (1 − M )]
< (M > 0).
aS AS (1 − M )
t
u
As a result of the theorem, the unique steady-state solution can be expressed
directly in terms of the strategies aS and aB , the attractiveness levels AS and
AB , and the migration constant M :
AS M [M + aS (1−M )]
s∗S =
[M + aB (1−AS )(1−M )][M + aS (1−AB )(1−M )] − AB aB AS aS (1−M )2
(7)
AB M [M + aB (1−M )]
s∗B = .
[M + aB (1−AS )(1−M )][M + aS (1−AB )(1−M )] − AB aB AS aS (1−M )2
(8)
Furthermore, we know that these values are in [0, 1] under our modeling assump-
tions without having to do further case analysis.
4.2 Steady-State Pool Utilities
Since there is a unique pair of steady-state pool sizes for each strategy profile,
we can find a steady-state equilibrium by assuming that the pool sizes and,
hence, the players’ utilities are given by Equations (7) and (8). In other words,
given a strategy profile (aS , aB ), we may write the utility of each pool under the
assumption that the relative sizes are the steady-state sizes s∗S and s∗B .
For pool S, we obtain
AS M [M + aS (1 − M )](1 − aB )
uS = − aS C, (9)
Denominator
and for pool B, we have
AB M [M + aB (1 − M )](1 − aS )
uB = − aB C, (10)
Denominator
where
Denominator = M + aS (1 − M − AB )][M + aB (1 − M − AS )]
+ aS aB [(1 − AS − AB )(1 − M )2 − AS AB ]. (11)
This formulation will permit us to determine all the steady-state equilibria
for the sequential game presented in Section 3 by finding Nash equilibria for a
single-shot two-player game with the above utilities.
4.3 Peaceful Equilibria
We begin our analysis of equilibria by determining the conditions under which
it is a stable strategy profile for each player to refrain from attacking the other
player.
Theorem 2. The strategy profile (aS , aB ) = (0, 0) is a Nash equilibrium just in
case
AB AS
C≥ . (12)
min{M, 1 − AB , 1 − AS }
Proof. Suppose that aS = 0. We want to characterize the conditions under which
aB = 0 is a best response. First, we express the utility of B in a steady state by
substituting aS = 0 into Equation(10), obtaining
AB [M + aB (1 − M )]
uB = − aB C . (13)
[M + aB (1 − M − AS )]
When B does not attack (aB = 0), her resulting steady-state utility is uB =
AB ; while if she attacks with full force (aB = 1) her utility becomes uB =
AB
1−AS −C. Because the utility function is analytic in aB , any intermediate attack
level can only be a utility-maximizing response strategy if the partial derivative
of uB with respect to aB evaluated at that specific attack level is zero.
Computing the first and second partial derivatives of uB with respect to aB ,
we obtain
∂uB AB AS M
∂aB
= 2 −C , (14)
[M + aB (1 − M − AS )]
and
∂ 2 uB −2AB AS M (1 − M − AS )
2 = 3 . (15)
∂aB [M + aB (1 − M − AS )]
Since the denominator of Equation (15) is always positive, the second deriva-
tive itself is of constant sign; and this sign is negative if and only if 1−M −AS > 0,
or equivalently M < 1 − AS . It is only in this case where the roots of the first
derivative will give relevant maximizing solutions for the attack level.
In the case M > 1 − AS , the roots of the first derivative will give minimizing
attack levels, and in the case M = 1 − AS the first derivative will be constant,
and hence the utility B will be maximized at either one of the endpoints of [0, 1],
or on the entire interval.
Setting the derivative from Equation (14) equal to zero and solving for aB ,
we obtain q
AB AS M
C −M
aB = . (16)
1 − AS − M
Now we consider two parameter cases.
– First, if M ≥ 1 − AS , then the maximizing attack level is one (or both) of
the two endpoints 0 or 1. In this case an optimal response is aB = 0 exactly
AB
when AB ≥ 1−A S
− C, or equivalently, when
AB AS AB AS
C≥ = .
1 − AS min{M, 1 − AS }
– Second, if M < 1 − AS , then the maximizing attack level will be zero if and
only if the point at which the derivative is zero is non-positive. Since we
are in the case 1 − AS < M , we may deduce from Equation
q (16) that the
AB AS M
analytically-maximizing aB is non-positive if and only if C − M ≥ 0.
This condition reduces to
AB AS AB AS
C≥ = .
M min{M, 1 − AS }
These two parameter cases exhaust all options; and we have shown that in
each case aB = 0 is a best response to aS = 0 if and only if
AB AS
C≥ .
min{M, 1 − AS }
To have (aS , aB ) = (0, 0) be an equilibrium, we also need aS = 0 to be a best
response to aB = 0. By symmetry, this will happen if and only if
AB AS
C≥ .
min{M, 1 − AB }
We conclude that a peaceful equilibrium exists if and only if
AB AS
C≥ .
min{M, 1 − AB , 1 − AS }
t
u
4.4 One-Sided Attack Equilibria
Our next special case to consider is when exactly one of the players attacks while
the other remains peaceful.
Theorem 3. The strategy profile (aS , aB ) = (0, 1) forms a Nash equilibrium if
and only if
AB AS
C≤ · min{M, 1 − AS } (17)
(1 − AS )2
Proof. First suppose that aB = 1. Then the utility of S is given by
uS = −aS C, (18)
and this quantity is clearly maximized for aS ∈ [0, 1] by taking aS = 0. So aS = 0
is always a best response to aB = 1.
Next suppose that aS = 0. We want to characterize the conditions under
which aB = 1 is a best response. From the previous special case analysis, we
have
AB [M + aB (1 − M )]
uB = − aB C .
[M + aB (1 − M − AS )]
Exactly as in the previous analysis, when aB = 0, we have uB = AB ; and if
AB
aB = 1, we get uB = 1−A S
− C. The same conditions applied to the derivative of
uB determine when the maximizing value of aB is reached at a boundary point
(0 or 1) or whether it may relate to where the derivative is zero at
q
AB AS M
C −M
aB = .
1 − AS − M
The derivative value is relevant if and only if M < 1 − AS .
In the case where M ≥ 1 − AS , a maximizing attack level is one of the
endpoints of [0,1]; and in this parameter case, the best response is aB = 1 if and
only if
AB AS AB AS
C≤ = · min{M, 1 − AS }.
1 − AS (1 − AS )2
In the case M < 1 − AS , the maximizing value is 1 if and only if the global
analytically-derived maximum is at least 1. This happens when
q
AB AS M
C −M
≥1
1 − AS − M
r
AB AS M
− M ≥ 1 − AS − M
C
AB AS M
≥ (1 − AS )2
C
AB AS M
≥C
(1 − AS )2
AB AS
C≤ · min{M, 1 − AS }
(1 − AS )2
Again these two parameter cases exhaust all options; and in each case (aS , aB ) =
(0, 1) is an equilibrium configuration if and only if
AB AS
C≤ · min{M, 1 − AS }.
(1 − AS )2
t
u
To have (aS , aB ) = (1, 0) be an equilibrium, we would need the condition
AB AS
C≤ · min{M, 1 − AB }.
(1 − AB )2
Of course both conditions may be simultaneously satisfied for C sufficiently
small, in which case both one-sided attack configurations will be equilibria.
5 Numerical Illustrations
5.1 The Peaceful Equilibrium
(a) AS = 0.2, AB = 0.3 (b) M = 0.5, C = 0.1
Fig. 1: Existence of the peaceful equilibrium (i.e., aS = 0 and aB = 0). Lighter
shaded areas represent parameter combinations where the peaceful equilibrium
exists.
Figure 1 shows the parameter combinations where the peaceful equilibrium
exists (i.e., aS = 0 and aB = 0 being best responses to each other). In Figure 1(a),
we fix the parameters AS = 0.2 and AB = 0.3, and we vary the parameters M
and C. The figure shows that, if both M and C are high, then the peaceful
equilibrium is possible; however, if either of these parameters is low, then there
can be no peace. The latter is especially important in the case of M , which
has no effect on the existence of the peaceful equilibrium once its value reaches
1 − Aj (0.7 in the figure). In practice, this means that both the pools and the
users have to act in order to reach the peaceful equilibrium: the pools have to
employ defensive countermeasures which increase C, while the users have to be
proactive in their mining-pool choice, which increases M .
In Figure1(b), we fix the parameters M = 0.5 and C = 0.1, and we vary the
parameters AS and AB . The figure shows that the peaceful equilibrium exists
if either one (or both) of the pools has a low attractiveness. In practice, this
means that a healthy competition between the pools, in which both of them try
to attract miners, is very beneficial: not only will it result in better deals for the
miners, but it may also bring peace.
5.2 One-Sided Attack Equilibria
(a) AS = 0.2, AB = 0.3 (b) M = 0.5, C = 0.1
Fig. 2: Existence of one-sided attack equilibria. Shades from darker to lighter:
no one-sided attack equilibrium, (aS = 1, aB = 0) forms an equilibrium, (aS =
0, aB = 1) forms an equilibrium, both form equilibria.
Figure 2 shows the parameter combinations where one-sided attack equilibria
exist (i.e., when pool i ∈ {S, B} playing ai = 1 and the other pool playing aī = 0
forms an equilibrium). In Figure 2, we fix the parameters AS = 0.2 and AB = 0.3,
and we vary the parameters M and C. The figure shows that one-sided attack
equilibria are more likely to exist when M is high but C is low. This means
that, to avoid a one-sided attack equilibrium, the pools must employ defensive
countermeasures that increase the cost of attack C. Furthermore, the figure
also shows that less attractive pools are more likely to play a one-sided attack
strategy (darker shaded middle section representing (aS = 1, aB = 0)). While
this may seem counterintuitive at first, it is actually very easy to explain. The
more attractive pool has more miners even without launching an attack; hence,
it is less inclined to dominate the other player with a marginalizing attack. The
less attractive pool, on the other hand, has a lot to gain from such an attack;
hence, it is more inclined to try to “steal” the miners of the more attractive pool.
In Figure 2(b), we fix the parameters M = 0.5 and C = 0.1, and we vary the
parameters AS and AB . The figure shows that, once a pool is highly attractive
to the miners, the other pool will have an incentive to launch a marginalizing
attack against it. While attacks are generally harmful to the Bitcoin ecosystem,
they have positive effects in this context, as they prevent one pool from growing
too large.
6 Conclusion and Future Work
In this paper, we proposed a game-theoretic model of attacks between Bitcoin
mining pools, which – to the best of our knowledge – is the first study to consider
long-term consequences. The analysis of our model has revealed a number of
interesting implications for making the Bitcoin ecosystem more viable. We have
seen that, in order to make the peaceful equilibrium viable, the unit cost of
attack and the miners’ base migration rate both have to be increased, and no
pool can have an overwhelming attractiveness. We have seen that these factors
also help preventing a marginalizing attack against one pool.
We limited the mining pools’ strategic choices to launching attacks and as-
sumed that the effects of defensive countermeasures are incorporated into the
unit cost of attack. In future work, we can extend this model by allowing the
pools to deploy additional defenses for some fixed cost, which decrease the ef-
fectiveness of attacks. As another direction, we can also extend the model by
allowing the pools to choose some of the parameters that affect their attractive-
ness levels. For example, a pool could choose to increase its fee, which decreases
its attractiveness but increases its utility for a given steady-state size. Finally,
in this paper, we modeled only two pools as strategic players, but we intend to
extend our work towards the case of multiple mining pools.
Acknowledgments
We thank the reviewers for their detailed feedback. This work was supported in
part by the National Science Foundation under Award CNS-1238959.
References
1. Bensoussan, A., Kantarcioglu, M., Hoe, S.: A game-theoretical approach for finding
optimal strategies in a botnet defense model. In: Alpcan, T., Buttyan, L., Baras, J.
(eds.) Decision and Game Theory for Security, Lecture Notes in Computer Science,
vol. 6442, pp. 135–148. Springer Berlin Heidelberg (2010)
2. Böhme, R., Christin, N., Edelman, B., Moore, T.: Bitcoin. Journal of Economic
Perspectives (forthcoming)
3. Christin, N., Grossklags, J., Chuang, J.: Near rationality and competitive equilib-
ria in networked systems. In: Proceedings of the ACM SIGCOMM Workshop on
Practice and Theory of Incentives in Networked Systems. pp. 213–219 (2004)
4. Clark, D., Konrad, K.: Asymmetric conflict: Weakest link against best shot. Journal
of Conflict Resolution 51(3), 457–469 (Jun 2007)
5. Fultz, N., Grossklags, J.: Blue versus red: Towards a model of distributed secu-
rity attacks. In: Proceedings of the 13th International Conference on Financial
Cryptography and Data Security (FC). pp. 167–183 (Feb 2009)
6. Goldfarb, A.: The medium-term effects of unavailability. Quantitative Marketing
and Economics 4(2), 143–171 (Jun 2006)
7. Grossklags, J., Christin, N., Chuang, J.: Secure or insure? A game-theoretic anal-
ysis of information security games. In: Proceedings of the 2008 World Wide Web
Conference (WWW’08). pp. 209–218 (Apr 2008)
8. Huang, D.Y., Dharmdasani, H., Meiklejohn, S., Dave, V., Grier, C., McCoy, D.,
Savage, S., Weaver, N., Snoeren, A.C., Levchenko, K.: Botcoin: Monetizing stolen
cycles. In: Proceedings of the 2014 Network and Distributed System Security Sym-
posium (NDSS) (2014)
9. Johnson, B., Laszka, A., Grossklags, J., Vasek, M., Moore, T.: Game-theoretic
analysis of DDoS attacks against Bitcoin mining pools. In: Böhme, R., Brenner,
M., Moore, T., Smith, M. (eds.) Financial Cryptography and Data Security (Pro-
ceedings of the 1st Workshop on Bitcoin Research), pp. 72–86. Lecture Notes in
Computer Science, Springer Berlin Heidelberg (2014)
10. Laszka, A., Felegyhazi, M., Buttyán, L.: A survey of interdependent information
security games. ACM Computing Surveys 47(2), 23:1–23:38 (Aug 2014)
11. Li, Z., Liao, Q., Blaich, A., Striegel, A.: Fighting botnets with economic uncer-
tainty. Security and Communication Networks 4(10), 1104–1113 (Oct 2011)
12. Litke, P., Stewart, J.: BGP hijacking for cryptocurrency profit (Aug 2014),
available at: http://www.secureworks.com/cyber-threat-intelligence/threats/bgp-
hijacking-for-cryptocurrency-profit/
13. Liu, P., Zang, W., Yu, M.: Incentive-based modeling and inference of attacker
intent, objectives, and strategies. ACM Transactions on Information and System
Security 8(1), 78–118 (Feb 2005)
14. Manshaei, M., Zhu, Q., Alpcan, T., Bacşar, T., Hubaux, J.P.: Game theory meets
network security and privacy. ACM Computing Surveys 45(3), 25:1–25:39 (July
2013)
15. Nakamoto, S.: Bitcoin: A peer-to-peer electronic cash system. http://bitcoin.
org/bitcoin.pdf (2008)
16. Plohmann, D., Gerhards-Padilla, E.: Case study of the miner botnet. In: Proceed-
ings of the 4th International Conference on Cyber Conflict (CYCON). pp. 345–360
(2012)
17. Schechter, S., Smith, M.: How much security is enough to stop a thief? In: Wright,
R. (ed.) Financial Cryptography, Lecture Notes in Computer Science, vol. 2742,
pp. 122–137. Springer Berlin Heidelberg (2003)
18. Schelling, T.: The Strategy of Conflict. Oxford University Press, Oxford, UK (1965)
19. Spyridopoulos, T., Karanikas, G., Tryfonas, T., Oikonomou, G.: A game theoretic
defence framework against DoS/DDoS cyber attacks. Computers & Security 38,
39–50 (Oct 2013)
20. Vasek, M., Thornton, M., Moore, T.: Empirical analysis of denial-of-service attacks
in the bitcoin ecosystem. In: Proceedings of the 1st Workshop on Bitcoin Research
(BITCOIN). pp. 57–71. Lecture Notes in Computer Science, Springer (March 2014)
21. Wu, Q., Shiva, S., Roy, S., Ellis, C., Datla, V.: On modeling and simulation of game
theory-based defense mechanisms against DoS and DDoS attacks. In: Proceedings
of the 2010 Spring Simulation Multiconference. pp. 159:1–159:8 (2010)
