pith. sign in

arxiv: 2604.06092 · v1 · submitted 2026-04-07 · 💻 cs.CR · cs.GT· econ.TH

Inertial Mining: Equilibrium Implementation of the Bitcoin Protocol

Manuel Mueller-Frank , Minghao Pan , Omer Tamuz This is my paper

Pith reviewed 2026-05-10 19:22 UTC · model grok-4.3

classification 💻 cs.CR cs.GTecon.TH
keywords inertial miningBitcoin protocolselfish miningequilibriumproof of workblockchain incentivesNakamoto consensusoff-path forks
0
0 comments X

The pith

Inertial mining turns the Bitcoin protocol into a stable equilibrium that yields a single longest chain.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces inertial mining, a rule that directs miners to follow the longest chain in the way Nakamoto originally described. This rule only modifies miner actions during off-path forks, leaving normal operation unchanged. When all miners adopt it, no individual miner can gain by withholding blocks or pursuing other strategies such as selfish mining. A reader should care because the original Bitcoin protocol allows profitable deviations that threaten the intended chain structure, while inertial mining removes that incentive problem without altering the consensus rules or the blockchain format.

Core claim

When miners follow inertial mining, they produce the single longest chain outcome intended by Nakamoto. Unlike the standard Bitcoin protocol, inertial mining constitutes an equilibrium assuming no miner controls more than half the mining power. Neither selfish mining nor any other deviation strategy is profitable for a rational miner. The protocol changes miner behavior only on off-path forks and can be implemented in Bitcoin without any modifications to its consensus mechanism or blockchain architecture.

What carries the argument

The inertial mining rule, which specifies precise continuation behavior only on forks that deviate from the main chain.

If this is right

  • Selfish mining and every other deviation strategy yield zero or negative profit.
  • The network produces exactly one longest chain as the stable outcome.
  • The change can be adopted by updating only the off-path fork-handling logic in existing mining software.
  • The equilibrium holds for any distribution of mining power below the 50 percent threshold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar inertial rules might stabilize other proof-of-work systems that currently permit withholding strategies.
  • Adoption would likely require miners to agree on the exact off-path continuation rule and update their software accordingly.
  • If real-world forks occur more often than assumed, the precise inertial specification could become a point of disagreement.
  • The approach shows that incentive compatibility can sometimes be restored by restricting behavior only in low-probability states rather than redesigning the core protocol.

Load-bearing premise

No single miner controls more than half the mining power and rational miners follow the inertial rule exactly on off-path forks without other strategic considerations.

What would settle it

A controlled simulation or network trace in which a miner holding less than half the power increases expected revenue by deviating from the inertial rule on a created fork.

Figures

Figures reproduced from arXiv: 2604.06092 by Manuel Mueller-Frank, Minghao Pan, Omer Tamuz.

Figure 1
Figure 1. Figure 1: Illustration of the definitions of apx, yq, cousinpx, yq and killerpx, yq. The block apx, yq is the closest ancestor of px, yq that lies on the longest chain C ˚ . The block cousinpxi , yiq is the unique block on C ˚ with the same depth as pxi , yiq. Suppose J “ 1. Then killerpx1, y1q “ cousinpx1, y1q and pw, zq is the killer of px0, y0q, px2, y2q and px3, y3q. Let Kt be the number of blocks killed by mt ,… view at source ↗
read the original abstract

The value of proof-of-work cryptocurrencies critically depends on miners having incentives to follow the protocol. However, the Bitcoin mining protocol proposed by Nakamoto (2008) and implemented in practice is well known not to constitute an equilibrium: Eyal and Sirer (2018) construct a profitable deviation called ``selfish mining'' which relies on strategically delaying disclosure of newly mined blocks rather than publishing them immediately. We propose inertial mining, a novel mining protocol. When miners follow inertial mining, they produce the outcome intended by Nakamoto, i.e., a single longest chain. But unlike the Bitcoin mining protocol, inertial mining constitutes an equilibrium (assuming no miner controls more than half of the mining power). Indeed, neither selfish mining nor any other deviation is profitable. Furthermore, inertial mining only changes miners' behavior in the event of off-path forks, and can be implemented in Bitcoin without any changes to its consensus mechanism or blockchain architecture.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes inertial mining, a strategy profile for Bitcoin miners that follows the Nakamoto protocol on the equilibrium path (producing a single longest chain) but specifies distinct behavior on off-path forks arising from network delays or simultaneous blocks. It claims that, assuming no miner controls more than 50% of hash power, this profile constitutes a Nash equilibrium in which neither selfish mining nor any other unilateral deviation is profitable, and that the protocol can be implemented in Bitcoin without altering its consensus rules or blockchain architecture.

Significance. If the equilibrium property is established rigorously, the result would address a central open problem in the game-theoretic analysis of proof-of-work systems: the non-equilibrium status of the original Bitcoin protocol under selfish mining. The practical advantage that the change is confined to off-path contingencies and requires no consensus-layer modifications would make the proposal directly relevant to protocol designers and implementers. The work also contributes a concrete example of how to close off profitable deviations while preserving the intended chain-selection outcome.

major comments (2)
  1. [§4] §4 (Equilibrium Analysis) and the statement of the main theorem: the claim that inertial mining is a Nash equilibrium requires showing that the prescribed strategy is optimal at every information set, including off-path histories in which forks arise from simultaneous blocks, network delays, or attempted deviations. The manuscript does not supply an explicit extensive-form game, payoff functions, or case analysis verifying that no miner can profit by switching chains, withholding blocks, or using a different tie-breaking rule in those contingencies; without this, the no-profitable-deviation assertion cannot be evaluated.
  2. [§3] Definition of the inertial rule (likely §3): the protocol is asserted to differ from Nakamoto only on off-path forks, yet the precise condition for chain selection or block publication when multiple tips exist is not formalized with reference to the miners' information partitions or continuation values. This leaves open whether the rule remains dominant-strategy optimal once a miner anticipates that others may also deviate in the same subgame.
minor comments (2)
  1. [Abstract and §1] The abstract and introduction should explicitly state the solution concept (Nash equilibrium in the extensive-form game) and the precise assumption on hash-power distribution rather than leaving it implicit.
  2. [§2] Notation for block heights, fork lengths, and publication delays should be introduced once and used consistently; several passages reuse the same symbols for different quantities.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments correctly identify opportunities to strengthen the formal presentation of the equilibrium analysis. We will revise the manuscript to incorporate more explicit modeling while preserving the core claims and implementation advantages of inertial mining.

read point-by-point responses

  1. Referee: [§4] §4 (Equilibrium Analysis) and the statement of the main theorem: the claim that inertial mining is a Nash equilibrium requires showing that the prescribed strategy is optimal at every information set, including off-path histories in which forks arise from simultaneous blocks, network delays, or attempted deviations. The manuscript does not supply an explicit extensive-form game, payoff functions, or case analysis verifying that no miner can profit by switching chains, withholding blocks, or using a different tie-breaking rule in those contingencies; without this, the no-profitable-deviation assertion cannot be evaluated.

    Authors: We agree that the equilibrium claim in §4 would be more rigorously established with an explicit extensive-form game. The current manuscript presents an intuitive argument that inertial mining coincides with Nakamoto on the equilibrium path and deters deviations such as selfish mining by prescribing specific off-path behavior, but it does not include a full game tree, payoff matrix, or exhaustive case analysis over all information sets. In the revised version we will add a formal definition of the game (including information partitions and continuation values) together with a case analysis of the principal off-path contingencies (simultaneous blocks, network delays, and attempted withholding). This will make the verification that no unilateral deviation is profitable under the <50% hash-power assumption directly checkable. The revision does not alter the paper's main result or implementation claims. revision: yes

  2. Referee: [§3] Definition of the inertial rule (likely §3): the protocol is asserted to differ from Nakamoto only on off-path forks, yet the precise condition for chain selection or block publication when multiple tips exist is not formalized with reference to the miners' information partitions or continuation values. This leaves open whether the rule remains dominant-strategy optimal once a miner anticipates that others may also deviate in the same subgame.

    Authors: We accept that the inertial rule in §3 requires a more precise formalization. The manuscript states that the rule matches Nakamoto on the equilibrium path and only alters behavior on off-path forks, but it does not explicitly tie the rule to information partitions or continuation values. In the revision we will supply an explicit definition of the inertial rule expressed in terms of miners' information sets and continuation values, clarifying exactly when chain selection or block publication deviates from the longest-chain rule. The added formalization will also show that the prescribed action remains a best response in the relevant subgames even when a miner anticipates possible deviations by others, under the maintained assumption that no miner controls more than half the hash power. revision: yes

Circularity Check

0 steps flagged

No circularity: inertial mining is a newly defined strategy whose equilibrium property is argued directly from game-theoretic assumptions.

full rationale

The paper introduces inertial mining as an explicit new strategy profile that coincides with Nakamoto's protocol on the equilibrium path and differs only on off-path forks. The claim that it forms a Nash equilibrium (no profitable unilateral deviation, including selfish mining) is presented as a direct consequence of the strategy definition plus the maintained assumption that no miner controls more than half the hash power. No equations reduce a fitted parameter to a renamed prediction, no self-citation chain supplies a uniqueness theorem, and no ansatz is smuggled in. The derivation is therefore self-contained and does not collapse to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claim rests on standard rational-agent assumptions in mining games plus the explicit 50% hash-power bound; the new protocol itself is the main added construct.

axioms (2)
  • domain assumption Miners are rational expected-reward maximizers.
    Standard assumption invoked for any equilibrium analysis of mining incentives.
  • domain assumption No miner controls more than 50% of total hash power.
    Explicitly required in the abstract for the equilibrium statement to hold.
invented entities (1)
  • Inertial mining protocol no independent evidence
    purpose: Defines a new strategy that miners follow to achieve equilibrium longest-chain behavior.
    Newly introduced rule set whose properties are asserted but not externally validated in the abstract.

pith-pipeline@v0.9.0 · 5459 in / 1261 out tokens · 38835 ms · 2026-05-10T19:22:49.254915+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    Bahrani and S

    M. Bahrani and S. M. Weinberg. Undetectable selfish mining. In Proceedings of the 25th ACM Conference on Economics and Computation, pages 1017--1044, 2024

    work page 2024

  2. [2]

    Biais, C

    B. Biais, C. Bisi \`e re, M. Bouvard, and C. Casamatta. The B lockchain F olk T heorem. The Review of Financial Studies, 32 0 (5): 0 1662--1715, May 2019. doi:10.1093/rfs/hhy095

    work page doi:10.1093/rfs/hhy095 2019

  3. [3]

    J. Bonneau. Why buy when you can rent? B ribery attacks on bitcoin-style consensus. In Financial Cryptography and Data Security, Lecture Notes in Computer Science, pages 19--26. Springer, 2016

    work page 2016

  4. [4]

    Brown-Cohen, A

    J. Brown-Cohen, A. Narayanan, A. Psomas, and S. M. Weinberg. Formal barriers to longest-chain proof-of-stake protocols. In Proceedings of the 2019 ACM Conference on Economics and Computation, pages 459--473, 2019

    work page 2019

  5. [5]

    E. Budish. Trust at scale: The economic limits of cryptocurrencies and blockchains. The Quarterly Journal of Economics, 140 0 (1): 0 1--62, 2025

    work page 2025

  6. [6]

    V. Buterin. Proof of stake faq. Blog post, 2017

    work page 2017

  7. [7]

    Eyal and E

    I. Eyal and E. G. Sirer. Majority is not enough: Bitcoin mining is vulnerable. Communications of the ACM, 61 0 (7): 0 95--102, 2018

    work page 2018

  8. [8]

    J. S. Gans and H. Halaburda. ``zero cost'' majority attacks on permissionless proof of work blockchains. Management Science, 70 0 (6): 0 4155--4165, 2024. doi:10.1287/mnsc.2023.02426

    work page doi:10.1287/mnsc.2023.02426 2024

  9. [9]

    Halaburda, G

    H. Halaburda, G. Haeringer, J. S. Gans, and N. Gandal. The microeconomics of cryptocurrencies. Journal of Economic Literature, 60 0 (3): 0 971--1013, Sept. 2022. doi:10.1257/jel.20201593

    work page doi:10.1257/jel.20201593 2022

  10. [10]

    O. J. Hall, S. Shiaeles, and F. Li. A study of E thereum’s transition from proof-of-work to proof-of-stake in preventing smart contracts criminal activities. Network, 4 0 (1): 0 33--47, 2024

    work page 2024

  11. [11]

    E. Heilman. One weird trick to stop selfish miners: Fresh bitcoins, a solution for the honest miner. In Financial Cryptography and Data Security Workshops. Springer, 2014. doi:10.1007/978-3-662-44774-1_12

    work page doi:10.1007/978-3-662-44774-1_12 2014

  12. [12]

    Huberman, J

    G. Huberman, J. D. Leshno, and C. Moallemi. Monopoly without a monopolist: An economic analysis of the bitcoin payment system. The Review of Economic Studies, 88 0 (6): 0 3011--3040, 2021

    work page 2021

  13. [13]

    J. D. Leshno and P. Strack. Bitcoin: An axiomatic approach and an impossibility theorem. American Economic Review: Insights, 2 0 (3): 0 269--286, 2020

    work page 2020

  14. [14]

    J. D. Leshno, E. Shi, and R. Pass. On the viability of open-source financial rails: Economic security of permissionless consensus. arXiv preprint arXiv:2409.08951, 2024

    work page arXiv 2024

  15. [15]

    S.-N. Li, C. Campajola, and C. J. Tessone. Statistical detection of selfish mining in proof-of-work blockchain systems. Scientific Reports, 14: 0 6251, 2024

    work page 2024

  16. [16]

    Nakamoto

    S. Nakamoto. Bitcoin: A peer-to-peer electronic cash system. 2008

    work page 2008

  17. [17]

    E. S. Pagnotta. Decentralizing money: Bitcoin prices and blockchain security. The Review of Financial Studies, 35 0 (2): 0 866--907, Feb. 2022. doi:10.1093/rfs/hhaa149

    work page doi:10.1093/rfs/hhaa149 2022

  18. [18]

    Distributed Approximation of Maximum Independent Set and Maximum Matching , booktitle =

    R. Pass and E. Shi. Fruitchains: A fair blockchain. In Proceedings of the ACM Symposium on Principles of Distributed Computing, pages 315--324, 2017. doi:10.1145/3087801.3087809

    work page doi:10.1145/3087801.3087809 2017

  19. [19]

    Sapirshtein, Y

    A. Sapirshtein, Y. Sompolinsky, and A. Zohar. Optimal selfish mining strategies in bitcoin. International conference on financial cryptography and data security, pages 515--532, 2016

    work page 2016

  20. [20]

    Schilling and H

    L. Schilling and H. Uhlig. Some simple bitcoin economics. Journal of Monetary Economics, 106: 0 16--26, 2019

    work page 2019

  21. [21]

    Solat and M

    S. Solat and M. Potop-Butucaru. Brief announcement: Zeroblock: Timestamp-free prevention of block-withholding attack in bitcoin. In Networked Systems. Springer, 2017. doi:10.1007/978-3-319-69084-1_25

    work page doi:10.1007/978-3-319-69084-1_25 2017