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An extension of a game theoretical model of users’ incentives in blockchain dark venues

A Waiting Time’s Perspective

Published onJan 14, 2023
An extension of a game theoretical model of users’ incentives in blockchain dark venues

Author: Tianyu Wu, Applied Mathematics and Computational Sciences, 

Class of 2023, Duke Kunshan University 

Supervisor: Prof. Luyao Zhang, Duke Kunshan University

Keywords: blockchain, game theory, waiting time, opportunity cost, user incentives, simulation

Disclaimer: This is the student Author, Tianyu Wu’s final project assignment in submission to Econ 211 Intelligent Economics: An Explainable AI Approach taught by Prof. Luyao Zhang at Duke Kunshan University, Autumn 2022.

Acknowledgments: I would like to express my gratitude to my ECON 211 instructor and supervisor, Prof Luyao Zhang, from Duke Kunshan University, who continuously provided insight and expertise, especially in the research methodologies part, that greatly assisted the research. We thank Yifan Chen and other students from Econ 211: Intelligent Economics, Fall 2022, for sharing their pearls of wisdom with us during the course of this research. Last but not least, we are immensely grateful to Prof Fan Zhang, Ph.D., from Yale University, for his great insights into the Ethereum blockchain mechanism that helps me remove a lot of barriers when building mathematical models.

Research Summary


The proposed research is aimed at a comprehensive theoretical analysis of different participants’ incentives in blockchain dark venues. Previous research has shown that, under economic incentives, the dark venue is at least partially adopted by miners and utilized by at least one arbitrageur. However, waiting time, independent from the economic incentives, is also intuitively influential to users’ ultimate decisions because it reflects the opportunity cost that the users will especially suffer from the transactions they undertake. Thus, we will extend the current game theoretical model by incorporating waiting time to simulate different participants’ decision-making processes, to explore whether miners and arbitragers will still prefer to adopt the dark venue after balancing economic incentives and the corresponding opportunity cost.

Intellectual Merit:

The proposed research extends the current game theoretical model to simulate how participants in the blockchain ecosystem would balance economic incentives and opportunity costs to finally decide the choice of transaction submission venues. It will also advance existing literature, including the application of the game theoretical model in the blockchain system, and the extension of market microstructure studies on dark venues into the blockchain scenario.

Practical Impacts:

The proposed research would help scholars and industry professionals to better understand the underlying incentives between the public and private channels, and further reveal whether the relay services, i.e., the dark venues provided by Flashbots and Eden Network, can really help solve (or mitigate) the MEV problems to maintain blockchain fairness and security.

Part I: Research Description

1. Introduction

1.1 Background, Motivation, and Application Scenario

A Peer-to-Peer (P2P) network is where nodes are interconnected to share resources with each other without relying on a centralized administrative system (Milojicic et al. 2003). When it is enabled in the blockchain, all the nodes, i.e., the “peers”, help maintain a complete replica of the records in the network, which ensures the accuracy and transparency of the data. However, the extreme transparency of the data is a double-edged sword, among which one of the most serious issues is that malicious players in the system could exploit the transparent information to execute attacks on the pending transactions (Daian et al. 2020). This phenomenon is now called the “Miner/Maximal-Extractable Value” (MEV) problem, which has been heavily discussed among both scholars and practitioners since it has indeed posed a great threat to blockchain security.

As shown in Figure 1, at present, two popular institutions, Flashbots and Eden Network, have put a lot of effort into looking into this MEV problem. Interestingly, they both put forward the idea of building a relay service, a private communication channel between users and miners, to try to solve and mitigate the MEV problem (Züst et al. 2021). Compared to the decentralized P2P network, the relay service, also called “the dark venue”, is instead centralized, and it enables users not to broadcast their transactions globally, but forward them directly to miners who also adopt the dark venues (Qin, Zhou, and Gervais 2022). In this way, users’ transactions will not be observed by malicious players, and they will not be worried about whether their transactions will be attacked or not anymore. 

Figure 1: Introduction to P2P Network and Dark Venues in the Blockchain

 (created by Whimsical)

Nowadays, we witness that a decent number of users and miners have been attracted to submit transactions via these dark venues or prioritize including those transactions into the blockchain. Confronting this phenomenon, it will be interesting to figure out the underlying incentives for different types of players to engage in dark venues instead of the P2P network. Some work has been looking into this research direction, among which Capponi et al. (2022) pioneered building a game theoretical model to simulate users’ behaviors in blockchain dark venues by analyzing their economic incentives in detail. But it is not enough to only include the economic incentives when simulating participants’ decision-making process. More specifically, it is equally important to take the opportunity cost into consideration. Liu et al. (2022)’s empirical research pointed out that the waiting time is also an important factor in describing users’ incentives in the blockchain ecosystem because it can be seen as the opportunity cost that users need to undertake for each transaction. This insight inspired us to extend Capponi et al. (2022)’s current game theoretical model by incorporating non-negligible waiting time, to comprehensively simulate participants' decision-making process in the venue selection (Figure 2).

Figure 2: Inspiration of this Proposed Research from Two Lines of Literature 

(created by Whimsical)

1.2 Research Question

Capponi et al. (2022) revealed that, under economic incentives, the dark venues would be at least partially adopted by miners and utilized by at least one of the two arbitragers. In this proposed research, where we incorporate another dimension, waiting time, to extend Capponi et al. (2022)’s game theoretical model, we are interested in whether the miners and arbitragers will still prefer to adopt the dark venue after balancing economic incentives and the corresponding opportunity cost.

This research question is not well answered by the existing literature because the current game theoretical model only took economic incentives as the factor to simulate participants’ behaviors in the blockchain ecosystem, thus our proposed research would further extend and enrich the current game theoretical model, to enable more accurate simulations. Moreover, this research question indeed counts because it would serve as a comprehensive benchmark for both scholars and practitioners to thoroughly understand the differences between P2P network and dark venues from the perspective of incentives for the participants.

2. Literature Review

This proposed research will contribute to three lines of literature, shown in Figure 3. First of all, current studies on market microstructure have analyzed how dark venues in the traditional financial market will impact market welfare and incentives of market participants from both behavioral and economic incentives (Buti, Rindi, and Werner 2017; Degryse, Van Achter, and Wuyts 2009; Zhu 2014), and our proposed research will further extend the analysis into the blockchain scenario with a comprehensive framework considering both opportunity cost and economic incentives. Second, a large number of economists have put a lot of effort into studying how the game theoretical model can be applied to simulate the realistic blockchain (Basu et al. 2019; Liu et al. 2019; Roughgarden 2020). Thus, this proposed research is able to extend the application scenario of the game theoretical models further to the comparative analysis of the participants’ behaviors in private and public channels of the blockchain. Last but not least, our research also contributed to the discussion of the popular MEV problem mentioned in the first section. Daian et al. (2020) first put forward this problem and point out its harm to the consensus stability in the blockchain. After the relay services, such as Flashbots, are increasingly adopted by the users, miners, and arbitragers to circumvent the MEV problem, Weintraub et al. (2022) queried the relevant data to quantify the extent to different players’ participation in this dark venue. Hence, our proposed research will further reveal the secrets of whether the dark venues could truly solve (or mitigate) the MEV problems, through an improved theoretical model and simulations.

Figure 3: This Proposed Research’s Contribution to Current Literature (created by Whimsical)

3. Methodology

In this proposed research, we will mainly use model and simulation as the methodology to answer the research question. 

As shown in Figure 4, our model is set up with three types of agents: miners, users, and arbitragers, who make decisions to join either of two transaction submission venues, the transparent P2P network, or the dark venues. Since our proposed research will include the waiting time into consideration, we will perform the simulation with altogether t rounds, with 3 periods for each round. 

During Period 1, assuming that the efforts for the miners to join dark venues or P2P network are costless, miners will first make a decision. Those who decide to join dark venues are able to see the transactions submitted to the dark venues. They will first prioritize the transactions in the dark venues to be included in the blockchain, with transaction fees from high to low, and then those in the P2P network, if they are able to mine the block during Period 3. On the contrary, for those who decide to stay in the P2P network, only those transactions that are broadcast globally can be observed by these miners. Similarly, if those miners are able to mine the block during Period 3, they will only prioritize the transactions broadcast in the P2P network to be included in the blockchain, with transaction fees from high to low. Suppose that the adoption rate, i.e., the percentage of miners who join the dark venues, is α%.

Figure 4: Model Setup (created by Whimsical)

After miners make the decision, users will then decide whether and where they will submit the transaction during Period 2. At this stage, there are three options for the users to choose from 1) submit the transaction to the P2P network; 2) submit the transaction to dark venues; 3) do not submit the transaction. This implies that there are two types of users that will be considered in the model: 1) frontrunnable users, whose pending transaction is likely to be attacked by the arbitrager; 2) non-frontrunnable users, where transactions are immune to any frontrunning trials.

Suppose in the last period, the miner who successfully mines the block can include at most B transactions into the blockchain, then in Period 2, without the loss of the generality, we suppose there are 1 frontrunnable user and B+1 non-frontrunnable users making the decisions, otherwise more transactions will only be discarded due to the limited capacity for each block. For each transaction, if it is included in the blockchain, naturally, the user will enjoy a benefit of v, apart from the transaction fee f that is priorly attached to this transaction. But for the frontrunnable transaction, if it is observed by the arbitrager and successfully front run, it would result in a loss of c from the frontrunnable user because instead, the arbitrager would earn the profit of c.

During Period 3, we assume there are two arbitragers who compete with each other to screen potential arbitrage opportunities as well as make decisions on which venues to send the order, to before determining which miners earn the right to record the transactions into the blockchain. The arbitrager has three options to broadcast their order 1) submit the order to the P2P network; 2) submit the order to the dark venues; 3) submit the order to both. Intuitively, if the arbitrage order is again submitted to the P2P network, then the other arbitrager would also observe this leaked information which leads to more competition between arbitragers.

Moreover, it is worth mentioning that no matter whether the user or the arbitrager who sends their transactions to the dark venue has an inherent execution risk because the execution is subject to the percentage of miners who also adopt the dark venues, i.e., the adoption rate α. Interestingly, users’ and arbitragers’ venue selection decisions will also, in turn, determine the adoption rate, because miners only care about which venues they can enjoy more revenues from mining. Figure 5 describes how this important exogenous variable, the adoption rate, affects or is affected by the participants’ decisions in detail, where the dimension of waiting time is assumed to heavily influence the users’ decisions since users care about the execution of their transactions, but the arbitragers only care about finding the best way to arbitrage for more profits.

Figure 5: Relationship between Adoption Rate α and Three Agents in the Model (Created by Whimsical)

Moreover, our model also relies on the following technical and behavioral assumptions, where assumptions 1-4 are in line with Capponi et al. (2022)’s existing game theoretical model assumptions, and assumption 5 is newly raised to describe how players in the ecosystem would value the opportunity cost along with the time:

  1. Miners who choose to join the dark venue will not leak any transaction information in that venue.

  2. All bids from users and arbitragers are distinct from each other to avoid any ties.

  3. All the miners have an equal probability of appending the new block during Period 3 for each round.

  4. During Period 3 for each round, if arbitragers send the order to both venues, then the transaction will be given the same nonce, thus at most, one of these two transactions will be implemented.

  5. Considering the effect of waiting time on users’ decisions, to model the users’ utility function, q(t) is naturally a decreasing function, where an approximation from the exponential distribution with the parameter μ is the most suitable, since, as a rational person, one will be reluctant to see his/her transaction pending, and the first several stages of pending dominate one’s decision.

4. Results

Before diving into the equilibrium analysis and simulations with continuous variables, we made some pilot results through a series of case studies, shown in Table 1.


Users’ choice of parameters


#1 (trivial)

λ = 0, i.e., only economic incentives

Partially adopted by miners and utilized by at least one arbitrageur.


λ = 1, i.e., weighing equally between economic incentives and opportunity cost, μ = 1


λ = 1, i.e., weighing equally between economic incentives and opportunity cost, μ = 0.5

Partially adopted by miners and utilized by at least one arbitrageur.


λ = 1, i.e., weighing equally between economic incentives and opportunity cost, μ = 1.5

Partially adopted by miners and utilized by at most one arbitrager.


λ = 0.5, i.e., weighing more on the economic incentives, μ = 1

Partially adopted by miners and utilized by at least one arbitrageur.


λ = 2, i.e., weighing more on the opportunity cost, μ = 1

Partially adopted by miners and utilized by at most one arbitrager.

Table 1: Case Studies by Pre-defining the Parameters of λ and μ

Table 1’s result can be easily determined by the following deductive logic: from Capponi et al. (2022)’s existing game theoretical model results that only take economic incentives into consideration, we realize that miners will partially adopt the dark venues, and at least one arbitrager will also utilize them. When adding a new dimension of opportunity cost, where players will balance the importance of economic incentives and opportunity cost, then the miners would still partially adopt, instead of fully or never adopting the dark venues because of the possibility that still exists for increasing their revenues of mining. At the same time, for the arbitragers, the main concern for them is to front run the transactions submitted to the P2P network by the users and also avoid any other arbitragers whenever submitting their orders. As the users are mainly concerned with the opportunity cost for their transactions, when they lay a higher emphasis on it, that is, with a larger λ or μ, their adoption rates for the dark venues will naturally be much higher (ceteris paribus for miners’ adoption rate), which would then decrease the arbitragers’ adoption rate, too.

At this stage, some patterns are indeed observable from Table 1, and here we documented several hypotheses to be tested via a more rigid mathematics proof or simulation, saved for future research: First, during the decision-making process, will users’ preference between economic incentives and opportunity cost, i.e., the choice of λ, dominate arbitragers' decisions? More specifically, will arbitragers quit the dark venues if the majority of users in the market care more about their opportunity costs rather than the economic incentives? Second, will the parameter μ that measures how the value of waiting time impacts users’ utility be dominant to arbitragers’ decisions? These two hypotheses will serve as our future directions to help us generate potential theoretical results for general interests.

5. Potential Impacts

5.1 Intellectual Merits

The proposed research explores extending Capponi et al. (2022)’s current game theoretical model by adding a non-negligible dimension, waiting time, to simulate how participants would balance between economic incentives and opportunity cost to finally decide the choice of transaction submission venues. However, our proposed model has its inherent limitations, that is, relying on a lot of assumptions to better simplify the model for simulations, which might cause a little bit of deviation from the real-world environment. For instance, our simulation does not follow the current transaction fee mechanism – EIP-1559, for we do not consider the base fee adjustment as well as the variable block size but directly adopt the legacy auction method, that is, the English auction.

But overall, this research is expected to become the benchmark for understanding the incentives difference between public and private channels, to better understand the information structure of the blockchain. It would also advance existing literature, such as adding to the economic and behavioral incentives analysis in the blockchain system via game theoretical model, as well as create a new scenario for market microstructure studies on dark venues.

5.2 Practical Impacts

The proposed research integrates two important factors that participants in the blockchain system concerned with into one game theoretical model, economic incentives and opportunity cost. The results are expected to help both scholars and practitioners to better understand the differences between public and private channels of the blockchain in detail. Moreover, this theoretical study will further reveal whether those relay services, i.e., the dark venues, can really help solve (or mitigate) the MEV problems to maintain blockchain fairness and security.

References (MyBib Folder: [URL])

Basu, Soumya, David Easley, Maureen O’Hara, and Emin Sirer. 2019. “Towards a Functional Fee Market for Cryptocurrencies.” SSRN Electronic Journal.

Buti, Sabrina, Barbara Rindi, and Ingrid M. Werner. 2017. “Dark Pool Trading Strategies, Market Quality and Welfare.” Journal of Financial Economics 124 (2): 244–65.

Capponi, Agostino, Ruizhe Jia, and Ye Wang. 2022. “The Evolution of Blockchain: From Lit to Dark.” ArXiv:2202.05779 [Q-Fin], February.

Daian, Philip, Steven Goldfeder, Tyler Kell, Yunqi Li, Xueyuan Zhao, Iddo Bentov, Lorenz Breidenbach, and Ari Juels. 2020. “Flash Boys 2.0: Frontrunning in Decentralized Exchanges, Miner Extractable Value, and Consensus Instability.” IEEE Xplore. May 1, 2020.

Degryse, Hans, Mark Van Achter, and Gunther Wuyts. 2009. “Dynamic Order Submission Strategies with Competition between a Dealer Market and a Crossing Network.” Journal of Financial Economics 91 (3): 319–38.

Liu, Yulin, Yuxuan Lu, Kartik Nayak, Fan Zhang, Luyao Zhang, and Yinhong Zhao. 2022. “Empirical Analysis of EIP-1559: Transaction Fees, Waiting Time, and Consensus Security.” ArXiv:2201.05574 [Cs, Econ, Q-Fin], January.

Liu, Ziyao, Nguyen Cong Luong, Wenbo Wang, Dusit Niyato, Ping Wang, Ying-Chang Liang, and Dong In Kim. 2019. “A Survey on Blockchain: A Game Theoretical Perspective.” IEEE Access 7: 47615–43.

Milojicic, Dejan, Vana Kalogeraki, Rajan Lukose, Kiran Nagaraja, Jim Pruyne, Bruno Richard, Sami Rollins, and Zhichen Xu. 2003. “Peer-To-Peer Computing.”

Qin, Kaihua, Liyi Zhou, and Arthur Gervais. 2022. “Quantifying Blockchain Extractable Value: How Dark Is the Forest?” IEEE Xplore. May 1, 2022.

Roughgarden, Tim. 2020. “Transaction Fee Mechanism Design for the Ethereum Blockchain: An Economic Analysis of EIP-1559 *.”

Weintraub, Ben, Christof Ferreira Torres, Cristina Nita-Rotaru, and Radu State. 2022. “A Flash(Bot) in the Pan: Measuring Maximal Extractable Value in Private Pools.” ArXiv:2206.04185 [Cs], June.

Zhu, Haoxiang. 2014. “Do Dark Pools Harm Price Discovery?” The Review of Financial Studies 27 (3): 747–89.

Züst, Patrick, Tejaswi Nadahalli, Ye Wang, and Roger Wattenhofer. 2021. “Analyzing and Preventing Sandwich Attacks in Ethereum.”

Part II: Supplementary Resources

[Google Sheets for the Supplementary Resources]

  1. Experts for Comments

To the best of my knowledge in the related field of this proposed research, I especially would like to ask for comments from the three professors listed below:

The reason why I chose the first two experts for comments is naturally, their co-authored theoretical paper is the main reference that I relied on. As I have discussed in the intellectual merit section of the research summary, this proposed research will extend their theoretical blockchain research result in the dimension of waiting time, thus I believe they are the experts that I am looking for amazing comments after my deriving some preliminary results. Moreover, Prof Fan Zhang is indeed an expert in computer science, concerning blockchain fairness and security in many different approaches and has great insights into the fundamental mechanism of Ethereum blockchain mechanism, thus his insight is more likely to inspire us to think about how our simulation result would benefit blockchain fairness and security in a desirable framework.

2. Resources for Further studies

To further move forward and accomplish this proposed research, I will mainly focus on studying the definition and application of the quasilinear utility function. Quasilinear utility function indeed enables the measurement of waiting time to be included in modeling and simulating user’s behavior while making decisions in the blockchain system. Since this concept is indeed one of the major topics of game theory, therefore the following relevant resources: corresponding chapters from an e-book The Economy, Chapter 2.7 from Game Theory II Coursera course, etc., would help. These resources would better help me understand whether and how the quasilinear utility function could be well applied to the model and simulation of this proposed research.

3. Seminar, Symposium, and Conference

This proposed research studies a small proportion of the giant phenomenon of blockchain congestion from the perspective of transaction fee mechanism, thus the following symposiums and seminars indeed match this topic and are beneficial for my understanding of this topic in a broader sense: ETHconomics @Devconnect, Luohan Webinar, etc. ETHconomics is actually a whole-day session presented by both researchers and developers in academia and industry, and its topic includes but is not limited to transaction fee mechanism, cryptoeconomics, game theory, etc., which is expected to largely broaden my understanding of the economics perspective of Ethereum blockchain. Luohan Webinar is where distinguished scholars throughout the world exchange their frontier views and present interdisciplinary studies on the digital economy, among which Prof Agostino Capponi’s “The Information Content of Blockchain Fees”, Prof Joshua Gans’s “A Solomonic Solution to Ownership Disputes: An Application to Blockchain Front-Running” are highly relevant so that I might spend more time in understanding their talks as well as their methodology and results before diving into this proposed research. 

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