Bitcoin After Block Rewards Junhyuk Lee
arXiv:2606.05503v1 [cs.CR] 3 Jun 2026
Texas A&M University, College Station, TX, USA [email protected]
Abstract. Bitcoin’s block reward is scheduled to decline to zero, raising concerns about whether the network can remain secure once miners rely solely on transaction fees. This paper seeks to identify the conditions under which largescale and persistent deviation from honest mining can arise. We analyze and compare the payoffs of honest and deviating miners in a sequential decision model, and identify a deviation threshold Gt at which honest mining ceases to be privately optimal. Around the 2024 Bitcoin halving, we show that current mining behavior does not exhibit large-scale or structural deviation. However, when the block reward is removed, the Gt criterion implies that deviation can arise even with a very small fraction of transaction fees. Finally, we evaluate three protocol-level mechanisms: Base Fee, Fee Floor, and an adaptive maximum block size rule, and show that their combination raises the deviation threshold and mitigates incentive breakdown in a fee-only regime. These results provide a practical benchmark for assessing Bitcoin’s security as block rewards disappear. Keywords: Bitcoin · mining incentives · deviation threshold · transaction fees · Byzantine fault tolerance · protocol design
1
Introduction
1.1
Motivation
Bitcoin was introduced by Satoshi Nakamoto as a decentralized alternative to traditional financial infrastructures [21], motivated in part by concerns regarding centralized control and discretionary intervention during the 2008 financial crisis. The system operates on a distributed network of nodes and maintains its security through the collective participation of miners, who contribute computational resources known as hashing power to validate transactions and prevent attacks such as double spending or unauthorized ledger modifications. The security of the Bitcoin network critically depends on these mining incentives. However, Bitcoin’s mining protocol dictates that block rewards halve approximately every four years in order to enforce long-term scarcity. This design implies a corresponding decline in miner revenue unless compensated by increases in external factors such as transaction fees or market demand. As block rewards diminish, the risk arises that miners may withdraw hashing power, thereby reducing the cost of attacks and threatening the network’s long-term security.
2
J. Lee
In the post-reward era, miner compensation must rely primarily on transaction fees and emerging sources of value such as Maximum Extractable Value (MEV). These components increase uncertainty in miner revenue and create additional strategic considerations for miners. An unstable reward structure can create misaligned incentives, encouraging miners to deviate from honest mining to maximize short-term gains. Such deviations may compromise consensus threshold and contribute to network security instability. These concerns motivate this research, which investigates how Bitcoin’s protocol and incentive mechanisms can evolve to sustain network security once block rewards are no longer available. Understanding and formalizing miner behavior in the postreward era is essential for evaluating the long-term viability of Bitcoin as the first digital currency and for preserving the financial autonomy that underlies its original design. 1.2
Problem Statement
The central question of this thesis is whether the Bitcoin network can remain secure when miner revenue is derived solely from transaction fees. In particular, we examine whether miners have sufficient incentives to continue honest mining in a fee-only environment and whether such conditions satisfy the Byzantine Fault Tolerance (BFT) stability threshold required for network security. 1.3
Objectives
This thesis studies whether Bitcoin can remain secure as block reward declines and miner revenue becomes increasingly dependent on transaction fees and potential private gains. The core objectives are as follows: 1. Formulate a per-block miner decision model that characterizes the choice between honest mining and deviation through the deviation threshold condition Gt ≥ ϕ(w)· Xt . 2. Empirically analyze the 2024 halving window (block heights 790,000–890,000) to examine how reductions in observable rewards Xt and changes in fee-market conditions affect miner profitability and deviation-consistent behavior, using block audit scores as a proxy. 3. Evaluate miner profit–stabilization mechanisms, including a base fee, a fee floor, and an adaptive maximum block size rule under a hypothetical fee-only regime, and identify policy combinations that keep the deviation rate below the BFT-style threshold. 1.4
Contributions
This thesis makes the following contributions to understanding Bitcoin miner behavior as block reward declines. 1. It shows that rational miner deviation can be characterized by a simple per-block threshold linking additional private gains to observable miner rewards, clarifying how reductions in Xt mechanically relax deviation constraints in a fee-dominated environment.
Bitcoin After Block Rewards
3
2. It reveals that although observable miner rewards and on-chain demand declined substantially, deviation-consistent behavior did not increase persistently while block rewards remained positive, indicating the stabilizing role of the remaining block reward. 3. It demonstrates that in a hypothetical fee-only regime, deviation becomes widespread even for small private gains, and proposes reward stabilization mechanisms, including a base fee, a fee floor, and an adaptive maximum block size, that mitigate massive deviation and help maintain BFT-style security thresholds.
2
Related Work
2.1
Miner Incentives and Deviation
Bitcoin’s security relies on miners being economically incentivized to extend the longest chain, yet prior work shows that this incentive structure becomes fragile as block rewards decline and mining revenue increasingly depends on transaction fees. Carlsten et al. [6] analyze a fee-dominated regime and show that high variance in transaction fees gives rise to incentive-compatible deviations such as fee-sniping and delayed mining, highlighting the stabilizing role of block subsidies in discouraging deviation. Complementing this view, Eyal and Sirer [13] demonstrate that strategic deviation can be profitable even without a majority of hashing power when network propagation is imperfect, as withholding and selective block release allow deviating miners to earn rewards exceeding their proportional share. Together, these results establish that reduced block rewards and propagation delays jointly weaken incentives for honest mining, motivating a formal analysis of deviation conditions. 2.2
MEV and Block Propagation Delay on Mining Incentives
Deviation gains arise when miners do not receive blocks and transactions at the same time. Daian et al. [9] show that small timing advantages in information propagation can be directly monetized through MEV, while Decker and Wattenhofer [10] empirically show that block propagation delay is common and is the primary cause of blockchain forks and orphaned blocks. As propagation delay increases, honest miners waste more work on blocks that do not enter the main chain, while miners who can exploit or tolerate such delays are less affected, widening the payoff gap between honest and deviating behavior. 2.3
Consensus Security
While the above works focus on miner incentives, consensus-security analyses formalize the conditions under which the Bitcoin protocol remains safe and live. Garay, Kiayias, and Leonardos develop the Bitcoin Backbone model [15, 16], which proves that the protocol achieves robust chain growth and chain quality as long as the adversarial hashing power remains below a given fraction of the total. Their framework highlights the importance of maintaining an honest majority and provides a probabilistic justification for using a Byzantine Fault Tolerance style threshold when evaluating network stability.
4
J. Lee
2.4
Relevant Bitcoin Improvement Proposals (BIPs)
As summarized in Table 1, several Bitcoin Improvement Proposals (BIPs) have explored changes to Bitcoin’s block-size limit, primarily to accommodate growing transaction demand. Early proposals such as BIP-100, BIP-101, and BIP-102 advocated increasing the maximum block size through fixed expansions or scheduled growth, focusing mainly on throughput and congestion relief [1, 17, 18]. Later proposals introduced adaptive mechanisms that adjust the block-size limit based on network conditions. BIP 103 and BIP 104 propose algorithmic rules tied to bandwidth growth or block difficulty, while BIP 107 adjusts the limit according to recent block utilization [19, 22, 25]. These approaches aim to improve scalability but do not explicitly consider how block-size dynamics affect miner deviation incentives. Table 1: Summary of Block-Size Related Bitcoin Improvement Proposals (BIPs) BIP Author(s) Category / Mechanism 100 Garzik et al. Miner voting / Dynamic sizing [18]
Key Idea Miners vote to adjust the maximum block size periodically. 101 Andresen Fixed schedule [1] Increase block size to 8 MB and double every two years to follow projected demand. 102 Garzik Fixed increase [17] One-time increase to a 2 MB block size. 103 Wuille Algorithmic / Bandwidth-based [25] Adjust block size according to long-term bandwidth growth (17.7% per year). 104 Khan Algorithmic / Difficulty-like rule [19] Block-size cap adjusts similarly to mining difficulty using 75% target utilization. 105 BtcDrak Algorithmic / Retargeting algorithm [4] Consensus-driven dynamic block-size retargeting mechanism with ±10% adjustment window. 106 Chakraborty Algorithmic / Automatic control [7] Automatically controlled block-size limit responding to previous block size or transaction-fee conditions. 107 Sanchez Algorithmic / Utilization-based [22] Block-size limit is adjusted based on recent block utilization trends. 109 Andresen Fixed + policy constraints [2] Increase block size to 2 MB and introduce signature-operation limits for validation stability.
3
Model Assumptions
3.1
Miner Computation and Pool Participation
We model each miner as an independent computational agent executing an interactive Turing machine (ITM). This abstraction captures the fact that mining is an ongoing, interactive process in which miners continuously receive information from the network, such as new blocks and transactions, and adapt their behavior accordingly. Miners may choose to participate in mining pools, but pool participation is treated as a voluntary decision made solely to maximize individual profit. Regardless of whether a miner operates solo or within a pool, rewards are assumed to be distributed proportionally to the miner’s contributed hashing power. As a result, joining a pool does not alter the miner’s underlying strategic objective, but only affects the aggregation and sharing of rewards.
Bitcoin After Block Rewards
5
Accordingly, we analyze strategic behavior at the level of individual miners and do not model pools as independent strategic agents. Collective pool behavior is treated as the aggregate outcome of individual miners’ decisions, and group-level strategy is therefore outside the scope of this model.
3.2
Rational Miners as Long-Lived Strategic Agents
We model miners as rational, long-lived strategic agents. Current mining decisions can influence future decision environments, and the economic consequences of mining behavior such as profit realization, operational costs, and risk exposures are determined along a miner’s own timeline rather than at an isolated block level. As a result, miners do not make decisions on a purely block-by-block basis, but instead account for how present actions affect future outcomes. For this reason, miner behavior is modeled using a Markov Decision Process (MDP), which is the standard framework for sequential decision-making in stochastic and evolving environments [23]. The MDP formulation allows mining strategies to be evaluated not only by their immediate payoff, but also by their impact on future states and expected profits, which is consistent with long-lived strategic behavior in blockchain systems. The objective of miner i is defined as "∞ # X Vi = E γ t Πi (St , ai (t)) ,
(1)
t=0
where t indexes block discovery window following the resolution of the fork competition for block t − 1, Πi (St , ai (t)) denotes the per-block profit, and γ ∈ (0, 1) is a discount factor. At block t, the network state observed by miner i is defined as St = Ft , Mt , hi , δt ,
(2)
where Ft denotes the transaction fee revenue observed at t, Mt denotes the realized miner-extractable value (MEV), hi is miner i’s share of total network hashing power, and δt represents the effective network propagation delay determined by block size and mining strategy. The action of miner i at block t is given by ai (t) ∈ {hon, dev},
(3)
corresponding to honest mining and deviating mining strategies, respectively. Miners evaluate mining strategies over long-term investment horizons, while decisions are made on a per-block basis. The MDP formulation is therefore not used to assume myopic block-by-block decision-making, but to justify it: given the Markov structure of the state and payoff, the optimal policy admits a per-block decision rule that maximizes expected long-term discounted payoffs for a long-lived strategic agent.
6
3.3
J. Lee
Honest Mining and Deviating Mining
At each block, miners choose between honest and deviating mining strategies. We assume that deviation does not alter a miner’s computational capability; that is, the miner’s hash rate remains unchanged across strategies, and only the propagation behavior differs. Honest mining refers to protocol-conformant behavior in which miners follow the Bitcoin protocol as specified. Honest miners extend the longest known valid chain, construct blocks using transactions from the public mempool, and immediately broadcast newly discovered blocks to the network. Operationally, this corresponds to repeatedly attempting to solve the PoW puzzle for the current chain tip and, upon success, diffusing the resulting block without delay, as specified by the standard Bitcoin protocol and its formalizations [21]. Deviating mining refers to any strategy that departs from this protocol-conformant behavior in order to increase private profit. Examples include block withholding, delayed block publication, and selective transaction inclusion to capture higher fees or other private gains. Such strategies intentionally deviate from immediate block propagation or standard block construction rules to improve expected payoff [13]. 3.4
Partially Synchronous Network Model
We assume a partially synchronous network model in the sense of Dwork, Lynch, and Stockmeyer [11], where there exists an unknown but finite upper bound on message delivery delay that eventually holds. This model captures realistic network behavior that lies strictly between fully synchronous and fully asynchronous communication. We model block propagation using an effective delay function δ(B), which depends on block size and network conditions. We assume that δ(B) ≤ ∆
for all block sizes B,
(4)
where ∆ denotes an unknown but finite network delay bound. This assumption guarantees eventual block propagation while allowing temporary forks and orphaned blocks induced by propagation latency. 3.5
Block Propagation Delay
We model block propagation delay as an increasing function of block size. Empirical measurements show that block propagation delay grows approximately linearly with block size due to bandwidth and validation constraints in the peer-to-peer network [8]. Let δ(B) denote the propagation delay of a block with size B. The delay is modeled as δ hon (B) = δ0 + κ · B, (5) where δ0 is the baseline block propagation delay and κ captures the incremental delay per unit of block size. Throughout the analysis, block size is measured in virtual bytes (vB), reflecting the SegWit-adjusted block weight.
Bitcoin After Block Rewards
7
For deviating mining strategies, we allow miners to intentionally delay block publication. Accordingly, the propagation delay under deviation is given by δ dev (B) = δ hon (B) + w,
(6)
where w ≥ 0 represents an intentional withholding period chosen by the miner. This additional delay increases the probability of orphaning and directly affects the expected payoff of deviating strategies. Additional examination of the withholding time w is provided in the Appendix. 3.6
Successful Mining and Propagation
We define the successful mining probability of miner i as the probability that a block discovered by the miner is not orphaned and is successfully incorporated into the main chain. This probability accounts for both block discovery and successful network propagation. Let hi denote miner i’s share of total network hashing power, and let ρ denote the orphan rate. The successful mining probability is then given by pi (t) = hi (1 − ρt )
(7)
The block discovery rate hi remains unchanged across strategies, as we assume that miners have identical computational capabilities under both honest and deviating mining. Also, the orphan rate ρ captures the probability that a competing block is discovered during the propagation window of a newly mined block. Consistent with prior analytical and empirical studies of Bitcoin mining, block discovery is modeled as a Poisson process with arrival rate λ [10]. If δ denotes the block propagation delay, the probability that at least one competing block is discovered during this interval is ρt = 1 − exp(−λδt )
(8)
Accordingly, a block is successfully mined only when no competing block is found during its propagation period. 3.7
Protocol Summary Mining for block t Parties. Individual miners {1, . . . , n}, each modeled as an interactive Turing machine (ITM). Input (to miner i). Network state St = (Ft , Mt , hi , δt ) and miner cost Ci . Execution. For each block t, miner i executes the following steps: 1. Observe. Observe the network state St , which summarizes the relevant information associated with the block discovery event at block t.
8
J. Lee
2. Decide. Choose a mining strategy for block t based on the observed state St , ai (t) ∈ {hon, dev}, corresponding to honest mining or deviating mining behavior. 3. Execute. Execute the chosen mining strategy. If ai (t) = hon, follow honest mining behavior. If ai (t) = dev, follow deviating behavior (e.g., withholding or selective transaction inclusion) as described in Section 3.3. 4. Propagate. Upon block discovery, broadcast the block. The block is subject to an effective propagation delay according to the delay model in Section 3.5, which probabilistically determines the block outcome (adopted or orphaned) as described in Section 3.6. Output. A realized block outcome (adopted or orphaned) and the corresponding realized payoff Πi (St , ai (t)) as defined in Section 3.2.
4
Problem Formulation
In this chapter, we formalize Bitcoin miners’ profit incentives by defining the total profit function Vi and the per-block payoff function Πi . We then characterize the additional deviation gain Gt required for deviation to become economically rational and derive the corresponding threshold condition under which deviation occurs. By controlling the magnitude of Gt , the fraction of miners for whom deviation is profitable can be reduced, thereby maintaining the network’s fault-tolerance threshold of 1/2 as established in the Bitcoin Backbone protocol [15, 16]. 4.1
Miners’ Profit
We model miners as rational agents who maximize long-run discounted profit [23]. The total profit of miner i is defined as "∞ # X Vi = E γ t Πi (St , ai ) , (9) t=0
where Πi (St , ai ) denotes the per-block payoff at network state St under action ai , and γ ∈ (0, 1) is a discount factor. Throughout this work, we set the discount factor per block interval (roughly 10 minutes) to γ = 0.991/144 ≈ 0.99993, so that the implied daily discount factor is 0.99. The per-block payoff function is defined as Πi (St , ai ) = pi · Xt − Ci ,
(10)
where pi is the successful mining probability of miner i, Xt is the total reward available in block t, and Ci represents the miner’s operational cost. The total reward Xt is given by Xt = Rt + Ft + Mt , (11)
Bitcoin After Block Rewards
9
where Rt is the block reward, Ft denotes transaction fees, and Mt captures miner extractable value (MEV). In the post-reward regime considered in this chapter, we assume Rt = 0, so that Xt = Ft + Mt . The cost term Ci represents the mining cost incurred in producing a block, including electricity expenses and hardware depreciation. Although miners are long-lived strategic agents with γ = 0.99993, both honest and deviating strategies share the same long-run value function, and the state is Markov. The optimal policy therefore reduces to a per-block decision rule, without assuming myopic behavior. At each block t, a miner chooses between honest mining and deviation by comparing the corresponding per-block profits. Let Πihon (t) and Πidev (t) denote the per-block payoffs under honest and deviating behavior, respectively. These are given by Πihon (t) = phon i · Xt − C i ,
(12)
Πidev (t) = pdev i · (Xt + Gt ) − Ci ,
(13)
where Gt represents the additional gain obtainable only through deviation, such as withholding or opportunistic forking.
Algorithm 1 Miner Decision at Block t 1: for each block t do 2: for each miner i do 3: if Πidev (t) ≥ Πihon (t) then 4: ai (t) ← deviation mining 5: else 6: ai (t) ← honest mining 7: end if 8: end for 9: end for
4.2
▷ forking or withholding
Additional Deviation Gain Gt
At each block t, miner i compares the per-block payoffs of honest mining and deviation. This per-round comparison is justified because the underlying value function Vi (·) is identical for honest and deviating strategies. The strategies differ only in the realized per-block payoff Πi , which allows the miner to directly compare Πihon (t) and Πidev (t) at each block. The miner i chooses deviation whenever Πidev (t) ≥ Πihon (t).
(14)
10
J. Lee
Substituting the payoff definitions yields hon pdev i (Xt + Gt ) ≥ pi Xt ,
(15)
dev hon pdev i Xt + pi Gt ≥ pi Xt ,
(16)
hon pdev − pdev i Gt ≥ (pi i ) Xt , hon dev p −p Gt ≥ i dev i Xt . pi
(17) (18)
We define the multiplicative term on the right-hand side as ϕ(w) ≜
phon − pdev i i , dev pi
(19)
which represents the relative success-probability penalty induced by deviation. The term ϕ(w) arises from the withholding time w, and increases as w grows. A larger withholding time reduces the miner’s effective success probability while increasing the orphan rate, thereby requiring a higher deviation gain Gt to make deviation profitable. Accordingly, we refer to ϕ(w) as the withholding penalty factor. The ϕ(w) will be dealt with in Further Discussion chapter later. Using this definition, the incentive condition can be written compactly as Gt ≥ ϕ(w) · Xt .
(20)
As shown above, the deviation threshold for Gt is proportional to both the withholding penalty factor ϕ(w) and the total available reward Xt . Deviation can therefore be discouraged either by reducing the deviation-only gain Gt in the left-hand side or by increasing threshold through a larger ϕ(w) or Xt in the right-hand side. However, ϕ(w) is primarily determined by miner-specific behavior and network conditions, such as withholding time and propagation delay, and is not directly controllable by the protocol. For this reason, we treat ϕ(w) as an external cause and focus our analysis on mechanisms that reduce Gt and raise Xt .
5
Empirical Data Analysis
5.1
Overview
The goal of this section is to examine how changes in observable rewards Xt = Rt + Ft + Mt affect the deviation threshold Gt ≥ ϕ(w) · Xt , and to inspect whether a reduction in Xt influences deviation mining and induces significant strategic changes by miners, including the halving event. 5.2
Empirical Setup and Measurement
Data source. Bitcoin on-chain data are obtained from blockchain.com, and block-level audit metrics are sourced from mempool.space.
Bitcoin After Block Rewards
11
Data type. From blockchain.com, we extract block-level statistics including total fees (in satoshis), total transaction size (vBytes), block weight, and mining pool identifiers. Block audit scores at each block height are obtained from mempool.space. Data calibration. Throughout the observation window, median values are used for Xt , Ft , fee rate (sat/vB), and block utilization ratios to mitigate the impact of outliers and to capture underlying trends in each metric. Experimental window. The 100,001-block dataset spans block heights 790,000 to 890,000, corresponding to a roughly ±50,000 block window around the 2024 halving at height 840,000. This period covers May 16, 2023 to March 29, 2025, and includes both pre-halving and post-halving regimes. Revenue and cost. All revenue and cost quantities are computed at a daily frequency. For each block, the miner reward consists of the block subsidy Rb , transaction fees Fb , and MEV Mb . Daily revenue for mining pool i is obtained by aggregating block-level rewards and multiplying by the BTC–USD price on day d. Mining costs are estimated using the annualised electricity consumption estimate GUESS(d) (TWh/year) reported by the Cambridge Bitcoin Electricity Consumption Index (CBECI). Following the CBECI methodology, which models electricity expenditure for industrial-scale mining operations using a constant global average electricity price [5], we assume an electricity price of pe = 0.05 USD/kWh. We convert annual consumption to a daily network-wide electricity cost by dividing by 365.25 and multiplying by pe . The resulting daily network cost is then allocated to each mining pool in proportion to the number of blocks it mines on day d. Daily profits are finally aggregated into monthly time series. X Daily BTC reward of pool i : Xi (d) = Rb + Fb + Mb , (21) b∈Bi,d
Daily revenue in USD :
Revi (d) = PBTC (d) Xi (d),
(22)
Daily electricity cost of pool i :
GUESS(d) pe , 365.25 Cnet (d) c(d) = , N (d) Ci (d) = c(d) ni (d),
(25)
Daily profit of pool i :
Πi (d) = Revi (d) − Ci (d).
(26)
Daily network electricity cost : Electricity cost per block :
Cnet (d) =
(23) (24)
Assumption in Max Extractable Value (MEV). Because there is no widely accepted empirical measurement of MEV in the Bitcoin network, we approximate Mt by conservatively scaling empirical MEV measurements from Ethereum to reflect Bitcoin’s more limited transaction structure and lower complexity [3, 14]. We model Mt using a zero-inflated lognormal distribution and generate 100,001 synthetic MEV samples, assigning one sampled value to each block in our analysis. MEV is treated as an exogenous component and is added to transaction fees when computing miner payoffs. This simplified treatment captures occasional private gains without making MEV a primary driver of the results.
12
J. Lee
Deviation indicator. Because miner deviation strategies are not directly observable from public blockchain data, we use the block audit score as a proxy for deviationconsistent behavior. The audit score measures the ratio of actual miner revenue to the expected revenue under a public-mempool fee-maximizing template. Lower audit scores indicate departures from public-mempool fee maximization, which may arise from the presence of private transaction fees or other forms of private gains Gt . Simply put, a low audit score implies that the miner did not follow the public mempool’s expected fee-maximizing protocol. Audit Score =
5.3
actual miner revenue in the block . expected revenue under the public-mempool fee-maximizing template (27)
Analysis Result
Decline of Rt and Ft in Xt Figure 1 shows a substantial decline in the total observable reward Xt following the 2024 halving, primarily driven by the reduction in the block reward Rt from 6.25 BTC to 3.125 BTC. The figure indicates that the sharp decrease in Xt is largely attributable to the halving of Rt , while the transaction fee component Ft exhibits only moderate variation over time. Figure 2 focuses exclusively on transaction fees and shows that the median transaction fee level declines in magnitude after the halving. Compared to the pre-halving period, post-halving fees display lower variability, suggesting a sustained reduction in the fee contribution to Xt .
Bitcoin After Block Rewards
13
Total Reward Xt = Rt + Ft + Mt Pre-Halving Median: 650.21 M sat Post-Halving Median: 319.74 M sat
X_t (Median) Pre-Halving Post-Halving Halving (2024-04-20)
700
Xt (Million sat)
600
500
400
300
200
100
0 2023-04
2023-07
2023-10
2024-01
2024-04
Date
2024-07
2024-10
2025-01
2025-04
Fig. 1: Miner’s revenue Xt changes over the 2024 halving.
Transaction Fees Ft Pre-Halving Median: 27.08 M sat Post-Halving Median: 7.24 M sat
Fees Median Pre-Halving Post-Halving Halving (2024-04-20)
140
Fees Median (Million sat)
120
100
80
60
40
20
0 2023-04
2023-07
2023-10
2024-01
2024-04
Date
2024-07
2024-10
2025-01
Fig. 2: Transaction fees Ft before and after the 2024 halving.
2025-04
14
J. Lee
Impact of Reduced Xt on Miner Profits Πi As shown in Figure 3, miner profit Πi drops sharply following the 2024 halving, reflecting the substantial reduction in block rewards Rt and transaction fee revenue Ft documented in the previous sections. In the post-halving period, profit recovery is concentrated among large mining pools with substantial hashing power, while smaller pools experience persistently depressed profit levels. This divergence indicates that the halving-induced revenue shock is not absorbed uniformly across miners; instead, scale plays a central role in determining both the speed and magnitude of post-halving recovery. These various profit outcomes imply that revenue reductions place greater economic pressure on smaller or less efficient miners. When such pressure persists, mining activity may gradually shift toward a smaller number of dominant pools, as larger miners are better positioned to absorb temporary profit shocks. Over time, this tendency toward concentration could weaken decentralization and pose risks to the long-run security of the network. Looking ahead, the 2024 halving reduced the block reward from 6.25 BTC to 3.125 BTC, which still remains relatively sufficient to cover mining costs for many participants. However, in a future fee-only regime where Rt = 0, these pressures would intensify unless they are offset by substantial increases in transaction fees or Bitcoin price. Moreover, in the presence of additional private gains Gt from deviation-consistent strategies, the deviation condition Gt ≥ ϕ(w) · Xt becomes easier to satisfy as Xt declines, particularly for miners with smaller hash rate shares and higher marginal costs. In what follows, we examine whether this reduction in miner profits is associated with increased deviation-consistent behavior, even while block rewards remain positive.
Profit by Miner (Top 7)
1e8 3.5
Pool Name Foundry USA AntPool ViaBTC F2Pool Luxor SECPOOL SpiderPool
Halving
3.0
Profit
2.5 2.0 1.5 1.0 0.5 0.0 2023-04
2023-07
2023-10
2024-01
2024-04
Date
2024-07
2024-10
2025-01
2025-04
Fig. 3: Monthly miner profits for major mining pools before and after the 2024 halving.
Bitcoin After Block Rewards
15
Shrinking On-Chain Demand after the Halving Beyond the changes in miner profits discussed above, the 2024 halving is also associated with a clear contraction in on-chain demand for Bitcoin block space, reflected in both lower block space prices and reduced utilization. As shown in Figure 4a, the median transaction fee rate (sat/vB) declines substantially in the post-halving period, indicating that block space becomes cheaper on average. This reduction in demand is further evidenced by changes in block utilization. Figure 4b reports the monthly block fill ratio, which drops significantly after the halving, implying that available block space is no longer fully demanded. Taken together, the simultaneous decline in block space prices and block utilization provides consistent evidence of shrinking on-chain demand in the post-halving regime, contributing to lower transaction fees and a reduction in the fee component of miner revenue.
16
J. Lee
Satoshi per vB Pre-Halving Median: 27.12 sat/vB Post-Halving Median: 9.56 sat/vB
Price per vB (Median) Pre-Halving Post-Halving Halving (2024-04-20)
140
Price per vB (sat/vB)
120
100
80
60
40
20
0 2023-04
2023-07
2023-10
2024-01
2024-04
Date
2024-07
2024-10
2025-01
2025-04
(a) Transaction fee rate measured in sat/vB over time.
100
Block Utilization Ratio: Weight/4MB Pre-Halving Median: 99.83% Post-Halving Median: 42.17%
Block Fill Ratio (Median) Pre-Halving Post-Halving Halving (2024-04-20)
Block Utilization (%)
90
80
70
60
50
40 2023-04
2023-07
2023-10
2024-01
2024-04
Date
2024-07
2024-10
2025-01
(b) Monthly block utilization measured by the block fill ratio.
Fig. 4: Post-halving decline in block space pricing and utilization.
2025-04
Bitcoin After Block Rewards
17
Audit Scores and Deviation-Consistent Behavior To examine whether a lower deviation threshold translates into observable changes in miner behavior, we analyze the block audit scores provided by mempool.space. Because miner deviation strategies are not directly observable from public blockchain data, the audit score is used as a proxy for deviation-consistent behavior. The audit score measures the ratio between the actual revenue earned by a miner and the expected revenue obtained by constructing a block using transactions from the public mempool under a standard fee-maximizing rule [20]. Lower audit scores therefore indicate departures from public-mempool-based fee maximization and are consistent with the presence of private transaction fees or other private incentives not captured by observable rewards Xt . Importantly, the audit score is highly sensitive to changes in the transaction fee market. When transaction demand or fee-setting incentives change abruptly, the fees actually earned by miners can differ from the fees predicted by a public-mempoolbased fee-maximizing template, even if miners do not intentionally engage in persistent deviation strategies. Such temporary mismatches are known to occur during periods of intensified ordinal and inscription activity, which introduce unusual transaction patterns and alter fee competition [24]. Similar effects can also arise during episodes of heightened Bitcoin price volatility, when transaction urgency and fee bidding behavior change rapidly [14]. Figure 5 presents the time series of the proportion of blocks with low audit scores. We do not observe a sustained or structural increase in low-audit score blocks following the halving event. Instead, fluctuations in audit scores are more closely associated with short-lived, event-driven disruptions in the fee market, such as changes in transaction demand induced by ordinal and inscription activity or bursts of price-driven transaction urgency.
0.30
Proportion of Blocks (Pr(Score < ))
0.25
Low Audit Blocks over Time Threshold ( )
= 90% (Score < 90%) = 95% (Score < 95%) = 98% (Score < 98%) Halving (2024-04-20)
Halving Event Event-driven
0.20
Event-driven
0.15
0.10
0.05
0.00 2023-04
2023-07
2023-10
2024-01
2024-04
Date
2024-07
2024-10
2025-01
Fig. 5: Time series of the proportion of blocks with low audit scores.
2025-04
18
5.4
J. Lee
Summary of Empirical Findings
The empirical analysis shows that the 2024 halving substantially reduced observable miner rewards by lowering the block reward Rt , transaction fees Ft , and on-chain demand, leading to lower miner profits and a relaxation of the deviation constraint. Despite this relaxation, we do not find evidence of a structural or persistent increase in deviationconsistent behavior in the post-halving period. In particular, audit scores remain broadly stable after the halving. If the halving had triggered widespread deviation, a sustained increase in low-audit blocks would be expected, but such a pattern is not observed. This stability suggests that the remaining block reward continues to play a stabilizing role under current conditions. Collectively, these findings indicate that the halving alone is not sufficient to induce large-scale deviation in the short run, but deviation risks are likely to become more pronounced as block rewards decline further toward a fee-only regime.
6
Solutions
6.1
Gt Threshold when Rt = 0
We consider a hypothetical fee-only regime by setting the block subsidy to zero (Rt = 0). Figure 6 shows the additional gain Gt required for deviation as a function of the observable reward Xt . The figure shows that when miner revenue comes only from transaction fees, the deviation threshold becomes very small. An additional gain, Gt , of just 0.17% of Xt is sufficient to make deviation attractive across all normalized hash-rate groups in our data, since transaction fees provide a much weaker revenue base than block rewards. This implies that miners remain honest today mainly because block rewards still dominate revenue, and that even small private gains may affect incentives as block rewards decline. 6.2
Rationale Over Solution Design
The analysis in the previous section shows that miner incentives become fragile as the block reward declines. This motivates a focus on the deviation threshold condition Gt ≥ ϕ(w) · Xt ,
(28)
which determines whether deviation yields higher returns than honest mining, as derived in the problem formulation chapter. While the empirical results indicate that short-term, event-driven factors currently explain most observed deviations, miners’ long-run behavior is fundamentally shaped by the level and stability of the observable reward Xt = Rt + Ft + Mt .
(29)
As the block reward Rt continues to decline through future halvings and transaction fee revenue Ft becomes more volatile and demand-dependent, miner revenue may weaken structurally, increasing the likelihood of deviation.
Bitcoin After Block Rewards
19
Deviation Threshold (Rt = 0) Threshold = 0.17% × Xt Pre-Halving (82K sat) Post-Halving (29K sat)
140
Deviation
Gt Threshold (K satoshi)
120 100 80 60 40
Honest
20 0
0
10
20
30
40
Xt (M satoshi)
50
60
70
80
Fig. 6: Additional deviation gain Gt required for deviation when Rt = 0.
To prevent deviation from becoming attractive too easily, there are two broad policy levers. One can reduce the additional gain from deviation Gt , or increase the right-hand side of the inequality by raising ϕ(w) or stabilizing Xt . Because Gt and ϕ(w) depend on strategy-specific and miner’s specific situation, both are difficult to control directly at the protocol level. We therefore focus on mechanisms that stabilize and support Xt , which is the most tractable policy variable. 6.3
Base Fee and Fee Floor
We consider a long-run regime in which the block reward converges to zero, such as after the year 2140 in Bitcoin, so that miner revenue is given by Xt = Ft + Mt .
(30)
with Mt ≈ 0. In a fee-only regime, declining demand or reduced transaction activity can drive Ft toward zero, threatening miners’ ability to cover fixed costs such as electricity and hardware depreciation. Our objective is therefore to preserve a minimum level of miner revenue while maintaining efficient congestion control. Base Fee Design The base fee is a congestion-control mechanism inspired by EIP1559 [12] that adjusts transaction prices in response to realized network utilization. Let bt denote the base fee in block t. Block utilization Ut is defined as the fraction of block capacity that is actually used, Wt Ut = , (31) Wmax where Wt denotes the realized block weight and Wmax = 4 MB is the protocol-imposed maximum under SegWit. Let U ∗ denote a target utilization level.
20
J. Lee
The base fee evolves according to a utilization-based feedback loop, Ut − U ∗ bt+1 = bt 1 + α · . U∗
(32)
where α > 0 governs the speed of adjustment. When block utilization exceeds the target, the base fee increases to dampen demand; when utilization falls below the target, the base fee decreases to encourage transaction inclusion. As a result, the base fee reflects congestion conditions induced by transaction demand in a disciplined manner, with U ∗ setting the target utilization and α controlling the adjustment speed. Fee Floor Design To ensure a minimum level of transaction fees during periods of low demand, we introduce a fee floor that imposes a lower bound on effective transaction fees independent of congestion conditions. Let F min denote the fee floor. The effective transaction fee is defined as Fteff = max(Ft , F min ),
(33)
and the effective miner reward becomes Xteff = Fteff + Mt .
(34)
Unlike the base fee, which primarily mitigates fee volatility through congestion control, the fee floor directly enforces a lower bound on miner revenue. As a result, the deviation condition becomes Gt ≥ ϕ(w) · Xteff . (35) with Xteff ≥ Xt holding by construction. This mechanism is particularly effective in low-demand regimes, where it prevents the deviation threshold from collapsing due to vanishing transaction fees and miner rewards. Overall, the base fee and the fee floor serve distinct but complementary roles. The base fee operates as a congestion-control mechanism, while the fee floor preserves a minimum security budget by enforcing a minimum level of miner revenue in a fee-only environment, as shown in Figure 7.
Bitcoin After Block Rewards
21
Base Fee Fee
Actual Transaction Fee
Fee Floor
Time
Fig. 7: Base fee and fee floor example.
6.4
Adaptive Maximum Block Size
We propose an adaptive maximum block size policy as a supplementary mechanism for controlling persistent network congestion and miner deviation. Rather than aiming to optimize throughput or stabilize utilization itself, the role of block size adjustment is to prevent congestion-driven incentives that increase the likelihood of miner deviation. When block utilization remains persistently high, competition for limited block space intensifies, making timely inclusion more expensive. As users are forced to bid higher per-vB fees, the value of selective inclusion and timing increases, thereby raising the potential gains from deviating strategies: Ut ↑ =⇒ congestion ↑ =⇒ per-vB price of block space ↑ =⇒ Gt ↑ =⇒ deviation incentive ↑ .
(36)
Conversely, when utilization remains persistently low, congestion diminishes and competition for block space weakens. As users no longer need to bid aggressively for transaction inclusion, transaction fees decline, reducing total miner rewards: Ut ↓ =⇒ congestion ↓ =⇒ Ft ↓ =⇒ Xt ↓ =⇒ deviation incentive ↑ .
(37)
Although the underlying mechanisms differ, both regimes relax the deviation condition by either increasing the left-hand side or reducing the right-hand side of the deviation inequality. Also, above mechanism explains why absolute bigger block size and higher network demand could not solve problem of miner’s deviation. Setting Adaptive Block Size Rule Instead of setting a fixed block size, we define a rule that adjusts the maximum block size in response to observed block utilization at the each epoch. The rule updates block size conservatively while imposing explicit lower and upper bounds, Bmin and Bmax , to prevent excessive variance in capacity. We set the minimum block size Bmin to 1 MB (4 million Weight Units) and the maximum Bmax to 2 MB (8 million Weight Units).
22
J. Lee
Bmin ≤ Bt ≤ Bmax .
(38)
Block size adjustments follow the utilization ratio relative to a target level. We reuse the utilization ratio Ut to compute its daily average Ūt over a rolling window of 144 blocks and compare this daily average to the utilization target U ∗ defined in the previous section. We introduce β as a proportional adjustment factor that controls the rate of block size change. Ūt =
t X 1 Ui , 144 i=t−143
∗ min{Bmax , (1 + β)Bt }, if Ūt > U , Bt+1 = max{Bmin , (1 − β)Bt }, if Ūt < U ∗ , Bt , otherwise.
(39)
(40)
Starting from an initial baseline, the maximum block size Bt evolves gradually within the interval [Bmin , Bmax ] using the proportional step β at each epoch. This conservative adjustment limits abrupt capacity changes while allowing the network to respond to persistent congestion. In doing so, the policy avoids excessive increases in deviation gains Gt caused by congestion-induced delay, and prevents fee collapse that would otherwise reduce effective miner rewards. 6.5
Simulation Results
We evaluate miner deviation behavior under combinations of three policy mechanisms: Base Fee, Fee Floor, and Adaptive Block Size, in a zero block reward setting (Rt = 0). Historical Bitcoin blocks from heights 790,000 to 890,000 are used to assess the effectiveness of these policies. For each block, miners compare profits from honest and deviating strategies using observed transaction fees, block size, MEV estimates, and miner cost data, consistent with the empirical data analysis in the previous section. Miner deviation is evaluated using the Gt threshold, and the deviation rate β is computed as the fraction of blocks in which miners choose deviation. Following the Bitcoin Backbone framework [15, 16], network stability requires the deviation rate β to remain below 50%. Detailed simulation assumptions, parameter grids, and robustness checks are reported in the Appendix. Figure 8 summarizes miner deviation outcomes across all policy configurations under a zero block reward scenario, with the additional gain fixed at 0.17% of Xt , where deviation becomes relatively easy, as discussed in Section 6.1. In policy configuration F, where no policy is applied, deviation becomes widespread even for this small additional gain, with the deviation rate exceeding 70%. This indicates that a fee-only regime without stabilization mechanisms is highly fragile to deviation. Introducing policy mechanisms substantially reduces deviation. Every configuration that retains the base fee, namely A, B, C, and D, remains within the BFT threshold, and the strongest performance is observed in A and C, where the base fee is paired with
Bitcoin After Block Rewards
23
Analysis by Policies Policy Configuration
Deviation Rate (%) 76.9%
80 Base Fee
Fee Floor
Adaptive
Deviation Rate (%)
Policy
70
A B C D E F
60 50
BFT Threshold (50%)
46.9%
40
26.0%
30
25.9%
20 10 0
2.0% A
Base Fee ON Base Fee OFF
1.9% B
C
Policy
D
E
F
Fixed: G_ratio = 0.17%, Fee Floor = 20M sat (when FF ON)
Fig. 8: Deviation rates under different policy combinations. the fee floor; extending this pairing with adaptive block size, as in A, produces only a marginal difference from C. Policy E, which omits the base fee, also stays within the threshold, but its deviation rate approaches the limit, confirming that the base fee is the dominant factor in preventing deviation. Policy F, with no policy applied, exceeds the threshold and illustrates the fragility of a fee-only regime without stabilization. These results show that stabilizing observable miner rewards is essential for maintaining security in a post-block-reward environment, with the base fee providing the foundational guarantee and the fee floor reinforcing it most effectively.
7
Further Discussion
In the problem formulation section, the withholding penalty factor ϕ(w) was defined as ϕ(w) ≜
phon − pdev i i , dev pi
(41)
which measures the relative difference in mining success probability between honest and deviating strategies. Under standard assumptions on block discovery and propagation, the mining success probability can be written as pi = hi · (1 − ρ),
(42)
where hi denotes miner i’s hash rate share and ρ denotes the orphan probability, reflecting the likelihood that a mined block fails to be successfully propagated. Substituting this expression into the definition of ϕ(w) yields hi (1 − ρhon ) − hi (1 − ρdev ) hi (1 − ρdev ) dev ρ − ρhon = , 1 − ρdev
ϕ(w) =
(43)
24
J. Lee
which shows that ϕ(w) is independent of the individual miner’s hash rate and is instead determined by differences in orphan probabilities across mining strategies. Since the orphan rate gap between honest and deviating mining strategies arises from the withholding time w, we can simplify the equation as follows.
ϕ(w) =
ρdev − ρhon 1 − ρdev hon
e−λδ 1 − e−λw = e−λ(δhon +w) 1 − e−λw = e−λw λw = e − 1.
(44) (45) (46) (47)
As a result, a systematic analysis of how orphan rates and the withholding time w interact with the deviation threshold Gt is left for future work.
8
Conclusion
This work began by examining miner profitability and its role in preventing strategic deviation in the long run. Bitcoin is the first decentralized currency that enables permissionless participation and financial autonomy without reliance on centralized authority. As such, the stability of its incentive structure is central to the success of this ongoing social and economic experiment. To date, the Bitcoin network has not experienced a hard fork driven by self-interested behavior of nodes or large-scale miner deviation. However, as block rewards decline and eventually vanish, miner incentives may weaken, potentially increasing the vulnerability of the network even before a pure fee-only regime is reached. A loss of confidence in network security could further amplify this risk. Our analysis shows that protocol-level mechanisms such as the base fee, fee floor, and adaptive block size can play an important role in stabilizing miner incentives by supporting observable rewards. While these mechanisms are not presented as definitive solutions, they highlight key design directions and raise important questions for stakeholders concerned with the long-term security and sustainability of Bitcoin. Acknowledgments. The author thanks his thesis advisor, Prof. Juan A. Garay (Department of Computer Science and Engineering, Texas A&M University), for his invaluable guidance and feedback throughout this work, including the comments received during the thesis defense. Disclosure of Interests. The author has no competing interests to declare that are relevant to the content of this article.
References
[1] Andresen, G.: BIP 101: Increase maximum block size. https://bips.dev /101/ (2015), accessed: 2026-02-15 [2] Andresen, G.: BIP 109: Two million byte size limit with sigop and sighash limits. https://bips.dev/109/ (2016), accessed: 2026-02-15 [3] Block Scholes: Ethereum staking deep dive: Analysing execution layer rewards & MEV (2024), block Scholes Research Note [4] BtcDrak: BIP 105: Consensus based block size retargeting algorithm. https: //bips.dev/105/ (2015), accessed: 2026-02-15 [5] Cambridge Centre for Alternative Finance: Cambridge bitcoin electricity consumption index (cbeci) methodology (2025), https://ccaf.io/cbnsi/cb eci/methodology, assumption 1: Average electricity price is 0.05 USD/kWh [6] Carlsten, M., Kalodner, H., Weinberg, S.M., Narayanan, A.: On the instability of bitcoin without the block reward. In: Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security (CCS). pp. 154–167. ACM (2016). https://doi.org/10.1145/2976749.2978408 [7] Chakraborty, U.: BIP 106: Dynamically controlled bitcoin block size max cap. https://bips.dev/106/ (2015), accessed: 2026-02-15 [8] Croman, K., Decker, C., Eyal, I., Gencer, A.E., Juels, A., Kosba, A., Miller, A., Saxena, P., Shi, E., Sirer, E.G., Song, D., Wattenhofer, R.: On scaling decentralized blockchains. In: Financial Cryptography and Data Security Workshops (FC). Lecture Notes in Computer Science, vol. 9604, pp. 106–125. Springer (2016). https://doi.org/10.1007/978-3-662-53357-4_8 [9] Daian, P., Goldfeder, S., Kell, T., Li, Y., Zhao, X., Bentov, I., Breidenbach, L., Juels, A.: Flash boys 2.0: Frontrunning, transaction reordering, and consensus instability. In: Proceedings of the IEEE Symposium on Security and Privacy (S&P). pp. 910–927. IEEE (2020). https://doi.org/10.1109/SP40000.20 20.00040 [10] Decker, C., Wattenhofer, R.: Information propagation in the bitcoin network. In: Proceedings of the IEEE International Conference on Peer-to-Peer Computing (P2P). pp. 1–10. IEEE (2013). https://doi.org/10.1109/P2P.20 13.6688704 [11] Dwork, C., Lynch, N., Stockmeyer, L.: Consensus in the presence of partial synchrony. Journal of the ACM 35(2), 288–323 (1988). https://doi.org/10 .1145/42282.42283 [12] Ethereum Improvement Proposals: Eip-1559: Fee market change for ETH 1.0 chain. https://eips.ethereum.org/EIPS/eip-1559 (2021), accessed: 2026-02-15 [13] Eyal, I., Sirer, E.G.: Majority is not enough: Bitcoin mining is vulnerable. In: International Conference on Financial Cryptography and Data Security (FC). pp. 436–454 (2014). https://doi.org/10.1007/978-3-662-45472-5 _28
26
J. Lee
[14] Flashbots: Quantifying maximal extractable value (2022), flashbots Research Note [15] Garay, J.A., Kiayias, A., Leonardos, N.: The bitcoin backbone protocol: Analysis and applications. In: Advances in Cryptology – EUROCRYPT 2015. Lecture Notes in Computer Science, vol. 9057, pp. 281–310. Springer (2015). https: //doi.org/10.1007/978-3-662-46803-6_10 [16] Garay, J.A., Kiayias, A., Leonardos, N.: The bitcoin backbone protocol with chains of variable difficulty. In: Katz, J., Shacham, H. (eds.) Advances in Cryptology - CRYPTO 2017 - 37th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 20-24, 2017, Proceedings, Part I. pp. 291–323. Lecture Notes in Computer Science, Springer (2017). https://doi.org/10.1 007/978-3-319-63688-7_10 [17] Garzik, J.: BIP 102: Block size increase to 2mb. https://bips.dev/102/ (2015), accessed: 2026-02-15 [18] Garzik, J., Harding, T., Johannsson, D.V.: BIP 100: Dynamic maximum block size by miner vote. https://bips.dev/100/ (2015), accessed: 2026-02-15 [19] Khan, T.: BIP 104: Block75 – max block size like difficulty. https://bips.d ev/104/ (2017), accessed: 2026-02-15 [20] Mempool Open Source Project: Block audit score and mempool audit ratio. http s://mempool.space/docs/api/rest (2025), section: get-block-auditscore. Accessed: 2026-02-15 [21] Nakamoto, S.: Bitcoin: A peer-to-peer electronic cash system. https://bitc oin.org/bitcoin.pdf (2008), accessed: 2026-02-15 [22] Sanchez, W.Y.: BIP 107: Dynamic limit on the block size. https://bips.d ev/107/ (2015), accessed: 2026-02-15 [23] Sutton, R.S., Barto, A.G.: Reinforcement Learning: An Introduction. MIT Press, Cambridge, MA, 2 edn. (2018) [24] Wiedenmann, A., Guettler, A.: Bitcoin ordinals and inscriptions: An analysis of bitcoin’s evolving network dynamics. The Journal of The British Blockchain Association 9(1) (2026). https://doi.org/10.31585/jbba-9-1-(1)20 26 [25] Wuille, P.: BIP 103: Block size following technological growth. https://bi ps.dev/103/ (2015), accessed: 2026-02-15
Bitcoin After Block Rewards
A
Additional Proofs on Assumptions and the Solutions
A.1
Gt Threshold and ϕ(w)
27
In this appendix, we formally show that the successful mining probability gap (phon − i dev hon ) appearing in the G threshold condition and the orphan rate gap (ρ − ρ ) in pdev t i ϕ(w) are non-negative. We assume: – Withholding time w ≥ 0, – Block arrival rate λ > 0 (Bitcoin targets one block per 600 seconds), – Hash share hi > 0. Orphan Rate The orphan probability under propagation delay δ is ρ(δ) = 1 − exp(−λδ).
(48)
Successful Mining Probability The successful mining probability of miner i is pi (δ) = hi (1 − ρ(δ))
(49)
= hi exp(−λδ).
(50)
Propagation Delay under Deviation For a deviating miner with withholding time w ≥ 0, δ dev = δ hon + w.
(51)
Gap in Successful Mining Probability phon = hi exp(−λδ hon ), i
(52)
pdev = hi exp(−λδ dev ) i = hi exp −λ(δ hon + w)
(53)
= phon exp(−λw). i
(55)
phon − pdev = phon 1 − exp(−λw) . i i i
(56)
(54)
Therefore, Since w ≥ 0 and λ > 0, 0 < exp(−λw) ≤ 1
⇒
1 − exp(−λw) ≥ 0.
(57)
Hence, phon − pdev ≥ 0. i i
(58)
The inequality is strict when w > 0. When w = 0, the deviating strategy is identical to honest mining, as no additional withholding delay is introduced. In this case, the deviation condition reduces to a trivial comparison, and the Gt threshold no longer carries substantive meaning. As a result, the gap in successful mining probability in Gt threshold is non-negative.
28
J. Lee
Gap in Orphan Rate ρhon = 1 − exp(−λδ hon ),
(59)
ρdev = 1 − exp(−λδ dev ) = 1 − exp −λ(δ
hon
(60)
+ w) .
(61)
Taking the difference, ρdev − ρhon = exp(−λδ hon ) − exp −λ(δ hon + w) = exp(−λδ
hon
) (1 − exp(−λw)) .
(62) (63)
Since exp(−λδ hon ) > 0 and 1 − exp(−λw) ≥ 0, ρdev − ρhon ≥ 0.
(64)
Thus, withholding time w weakly increases the orphan probability while weakly decreasing the successful mining probability. Consequently, the orphan rate gap underlying ϕ(w), as discussed in the Further Discussion chapter, is non-negative.
B
Simulation Parameters and Data Sources
B.1
Reproducibility
All code, configuration files, and data-processing scripts needed to reproduce the experiments and figures in this thesis are publicly available in the project repository: https://github.com/xodn348/BTC_EXP The repository includes (i) end-to-end data processing pipelines, (ii) simulation code and default configuration files, and (iii) scripts to regenerate the main experimental outputs. B.2
Network and Protocol Parameters
Bitcoin After Block Rewards
29
Table 2: Network and Protocol Parameters Parameter
Symbol Value
Block arrival rate λ Discount factor γ Base network delay δ0 Delay per MB κ Withholding delay w
B.3
Source/Rationale −1
1/600 s Bitcoin protocol (10-minute target) 0.99993 Standard in repeated-game models 742 ms KIT invstat.gpd (50th percentile) 26.40 ms/MB Linear regression on propagation data 1.0 s Eyal & Sirer (2014)
Policy Mechanism Parameters
Table 3: Policy Mechanism Parameters Parameter
Symbol Value
Base fee adjustment speed α Initial base fee b0 Target utilization U∗ Block size adjustment step β Minimum block size Bmin Maximum block size Bmax
B.4
Description
0.125 EIP-1559–inspired update rate 20 sat/vB Median observed fee rate 0.80 Target block utilization 0.10 Per-epoch adjustment rate 1 MB Lower bound for adaptive sizing 2 MB Upper bound for adaptive sizing
MEV Modeling Parameters
MEV is generated using a zero-inflated lognormal distribution, where a fraction of blocks contain zero MEV and non-zero realizations are drawn from a lognormal distribution parameterized by (µ, σ). The parameter values are adapted from empirical MEV studies in Ethereum and the BlockScholes modeling framework, and rescaled to reflect Bitcoin’s lower MEV prevalence [3].
Table 4: MEV Distribution Parameters Parameter
Value Source/Rationale
Zero-inflation rate 0.80 Adjusted from Ethereum MEV measurements Lognormal mean (µ) 14.9 Calibrated to match target MEV magnitude Lognormal std. dev. (σ) 1.8 Captures long-tail behavior Maximum MEV 0.1Rt Conservative upper bound
30
B.5
J. Lee
Data Sources and Processing
Simulations are conducted using the top 13 mining pools by average hash rate share, covering approximately 99.5% of the total network hashing power. Pool-level mining costs are computed using CBECI’s annualised electricity consumption estimate (GUESS), converted to a daily network-wide cost via an assumed electricity price, and then allocated to each pool based on the number of blocks it actually mined on that day.
Table 5: Simulation Data Sources and Processing Category
Data
Source
On-chain blocks Block height, fees, vbytes, tx count, pool ID Blockchain.com API Price data BTC–USD daily price Yahoo Finance Mining costs Annualised electricity consumption (TWh/year) Cambridge CBECI Pool shares Daily pool hashrate shares Blockchain.com API MEV (synthetic) MEV samples (sat, USD) Parameter-based model Propagation delay Block propagation statistics KIT invstat.gpd