=Paper=
{{Paper
|id=Vol-3791/paper20
|storemode=property
|title=Incentive Compatibility In Consensus Protocols And DAOs: A Game-Theoretic Approach
|pdfUrl=https://ceur-ws.org/Vol-3791/paper20.pdf
|volume=Vol-3791
|authors=Angelo Murano,Bruna Bruno,Vincenzo Vespri
|dblpUrl=https://dblp.org/rec/conf/dlt2/MuranoBV24
}}
==Incentive Compatibility In Consensus Protocols And DAOs: A Game-Theoretic Approach==
https://ceur-ws.org/Vol-3791/paper20.pdf
Incentive Compatibility In Consensus Protocols And DAOs:
A Game-Theoretic Approach
Angelo Murano1,* , Bruna Bruno2 and Vincenzo Vespri3
1
Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano SA
2
Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano SA
3
Università degli Studi di Firenze, Viale Morgagni, 67/a I-50134 Firenze
Abstract
This study aims to explore blockchain technologies by integrating the economic perspective and the computer
science approach, with a special focus to economic applications where non-cryptocurrency blockchain are usually
adopted. This interdisciplinary approach may help to create blockchain solutions that are both technically
robust and economically sustainable. For non-cryptocurrency permissioned blockchains to be widely adopted in
various business and organizational applications, it is important to view nodes as economic agents rather than
abstract entities. These agents exhibit behaviour driven by their individual interests. Therefore, the consensus
protocol must be designed to consider effective incentive compatibility constraints. The theoretical approach is
based on the evaluation of incentive compatibility constraints for both honest and faulty nodes, according two
different remuneration schemes. By extending the principles of incentive compatibility from non cryptocurrency
blockchains to organizational structures modelled as DAOs, we explore how decentralized governance models
can benefit from blockchain-based incentive mechanisms while addressing the inherent economic risks and legal
challenges.
Keywords
Non-cryptocurrency, Incentives compatibility, PBFT
1. Introduction
Blockchain technologies have received significant attention from economists, especially in the study of
non-fiat currencies and their impact on traditional monetary systems. Non-fiat currencies are those that
are not issued or regulated by a central government and lack physical backing like traditional currency.
Bitcoin [28] is an example of a decentralised cryptocurrency that operates on the blockchain. Unlike
traditional currencies, it is not issued or regulated by a central bank or government. The acceptance
and validity of the blockchain system depend on the network of users participating in it and their trust
in the security and integrity of the underlying technology.
However, it is important to acknowledge that blockchain is a valuable technology with different
applications, exerting significant influence across various sectors including environment, education,
health, financial services, trading, utilities, manufacturing, as well as both private and public adminis-
trations. Beyond its origins in cryptocurrency, blockchain technology offers innovative solutions in
numerous fields, such as supply chain management, healthcare, and digital identity management. For
instance, in supply chain logistics, blockchain provides a transparent and tamper-proof record system
that enhances traceability and accountability, thereby reducing fraud and ensuring product authenticity
[1]. In the healthcare sector, blockchain can secure and streamline the sharing of medical records,
improving patient outcomes and safeguarding data privacy [12]. Similarly, digital identity systems
based on blockchain technology can offer a decentralized and secure platform for identity verification,
reducing identity theft and enhancing user control over personal information [15].
DLT 2024 - 6th Distributed Ledger Technology Workshop
*
Corresponding author.
$ amurano@unisa.it (A. Murano); brbruna@unisa.it (B. Bruno); vincenzo.vespri@unifi.it (V. Vespri)
https://rubrica.unisa.it/persone?matricola=052051 (A. Murano); https://docenti.unisa.it/003707/home (B. Bruno);
https://people.dimai.unifi.it/vespri/ (V. Vespri)
0009-0006-2554-1921 (A. Murano); 0000-0001-6767-1069 (B. Bruno); 0000-0002-2684-8646 (V. Vespri)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
Central to the success of these non-cryptocurrency applications are the incentive mechanisms that
encourage participation and ensure the integrity of the system. These mechanisms often involve
rewarding participants with tokens or digital assets for their contributions to the network, such as
validating transactions or providing storage space, which helps maintain the high-security standards
necessary for these applications. For example, in blockchain-based supply chains, participants including
suppliers and transporters may receive tokens as incentives for maintaining accurate and timely data
entries, which are crucial for the effectiveness and reliability of the supply chain [17]. This form of
incentivization not only promotes system integrity but also fosters a cooperative environment where
all stakeholders are motivated to contribute positively.
This study aims to explore blockchain technologies in-depth, including the integration of fundamental
blockchain features within economic theory and applications. It is important to note that applying an
economic perspective to the functioning and attributes of blockchain may provide valuable insights
to the extensive body of literature that has predominantly evolved within the realm of computer
science. It is important to recognise the unique approach of economics in addressing problems and in
choices making, which differs from the hypotheses and objectives typically used in computer science
investigations.
Contribution of economic theory to computer science practice on blockchains is especially relevant
in the realm of some potential applications of blockchain technologies where free-riding or other
opportunistic behaviours may emerge. In the economic perspective, blockchain nodes are individuals,
whose behaviours and objectives must be considered.
In contexts where blockchain is used outside of cryptocurrency - such as in digital supply chains, public
record management, or consensus models for collective decision-making - these issues become particu-
larly pronounced. Free-riding and opportunistic behaviors are crucial concerns in non-cryptocurrency
blockchain applications due to the decentralized nature and lack of a central authoritative entity which
in traditional systems often acts as a monitor or enforcer. For example, in a blockchain-based digital
supply chain, a participant may under-report sales to evade fees or engage in selective misreporting
to gain competitive advantages. Without the direct oversight typical in centralized systems, such
actions can undermine the integrity and trust upon which the blockchain operates. Real-world data
and theoretical analysis show that free riding can lead to significant inefficiencies and increase the cost
of maintaining blockchain systems, potentially making them economically unviable without proper in-
centive mechanisms in place [30]. Furthermore, other studies [7] demonstrate through simulations how
differing incentives can lead to varied degrees of participation and honesty among nodes, highlighting
the fragile balance required to sustain cooperative behaviour in blockchain networks.
Agents often face complex and dynamic situations that involve rational, emotional, and social
elements. Economics takes a unique approach to problem-solving and decision-making, based on a
deep understanding of the behaviour, dynamics, and interactions among agents. In contrast, computer
science is primarily focused on system design, algorithm optimization, and information management.
The issues tackled in computer science are usually well-defined and structured, governed by clear
logic and rules. The primary goal is to ensure the efficiency and accuracy of technical solutions. The
integration of these perspectives is crucial in the development of blockchain solutions that are not only
technically robust but also economically sustainable.
The traditional paradigm in distributed computation categorises agents as either ’good’ or ’bad’ [2].
In this scenario, individuals who follow the prescribed protocol are considered ’good’, while those who
use any means necessary to disrupt it are considered ’bad’. Research has shown that the protocol is
successful when a given percentage of individuals take on the ’bad’ role, highlighting the binary nature
of traditional computational perspectives.
To understand the ethical and security challenges presented by blockchain technology, it is useful
to analyse a case study that clearly demonstrates the complex relationship between technological
vulnerabilities, ethical concerns, and the involvement of malicious actors in shaping the development
of blockchain technology. In 2016, a hacker exploited a vulnerability in the code of the Decentralised
Autonomous Organisation (DAO) and transferred 3.6 million Ether, valued at approximately 50 million
dollars, into a personal account. The DAO was a pioneering initiative in blockchain-based governance,
aiming to democratise investment in Ethereum-based start-ups by enabling investors to allocate funds
to proposed projects using Ethers. The breach not only highlighted vulnerabilities within such platforms
but also sparked a controversial debate over the blockchain’s immutable nature and the ethical principle
of ’code is law’. The Ethereum community’s decision to rectify the theft through a network fork resulted
in significant fallout, including the emergence of Ethereum Classic and a revaluation of hard and
soft fork applications in the blockchain domain [25]. Further complicating the situation, the hacker
communicated directly with the Ethereum and DAO communities after the attack through an open letter.
The hacker defended the extraction of funds as legal under the jurisdiction’s laws, exploiting a loophole
in the DAO’s smart contract code. The hacker claimed that exploiting a systemic flaw was not a crime
and threatened legal action against any attempts to recover the stolen Ether. This increased tensions
within the community. In addition, the hacker’s efforts to sabotage soft fork proposals by offering
significant bribes to Ethereum miners highlight the difficulties in implementing technical solutions
to ethical violations. This controversial exchange resulted in the abandonment of a soft fork due to
identified vulnerabilities, ultimately requiring a hard fork. The division of the Ethereum blockchain into
Ethereum (ETH) and Ethereum Classic (ETC) resolved the immediate crisis and sparked a significant
discussion on censorship resistance and the principle of blockchain immutability [37].
Other contributions draw attention to the contrasting methodologies employed in computer science
and economics within the context of blockchain technology [8]. In computer science and modern
cryptography, assumptions typically concern the actions of agents, which are categorized as either
honest or faulty nodes. Honest nodes consistently exhibit honest behaviour, while faulty nodes engage
in misconduct. In contrast, economists tend to make assumptions about primitives, such as agents’
utility functions, and subsequently analyse their strategic behaviours within equilibrium frameworks
[8]. The expansive domain for economic exploration within blockchain technology becomes particularly
evident when addressing conditions of asymmetric information. In other studies it is emphasized the
substantial latitude available for manoeuvring within this nascent field of research [5]. Furthermore,
some works accentuate the disparity between computer science hypotheses and economic traditions
regarding individual behaviour [4]. The authors emphasise the contrast between the game-theoretic
approach, which assumes rationality among all players or processes, and the conventional computer
science approach. In the latter, some processes are deemed correct and expected to adhere to the
specified protocol, while others are designated as Byzantine, introducing a nuanced perspective on
individual behaviour [4].
A practical example of this dynamic is Dogecoin, a cryptocurrency conceived purely ironically in 2013.
Contrary to the rational expectations of game theory, Dogecoin gained popularity and market value
despite its unconventional origin [34]. From an economic perspective, the existence of cryptocurrencies
such as Dogecoin underlines how psychological and social factors can influence economic valuation.
This highlights that, in blockchain, economic valuation cannot completely disregard human behavioural
dynamics, adding a complex dimension to the relationship between information technology and eco-
nomics in this emerging field. The term ’Byzantine’ originates from the Byzantine generals’ problem
[20], a theoretical framework that illustrates the complexity of coordinating distributed processes when
certain entities may disseminate erroneous information or exhibit malicious behaviour.
In the context of blockchain and consensus protocols, it is crucial for nodes to adhere to the established
protocol to ensure network consistency and security. The use of the ’Byzantine’ concept recognises the
possibility of non-compliant actions by certain nodes. Therefore, it is necessary to develop consensus
algorithms that can handle such deviations in behaviour, ensuring the security and integrity of the
system, even in the presence of defective or malicious nodes.
A specific investigation delves into numerous economic implications and challenges within the realm
of economic science [8]. However, this discussion focuses mainly on the domain of cryptocurrency
blockchain. Therefore, it is necessary to conduct an extensive economic analysis of non-cryptocurrency
blockchain, which is highly relevant for economic applications. It is important to note that non-
cryptocurrency blockchains differ significantly from their cryptocurrency counterparts in terms of
explicit monetary incentives. Given the relevance of non-cryptocurrency blockchain for economic
application, economic perspective of non-cryptocurrency blockchain must be investigated.
Non-cryptocurrency applications of blockchain are often private blockchains, due to limitations
observed in public blockchains when applied to industrial use cases. It is important to note that public
blockchains exhibit constraints that prove detrimental to corporate and administrative applications [9].
Another research has highlighted that incentive-compatible mechanisms widely employed in existing
blockchain systems are not seamlessly applicable to non-cryptocurrency scenarios [24]. This statement
highlights the need for consensus protocols in business applications that use non-cryptocurrency
blockchains. Though non-cryptocurrency blockchains are already employed in some business applica-
tions, wider organizational structures with multiple agents interacting on the blockchain (e.g. in public
administration) may exhibit more occurrences of opportunistic behaviours to be prevented. Therefore, it
is necessary to investigate incentive-compatible mechanisms enforcing non crypto-currency consensus
protocols for suitable economic applications.
Contribution of the paper is the analysis of different rewarding schemes ensuring the compatibility of
incentives for self-interest agents maximizing their utility functions. As also emphasised recently [24],
to our best knowledge the work carried out to develop this approach is still limited. Paper is organized
as follows. The next Section presents a brief overview of economic literature investigating blockchain
technologies. The model description in following section, while the last Section summarizes concluding
remarks.
2. Overview on previous works
In blockchain, incentivization mechanisms play a crucial role in motivating miners to participate in
validation processes, contributing significantly to the system’s overall maintenance. While effective in
cryptocurrency applications, extending existing blockchain systems to non-cryptocurrency domains
with distinct business models poses a significant challenge. The lack of effective incentives can be a
significant obstacle to the wider adoption of blockchain technology.
An incentive compatibility mechanism characterizes a process where participants would not find
it advantageous to violate the rules of the process [11]. In the blockchain domain, it ensures that the
consensus protocol will induce validators to exhibit honest and cooperative behaviour, adhering to the
prescribed protocol rules and actively contributing to the overall security and stability of the network.
Incentives may be in economic form, such as cryptocurrency rewards or penalties for misconduct, or
in reputational form, encompassing trustworthiness scores or penalties for any deviations from the
prescribed behaviour.
The main challenge is to create an incentive mechanism that can be easily adapted to non-
cryptocurrency applications. The need for a compatible incentive mechanism becomes evident in
the context of non-cryptocurrency blockchain applications that employ ’Voting-based’ consensus al-
gorithms. Research in this domain is currently limited and mostly focused on incentive mechanism
design for cryptocurrency applications that use ’Proof-based’ consensus algorithms [24].
It is essential to point out that the ‘Voting-based’ consensus algorithms are used also in cryptocurrency
permissionless blockchains, where probabilistic consensus can be achieved due to the large numbers of
not identified nodes. In this framework, fast probabilistic consensus protocols [32] [27] may reduce the
high communication costs associated to voting mechanisms at a large scale, without incurring in PoW
energy costs [33].
It is important to note that studies on incentive compatibility for Proof-of-Work (POW) are not
applicable to PBFT-based blockchains due to the inherent disparity between ’Proof-based’ and ’Voting-
based’ consensus algorithms. Bridging this gap is essential for the successful integration of blockchain
technology into non-cryptocurrency applications.
To tackle the issue of creating digital identities on a blockchain and implementing an anonymous
voting system to prevent the creation of fictitious digital identities, an approach is needed where the
blockchain is used as an immutable ledger to securely store digital identities [19]. In this context,
smart contracts should define rules and conditions related to digital identities. For example, they could
establish the requirement for verification by a central authority to create or update an identity [14].
To ensure anonymity in the voting system, cryptographic techniques such as zero-knowledge proofs
and homomorphic encryption voting systems can be implemented. These techniques allow participants
to prove they possess a certain piece of information without revealing it. The proposed zero-knowledge
proofs and homomorphic encryption voting system ensures integrity, confidentiality, and authenticity
of votes while preserving voter anonymity [26]. Smart contracts manage the voting process and record
each vote on the blockchain. Cryptography ensures the anonymity of votes, while smart contracts
verify their validity.
To prevent the creation of fictitious digital identities, identity validation is performed before voting
through offline verification procedures or interactions with certification authorities. Anti-Sybil mecha-
nisms can be implemented that require approval by existing members before assigning an identity [31].
In addition, smart contracts define approval rules for adding new identities to the registry, requiring
the consent of the majority of authorised actors [35].
Despite the shift in consensus algorithms from ’Proof-based’ to ’Voting-based’, concerns about
incentive compatibility remain, particularly in PBFT-based consensus algorithms. Free riders within
peer-to-peer networks exploit benefits to the detriment of honest actors [16]. Therefore, it is necessary to
have robust mechanisms that can identify and penalise instances of free-riding behaviour. Our proposed
approach, based on an economic framework, examines the complex study of participant interactions
and strategic behaviours. The establishment of robust and efficient consensus mechanisms, critical
for the optimal functioning of blockchain systems, relies heavily on the behaviour of participating
entities [23]. However, in distributed systems, especially in open and permissionless settings, it is
impractical to assume consistent proactive and honest behaviour from every participant. Therefore, the
integration of incentives plays a crucial role in attracting participants and motivating them to adhere to
desired behaviours. However, incompatible incentive mechanisms have been identified as a source of
compromise to honesty, leading to significant security vulnerabilities.
Acknowledging the rational and self-interested nature of participants, the design of incentive mecha-
nisms rooted in game theory proves indispensable in addressing these challenges [36]. The efficacy
of such mechanisms in blockchain systems hinges on the ability to attract a substantial number of
participants and motivate them to act proactively. Some studies underscore the positive impact of game
theory-based incentives in enhancing the overall effectiveness of blockchain systems [38]. Although
game theory has made significant contributions to incentive mechanism design in blockchain systems,
limited efforts have been directed towards applying it to PBFT consensus algorithms.
Several studies have analysed the compatibility problems that arise in blockchain systems based
on Practical Byzantine Fault Tolerance (PBFT), particularly regarding incentive mechanisms. One
notable endeavour, the SmartCast system [18], has systematically scrutinized compatibility challenges
through the introduction of an off-chain incentive mechanism. However, this solution’s efficacy
becomes impractical when applied to applications functioning within a public peer-to-peer network.
This is particularly exemplified by certain Internet of Things (IoT) applications [21]. In a distinct
proposal, Solidus [3], attempts to address incentive compatibility by presenting a consensus algorithm
meticulously designed for Proof-of-Work (POW)-based PBFT. However, it still relies on Proof of Work
(POW) for selecting proposers, limiting its use to cryptocurrency contexts rather than a wider range
of non-cryptocurrency applications. An alternative scholarly work (Wei et al, 2020) undertakes a
game theory-based analysis with a principal focus on the incentive structure of the business model.
Unfortunately, this study does not provide a detailed analysis of the complex dynamics involved in
maintaining the blockchain network. It focuses mainly on incentives related to business considerations.
Concurrently, efforts have been made to enhance Practical Byzantine Fault Tolerance (PBFT) consensus
algorithms by incorporating reputation-based methods [8], [22]. Although these algorithms significantly
contribute to enhancing the PBFT consensus mechanism, their primary objective is to formulate efficient
selection methods rather than directly addressing the challenges associated with incentive compatibility
intrinsic to PBFT-based blockchain architectures. The need for comprehensive solutions that address
both consensus efficiency and incentive compatibility remains a significant concern in the developing
field of blockchain research.
3. Model
A fundamental process in blockchain functioning is the broadcast process, implying a problem concern-
ing asymmetric information, because nodes have private information on the content of the message they
receive. The message content may be a dichotomous variable, as in the retrieve/attack message of the
Byzantine problem [20]. In a non-crypto blockchain used for certification purposes (e.g. competencies,
property rights, characteristics of individuals or things), the dichotomous choice may be a true/false
alternative. Consequently, a faulty or malicious node may have incentives in wrong certifications,
namely in certificating features that are not true by broadcasting a message different from the one
received. A self-interested validator may adopt opportunist behaviours, intentionally fraudulent. An
honest node broadcasts the same message received. Utility functions for honest and malicious nodes
may be different, and the incentive compatibility constraints for each type of node should be verified.
To comprehend and enhance the incentive mechanisms in such a system, an understanding of game
theory is indispensable. The concept of Nash equilibrium, where each player, recognizing the strategies
of others, has no incentive to unilaterally change their strategy, can be crucial in designing systems
where honest behaviour is maintained [29]. Furthermore, incorporating dominant strategies, which
are advantageous regardless of others’ actions, can ensure that maintaining honesty becomes the
most beneficial course of action for all nodes. By aligning these utility functions with game theory
principles such as Nash equilibrium and dominant strategies, blockchain systems can be designed to be
robust against dishonest behaviours and ensure a more secure and trustworthy environment for all
participants.
Consider a non-crypto currency blockchain with n=3k+1 validators, including faulty/malicious
validators who are interested in blockchain malfunctioning and/or wrong certifications. In a PBFT
protocol the block is accepted if at least 2k+1 validators agree on the block acceptance. The agreement
of each validator i is expressed through a vi vote (i=1,..,n), which can take two values:
{︃
1 if the validator agree,
𝑣𝑖 =
0 if the validator does not agree.
Therefore, the final decision on the block can be summarized as follows:
𝑛
∑︁ 2
If 𝑣𝑖 > 𝑛, the block is accepted (1)
3
𝑖=1
𝑛
∑︁ 2
If 𝑣𝑖 ≤ 𝑛, the block is rejected (2)
3
𝑖=1
This simplified representation of a PBFT protocol can be used to represent validators’ behaviours
when receiving a block proposal message M=(x,y), where x is the content and y is the result of the
voting process on the message. In details:
{︃
1 if the content is true
𝑥=
0 if the content is false
{︃
1 if the message reaches the consensus
𝑦=
0 if the message does not reach the consensus
Furthermore, we assume that a reputation grade 𝑤𝑖 > 0, or another kind of non-monetary reward, is
associated with the voting (consensus) process.
Finally, we assume that the validators have the capability to access the data encapsulated within the
blocks, although they cannot comprehend or interpret the contents. From the validator’s perspective,
the data housed within the block is simply construed as uncomplicated sequences of bits, even singular
bits. The validator’s role is emphasised as impartial through this abstraction, with a focus on the binary
nature of the data rather than its semantic interpretation.
The following behavioral hypotheses are adopted:
H1: Validators can read the message content as either true or false.
H2: Honest validators receive positive satisfaction from the acceptance of true blocks and the
rejection of false blocks (with the same satisfaction gain). Faulty validators receive positive satisfaction
from the acceptance of false blocks and the rejection of true blocks (with the same satisfaction gain).
H3: Honest and faulty validators receive positive satisfaction for positive 𝑤𝑖 .
With these basic elements of the blockchain, we can describe the utility functions of the faulty (𝐹 )
and honest (𝐻) validators as 𝑈𝐻,𝐹 = 𝑓 (𝑀, 𝑤𝑖 ) as follows:
⎧ {︃
⎪ (1, 1) a true message is accepted
′ 𝜕𝑈𝐻 ⎨> 0 if 𝑀 =
𝑈𝐻𝑀 = (0, 0) a false message is rejected
𝜕𝑀 ⎩⎪
= 0 otherwise
(3)
⎧ {︃
⎪ (1, 0) a true message is accepted
′ 𝜕𝑈𝐹 > 0 if 𝑀 =
⎨
𝑈𝐹𝑀 = (0, 1) a false message is rejected
𝜕𝑀 ⎪
= 0 otherwise
⎩
(4)
In other words, these utility functions imply that the honest validator receive positive satisfaction
from the acceptance of a true block or the rejection of a false block, whereas the faulty validator
pursue the objective of false blocks accepted and true blocks rejected (H2). Note that the derivatives
are independent from the vote expressed by the validators. As to the reputation grade, both types of
validators have increasing utility from the reward (H3).
𝜕𝑈𝐻
𝑈 ′ 𝐻𝑤 = >0 (5)
𝜕𝑤
𝜕𝑈𝐹
𝑈 ′ 𝐹𝑤 = >0 (6)
𝜕𝑤
The remuneration scheme is fundamental in a mechanism design. Two different remuneration
schemes will be analysed, considering alternatively the payoffs for all types of validators at the end of
the consensus process (final payoffs), or in the while the consensus process is still running (expected
payoffs), that is when the y value is still unknown. The final payoffs are described in a strategic form, the
matrix representation of a simultaneous (two players) non cooperative game, with the validator as row
player and the State of Nature as column player, who selects the message proposed to validator. Payoffs
are quantified only for the row player (the validator). Validator has two available actions/strategies
when receiving the message, corresponding to the values (1,0) chosen for the vote expressed with vi.
For each strategy, resulting payoffs are reported according to the different States of Nature.
Scheme A: the validator receives a positive 𝑤𝑖 if the block is accepted, and the vote is 𝑤𝑖 = 1.
No reward is provided for the ’no agreement’ vote.
Final payoffs – Scheme A
Table 1
Honest Nodes
Actions/Messages M=(1,1) M=(1,0) M=(0,0) M=(0,1)
′
𝑉𝑖 = 1 U’𝐻 𝑀 + 𝑈𝐻 𝑤
U’𝐻𝑀 U’𝐻 𝑤
𝑉𝑖 = 0 U’𝐻 𝑀 U’𝐻 𝑀
Table 2
Faulty Nodes
Actions/Messages M=(1,1) M=(1,0) M=(0,0) M=(0,1)
𝑉𝑖 = 1 U’𝐹𝑤 U’𝐹 𝑀 U’𝐹𝑀 + 𝑈𝐹′ 𝑤
𝑉𝑖 = 0 U’𝐹𝑀 U’𝐹𝑀
Table 1 shows the payoffs for honest nodes in Scheme A. It is easy to verify that the agreement
vote (𝑣𝑖 = 1) is a dominant strategy, even for false messages (when 𝑀 = (0, 0) or (0, 1)). Similarly,
for faulty nodes (Table 2), the 𝑣𝑖 = 1 vote is a dominant strategy. This is because the reward scheme
only awards validators who agree on the block. Furthermore, one can observe that for honest nodes,
rejecting a false block is preferred to acceptance only if 𝑈𝐻′ ′
𝑀 > 𝑈𝐻 𝑤 , whereas for faulty nodes, the
condition to prefer rejecting a true block to its acceptance is that 𝑈𝐹 𝑀 > 𝑈𝐹′ 𝑤 .
′
Nevertheless, when validators give their vote, they know if the message content is true or false,
but they do not know if the message will be added to the block or not. They may have an expected
result 𝑦 𝑒 , based on the PBFT hypothesis on faulty/honest nodes. If all faulty nodes vote 0 to true blocks
and 1 to false block, whereas honest nodes do the opposite, the probability of each type of block to be
accepted/rejected are:
2
𝜋(1, 1) = 𝜋(0, 0) = 𝑛 + 1 (7)
3
1
𝜋(1, 0) = 𝜋(0, 1) = 𝑛 − 1 (8)
3
If they can recognize the true/false feature of the block, they can evaluate the expected payoffs as
follows. Let’s consider first the expected values for honest nodes (𝐸𝑉𝐻 ) when facing true blocks.
{︃
′
( 23 𝑛 + 1)(𝑈𝐻𝑀 ′ ) if 𝑣 = 1
+ 𝑈𝐻𝑤
𝑒 𝑖
𝐸𝑉𝐻 (1, 𝑦 ) = ′
(9)
( 23 𝑛 + 1 − 1)𝑈𝐻𝑀 if 𝑣𝑖 = 0
From the above descriptions it easy to verify that 𝑣𝑖 = 1 is a dominant strategy for an honest node
facing true blocks. When coming at false blocks, the expected values are as follows.
{︃
′
( 32 𝑛 + 1 − 1)𝑈𝐻𝑀 ′
+ ( 13 𝑛 − 1 + 1)𝑈𝐻𝑤 if 𝑣𝑖 = 1
𝐸𝑉𝐻 (0, 𝑦 𝑒 ) = 2 ′
(10)
( 3 𝑛 + 1)(𝑈𝐻𝑀 ) if 𝑣𝑖 = 0
Here, one would expect that 𝑣𝑖 = 0 is a dominant strategy, but this is true only if
𝑈 ′𝐻𝑀 1
′
> 𝑛 (11)
𝑈 𝐻𝑤 3
Consequently, the false block rejection will depend on the specific form of the utility function (the
validator preferences for honest behaviours and reward) and on the blockchain dimension n.
When coming at the expected values for faulty nodes (𝐸𝑉𝐹 ), for true blocks it would be:
{︃
𝑒 ( 23 𝑛 + 1 + 1)𝑈𝐹′ 𝑤 + ( 13 𝑛 − 1 − 1)𝑈𝐹′ 𝑀 if 𝑣𝑖 = 1
𝐸𝑉𝐹 (1, 𝑦 ) = (12)
( 13 𝑛 − 1)𝑈𝐹′ 𝑀 if 𝑣𝑖 = 0
The faulty should vote 0 to the true block, but this is true only if
𝑈 ′𝐹 𝑀 2
′
> 𝑛+2 (13)
𝑈 𝐹𝑤 3
Whereas for false blocks, the faulty node has a dominant strategy in 𝑣𝑖 = 1
{︃
𝑒 ( 13 𝑛 − 1)(𝑈𝐹′ 𝑀 + 𝑈𝐹′ 𝑤 ) if 𝑣𝑖 = 1
𝐸𝑉𝐻 (0, 𝑦 ) = (14)
( 13 𝑛 − 1 − 1)𝑈𝐹′ 𝑀 if 𝑣𝑖 = 0
The results for the scheme put in evidence that for honest nodes facing true blocks the agreement
strategy is always dominant, but the rejection of false blocks is conditioned. Vice versa, faulty nodes will
always agree on false blocks, but the rejection of true blocks is conditioned. In this scheme more likely
could happen that both true and false blocks will be accepted, compared with the previous scheme.
Scheme B: the validator receives a positive wi when the block is accepted and the vote is
vi=1; a positive wi is also received when the block is rejected, and the vote is vi=0.
Final payoffs – Scheme B
Table 3
Honest Nodes
Actions/Messages M=(1,1) M=(1,0) M=(0,0) M=(0,1)
′ ′ ′ ′
𝑉𝑖 = 1 𝑈𝐻 𝑀 + 𝑈𝐻𝑤 𝑈𝐻 𝑀
𝑈𝐻 𝑤
′ ′ ′ ′
𝑉𝑖 = 0 𝑈𝐻 𝑀 𝑈𝐻 𝑤
𝑈𝐻 𝑀 + 𝑈𝐻𝑤
Table 4
Faulty Nodes
Actions/Messages M=(1,1) M=(1,0) M=(0,0) M=(0,1)
𝑉𝑖 = 1 𝑈𝐹′ 𝑤 𝑈𝐹′ 𝑀 𝑈𝐹′ 𝑀 + 𝑈𝐹′ 𝑤
𝑉𝑖 = 0 𝑈𝐹′ 𝑀 + 𝑈𝐹′ 𝑤 𝑈𝐹′ 𝑤 𝑈𝐹′ 𝑀
There is no dominant strategy both for honest and faulty nodes. The expected payoffs for honest
nodes are as follows.
{︃
′
( 32 𝑛 + 1)(𝑈𝐻𝑀 ′ )
+ 𝑈𝐻𝑤 if 𝑣𝑖 = 1
𝐸𝑉𝐻 (1, 𝑦 𝑒 ) = 2 ′ 1 ′
(15)
( 3 𝑛 + 1 − 1)𝑈𝐻𝑀 + ( 3 𝑛 − 1 + 1)𝑈𝐻𝑤 if 𝑣𝑖 = 0
It easy to verify that 𝑣𝑖 = 1 is a dominant strategy for an honest node facing true blocks. When
coming at false blocks, the expected values are as follows.
{︃
′
( 32 𝑛 + 1 − 1)𝑈𝐻𝑀 ′
+ ( 13 𝑛 − 1 + 1)𝑈𝐻𝑤 if 𝑣𝑖 = 1
𝐸𝑉𝐻 (0, 𝑦 𝑒 ) = 2 ′ ′
(16)
( 3 𝑛 + 1)(𝑈𝐻𝑀 + 𝑈𝐻𝑤 ) if 𝑣𝑖 = 0
It easy to verify that 𝑣𝑖 = 0 is a dominant strategy for an honest node facing false blocks. When
coming at the expected values for faulty nodes (𝐸𝑉𝐹 ), for true blocks it would be:
{︃
( 23 𝑛 + 1 − 1)𝑈𝐹′ 𝑤 + ( 13 𝑛 − 1 − 1)𝑈𝐹′ 𝑀 if 𝑣𝑖 = 1
𝐸𝑉𝐻 (1, 𝑦 𝑒 ) = (17)
( 13 𝑛 − 1)(𝑈𝐹′ 𝑀 + 𝑈𝐹′ 𝑤 ) if 𝑣𝑖 = 0
The faulty should vote 0 to the true block, but this is true only if
𝑈 ′𝐹 𝑀 1
′
> 𝑛+3 (18)
𝑈 𝐹𝑤 3
Whereas for false blocks, the faulty node has a dominant strategy in vi=1, if the same conditions as
above holds.
{︃
( 13 𝑛 − 1)(𝑈𝐹′ 𝑀 + 𝑈𝐹′ 𝑤 ) if 𝑣𝑖 = 1
𝐸𝑉𝐹 (0, 𝑦 𝑒 ) = 1 ′ 2 ′
(19)
( 3 𝑛 − 1 − 1)𝑈𝐹 𝑀 + ( 3 𝑛 + 1 + 1)𝑈𝐹 𝑤 if 𝑣𝑖 = 0
If the remuneration scheme award both positive votes for accepted blocks and negative votes for
rejected blocks, the honest validators are always incentivized to behave correctly, by agreeing on true
blocks and rejecting the false ones. On the other hand, the faulty nodes will be incentivized to behave
maliciously only if their motivation is strong enough, that is if
𝑈 ′𝐹 𝑀 1
′
> 𝑛+3 (20)
𝑈 𝐹𝑤 3
The above discussion shows that the remuneration scheme A is not incentive compatible for honest
nodes, who will prefer to accept also false blocks, also when considering expected payoffs. The false
block rejection will depend on the validator’s preferences and on the blockchain dimension. Conversely,
the second scheme (B) is incentive compatible for honest nodes, who have a dominant strategy in
accepting true blocks and rejecting false blocks. Faulty nodes will always agree on false block acceptance,
whereas the rejection of true blocks is conditioned.
4. Concluding remarks
This paper starts from the idea that, to widely employ non-cryptocurrency permissioned blockchain in
business and organizational applications, it is important to consider that nodes are not abstract entities,
but economic agents who behave pursuing their own interests.
With this perspective, the consensus protocol must be designed considering effective incentive
compatibility constraints, at least for those currently defined as honest nodes. The proposed analysis
of individual behaviour is based on two different types of agents, with different objective function. It
compares two different remuneration schemes, showing that the design of mechanisms may be relevant
for addressing individual choices toward protocol integrity and secure network environment.
The model we consider for incentive compatibility constraints is inherently static, as the decision-
making of each validator is contingent solely upon its conduct within the current round, without regard
to its past performance or prevailing network conditions. Such a static approach may entail potential
inefficiencies and vulnerabilities within the system. Transitioning from a static to a dynamic model
necessitates several adjustments to accommodate the evolving dynamics of validator participation,
potentially fortifying the consensus system within the framework of voting-based blockchains. The dy-
namic model permits the system to adapt to changing conditions or alterations in validator engagement,
thereby enhancing the system’s resilience and adaptability.
Within the dynamic model, the reputation of validators becomes subject to fluctuations over time
based on their actions and performance. Active participation, consistent voting patterns, and construc-
tive contributions to consensus can lead to an augmentation of a validator’s reputation. Conversely,
unethical behaviour, erratic voting, or inadequate participation may result in a deterioration of rep-
utation. Dynamic reputation is contingent upon metrics such as voting consistency, adherence to
consensus commitments, and overall participation. This framework creates incentives for validators to
cultivate a positive, enduring reputation, fostering trust within the network.
Furthermore, the dynamic model takes into consideration the interactions among validators during
various consensus phases. For instance, a validator altering its vote during the preparation phase
without a valid justification might incur a reputation penalty. Interactions among validators can shape
the interpretation of votes and decisions during the consensus process.
Should the incentive compatibility constraints model evolve into a dynamic framework, mechanisms
could be implemented to assess the trustworthiness of each validator based on historical behaviour,
network participation, quality of service, reputation, and other relevant factors. This augmentation
could bolster the security and efficiency of the protocol, rendering it more resistant to potential attacks
and scalable. The dynamic model, by incorporating the evolution of validators’ actions and decisions
over time, introduces a temporal and adaptive component into the consensus system. Time assumes
a pivotal role in this model, as past and present actions of validators influence their reputation and,
consequently, their future rewards, thereby encouraging consistent and reliable conduct over time.
Situations characterized by a framework akin to the one recently delineated find representation in
game theory through repeated games [6]. However, despite existing literature applying game theory to
blockchain, none of these works delve into the interaction among validators using an infinitely repeated
game theoretical approach to address the issue. Infinitely repeated games provide insights into the
structure of individual behaviour when interacting repeatedly, as the outcomes of prior interactions
influence their future behaviours. Such games contribute to reinforcing cooperative behaviour [10]. In
this form of interaction, players strive to optimize rewards in both the short and long term.
Within an infinitely repeated prisoner’s dilemma, the "grim trigger strategy" has been identified as
a mechanism to encourage cooperation among players [13][6]. In a grim strategy, a player initiates
cooperation (playing honestly) and continues to cooperate unless the other player deviates (acting
maliciously) at a certain point. This could potentially represent a future evolution within the context
of our work, as a player must commence with cooperation and persist in cooperation to build their
reputation.
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