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Incentivizing Energy Efficiency and Carbon Neutrality in Distributed Computing Via Cryptocurrency Mechanism Design

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Cryptocurrencies and blockchain technologies have exploded in popularity in recent years. However, a huge amount of energy is consumed by many of these cryptocurrencies, exacerbating climate change, and most of the energy is used for computations which have no value other than securing the blockchain. In light of this fact, a number of cryptocurrencies were created with the intent of being secured by, or rewarding, useful computations. In this paper, we design a reward mechanism for the distributed computing platform BOINC (the Berkeley Open Infrastructure for Network Computing), which is used by many universities and other institutions around the world. The reward mechanism is separate from, but intended to be implemented on top of, a cryptocurrency protocol. The mechanism achieves the three main intended goals of the BOINC credit system via a multi-dimensional generalization of the proportional allocation reward mechanism of Bitcoin, and an anti-cheating mechanism adapted from an existing BOINC cheat-detecting mechanism. Since the generalization must be approximated , we introduce several ways of doing so, and analyze the game theoretic aspects of the approximation. One of these approximations requires a hardware profiling database, which has many use cases well beyond BOINC or cryptocurrencies. We also explore market mechanisms that balance the preferences of BOINC users with the strengths of their machines using the well-known Top Trading Cycles algorithm and some of its generalizations, with the effect of increasing network output and energy efficiency. Additionally, we explore avenues to make the cryptocurrency carbon-neutral. We conclude with an overview of some existing cryptocurrencies that currently reward distributed computing.
Incentivizing Energy Efficiency and Carbon Neutrality in
Distributed Computing Via Cryptocurrency Mechanism
Levi Rybalov
January 1, 2022
Cryptocurrencies and blockchain technologies have exploded in popularity in recent years.
However, a huge amount of energy is consumed by many of these cryptocurrencies, exacer-
bating climate change, and most of the energy is used for computations which have no value
other than securing the blockchain. In light of this fact, a number of cryptocurrencies were
created with the intent of being secured by, or rewarding, useful computations. In this paper,
we design a reward mechanism for the distributed computing platform BOINC (the Berkeley
Open Infrastructure for Network Computing), which is used by many universities and other
institutions around the world. The reward mechanism is separate from, but intended to be
implemented on top of, a cryptocurrency protocol. The mechanism achieves the three main
intended goals of the BOINC credit system via a multi-dimensional generalization of the pro-
portional allocation reward mechanism of Bitcoin, and an anti-cheating mechanism adapted
from an existing BOINC cheat-detecting mechanism. Since the generalization must be approx-
imated, we introduce several ways of doing so, and analyze the game theoretic aspects of the
approximation. One of these approximations requires a hardware profiling database, which has
many use cases well beyond BOINC or cryptocurrencies.
We also explore market mechanisms that balance the preferences of BOINC users with the
strengths of their machines using the well-known Top Trading Cycles algorithm and some of its
generalizations, with the effect of increasing network output and energy efficiency. Additionally,
we explore avenues to make the cryptocurrency carbon-neutral. We conclude with an overview
of some existing cryptocurrencies that currently reward distributed computing.
1 Introduction
There are currently thousands of cryptocurrencies in existence, and the technology is under heavy
scrutiny by critics who call attention to the environmental impact of many Proof-of-Work (PoW)
based coins, most notably Bitcoin. A common criticism of PoW coins like Bitcoin is that the
SHA-256 hashing algorithm it uses has no useful output. A number of cryptocurrencies have been
designed to deal with this issue. In this paper, we design a reward mechanism for distributed
computing using the distributed computing platform BOINC, the Berkeley Open Infrastructure for
Network Computing.
BOINC is a middleware that connects projects with crunchers. Projects are often run by
institutions like universities, but can also be run by individuals, companies, or anyone with a server
and enough willpower. They send computational workloads to volunteers, called crunchers, who
use their hardware to crunch the computational workloads. All details regarding BOINC necessary
for this paper will be explained in Section 1.1. Despite the focus on BOINC, the methods laid out
in this paper can also be applied to other distributed computing platforms.
Before jumping into the details of how to design such a cryptocurrency, there are some perti-
nent questions that must be answered as to why such a cryptocurrency should be created at all.
The first question is, what does this currency do that cannot be done with fiat? Indeed, with a
number of countries exploring government-approved digital currencies, what is even the purpose of
decentralized cryptocurrencies?
One of the primary benefits of a cryptocurrency – and the one that will be most leveraged in this
paper – is that its stakeholders have the ability to determine the incentive mechanisms underlying
it; certain behaviors can be encouraged, and others discouraged. This is enabled by the numerous
checks and balances core to the philosophies underlying cryptocurrencies – the intertwined and
overlapping power centers, like core developers, miners (in PoW based coins) or stakers (in Proof-
of-Stake (PoS) based coins), merchants and individual users, and payment services – combined with
the implicit voting inherent to forking a blockchain in order to update or change the direction of
the coin [8], as well as exchanges, which wield significant power in the cryptocurrency world.
Since BOINC is being used as a foundation for this reward mechanism, it is also worth asking
what the cryptocurrency that enables these rewards could do that already-existing BOINC credits
cannot. Primarily, a cryptocurrency that rewards BOINC computations would enable transferabil-
ity and fungibility, especially in light of the fact that BOINC projects each have their own credit
reward schemes. It is generally acknowledged that mechanism design – the study and design of
incentive structures – is much easier with money than without money, and that transferable utility
(via the ability to use the cryptocurrency as payment) changes (but mostly expands) the range of
possible incentive structures that can be designed.
If we wanted to translate the traditional PoW reward mechanism to a distributed computing
setting, the mechanism for the latter would reward crunchers proportionally to their computational
contributions – what a cruncher puts into the network, they get out of it in cryptocurrency. How-
ever, there is a major difference between traditional PoW mining and computations in distributed
computing. In the former, there is only a single function – in the case of Bitcoin, it is the SHA-
256 hashing algorithm. All hardware will have a fixed hashrate on the algorithm, all else being
equal. However, in distributed computing, there is a wide variety of types of computations, and
the performance of different types of hardware on these computations does not vary linearly with
the properties of the hardware.
1.1 BOINC Preliminaries
What follows is a brief overview of BOINC, particularly the elements of it that are necessary for
understanding this paper; see Anderson’s paper [3] for a comprehensive overview.
BOINC is an open-source middleware that connects two main groups: the first group consists
of projects, which are individuals or institutions that need a large amount of computation that
can be divided into smaller, discrete workloads called jobs; the second consists of individuals called
crunchers who perform those computations with their hardware and return the results to the project
Projects often subdivide the types of computations they need completed into applications. Ap-
plications are usually a program, with each job of that application having different inputs to that
program. A project could have some applications that only run on CPUs, and others that are
GPU-based and requires very little CPU time. This paper will base the reward mechanism on
applications, rather than projects, for reasons that will become clear in later sections.
Since crunchers can either intentionally or unintentionally return bad results, projects will often
send out multiple copies of the same job, which are called instances of that job; each instance is
also referred to as a Work Unit (WU). If a project determines that the results of a WU are valid,
which is often based on finding a quorum of results that agree with each other within some bounds
determined by the project, then that cruncher is awarded credits for completing that WU. The
bounds are necessary because the multitude of different hardware vendors and operating systems
can produce different results despite running the same programs with the same inputs, even when
the results should be deterministic. The credits are non-transferable and non-fungible, and are used
as an incentive to retain crunchers. The goals of the credit system are reproduced from [3] below:
be device neutral: similar jobs (in particular the instances of a replicated job)
should get about the same credit regardless of the host and computing resource on
which they are processed;
be project neutral: a given device should get about the same credit per time
regardless of which project it computes for;
resist credit cheating: i.e. efforts to get credit for jobs without actually processing
In practice, project administrators choose their own credit-granting schemes, and reward differ-
ent amounts of credits for the completion of different WUs based on a variety of factors. Oftentimes,
the credit systems implemented by projects fail to achieve the first goal of the credit system. Since
the actual number of credits awarded is also decided arbitrarily by the projects, the second goal
has not at all been met, with some projects awarding orders of magnitude more credits for approx-
imately the same amount of computation. Likewise, the third goal has not been achieved – the
most recent credit design [1] is known to have vulnerabilities that allow crunchers to receive much
more credit than they should.
Every cruncher has an identifier called the cross-project identifier (CPID), which is an identity
that represents a cruncher and their registered machines across all BOINC projects. Under the
current credit scheme, newly awarded credits for every project are factored into both the total
credit, and a number called the Recent Average Credit (RAC), which is a measure of the most
recent contribution of a CPID to a given project. The maximum RAC that a machine can achieve
is equal to the maximum number of credits that that machine can achieve in a day. The RAC
function has a half-life build-up/decay time of one week, meaning that after one week of consistent
crunching, the machine would achieve 50% of its maximum possible RAC, after two weeks, 75%,
etc. Likewise, if a machine stops crunching at any moment in time, then after one week, its RAC
would be 50% of what it was at the time that the machine stopped crunching, after two weeks,
25%, etc. See Appendix A for a code snippet simulating RAC build-up.
1.2 The Bitcoin Allocation Rule and Its Properties
In [6], Chen, Papadimitriou, and Roughgarden establish an axiomatic approach to block rewards in
the Bitcoin blockchain, demonstrating that the current proportional allocation rule used by Bitcoin
is the unique allocation rule that satisfies a number of properties: non-negativity, symmetry, (strong)
budget balance, sybil-proofness, and collusion-proofness. The definition of proportional allocation
is simple: for a miner i, the expected rewards pifrom a single block is their hash rate hidivided
by the hashrate of all miners:
where [n]=0,1,...n, where nis the number of miners. We will re-use this latter notation in the
rest of this paper. Note that the expected rewards for a miner are linear in that miner’s hash rate.
The non-formal definitions of the aforementioned desired properties are reproduced below:
Non-negativity. Expected rewards (the pi’s) should be nonnegative. That is, the
protocol cannot require payments from miners.
Budget-balance. The protocol cannot be “in the red,” meaning the sum of expected
block rewards cannot exceed the unit of block reward available. Strong budget-
balance insists that the entire unit of block reward is allocated, while weak budget-
balance allows the protocol to withhold some of the block reward from miners.
Symmetry. The allocation rule should not depend on the names of the miners (i.e.,
their public keys), only on their contributed hash rates.
Sybil-proofness. No miner can possibly benefit by creating many accounts and
splitting its mining power among them.
Collusion-proofness. Two or more miners cannot benefit by pooling their mining
resources and somehow splitting the proceeds. This property has different variants
depending on what types of payments between colluding miners are permitted; see
Section 2.2.
The authors made the above claims for the case of risk-neutral miners. They also examined the
cases for risk-averse (and risk-seeking) miners, and supplied corresponding possibility and impossi-
bility results.
1.3 Results
In this paper, we introduce an analogue of the proportional allocation rule used in Bitcoin for a new
BOINC credit system, addressing the first two goals of the credit system described above, as well as
introduce anti-cheating mechanisms for risk-neutral crunchers (with results that can be extended
for risk-seeking crunchers), addressing the third goal of the credit system. We demonstrate how
this new proportional allocation rule and the anti-cheating mechanisms incentivize energy efficiency,
and describe a path towards carbon-neutrality.
In Section 2, a generalization of the proportional allocation rule used in Bitcoin to the multi-
dimensional case is constructed via an Equivalence Ratio between applications, which can be seen
as an exchange rate between the WUs of different applications. This mechanism is also shown to
incentivize energy efficiency. However, calculating the parameters necessary for the generalization
is difficult in practice, so approximations must be made. The necessity of an approximation makes
it so that sybil-proofness and collusion-proofness are no longer automatically guaranteed by the
construction of the rule, and so these properties must be recovered.
In Section 3, a linear regression approach is used to approximate the Equivalence Ratio, and
some issues and game-theoretic aspects of the approximation are examined.
In Section 4, another approach to approximating the Equivalence Ratio based on benchmarks
of every machine on the network is introduced. This approach offers a wide variety of benefits
extending far beyond cryptocurrencies and BOINC, but suffers from the inherent unreliability and
unverifiability of benchmarks returned by crunchers.
In Section 5, an anti-cheating mechanism is introduced, which addresses the third intended goal
of the BOINC credit system. However, a number of open problems remain, especially regarding
the possibility of projects cheating by recognizing fake results as legitimate.
In Section 6, the diverging incentives caused by the difference between applications that reward
crunchers the most and the applications that those crunchers most prefer is addressed by a reduction
to a market problem and invoking the Top Trading Cycle mechanism and its generalizations, with
the most general case remaining an open problem.
In Section 7, opportunities to offset carbon emissions and imbue sustainability into the fabric
of the cryptocurrency are explored.
In Section 8, a brief overview of some existing cryptocurrencies which reward distributed com-
puting is provided.
2 Equivalence Ratio
In this section we develop an allocation rule, which is a function that maps the set of crunchers’
computational contributions to a reward vector. Since each cruncher can control more than one
machine, we frame the problem in terms of machines rather than crunchers. This can also be viewed
as each cruncher being broken up into multiple crunchers, each of whom owns one machine.
For any given machine under constant utilization, any given application with a stationary prob-
ability distribution over parameter inputs, the runtimes of WUs from that application and on that
machine follow some probability distribution. By the Law of Large Numbers and the Central Limit
Theorem, the sample average of the runtimes converges to the true average of the runtimes. Thus,
if rewards are based purely on the number of completed WUs of a particular application, then those
rewards would converge to what they would have been if the rewards had been based on the run-
times of the individual WUs. For that reason, the allocation rule rewards the same amount for each
WU from a given application, which in expectation accomplishes the first goal of the BOINC credit
system. While the remainder of this paper could be easily modified to account for the credit schemes
that projects currently use, dealing with WUs removes many of the complications and potential
manipulations involved with calculating credit, without losing any necessary properties. However,
an allocation rule based on this principle is still susceptible to manipulations via cherry-picking
attacks, which is discussed in Section 5.
2.1 Setting
We can now formalize the setting: stripping down the environment of BOINC to the essential
components, the setting consists of a set of machines M={m1, m2, . . . , m|M|}, and a set of appli-
cations A={a1, a2, . . . , a|A|}. Every machine mMis assumed to have the necessary components
required for crunching WUs (a CPU, RAM, main memory etc.), and may have co-processors like
In order to construct the allocation rule, we need to formally define the computational contri-
bution of a machine. Since each machine can crunch different applications, the concept of a single
hash rate like in Bitcoin – or more generally, a single function – no longer applies. Rather, the
computational contribution of a machine mican be represented as a tuple wi= (wi1, wi2,...wi|A|),
where wij is the WU contribution of machine mito application ajmeasured in some arbitrary time
unit. For simplicity, we can assume that this time unit is equivalent to the length of the period
for which crunchers are rewarded (this is analogous to the time between blocks in Bitcoin, which
is designed to be about 10 minutes), which we will call the reward period. Then, the contribution
of every machine to every application can be summarized in an |M|×|A|matrix W, which is the
input data to the allocation rule.
The output of the allocation rule is a reward vector r= (r1, r2, . . . , r|M|), where riis the reward
allocated to machine mi. In the literature, the reward vector is usually called the payoff vector.
Note that in contrast to the randomized reward allocation rule used in Bitcoin, rewards in this
setting will be deterministic. For now, the reward mechanism should be thought of as being an
independent layer on top of some underlying blockchain protocol, whether PoW, PoS, or some other
protocol – this topic will be explored more in Section 5.
Throughout the remainder of this paper, when machines are referred to as having agency, it
should be assumed that the machine’s owner is the one making the strategic decisions.
2.2 Allocation Rule
The approach to constructing the allocation rule and reward vector is straightforward: find an
equivalence between the WUs of different applications, and use the equivalence to determine a
universal computational contribution for each machine, analogous to the hash rate in Bitcoin.
In order to construct the equivalence, we need to define the WU completion rate, which is a
measure of how many WUs a machine can crunch on an application.
Definition 2.1. Let ρi
jbe the WU completion rate for machine mion application aj. The WU
completion rate ρi
jis the maximum number of WUs that machine mican accomplish on application
ajin a reward period.
In practice, BOINC crunchers can set limitations on how much their machines crunch. For
the purposes of this paper, machines that are not operating at full capacity can be viewed as less
powerful machines operating at full capacity, or virtual machines that have only subsets of their
hosts’ original computational power available (in fact, a number of BOINC applications run in
virtual machines, which negatively impacts performance).
The next step is to define the Equivalence Ratio (ER) of a machine. The ER for a machine is a
tuple consisting of the machine’s WU completion rates on each of the applications. Extending the
notation of ρi
j, denote by ρithe tuple representing the ER for mi. Then, for every i[|M|],
Definition 2.2.
ρi:= (ρi
1, ρi
2, . . . , ρi
However, ρi0obtained by normalization on another mi06=miwould yield a different ER (as-
suming that miand mi0are not identical machines). The next step is to extend the notion of ER
beyond a single machine.
Beginning with the base case, the ER for a set of machines M={m1}is calculated by consid-
ering how much this set can accomplish on each application – in this case, the ER is ρ1. Extending
this logic to two machines, the ER of a set of two machines M={m1, m2}is calculated by con-
sidering how much m1and m2can accomplish on each application together – this would be the
element-wise sum of ρ1and ρ2. Generalizing, the network-wide ER can be calculated by considering
how much the set of all machines in Mcan accomplish on each application together. Extending
notation again, we denote by ρjthe jth component of network-wide ER.
Definition 2.3. The jth component of the network-wide Equivalence Ratio is
Extending notation one last time, we denote by ρthe network-wide ER.
Definition 2.4. The network-wide Equivalence Ratio ρcan be written as
ρ:= (ρ1, ρ2, . . . , ρ|A|)
The network-wide ER ρis the basis for the allocation rule. Let Rbe the total amount of rewards
allocated to machines during a reward period. The allocation rule is described in Algorithm 1. There
is a direct analogy to the proportional rule from Bitcoin: the total normalized WU contribution Ni
is analogous to hi, the hash rate, and the total normalized WU contribution T N W U is analogous
to Pj[n]hj. In contrast to the hash rate, notice that Niis a dimensionless number.
Note that most advanced hardware is located in the industrialized world, a large portion of which
experiences relatively cold winters. It is very common for BOINC crunchers to use their machines as
alternative heat sources during these winters, meaning that there will be more total computational
power on the network during winters than at other times of the year. If the rate of cryptocurrency
minting remains these same throughout the year, then it will be more profitable to crunch during
summer in the northern hemisphere (where the vast majority of the industrialized world, the world’s
population, and hardware is located), which would lead to a lot of wasted potential.
However, if the rewards are scaled to reflect the amount of computational power on the network,
a cruncher could receive the same rewards with the same hardware whether or not they crunch in the
summer or the winter. Since crunching during the winter offsets expenditures on heating, that would
make it more economical and energy efficient to crunch in the winter. Making a further analogy
to the generalized proportional allocation rules discussed in [6], Rtakes the place of c(Pj[n]hj),
which is some function of the sum of all the hash rates on the network. In particular, if we
set R=T N W U , then each cruncher is rewarded the same fixed amount for every WU of some
application that they complete. Unlike in Bitcoin, this would mean that the total rewards would
be proportional to the computational power on the network.
A natural question to ask about this mechanism is what kind of behavior it incentivizes. It
turns out that using the ER to derive a universal BOINC credit incentivizes crunchers to direct
their hardware towards applications on which they are strongest relative to the network-wide ER,
if they can only crunch one application at a time. A generalization of this idea also applies to the
case where a machine can crunch more than one application at a time. This is an energy-efficient
mechanism in the sense that if the incentives are followed, WUs are going to be crunched by machines
which will accomplish them with lower energy consumption relative to the other machines on the
network. Note that crunchers may have pro ject or application preferences of their own, which would
lead them to not pursue this financial incentive. This contradiction, and attempts to resolve it, is
explored more deeply in Section 6.
Theorem 1.The Reward Assignment Algorithm incentivizes directing machines towards applica-
tions on which they are strongest relative to the network-wide ER, if the machines can only crunch
one application at a time.
Algorithm 1 Reward Assignment Algorithm
Network-Wide ER ρ
WU completion matrix W
Total network-wide rewards R
|M|×|A|normalized WU matrix N
T N W U , the Total Normalized Work Units (network-wide)
ri, the rewards that mireceives in the time interval
Calculate Nand T N W U
Initialize T N W U = 0
for i = 1 . . . |M|do
for j = 1 . . . |A|do
Nij =Wij j
T N W U =T N W U +Nij
end for
end for
Calculate reward vector
for i = 1 . . . |M|do
end for
Proof. By definition, maximizing rewards for miis maximizing ri. From the definition of ri,
Pj[|A|]Nij , and by extension T N W U, are the factors that can be directly affected by a cruncher.
As in the algorithm, let Ni=Pj[|A|]Nij , and let T N W Ui=T N W U Nibe the T N W U
without the contribution of mi. Then, T N W U =T N W Ui+Ni, and
T N W Ui+Ni
riis an increasing function in Ni, meaning that maximizing Nialso maximizes ri. Since Nij =
Wij j, and the ER is given, this in turn means that Wij is the only factor that can be modified
by the owner of miin order to maximize their reward.
The maximum possible Wij for machine miand all applications aAis in fact ρiby definition.
So maximizing Nibecomes finding the application a0that maximizes ρi
a0a0, i.e.
Thus, given some ER, the maximum possible reward for any machine that can only crunch
one application at a time is realized by dedicating all of that machine’s computational power to
the application for which the machine’s ER has the largest relative difference compared to the
network-wide ER.
Corollary 1.If a machine cannot maximize its rewards by crunching only a single application, then
it can maximize its reward by crunching two or more applications which together yield a higher
normalized WU contribution than the application on which it is strongest relative to the ER.
Remark. For a single machine that has both a CPU and GPU, the ER equally values CPU and GPU
calculations. Since GPUs with strong FP64 capabilities tend to be more expensive, and increases
in FP64 performance do not always scale at the same rate as increases in FP32 performance, FP64
can in a sense be considered a different type of computation. Then, a single machine actually also
equally values CPU, GPU FP32, and GPU FP64 calculations. There is a similar phenomenon on a
network-wide level, but with the added twist that all machines will have CPUs, but not all machines
will have GPUs.
The “value” of CPU calculations relative to the “value” of GPU calculations depends on a
number of factors and is ultimately subjective. For that reason, it may be preferable to consider
ERs for CPU and GPU applications separately, and then manually choose an equivalence between
the two classes – for example, by considering the average power consumption of recent CPUs and
recent GPUs. This would have the effect of roughly valuing the computational output per unit of
energy equally between CPUs and GPUs, thus directing focus on the most energy-efficient way of
programming an application.
Remark. The ER can change based on the machines that are on the network and the applications
available to be crunched. An interesting implication of this fact is that there is no intrinsic measure
of computational power like FLOPs or runtime that describes the equivalence of computational
requirements between applications; rather, the equivalence is relative to the machines that are on
the network at any given moment in time. This novel approach diverges substantially from prior
attempts to solve this problem, such as in the current credit systems used by many projects, the
most recent iteration of the recommended BOINC credit system [1], or in [4], where Awan and
Jarvis proposed a new type of fixed credit based on a diverse set of benchmarks.
However, determining ρis impractical for a number of technical reasons, the most obvious and
disqualifying of which is the difficulty or even impossibility of getting trustworthy benchmarks of
every machine crunching every application. Even if getting trustworthy benchmarks was possible,
another major issue is the difficulty of running enough WUs of each application in order to get
an accurate WU completion rate. For some machines, a single WU from some applications can
take days or even weeks. For that reason, the rest of this paper will deal with acquiring approxi-
mations of these values. We propose two main methods for achieving this. The first method can
be implemented using only minor modifications to the data currently openly provided by BOINC
projects; in particular, the data needed would be a breakdown of a CPID’s RAC to more granular
data. The second method requires the construction of a hardware profiling database, which could
be implemented using lightweight benchmarks of each machine on the network.
Due to the fact the ER must be approximated, there must be assurances on how the actions of one
machine or a group of machines can be used to alter the ER for the benefit of the machine(s). Two
of the main desirable properties discussed in Section 1.2, sybil-proofness and collusion-proofness,
can now be framed in the following way:
Sybil-proofness : no machine can benefit by creating more than one CPID and splitting its
computational resources among those CPIDs.
Collusion-proofness: two or more machines cannot benefit by pooling their resources and split-
ting the rewards in some manner.
The other desirable properties – non-negativity, symmetry, and budget balance – are satisfied
by construction.
3 The Linear Regression Approach
In order to build an intuition for the linear regression approach to approximating the ER, we will
use an example involving a sample universe of two applications: A={a1, a2}. After introducing
some assumptions, we will go through the process of approximating the ER, and afterwards address
each of the assumptions and their consequences in a real-world implementation of this approach.
In this universe, machine mican crunch either one application exclusively, the other applica-
tion exclusively, or a combination of both applications. All three of these settings can be chosen
by assigning different priorities (called “Resource share” in the BOINC Manager) to each of the
Consider a priority vector (ui,1ui), ui[0,1] for mi, which denotes the fraction of resources
that goes to each application. One of the simplifying assumptions we will make is that a linear
decrease in the WU output for one application implies a linear increase in the output of another
application; that is,
wi= (ρi
1·ui, ρi
2·(1 ui))
An example of a line defined by wiranging between 0 and 1 is shown in Figure 1.
Using this model, we begin with a set of machines crunching these two applications, each with
its own computational contribution w, which is visualized as a point in two-dimensional space.
Next, we run a linear regression on the points. Finally, we take the resulting best-fit line, place a
line of equal slope through each of the points, and use these lines to make an approximation b
ρifor every mi. See Figure 2 for a visualization of this process.
Calculating the points of intersection with the axes is straightforward. Suppose that for each
mi, the standard-form equations describing ρiare c1a1+c2a2+ei= 0, with c1, c2determined by
Figure 1: Example of a line defined by an ER of (100,50) for machine mi, with uiabove, and wi
below, their corresponding points.
Figure 2: The Linear Regression approach for approximating the ER. On the left is a set of points
wifor each machine mi, i = 1,..., 20, which plot the computational contributions of each machine.
In the center is the regression line. On the right are lines with the same slopes as the regression line,
passing through each of the points. The intersection of each line with the axes is an approximation
ρiof ρi, from which an approximation bρof ρcan be calculated.
the linear regression, and an intercept eifor each machine mi. Then, the network-wide ER would
In three dimensions, the logic generalizes: the regression will define a plane, and the three points
where the plane intersects the axes are the corresponding values in the ER. Generalizing further
to |A|>3 applications, every wican be mapped to its own (|A| − 1)-dimensional hyperplane,
with the intersection of that hyperplane with the |A|axes constituting the b
ρi. By Definitions 2.3
and 2.4, adding up these points of intersection application-wise would yield an approximation of
the network-wide ER. Since the b
ρiare all constant multiples of the points of intersection of the
(|A| − 1)-dimensional hyperplane with the |A|axes, the sum would also be a constant multiple of
the points of intersection, and therefore, only the output of the linear regression need be considered,
just as in the preceding 2-dimensional example.
When running linear regression in practice, one application is written as a linear function of the
a1=c1+c2a2+c3a3+. . . c|A|a|A|
Then bρ1=c1, and
i6= 1
Factoring out c1,
bρ= (1,1
In Figure 3 are the results of the linear regression approach applied to data obtained from the
Gridcoin network (a cryptocurrency that rewards crunchers for their BOINC contributions) for a
subset of BOINC projects. However, the available data was limited to a list of CPIDs by their
RACs on every project, and had to be whittled down. In order to implement the linear regression
approach properly, this data would need to be even more fine-grained by machine (rather than by
CPID), by application (rather than by project), and by number of WUs completed (rather than
by RAC); these data can only be acquired via changes to the BOINC source code. Since that data
is not currently available, only a subset of the data that likely corresponds to individual machines
crunching one application was used. See Appendix B and the corresponding Jupyter notebook for
a detailed description of how the data was handled and why the granular data is needed.
Figure 3: Actual ER Calculation
3.1 Shortcomings of the Linear Regression Approximation
3.1.1 Inaccuracies in Estimation
The assumption that under constant hardware utilization a linear decrease in output for one ap-
plication implies a linear increase in output for another application is not necessarily correct. For
example, by varying u, it is possible that some threshold of cache, RAM, or other resource is
reached which results in a non-linear change or discontinuity in the amounts of each application
that a machine could crunch. Furthermore, the underlying relationship could even be a (piecewise)
curve. Another assumption made in the linear regression approach was that, in addition to the
relationships being linear, the hyperplanes defining the ER for each machine had the same coeffi-
cients, which also is not necessarily correct. The following theorem bounds the error between ρand
bρas determined by the linear regression.
Theorem 2.The application-wise absolute difference between the true ER and the estimated ER
defined by the linear regression is bounded above by the number of machines times the maximum
application-wise absolute difference for any machine mibetween ρiand ρ:
|ρacρa|≤|M| · max
Proof. By Definition 2.3,
We also have that
=⇒ |ρacρa|=|X
≤ |M| · max
In practice, the estimated ER will be much more accurate, because it will not be the case that
the difference between c
aand ρi
awill have the same (maximum) magnitude and direction for all
Another key issue is the fact that if one application uses only a CPU, while the other uses a bit
of the CPU and mostly the GPU, then the linear relationship described earlier will not hold even
approximately. For example, in the two-application universe, while it will be true that the b
ρifor a
particular machine on those applications will intersect the axes, the relationship will not be linear
– it will look more like a rectangle with a rounded upper right corner. If there are enough such
machines, then the linear regression over all the machines on the network will not even intersect
the two axes on their positive rays.
As was discussed in Section 2, this problem can be addressed by separating the CPU applications
and GPU applications, determining the ERs within these classes, and then choosing an equivalence
between the two classes of computations. This process can also be done at an even more granular
level – for example, by splitting the GPU class into FP32 and FP64 applications.
A further potential shortcoming of the linear regression approach is that converging on an
accurate estimate of the ER in the limit of a large number of machines requires that the distribution
of the computational powers of the machines crunching each of the applications is identical – i.e.
that for each application, each machine crunching it is i.i.d. from the same distribution as all the
other applications. However, there is no way to know whether this is in fact the case, or to create
such a situation.
Setting aside things like driver versions, if we assume that there are no WUs that are too simple
to run on any hardware, then we can conclude that WUs can only not run on some hardware
if those WUs are too demanding. For example, consider again a two-application universe, and
suppose that WUs from one application require more RAM or GPU VRAM than many machines
have, and the other application supports all machines. A linear regression would roughly lead to
an approximated ER that equates the averages of the outputs on the two applications. However,
since only more powerful machines can crunch the more restrictive application, then the more
powerful machines that crunch the more restrictive application will be undervalued. This would
incentivize application developers to develop applications that can be run on as many different
types of machines as possible.
Overall, while this may not affect strategic aspects regarding sybil-proofness and collusion-
proofness, it does add another error to the way that the ER is being calculated.
3.1.2 Game-theoretic Analysis
It is possible that sybil-attacks or collusion-attacks could also be used to alter the ER. These types
of attacks can be partially mitigated by using weighted linear regression, where the weights could
be, for example, the computational contribution of the points. An initial ER could be calculated
by standard linear regression, which is the same as every point having the same weight. Then,
each machine could be weighted by its Ni, and the weighted linear regression could be iterated
until convergence. It is not clear whether this method of calculating the ER is sybil-proof and/or
In recent game-theoretic literature, there has been a focus on determining the computational
tractability of computing moves that would benefit agents. It is possible that the weighted linear
regression approach, or even the linear regression approach without weights, would have so many
agents and so many unpredictable factors that computing moves that maximized rewards even in
expectation could be in some complexity class that in practice renders attempts at manipulation
unfeasible. Furthermore, by Theorem 1 and by construction of the weighted linear regression, any
such manipulations intended to award the manipulator(s) higher rewards would still ultimately
incentivize the manipulator(s) to be energy efficient. Whether or not the linear regression approach
is sybil-proof or collusion-proof either by the definitions given earlier, or because finding beneficial
strategies is computationally intractable, remains an open problem.
The remainder of this section is a game-theoretic analysis of what would happen if every cruncher
chose to crunch applications which maximized their rewards based on the most recent ER. The case
where crunchers do not act solely in their financial interests is addressed in Section 6. While it is not
necessary for understanding the rest of the paper (since the underlying assumption that crunchers
will only act in their financial interest almost certainly will not hold), some readers may find it
Recall that we begin with the input W, from which the ER is calculated. Suppose then that
every machine tries to maximize its reward for the next reward period based on the ER from the
current reward period. Recall that the true slopes of the ERs, ρi, may not all be constant multiples
of bρor ρitself – see Figure 4 for an example of the general case of the example ERs from Figure
2. Thus, depending on W, it may be possible that bρis changed during its next calculation such
that machines have an incentive to switch to a different application (note that if in fact the slopes
were all constant multiples of bρ, then switching from one application to another would not change
bρ, since wiwould remain the same distance away from bρfor any ui).
Figure 4: General ERs
A two-dimensional implementation of this game can be found in Appendix C. We were able to
find instances which converged to a stable ER, diverged, and cycled. The latter finding indicates
that even in this simple scenario, a pure-strategy Nash equilibrium does not exist. For much larger
numbers of machines, which would be the case in practice, we did not find an instance that did not
In theory, a mixed-strategy Nash equilibrium should exist. There are a finite number of crunch-
ers, and each each cruncher has a finite (although quite large) set of strategies, which are the
various combinations of WUs from different applications that they can crunch in a reward period.
The simulations just described were repeated games in which crunchers were choosing the best
possible strategies based on the results of the previous game.
In practice, a number of the assumptions necessary for a Nash equilibrium to be reached would
not be fulfilled. As already mentioned, not all crunchers will try to maximize their rewards. Even
if we kept fixed the contributions of crunchers who chose not to maximize their rewards, and only
considered crunchers who did, there are a number of reasons that the game would still not converge
to a Nash equilibrium. First, there is no guarantee that each remaining cruncher would execute
their strategy flawlessly. Furthermore, it is not the case that a deviation of a cruncher would not
cause deviations by other crunchers. Additionally, due to the dynamic nature of the network, with
hardware coming online and offline, along with crunchers switching which applications they crunch
for other reasons, computing the best possible strategy (or even a profitable one) might be in a
complexity class that is not possible to compute.
4 Hardware Profiling Database
Like the linear regression approach introduced in Section 3, the basis of the Hardware Profiling
Database (HPD) approach to estimating ρrelies on obtaining predicted ERs for each machine. At
the core of the HPD method is a set of benchmarks B={b1, b2, . . . , b|B|}, which are a variety
of tasks that measure processor speeds, RAM size and bandwidth, bandwidth between CPU and
GPU, available disk space, etc.; another attempt to construct a new BOINC credit by Awan and
Jarvis also included a larger number of benchmarks than the current BOINC system uses [4]. As
before, machines running at a fraction of their capacity can be viewed as either weaker machines or
virtual machines. Machines without GPUs would have their GPU benchmarks equal to 0. Then,
the HPD is an |M| × |B|matrix H, where Hij is the output of machine mion benchmark bjB.
With Hin hand, it is possible to map a machine’s benchmarks to its expected output on each
application. Define a family of functions F={f1, f2,...f|A|}. Each function fFmaps the
benchmarks bof a machine mto an expected output on an application a. Let the benchmarks for
machine mibe bi. Then,
fa:bi7→ c
a,aA, m M
With c
ain hand for all machines and all applications, and using Definitions 2.3 and 2.4, bρcan
be easily constructed.
The family of functions Fcan be any of a variety of traditional statistical methods or machine
learning algorithms. Below we list some costs and benefits of linear regression, non-linear regressors,
and neural networks, but these methods are non-exhaustive and should be considered merely as
Linear regression would provide weights (coefficients) for each benchmark, which helps explain
which aspects of a machine are important in determining its b
ρi. However, it is limited by its
linearity, which can be somewhat compensated for by feature engineering. Additionally, linear
regression naturally supports multiple outputs. Further improvements can be made on top of linear
regression, such as generalized linear models, or in the case that cruncher-supplied data is being
used to make the mappings, using linear regression first on the machines that only crunch single
applications, and once an equivalence ratio is found, adjusting it based on the machines that crunch
multiple applications.
Non-linear regressors, such as random forests or gradient boosting, would also provide the
relative importances of benchmarks in predicting outputs. Unlike linear regression however, they
can find how non-linear relationships between the benchmarks determine a machine’s performance
on a given application. One downside of this approach is that these algorithms (at least, the ones
mentioned) are not inherently multi-output. Multi-output wrappers do not take into account how
the predictions for each component of the output depend on each other, and so only data involving
single outputs (applications) could be used.
Neural networks are both multi-output and non-linear; however, they do not provide an easy
way of determining relative importances of particular inputs.
4.1 Implementation Challenges
There are two main implementation issues in the HPD approach. The first issue is the difficulty of
getting reliable benchmarks from each machine, or a statistically large enough sample of machines.
The second issue is determining F.
Suppose that these benchmarks were obtained by sending WUs from a BOINC project. With the
assumption that the benchmarks would either be posted to the blockchain or verified in some other
way by the network, the verification that the benchmarks are in fact what crunchers submitted them
as is trivial. The issue is that WUs are sent as binaries, and should be treated as being completely
reverse-engineerable, in the sense that one could recover the original code that constructed the WU.
From this point, the WU could be manipulated – a cruncher could run any code, and submit any
result, that they wanted to. Unlike deterministic BOINC WUs, there is currently no way to verify
whether the results from a benchmarking WU are honest, barring obvious and extreme cases. The
most obvious method of ensuring that such benchmarks would be trustworthy – trusted computing
– is unpopular because of privacy concerns.
From a computational standpoint, determining Fis trivial. The troubles arise when determining
which benchmarks and application outputs are used. In a more centralized setting, there could be a
set of trusted machines, and their outputs could be used to determine F. These machines and their
trustworthiness could be a combined effort of crunchers and projects, or some cryptographically
secure approach. An alternative approach is to operate under the assumption that some subset of
the benchmarks is unreliable, and try to deal with that problem as best as possible. Additionally,
the benchmarks in Hcould also be combined with Win order to approximate the ER, in a similar
way to what was done with linear regression, but with the addition of this new and useful dataset.
4.2 Benefits of the HPD
This database would have a wide variety of uses beyond trying to approximate the ER, and even
beyond cryptocurrencies and BOINC.
First, projects would be able to use benchmarks and outputs to design their applications in
ways that take advantage of potential optimizations, and plan the structure of future applications
based on existing hardware or anticipated new hardware on the market.
Non-BOINC researchers could use the benchmarks for similar purposes as the project admin-
istrators – data-mine the database to find interesting patterns that could yield new insights into
hardware evolution over time, which has the added advantage of attracting new researchers to the
BOINC community. For example, in [9] and [10], Al-Qawasmeh et al. take a database with the same
form as Hand define three measures on the set of machines: Machine Performance Homogeneity
(MPH), Task-Machine Affinity (TMA), and Task Type Difficulty Homogeneity (TDH). These mea-
sures are used to characterize heterogeneous computing environments to provide insights on how
to optimize the use of the machines.
Crunchers could use the database to inform themselves about future hardware purchases, as
well as use the information to fully predict the ERs of their machines, which is useful in addressing
the tradeoff between crunching their favorite applications and the most profitable applications, as
addressed in Section 6.
Companies or other institutions could use the database for their own hardware-purchasing de-
cisions, and even use the network and benchmarking projects to send out their own benchmarking
WUs while paying crunchers for their services with the cryptocurrency, creating a source of demand.
5 Anti-Cheating Mechanism
Many projects have multiple instances of a job sent to different machines in order to detect cheating
or bad results due to numerical instabilities across different platforms and types of hardware. Some
projects use adaptive replication [2] of jobs, in which trust is established between an application
and a machine, resulting in that machine being sent jobs with fewer and fewer replicated instances
from that application, as long as the WUs are consistently returned with reliable results by that
particular machine. If results are found to be invalid, then the host has to start again from zero
Both the adaptive replication policy and the reward mechanism introduced in Section 2 can
be modified to reward a machine more for being trusted and consequently crunching jobs with
fewer replicated instances. The modified reward mechanism would discourage cheating by making
it as least as profitable (more profitable) to be honest as it is to cheat, making honesty a weakly
(strongly) dominant strategy, and consequently lead to gains in energy efficiency.
This anti-cheating mechanism can be implemented on the project-level and/or the network-
level. In the former case, the incentive for crunchers would be to minimize the number of replicated
instances that they crunch, but otherwise there would be no inter-project effects. In the latter case,
the energy efficiency of the network can be improved even further. Machines have an incentive to
crunch jobs that have the fewest number of replications, and projects have an incentive to have
their workloads computed quickly, want to attract the aforementioned machines, and so likewise
have an incentive to reduce the number of replications. This would create a system where projects
attempt to attract crunchers and compete with each other by reducing replication. Whether or
not this is a good dynamic to introduce to the network is beyond the scope of this paper, but the
possibility should be seen as one of a number of parameters that can be changed to incentivize
different types of behavior.
Such a mechanism could also instigate new research into cross-hardware, cross-platform, cross-
operating system numerical stability. Interestingly, it may also whittle down the machines that
crunch particular applications to only those that consistently return reliable results (which are
the machines that are numerically stable on those application/platform pairs), in a sense sorting
the machines by their ability to return correct results, if the cause of incorrect results is because
of some numerical instability. Note that this latter effect might undermine one of the underlying
assumptions of the linear regression approximation of the ER by making the distribution of machines
across applications different.
5.1 Past
Assume that a project can temporarily rescind credit for, and verify through additional replication,
the results of previously approved WUs. The case where only future WUs can be additionally
replicated is addressed in the subsequent section.
Consider a machine mithat has a stream of incoming WUs from some application a. Let wj
be the jth WU sent to miafter some initial starting time. Let tjbe the number of instances of the
job corresponding to wjthat were sent to different crunchers. Let rj=rbe the constant reward
given to the cruncher for completing a WU from application a.
In order to disincentivize cheating, the expected reward from being honest must be greater than
or equal to the expected reward from cheating. One natural mechanism to try would be to scale
the rewards for the completion of a given WU by the inverse of the number of instances of that job,
so that rj=r/tjrather than rj=r. This can be interpreted as granting a total reward rfor every
job, with every machine that crunched one of the instances of that job getting its equal share of
the reward. If tj= 1, then miis the only one that crunched wj, and so rj=r, and migets all of
the reward; if tj= 2, then migets half of the reward, etc.
Consider the simple case where either there is one copy or there are two copies of wj, i.e.
tj= 1 or tj= 2 – the following results generalize easily. Towards the aim of discouraging cheating,
project administrators replicate wjwith some probability p. We assume that the event that a job
is replicated is independent of the event that micheats, and that michooses to cheat using some
strategy of when to cheat and when to return legitimate results.
Since every job has probability pof being replicated independently of other jobs being replicated,
it is only necessary to consider a subset of the WUs that micrunches – in particular, those for
which micheats. Since these WUs had probability pof being verified when considering all the
WUs, they still have probability pwhen considering only the subset. Thus, the strategy by which
micheats is not relevant. Since the probability of a WU being replicated is Ber(p), the number of
instances sent until one is replicated is Geom(p), and so in expectation miwill be caught after 1
WUs, implying that there will be 1
p1 occurrences of micheating and not getting caught.
The goal is to have the loss in rewards be greater than or equal to the expected rewards from
cheating. Let nbe the number of verified WUs. Now, the loss
L= (rewards if past nWUs were not verified) - (rewards if past nWUs are verified)
and the rewards from cheating
C= rewards if past WUs were not verified while machine was cheating and was not caught
and we want LE[C]. When considering the results of WUs which are being retroactively
replicated, it will be assumed that they are all verified as being legitimate results – otherwise, this
entire process of replicating already accepted WUs can be applied recursively.
Invoking the memorylessness property of the geometric distribution, a sequence of cheat results
– the aforementioned subset – will be caught in expectation 1
pWUs into the sequence. Thus,
p1) r
LE[C] =n
It is important to note that this mechanism will not catch all illegitimate results submitted by
a cruncher; rather, it will financially disincentivize cheating in expectation, since crunchers might
still be able to profit by cheating in some cases, but not on average. Increasing nbeyond the lower
bound found above can be used to make cheating an arbitrarily bad strategy.
5.2 Future
When only future WUs can be additionally replicated, the goal is still similar to the one in the
past case: to have the expected rewards under an honest strategy be greater than (or equal to) the
expected rewards under a cheating strategy.
Since a machine could get to the point where it is sent the fewest number of replicated instances,
cheat for as long as it can, and then stop crunching the application once it is caught, it could profit
from cheating as long as it is not possible to retroactively verify WUs. For the future setting, we
introduce a deposit d, which can either be in the form of non-replicated WUs which are treated
as replicated WUs (meaning that the cruncher receives a lower reward than they should have for
crunching those WUs), or some amount of the cryptocurrency itself. If the deposit is withdrawn or
withheld, it must be rebuilt if the machine/cruncher wants to continue running that application.
The idea is that a machine can gain at least as much by taking its deposit and discontinuing
crunching that application as it can by cheating.
Let nand rbe as they were in the prior case, and as before, consider only the WUs for which
the machine is cheating. Consider some starting WU w1, during which miis already under the
WU verification scheme with minimum replication, and assume that miis cheating starting when
they received w1. Let wjbe the WU for which miwas caught returning bad results. Since by
construction we are only considering WUs for which miis cheating, w1. . . wj1have no other
instances – that is, miis the only one crunching them.
Up until wj,mihas gained (j1)rfrom cheating, so the loss for mimust be greater than (or
equal to) (j1)r; setting d(j1)raddresses this problem. Since we do not know jahead of
time, we can set dE[(j1)r] = ( 1
p1)r. Once miis caught, the deposit is lost, and it must be
rebuilt in the same manner it was constructed.
5.3 Cherry-picking Attacks
Under the most recent iteration of the BOINC credit system [1], there is an attack vector by which
a cruncher only crunches shorter WUs and abandons longer WUs, called a cherry-picking attack.
This would still be an attack vector in the reward mechanism described in this paper. There are two
main sources of information that a potential attacker might have: 1) the estimated WU runtime
as given by projects, and 2) empirical estimates of the mean, percentiles, variance, and potentially
other statistics of the runtimes on the attacker’s machine.
The problem caused by the first source of information can be solved by simply not providing
crunchers with such detailed information; perhaps an average runtime and variance would suffice.
The second problem is trickier. For a given machine, job runtimes from a particular application
will come from some probability distribution X. In order to eliminate the cherry-picking attack, X
would have to come from some family of distributions where, given the elapsed time t,Xis such
that E[Xt|t]E[X] for all t.
For example, if the runtime distribution is bimodal, then the time difference between the argu-
ments of the modes would need to be larger than the time difference between zero and the minimum
of the arguments of the modes. If the latter condition does not hold, then several WUs could be
wrapped together into a single WU such that the distribution of runtimes becomes unimodal, al-
though this would still not necessarily satisfy E[Xt|t]E[X] for all t. For example, a unimodal
distribution skewed strongly to the left might still not meet the necessary criterion.
Alternatively, some form of punishment could be used to dissuade crunchers from using this
attack. For example, the number of WUs sent to a machine could be reduced if too many WUs
are returned as abandoned. Another form of punishment could be higher levels of replication for
hosts that have returned many abandoned tasks. Additionally, it is worth noting that use of the
cherry-picking attack could actually damage the cruncher because of its effect on the ER (in the
linear regression approach), but it is likely not to matter on the scale of a single machine.
5.4 Project Collusion
While the crunchers can be incentivized to not cheat, the projects must also be disincentivized from
cheating. Consider a project that wants to confirm that some particular machine(s) completed
more WUs than they actually did. Split the machines on the network into two groups: machines
knowingly colluding with the project, and all other machines. If the WU distribution is randomized
in a trustworthy manner, then it would be impossible to favor any machines in the second group. For
example, the WUs could be randomly distributed by relying on randomized data from the blockchain
to determine which WUs go to which machines, with that process carried out and verified on the
blockchain. To know whether or not a job should be replicated, projects could be told to which
machine the WU was sent after it was already sent, after determining whether the result should be
accepted at all, or at some other point. The randomization process just described might interfere
with the methods that BOINC projects use to schedule jobs (in particular, projects need to know
which platform a machine uses, since there are different application versions for different platforms);
see [3]. However, the fact that machines are incentivized to crunch the applications on which they
are most efficient somewhat counterbalances this interference. It may also be possible to reconcile
these problems with further research – the purpose here is merely to demonstrate feasibility.
However, it would still be possible to favor colluding machines by simply having them return
a result from a set of results that the project automatically recognizes as legitimate, but knows is
actually being returned from a colluding machine. It may be the case that the only way to deal
with this problem is by having some sort of trustworthy oversight over which types of WUs are
considered legitimate results – however, this problem is beyond the scope of this paper. (Note that
there is nothing wrong with a (non-profit) project using its own machines to crunch WUs.)
5.5 Open Problems
There remain a number of open questions about this anti-cheating mechanism. Does the expected
payoff from cheating increase with the number of machines under the control of the attacker? What
fraction of the machines on the network (or crunching a particular application) would an attacker
have to control in order to have a higher expected payoff from cheating than from being honest?
What would happen if some fraction of WUs were sent on to triple verification – could this mitigate
large scale attacks?
In another vein, the anti-cheating mechanism as described only works properly for WUs that
have deterministic results. It is an open problem on how to disincentivize cheating for WUs with
non-deterministic results.
Note another interesting open problem: the project collusion issue is one of the main obstacles
in constructing a true Proof-of-Research coin, which is why both Curecoin and Gridcoin have their
reward mechanisms built on top of PoS protocols. One possible solution, theoretically but not
practically, is to have every project for which a cruncher can receive rewards publish data on the
blockchain – in particular, how jobs are created, to which machines they are distributed, the results,
and by what rules the results are accepted or rejected. Beyond obvious privacy concerns, this would
also be an enormous amount of data to be stored, in addition to the fact that it could potentially
make exploits much easier.
6 Top Trading Cycles
Despite financial incentives, crunchers might still choose to crunch less profitable applications. In
order to balance the efficiency of a machine and the desires of its owner, we offer two restricted
formulations of this problem and describe how they can be solved using the Top Trading Cycles al-
gorithm and its generalizations, and conclude with a general formulation describing a more realistic
version of the resource allocation problem.
6.1 Intuition
Suppose that two crunchers each have a machine of their own, with the following ERs:
Application 1 Application 2
Machine 1 ρ1
1= 100 ρ1
2= 50
Machine 2 ρ2
1= 50 ρ2
2= 100
A preference profile is a set of binary relations between each application where aibmeans
that iprefers ato b. We will assume that preference profiles are not complete – that is, it is not
necessary that all applications are in the relation. However, they are transitive: if aband bc,
then ac.denotes a weak preference, while denotes a strong preference.
Now suppose that the two preference profiles for the two crunchers are 1=a21a1and 2=
a12a2. Cruncher 1 can offer to crunch their less preferred application in exchange for cruncher
2 crunching cruncher 1’s more preferred application, while keeping the rewards for crunching the
less preferred application, and vice versa. This way, both crunchers can have their more preferred
applications crunched while still utilizing the full potential of their machines, and realizing an overall
better outcome.
Consider another example with three crunchers, each with their own machine:
Application 1 Application 2 Application 3
Machine 1 ρ1
1= 100 ρ1
2= 50 ρ1
3= 25
Machine 2 ρ2
1= 50 ρ2
2= 25 ρ2
3= 100
Machine 3 ρ2
1= 25 ρ2
2= 100 ρ3
3= 50
with preference profiles 1=a31a21a1,2=a22a12a3, and 3=a13a33a2. It
is easy to see that having each cruncher crunch their machine’s most profitable application results
in amounts of WUs of each application being crunched that is more preferred by each cruncher
than if they each crunch their favorite application.
What follows is an informal explanation of desired properties of mechanisms in the field of
resource allocation problems.
In both examples, it is clear that every cruncher, in terms of their application preferences,
prefers the outcome where they each crunch their most profitable applications than if they each
crunched their favorite applications. We can say that the former assignment Pareto dominates the
latter assignment, meaning that every cruncher at least weakly prefers the former assignment to
the latter, and at least one cruncher strictly prefers it. An assignment is Pareto efficient if it is not
Pareto dominated by any other assignment.
We also introduce the notion of the core. The core is the set of assignments in which no subset
of the agents can deviate and arrive at a better assignment for that subset. As explained in [7],
an assignment in the core also implies individual rationality – that is, no cruncher can possibly
do worse by participating in the mechanism than they could on their own – and Pareto efficiency,
since if an assignment is in the core, and it is Pareto dominated by some other assignment, then all
the agents would have an incentive to deviate to the other assignment, meaning that the original
assignment was not in the core.
The final concept is strategyproofness, which is a property that means no cruncher can benefit
by misreporting their preferences.
We can now define the first part of the problem: each machine mihas an endowment, which
in this case is ρi. The second part of the problem is traditionally the set of preferences profiles
= (1,2,...,|M|). However, the preference profiles over applications need to be modified to
be preference profiles over the machines.
Note that the mechanics of the exchange in the context of BOINC or blockchains/cryptocurrencies
are not discussed here; the purpose of this section is to explore the feasibility of the underlying
6.2 Reduction to Top Trading Cycles
We begin with the simplest possible reduction: suppose that each cruncher only owns one machine,
and each machine can only crunch one application at a time.
In order to transform preferences over applications to preferences over machines, the application
preferences just introduced need to be refined. Let the new preference profile over applications
ifor machine mibe an ordering over amounts of WUs from applications in decreasing order of
preference. For example, consider the second machine from the second example above. One possible
new preference profile for m2is 2=a25
3. This new preference profile over applications
indicates that m2would prefer to trade for more than 25 WUs of a2, which is logical, since it is
m2’s most preferred application, and they can only achieve 25 WUs alone. If there are no machines
available for trade that can achieve that many WUs, either because no other cruncher is offering
that, or because all the machines that could achieve that number of WUs were already traded for,
then m2’s subsequent preference is more than 50 WUs of a1– which again, m2cannot achieve on
its own. Finally, m2would prefer more than 100 WUs of a3– which again, m2cannot achieve on
its own.
Another possible sequence is 2=a75
3. This is the same as the previous
sequence, except with an additional preference at the beginning, indicating that m2prefers more
than 75 WUs of a1to more than 25 WUs of a2. Note that the assumption so far has been that
these minimums are exclusive. If the minimums are not exclusive, then a machine might end up
preferring itself to any other machine – this is easily taken care of by TCC and its generalizations.
Now these new preferences over amounts of WUs need be transformed into preferences over
machines. Consider the following transformation: for the first element in a preference profile, the
subset of machines satisfying the corresponding minimum can be sorted in decreasing order of
their performances on that application, appended to the final list of preferences over machines, and
removed from the pool of machines; this is the first subset of preferences for the machine. This
process would be repeated for every element in the preference profile. Note that a cruncher can
have two or more separate values for a single application, as long as those ranges are separated by
at least one other application (otherwise they could be reduced to the minimum among them), as
just illustrated with the second possible sequence. This would create an ordering over all machines
with no repeats, and is formalized in Algorithm 2.
This transformation makes the problem setting the same as the traditional setting for TTC
(a housing market), and so the original TTC algorithm can be invoked. TTC is known to be
core-selecting and strategyproof.
6.3 Multiple Endowments
A natural extension of this formulation is to the case where crunchers can own more than one
machine, which of course is the case in reality. This is a well-researched area with a plethora of
useful results, some of which we describe below.
First, S¨onmez showed in [11] (Corollary 2) that in indivisible goods economies, if there is at least
one agent that owns more than one good, then there is no allocation rule that is both core-selecting
and strategyproof (the author originally used Pareto-efficient and individually rational in place of
core-selecting, but the statement follows simply when taking into account that the core is both
Pareto-efficient and individually rational, as already described).
In [7], Fujita et al. present a generalized version of the housing market problem with condition-
ally lexicographic preferences over bundles of objects, and introduce an algorithm called Augmented
Algorithm 2 Preference Assignment Algorithm for mi
M,ρjj[|M|], and preference profile over amounts of WUs i
Preference profile over machines 0
i= [ ]
for k= 1 . . . len(i)do
amount, application = i[k]
satisfying = {i[|M|]|ρi
application >amount}
i.append(satisfying machines sorted in decreasing order of performance on application)
remove satisfying machines from M
end for
Top Trading Cycles (ATTC), which they showed is also core-selecting, and prove that finding ma-
nipulations is NP-complete. These manipulations include lying about preferences, splitting endow-
ments, and hiding endowments.
In [5], Aziz presents the Fractional Top Trading Cycle (FTTC) algorithm. The setting is one
in which agents can own more than one house or fractions of houses, and the author proceeds
by breaking each agent into subagents for each of the houses, where each subagent owns what its
superagent owned. We reproduce a theorem from that paper below:
Theorem 6 [Aziz]. For the housing markets with strict preferences, discrete but multi-unit
endowments, FTTC is equivalent to the ATTC mechanism.
A corollary is that FTTC is also as hard to manipulate as ATTC.
6.4 General Setting/Open Problem
A further extension of the problem is to the case where crunchers can offer to crunch more than
one application. Beyond this, there are also generalizations where crunchers can have non-strict
preferences, or even preference classes, over some ranges of WUs of different applications; however,
the latter generalizations are beyond the scope of this paper.
We will re-use the assumption from Section 3 that given a constant level of use for a given
machine, a linear decrease in the output of one application implies a linear increase in the output
of another application (keeping in mind the separation between CPU and GPU applications).
Formalizing this, let the non-negative weights for each application be g= (g1, g2,...g|A|), and
Pi[|A|]gi= 1. Then the endowment of a machine miis gρi, where denotes element-wise
multiplication. The preference profiles remain as they were previously. This setting differs from the
aforementioned multi-endowment and fractional settings in that the endowment here is not only
multi-dimensional, but the endowments and the allocations can be any arbitrary fraction of every
application, as long as the fractions sum to 1. Note the critical fact that the linear increase/decrease
property does not hold in general, and especially does not hold when taking into account CPU and
GPU applications. The purpose here is to present the most basic extension of this problem.
In [12], Yu and Zhang drop entirely the idea of trading cycles, and instead propose describing
the trades in terms of parameterized linear equations. This approach may be much better suited
for the general setting we have here, since it simplifies the complex problem of deciding who trades
what with whom, and the linearity of the equations might make it much easier to handle the linearly
balanced endowments.
There may also be entirely different approaches that are better suited to solving this prob-
lem. For example, genetic algorithms and other techniques intended to optimize use of distributed
computing resources could find allocations that satisfy certain social welfare functions better than
generalizations of TTC or other market mechanisms. However, the other approaches may not have
properties such as core-selection.
7 Carbon Neutrality
There remains a fundamental issue: what about the carbon emissions caused by running the hard-
ware? Also, who would buy this cryptocurrency, and thus pay for these computations?
There are a number of ways that this cryptocurrency could be made carbon-neutral. First,
renewable energy sources create excess energy on the grid during peak sunlight and wind hours,
which do not coincide with peak energy usage hours, meaning that the excess energy must be stored
for later use – this is the most critical issue facing renewable energy integration today. Distributed
computing can convert excess electricity into heat, which can be stored in water heaters or household
thermal energy storage systems for later use.
Second, for-profit companies which need distributed computing can buy this cryptocurrency in
order to mediate transactions by paying crunchers for use of their hardware, which would boost the
price and liquidity of the currency (price is affected by mediation [8]). Since some machines may
be used to heat air and/or water for immediate use (not energy storage), if there are no tasks from
companies that are willing to pay, but the heat is still needed, then the machines can fall back on
the volunteer projects.
Relatedly, is a platform connecting hosts owning hardware suitable for deep learning
with willing buyers. Until recently, many hosts had their machines defaulting to Ethereum
mining when their hardware was not in use for deep learning – this mining can easily be replaced
with BOINC tasks. There are also companies which construct machines specifically for the recovery
of waste heat from computing, like Qarnot, which offers opportunities for future cooperation.
Additionally, there are a number of BOINC projects dedicated to matters of environmental
concern – the most notable is, but there have also been projects in the past
that attempted to improve solar panels, and there are many more opportunities in this area.
Finally, crunchers could donate some of their currency – potentially profits, after paying for elec-
tricity – to foundations that protect land from deforestation, or responsible reforestation projects.
Then, those who are looking to offset their carbon emissions can buy the currency from the foun-
dations/projects, granting the latter the fiat currency necessary to buy and protect the land. Al-
ternatively, crunchers can invest their profits in other forms of carbon reduction, green energy, or
sustainable and recyclable hardware, which is a relatively new but growing field.
8 Current State of Distributed Computing Cryptocurren-
There are a number of cryptocurrencies that reward distributed computing. Curecoin rewards
computations for Folding@home, a distributed computing platform for protein folding. Primecoin
is a PoW coin where the PoW computations are searches for prime numbers. Obyte rewards
computations for World Community Grid, a BOINC project, but does not reward other projects.
Gridcoin rewards a subset of BOINC projects suited to its distribution mechanism. Since Gridcoin
is the only cryptocurrency currently rewarding more than one BOINC project, we will use it as an
example of how blockchains and their economics develop.
Gridcoin was started in 2013 as a fork of Blackcoin, which itself was a fork of Peercoin, which
itself was a fork of Bitcoin. It was started as a PoW coin using the scrypt hashing algorithm, with a
layer on top of the PoW protocol that rewarded BOINC contributions. Beginning in 2014-2015, the
coin was forked, the protocol was eventually shifted to PoS, and the GRC in existence at the time
was inflated to 340 million GRC. The PoS protocol awarded every UTXO that staked a block
with a percent yield based on the age of the UTXO, as well as rewards for that person’s BOINC
contributions. Another fork in 2018 resulted in constant block rewards, rather than a percent return
based on age, with 14 million GRC being minted per year – 25% going to the PoS block rewards,
and the remaining 75% going to rewards for BOINC computations. There is strong evidence to
suggest that most of the GRC at the time of the 2014-2015 fork remains in the hands of those who
were present before the fork.
As of this writing, there is a total of 455 million GRC in existence; of that, there is 30 million
GRC in the Foundation wallet. The Foundation wallet is a multi-signature wallet entrusted to
long-standing members of the community and is intended to be used for enhancement or promotion
of Gridcoin. It is still almost entirely derived from the initial fork, since developers have been
reluctant to accept payment for their contributions. Many Gridcoin crunchers still send donations
to the Foundation, despite the current rate of minting being small relative to the initial 340 million
GRC after the initial fork.
While Gridcoin experienced a rapid rate of growth during the 2017 cryptocurrency boom, it
plummeted in activity and price during the subsequent decline. It has experienced a not-nearly-as-
large amount of growth in price and users since the 2020 cryptocurrency boom, despite the absolute
size of the 2020 boom far eclipsing the size of the 2017 boom. Correspondingly, Gridcoin also has
mostly empty blocks and a very low velocity of money. However, it has one of the highest rates
of development in the entire cryptocurrency industry, and many non-core developers regularly add
new uses or external features to the blockchain, indicating the viability of such cryptocurrency
9 Conclusion
Unfortunately, BOINC has stalled in growth. As Anderson noted in [3],
The original BOINC participation model was intended to produce a dynamic and grow-
ing ecosystem of projects and volunteers. This has not happened: the set of projects has
been mostly static, and the volunteer population has gradually declined. The reasons
are likely inherent in the model. Creating a BOINC project is risky: it’s a significant
investment, with no guarantee of any volunteers, and hence of any computing power.
Publicizing VC [Volunteer Computing] in general is difficult because each project is a
separate brand, presenting a diluted and confusing image to the public. Volunteers tend
to stick with the same projects, so it’s difficult for new projects to get volunteers.
At the same time that BOINC has stalled, cryptocurrencies have exploded in popularity, with
the computational power and resources devoted to the latter far outstripping the former. The stall
in growth has come despite a massive increase in the amount and availability of computing power
among consumers. There is a good opportunity to direct this ever-increasing amount of computa-
tional power towards more useful work; cryptocurrencies and BOINC can play a big role in this. As
for the problems just quoted, a minor modification to the reward mechanism could create a floor of
computational power for each application/project, guaranteeing that if it receives below a certain
amount of normalized WU contributions, then crunchers of that application/project are rewarded
more, thus incentivizing crunchers to crunch them. Ideally, this would create a situation where no
application/project falls below that minimum. Likewise, it is possible to create a cap on the amount
of total rewards for all crunchers that can be awarded for crunching a particular application/project;
this upper bound would incentivize crunchers to crunch less popular applications/projects in case
a particular application/project became extremely popular.
10 Acknoledgements
The author would like to thank James C. Owens, Cy Rossignol, Itai Feigenbaum, and Vitalii
Koshura for useful comments and discussions.
[1] A new system for runtime estimation and credit.url:
[2] Adaptive replication.url:
[3] D. P. Anderson. “BOINC: A Platform for Volunteer Computing”. In: J Grid Computing
(2020), pp. 99–122.
[4] M. S. K. Awan and S. A. Jarvis. “MalikCredit - A New Credit Unit for P2P Computing”. In:
2012 IEEE 14th International Conference on High Performance Computing and Communica-
tion & 2012 IEEE 9th International Conference on Embedded Software and Systems (2012),
pp. 1060–1065.
[5] H. Aziz. “Generalizing Top Trading Cycles for Housing Markets with Fractional Endow-
ments”. In: arXiv preprint arXiv:1509.03915 (2015).
[6] X. Chen, C. Papadimitriou, and T. Roughgarden. “An Axiomatic Approach to Block Re-
wards”. In: Proceedings of the 1st ACM Conference on Advances in Financial Technologies.
2019, pp. 124–131.
[7] E. Fujita et al. “A Complexity Approach for Core-Selecting Exchange under Conditionally
Lexicographic Preferences”. In: Journal of Artificial Intelligence Research (2018).
[8] A. Narayanan et al. Bitcoin and Cryptocurrency Technologies. 2016.
[9] A. M. Al-Qawasmeh, A. A. Maciejewski, and H. J. Siegel. “Characterizing Heterogeneous
Computing Environments using Singular Value Decomposition”. In: 2010 IEEE International
Symposium on Parallel & Distributed Processing, Workshops and Phd Forum (IPDPSW).
IEEE. 2010, pp. 1–9.
[10] A. M. Al-Qawasmeh et al. “Characterizing Task-Machine Affinity in Heterogeneous Com-
puting Environments”. In: 2011 IEEE International Symposium on Parallel and Distributed
Processing Workshops and Phd Forum. IEEE. 2011, pp. 34–44.
[11] T. S¨onmez. “Strategy-Proofness and Essentially Single-Valued Cores”. In: Econometrica 67.3
(1999), pp. 677–689.
[12] J. Yu and J. Zhang. “Efficient and fair trading algorithms in market design environments”.
In: arXiv preprint arXiv:2005.06878v3 (2021).
A RAC Code
This code, which can also be found here, has been slightly modified from the original code, which
can be found here.
#include <iostream>
#include <math.h>
#define SECONDS_PER_DAY 86400
using namespace std;
void update_average (
double work_start_time, // when new work was started // (or zero if no new work)
double work, // amount of new work
double half_life,
double& avg, // average work per day (in and out)
double& avg_time, // when average was last computed
double& fakeTime // new, for simulations
) {
//double now = dtime();
double now = fakeTime;
if (avg_time) {
double diff, diff_days, weight;
diff = now - avg_time;
if (diff<0) diff=0;
diff_days = diff/SECONDS_PER_DAY;
weight = exp(-diff*M_LN2/half_life);
avg *= weight;
if ((1.0-weight) > 1.e-6) {
avg += (1-weight)*(work/diff_days);
else {
avg += M_LN2*work*SECONDS_PER_DAY/half_life;
else if (work) {
// If first time, average is just work/duration
cout << "avg_time = " << avg_time << "\n";
cout << "now = " << now << "\n";
double dd = (now - work_start_time)/SECONDS_PER_DAY;
avg = work/dd;
avg_time = now;
int main() {
double RAC = 0;
double timeOne = 1;
double timeTwo = 1;
double totalCredit = 0;
double timeInterval = 3600; // new; time in seconds between each RAC update
double work_start_time = 0; // when new work was started // (or zero if no new work)
double work = 200; // amount of new work
double half_life = 604800;
double& avg = RAC; // average work per day (in and out)
double& avg_time = timeOne; // when average was last computed
double& fakeTime = timeTwo; // new; for simulation
for (int i=0; i<1500; i++) {
if (1) {
if (i % 24 == 0) {
cout<<"week " << i/168 + 1 << ", day " << (i/24)%7 + 1 << ";
current hour = " << i << "; ";
cout<<"totalCredit = " << totalCredit << "; ";
cout<<"RAC = "<< RAC << "\n";
fakeTime += timeInterval;
update_average(work_start_time, work, half_life, avg, avg_time, fakeTime);
totalCredit += work;
cout<<"Final totalCredit = " << totalCredit << "\n";
cout<<"Final fakeTime = " << fakeTime << "\n";
cout<<"Final RAC = "<< RAC;
return 0;
B ER Approximation
The code for the ER approximation can be found here.
As described in Section 1.1, the computational contribution of all of the machines under a
cruncher’s CPID to a project is grouped into a single measurement called RAC, which is the
only statistic currently provided to the public by projects. Increasingly granular disaggregation
of the RAC yields an increasingly accurate estimate of the ER. There are three main levels of
1) Machine-level disaggregation. A single CPID can contain many machines, and those that do
can be massive outliers in the data and completely distort the regression, to the extent that bρmight
not even intersect the axes on their positive rays.
2) Application-level disaggregation. Most projects have multiple applications, each of which has
different computational requirements. Projects can normalize their credits across their respective
applications – however, even a single project not normalizing their credits across their applications
would prevent an accurate estimate of ρ. Furthermore, even if every single project normalized their
credits across their applications, the current credit computations are still flawed, which is one of the
motivations for basing the ER on WUs completed, rather than credits. Additionally, normalizing
based on all of the machines on the network provides a much larger sample size than normalization
based on a small subset of hardware to which projects have access.
3) WU-level disaggregation. A number of revisions to BOINC’s credit-granting mechanism have
been explored, see [1] for an example. However, the difficulty involved in taking into account all
the factors about the machines means that there have been no comprehensive solutions for this
problem yet. As explained in Section 1.1 and the preceding paragraph, this is why WU-level data
is preferred.
C Game-Theoretic Simulations
The code for the game simulations can be found here.
Figure 5: Game that converges
Figure 6: Game that diverges
Figure 7: Game that cycles
ResearchGate has not been able to resolve any citations for this publication.
Full-text available
“Volunteer computing” is the use of consumer digital devices for high-throughput scientific computing. It can provide large computing capacity at low cost, but presents challenges due to device heterogeneity, unreliability, and churn. BOINC, a widely-used open-source middleware system for volunteer computing, addresses these challenges. We describe BOINC’s features, architecture, implementation, and algorithms.
Conference Paper
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Many computing environments are heterogeneous, i.e., they consist of a number of different machines that vary in their computational capabilities. These machines are used to execute task types that vary in their computational requirements. Characterizing heterogeneous computing environments and quantifying their heterogeneity is important for many applications. In previous research, we have proposed preliminary measures for machine performance homogeneity and task-machine affinity. In this paper, we build on our previous work by introducing a complementary measure called the task difficulty homogeneity. Furthermore, we refine our measure of task-machine affinity to be independent of the task type difficulty measure and the machine performance homogeneity measure. We also give examples of how the measures can be used to characterize heterogeneous computing environments that are based on real world task types and machines extracted from the SPEC benchmark data.
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Conference Paper
Proof-of-work blockchains reward each miner for one completed block by an amount that is, in expectation, proportional to the number of hashes the miner contributed to the mining of the block. Is this proportional allocation rule optimal? And in what sense? And what other rules are possible? In particular, what are the desirable properties that any "good" allocation rule should satisfy? To answer these questions, we embark on an axiomatic theory of incentives in proof-of-work blockchains at the time scale of a single block. We consider desirable properties of allocation rules including: symmetry; budget balance (weak or strong); sybil-proofness; and various grades of collusion-proofness. We show that Bitcoin's proportional allocation rule is the unique allocation rule satisfying a certain system of properties, but this does not hold for slightly weaker sets of properties, or when the miners are not risk-neutral. We also point out that a rich class of allocation rules can be approximately implemented in a proof-of-work blockchain.
Core-selection is a crucial property of rules in the literature of resource allocation. It is also desirable, from the perspective of mechanism design, to address the incentive of agents to cheat by misreporting their preferences. This paper investigates the exchange problem where (i) each agent is initially endowed with (possibly multiple) indivisible goods, (ii) agents' preferences are assumed to be conditionally lexicographic, and (iii) side payments are prohibited. We propose an exchange rule called augmented top-trading-cycles (ATTC), based on the original TTC procedure. We first show that ATTC is core-selecting and runs in polynomial time with respect to the number of goods. We then show that finding a beneficial misreport under ATTC is NP-hard. We finally clarify relationship of misreporting with splitting and hiding, two different types of manipulations, under ATTC.
Conference Paper
The peer-to-peer (P2P) paradigm operates in an uncontrolled voluntary environment and, as the shared resources contributed by the general public have no legal obligation with respect to resource provisioning, an incentive based approach is needed to motivate and encourage participants to continue with the project and attract new volunteers. The Berkeley Open Infrastructure for Network Computing (BOINC), one of the most widely used middleware platforms in P2P systems, has devised an Accounting System to provide incentives to the project participants in order to attract more participants and retain existing ones. The primary incentive provided by the BOINC Accounting System is the award of 'credits', which themselves have no monetary value. However, one of the problems with the credit system is the use of significantly inconsistent performance results obtained from the Dhrystone and Whetstone benchmarks for calculating credits. This study analyses the existing BOINC Credits calculation system and proposes a new credit unit - MalikCredits - based on a more consistent synthetic lightweight benchmark - MalikStone, which is specifically designed for the dynamic challenges of the P2P paradigm. The results of this newly proposed credit calculation system have highlighted its superiority over the BOINC Credit calculation system in terms of consistency.
IN THIS PAPER WE SEARCH for solutions to various classes of allocation problems. We hand results pertaining to housing markets are much more encouraging. Roth 1982b shows that in the context of housing markets the core correspondence, which is shown to Ž. Ž . be single-valued by Roth and Postlewaite 1977 , is strategy-proof. Moreover Ma 1994 shows that it is the only solution that is Pareto efficient, individually rational, and strategy-proof.
Generalizing Top Trading Cycles for Housing Markets with Fractional Endowments
  • H Aziz
H. Aziz. "Generalizing Top Trading Cycles for Housing Markets with Fractional Endowments". In: arXiv preprint arXiv:1509.03915 (2015).
Bitcoin and Cryptocurrency Technologies
  • A Narayanan
A. Narayanan et al. Bitcoin and Cryptocurrency Technologies. 2016.
Efficient and fair trading algorithms in market design environments
  • J Yu
  • J Zhang
J. Yu and J. Zhang. "Efficient and fair trading algorithms in market design environments". In: arXiv preprint arXiv:2005.06878v3 (2021).