# · Junhyuk Lee · #1 · 21 Jun 2026, 05:02 UTC

Howdy.

I have written paper about Bitcoin;s security when the Bitcoin block rewards approach zero near 2140.

The original motivation of this paper is worrying about reducing block rewards and miner’s profitability because of halving, while the price could not doubling in every four years.

Problem :

How the Bitcoin’s security changes when the Bitcoin rewards is 0 and how could we prevent the miner’s deviation.

Findings :

1. Found G_t threshold which could determine whether miner’s deviation is happening or not

2. The massive deviation is not happening only because of block rewards

3. In a pure fee-only regime, deviation can become rational even at 0.17% of transaction fees

4. In a pure fee-only regime, base fee, fee floor and adaptive block size could help to alleviate massive deviation of miners.

My suggestion would not be a only solution fot the problem but I would be appreciate if this incur the other brilliant scientists’ inspiration.

The link is in below.

[2606.05503] Bitcoin After Block Rewards

Thank you.

Sincerely,

Junhyuk Lee

# · Cameron · #4 · 14 Jul 2026, 10:17 UTC

A Reply to “Bitcoin After Block Rewards” (Lee, 2026)

Re: Bitcoin After Block Rewards, J. Lee, Texas A&M University (arXiv:2606.05503v1)

1. Introduction and Summary of the Paper’s Contribution

The paper addresses one of the most consequential open questions in the study of permissionless consensus: whether the Bitcoin network can remain secure once the block subsidy has declined to zero and miner compensation derives solely from transaction fees. The author formalizes the miner’s per-block choice between honest and deviating behavior, derives the deviation threshold condition G_t ≥ φ(w) · X_t, and shows that the withholding penalty factor φ(w) = e^(λw) − 1 is independent of an individual miner’s hash-rate share, depending only on the orphan-risk differential induced by withholding. The empirical analysis of the 100,001-block window surrounding the 2024 halving is careful and welcome: it documents a substantial contraction in observable rewards, fees, and on-chain demand, yet finds no persistent increase in deviation-consistent behavior as proxied by block audit scores. The simulation results are sobering — in a fee-only regime, an additional private gain of merely 0.17% of X_t suffices to make deviation attractive across all hash-rate groups, and absent intervention the deviation rate exceeds 70% of blocks, well beyond the Backbone-style honest-majority requirement. The proposed remedies — an EIP-1559-inspired base fee, a fee floor, and an adaptive maximum block size — are evaluated systematically, and the finding that the base fee is the dominant stabilizing factor is a useful and insightful result.

These are genuine contributions. The threshold formulation is simple enough to serve as a practical benchmark, the empirical window is well chosen, and the policy evaluation is disciplined. This reply is offered in a constructive spirit, but its central contention is direct: the paper’s game-theoretic foundations are not realistic as a description of the end game. The model admits only one class of strategic actor — miners deciding block-by-block whether to deviate — and thereby excludes the players with by far the largest payoffs at risk: the parties with a stake in Bitcoin’s stable operation. Once those players are admitted, the paper’s fee-only catastrophe is no longer the equilibrium of the game; it is the worst case of the game — the outcome that obtains only if every stakeholder irrationally declines to defend their own stake. Correspondingly, the paper’s remedy — sustaining a permanent, always-on hashrate fleet through protocol-engineered fee revenue — should be recognized for what it is: the provision of minimum security at maximum cost. This is a fundamentally conservative and pessimistic construction. It is valuable precisely as such — a rigorous lower bound on post-subsidy security — but it should be stated as a limit model, not presented as a forecast of how a rational ecosystem will actually behave.

2. The Central Limitation: An Incomplete Game, and a Limit Model Presented as a Forecast

2.1 The missing players

The model fixes each miner’s hash-rate share h_i and stipulates that deviation “does not alter a miner’s computational capability.” The only strategic variable is propagation behavior; the only players are miners; and the miners’ stakes are bounded by per-block rewards. There is consequently no entry or exit margin, no capacity that responds to price or threat conditions, and no distinction between hashrate that is deployed and hashrate that is available. Security is assessed statically: the fraction of deviating blocks must remain below one half.

This player set is not merely simplified — it omits the actors whose incentives dominate the end game. Exchanges, custodians, ETF issuers, payment processors, and large holders carry stakes in Bitcoin’s stable operation that exceed the per-block sums over which miners strategize by many orders of magnitude: a successful deep reorg or double-spend regime does not cost these parties a block reward, it impairs the entire asset on their balance sheets and the viability of their businesses. In game-theoretic terms, the paper analyzes an equilibrium of a game from which the players with the largest payoffs at risk have been deleted. An equilibrium of the truncated game is not an equilibrium of the actual game: any outcome in which the chain becomes attackable at a private gain of 0.17% of X_t immediately creates an overwhelming incentive for stakeholders to intervene — and rational, deep-pocketed actors facing existential losses do not, in equilibrium, decline to spend a small fraction of their stake to prevent them.

2.2 Surge defense as the cost-minimizing provision of security

Once the stakeholders are in the game, the form their intervention takes follows from ordinary cost minimization. Security is needed continuously at a low level (ordinary block production must proceed and casual attacks must be unprofitable) but at a high level only rarely (when an attack is actually mounted). The cost-minimizing way to provision a good with that demand profile is well understood across domains — electricity systems pair cheap base-load generation with peaking reserves; states pair standing forces with mobilizable reserves; firms pair retained risk with contingent insurance. Applied here, the rational structure of end-game mining is dynamic and stratified:

(a) A minimal-cost base load. A steady-state stratum of hashrate sized to the fee revenue actually available — operating on marginal or stranded energy, covering ordinary block production, and setting the visible entry price for any attack.

(b) A dark, elastic surge capacity of unknown size. Reserve capacity — whether warehoused hardware, standby contracts with operating facilities, or committed rental capacity — held by the stakeholders identified above, and activated only upon detection of an attack such as an attempted double spend. Critically, this capacity is deliberately opaque: the defense reveals only the resources sufficient to defeat a given attack, never its full extent.

The surge defense is therefore not an exotic add-on to the paper’s model; it is the natural cost-minimization strategy of the very actors the model excludes. The strategic logic is deterrence through ambiguity: an attacker facing defensive capacity of unknown magnitude cannot compute the expected cost of an attack, and the visible base load no longer bounds the attacker’s required expenditure. This directly severs the link — central to the paper’s pessimistic fee-only results — between the observable reward X_t and the cost of subverting the chain. The funding logic is a Coasean resolution of the security-budget externality: the parties who internalize the largest losses from a successful reorg are the natural purchasers of the security that fee-paying transactors underprovide, and none of it need flow through the protocol.

2.3 Lee’s proposal as minimum security at maximum cost

Seen against this structure, the paper’s remedy occupies a specific and extreme corner of the design space. Its mechanisms — base fee, fee floor, adaptive block size — exist to guarantee that protocol-mediated revenue is always sufficient to pay a permanent, always-on, fully visible hashrate fleet. That is the most expensive possible way to provision security: peak-threat capacity is purchased continuously, around the clock, in perpetuity, financed by fees extracted from every transactor, whether or not any attack is ever attempted — a standing army sized for a war that almost never comes, with no reserve system. And what this maximum expenditure purchases is minimum security: the entire objective of the policy analysis is to keep the deviation rate just below the 50% Backbone threshold — the weakest guarantee under which the protocol’s security theorems still hold at all.

Minimum security at maximum cost is not a design flaw if it is understood correctly: it is a limit model. It answers the question “what must the protocol alone guarantee if no other actor ever defends the network?” — the fully pessimistic, fully conservative bound in which every stakeholder free-rides completely and security must be purchased entirely through the fee mechanism. That bound is genuinely useful: it prices the floor, and the paper’s finding that a base fee is the dominant lever for sustaining it is a real contribution. But the paper does not present itself as a limit model. It presents the fee-only collapse as the projected end state and protocol revenue engineering as the remedy, when both are artifacts of having assumed away the rational responses of every actor with something to lose. The correct framing — which this reply urges the author to adopt explicitly — is that the paper establishes the lower bound of the end-game security problem, while the equilibrium of the full game, with stakeholders included, lies strictly above it.

2.4 Open problems of the stratified model

It must be acknowledged that the stratified model carries its own difficulties, which any formalization would need to confront: idle ASIC capital depreciates whether or not it runs, creating pressure to deploy or sell reserve capacity; difficulty adjustment calibrated to a thin base load lowers the attacker’s entry cost along with the defender’s; each exercised defense leaks a lower bound on reserve size, inviting probing attacks; the sponsor coalition faces a free-rider problem, mitigated but not eliminated by its small size and repeated interaction; and deterrence through cost-uncertainty does not bind an attacker indifferent to profit. These are open research questions — but they are the questions the end game actually poses, and the paper’s truncated player set prevents it from asking them.

3. Witnessing as the Precondition of Elastic Defense

It is essential to be precise about the logical relationship between the mechanisms introduced above, because they do not stand in parallel: they form a strict dependency chain, and the chain’s first link is epistemic, not economic.

A reactive defense is conditional on detection. Dark capacity of unknown size deters nothing if it never learns that an attack is underway; the defenders can only surge to protect a chain they know the victim is trusting. Detection, in turn, is supplied by witnessing — the victim’s counterparties and the broader network observing, in real time, the blocks on which settlement is being predicated. It follows that an eclipse or Sybil attack does not need to contend with the elastic reserve at all: it routes around it, by ensuring the attack is never seen. An attacker who isolates a victim’s network view can feed it a cheap privately mined chain, and from the perspective of the honest network nothing anomalous is occurring — no conflicting history is visible, no trigger fires, no surge is mounted. The attack defeats both security strata simultaneously: the thin base-load work is inexpensive to outpace in private, and the reserve capacity — the entire deterrent — is blind.

This is why the eclipse attack, a bounded nuisance in the subsidy era, becomes the dominant attack vector in the late game. Under a large subsidy, work is so expensive that an eclipsed victim can be defrauded only of amounts small relative to the cost of the private chain; eclipse resistance can therefore live quietly in peer-selection heuristics, invisible to the trust calculation. In a fee-only regime with a minimal base load, the cost of fabricating a plausible-looking chain collapses along with X_t — this is the paper’s own central result, viewed from the network layer — while the cost of an eclipse attack, which is a network-position attack rather than a hashpower attack, does not scale with the security budget at all. The weaker the blocks, the more damaging the eclipse.

A reactive defense is furthermore only as good as its response window, and here the paper’s implicit settlement model — fast, fixed-depth finality — works against any dynamic scheme. Two mechanisms close the gap, and both must be understood as serving the witnessing requirement identified above.

Value-scaled confirmation depth. Confirmation requirements should scale with the value at risk, so that the attacker’s cost grows with depth while the prize remains fixed — and, equally important, so that the detection-and-response window widens for exactly the transactions worth attacking. This practice already exists informally at every major exchange; formalizing it acknowledges that base-layer Bitcoin is evolving into a settlement layer for large, patient transactions, with speed provided by higher layers.

Witnessed propagation as a validity condition. The more consequential proposal is a requirement that blocks be demonstrably relayed to the network — witnessed by a threshold of independent relays at approximately the time of mining — before they are accepted, i.e., a resubmission or attestation requirement mediated by a relay network. The effect on the paper’s own threat model is decisive: a privately mined chain, however much work it embodies, is invalid because it was never witnessed. Withholding-based double spends, selfish mining, and fee sniping are not merely re-priced; they are removed from the strategy space. In the paper’s notation, this does not raise φ(w) — it renders the withholding strategy inadmissible, forcing any attack into the open where a reactive defense can observe and answer it block by block.

The costs of this mechanism should be stated with equal candor. A relay attestation requirement reintroduces a trusted component into consensus: relay operators can censor, and observation-based validity is inherently subjective — two honest nodes with different network views may permanently disagree, and newly syncing nodes must trust a record of what was witnessed. These costs can be distributed (threshold attestation across a jurisdictionally diverse federation) but not eliminated; this is the known, fundamental price of trading reorg-ability for finality in a permissionless setting. A softer deployment — in which economically significant nodes and the sponsor coalition adopt witnessed-propagation as policy, refusing to credit unwitnessed blocks or honor deep reorgs of witnessed history, while consensus itself remains objective — captures most of the security benefit while confining trust to the settlement endpoints where it already resides.

4. The Underlying Reframing: Confirmation as Two Conjunctive Conditions

Taken together, these mechanisms amount to a reformulation of what finality in a post-subsidy Bitcoin actually consists of, and it is worth making the reformulation explicit, because the paper’s framework implicitly assumes the original, fused model.

The central claim is that confirmation is irreducibly two things — not accumulated work with an optional sanity check, but two co-equal, conjunctive conditions, the failure of either of which voids finality:

The cost of the blocks. Accumulated proof-of-work: evidence that the history was expensive to produce and would be expensive to rewrite. This is the economic component, and it is the quantity the paper’s threshold condition governs.
Knowledge that the network is aware of the blocks. Verified common knowledge: assurance that the history one is trusting is the history one’s counterparties, the relay network, and the elastic defense are also watching. This is the epistemic component, and it is what arms the surge capacity of Section 2.

In the subsidy era these two conditions travel together so reliably that the second is invisible: work is expensive enough that any sufficiently heavy chain one observes is almost certainly the public chain. In the fee-only era that coupling breaks. Chain weight becomes cheap to fabricate, and a heavy-looking chain shown to an isolated observer proves nothing. The second condition must then be established independently — and it cannot be established from chain data, because it is not a property of the chain. It is a property of the observer’s position in the network.

Proof-of-work answers one question: was this history expensive to produce? It is a costliness proof, and its assurance is forward-looking and economic — a bet that the work embodied in the chain will remain valuable enough that rewriting it stays irrational. The paper itself is, at bottom, an analysis of the conditions under which that bet stops paying. But work alone has never identified which expensive history is authoritative; it identifies only the most expensive history a given observer has seen — which is precisely why eclipse attacks succeed, and why the community’s actual crisis resolutions (the 2010 overflow incident, the 2013 fork) were social decisions that work subsequently ratified.

The second question — is this history common knowledge? — is answered not by work but by witnessing: verification that the network at large, including the parties one settles with, has seen the blocks in which one is placing trust. A double spend is, structurally, an attack on common knowledge; a witnessing requirement makes the attack impossible rather than expensive.

The proposal, then, is a deliberate re-weighting. In the subsidy era, work supplies effectively all of finality and the social layer is an unspoken backstop. In the post-subsidy era, work should be understood as the price signal — the entry fee that keeps attacks costly and disciplines miners at the margin, exactly the quantity the paper’s threshold condition governs — while witnessed common knowledge becomes the finality mechanism for transactions of consequence. Settlement assurance decomposes into (i) trust in the future relative value of the work securing the chain, and (ii) verification that one’s counterparties and the broader network agree on that chain. Each component is then provisioned by the party best suited to supply it: miners sell costliness; the stakeholder and relay network provides common knowledge.

4.1 The New Security Assumption, Stated Negatively

This yields what is proposed here as the explicit, late-game security assumption of a post-subsidy Bitcoin: a client is secure only if it is well connected to the Bitcoin network — and security fails if it is eclipsed. The assumption is deliberately stated in its negative form, because that is how the client must operationalize it. Nakamoto consensus treats connectivity as a liveness assumption: a partitioned node falls behind but never accepts a false history it could not detect upon reconnection, because work is too costly to counterfeit. The model proposed here necessarily promotes connectivity to a safety assumption: an eclipsed client in a low-work regime can be fed a finalized-looking false history at low cost, so well-connectedness becomes load-bearing for correctness, not merely for freshness.

The corresponding design requirement is that the client must model this assumption in its trust calculation rather than rely on it silently. Conceptually, the trust a client assigns to a block becomes a conjunction of the form

trust(block) = f(accumulated work) × P(client is well connected and not eclipsed),

with the essential property that no quantity of observed work compensates for a connectivity estimate near zero. A client unable to establish that it is genuinely embedded in the honest network should decline to finalize regardless of the weight of the chain it observes. Operationally, the connectivity term must be actively estimated, not assumed: diversity of peers across autonomous systems, network paths, and jurisdictions; out-of-band comparison of chain tips against independent relays and counterparties; attestation from the witnessing federation of Section 3; and alarm conditions when the client’s view diverges from, or cannot be corroborated by, independent channels. Eclipse resistance thereby moves from an implementation detail of peer selection to a first-class, quantified input to settlement — which is precisely where it must sit once the work term alone can no longer carry finality.

The new critical surface of this model must also be named: the security of the witnessing layer is exactly the security of the witnessing graph. If witness selection remains open, plural, and jurisdictionally diverse, corrupting common knowledge requires corrupting a visible plurality of the ecosystem — arguably harder and more detectable than renting hashpower. If it drifts toward a handful of default providers, a trusted third party has been rebuilt by habit. Resisting that drift is a design and governance problem in its own right — and under the negative assumption above, it is now a safety-critical one.

5. Recommendations

In light of the above, the following extensions are respectfully suggested, either for the present line of work or for future research building on it:

State the model explicitly as a limit case. The fee-only analysis should be framed as the pessimistic lower bound of the end game — the security obtainable if no stakeholder ever defends the network and the protocol alone carries the burden — rather than as a forecast. The corresponding upper region of the design space, in which rational stakeholders provision defense as cost minimization, should be identified as the open problem.
Endogenize hashrate and complete the player set. Extend the model to allow entry, exit, and latent capacity, and admit stakeholders — whose payoffs at risk dominate miners’ per-block rewards — as strategic actors. The deviation threshold should be re-derived in a setting where the attacker’s required expenditure is bounded by unobservable reserve capacity rather than by X_t.
Model the attacker–defender game. The paper’s threat model — individually rational per-block deviation — should be complemented by an explicit reorg/double-spend adversary and a defender reaction function, including detection lags, activation costs, and the information revealed by each defended attack.
Treat confirmation depth as a policy variable. The interaction between value-scaled settlement depth, the detection window, and the deviation threshold is analytically tractable within the paper’s own Poisson framework and would materially strengthen the policy section.
Evaluate witnessed-propagation regimes. Both the consensus-rule variant and the settlement-policy variant of relay attestation deserve formal treatment, including their failure modes (censorship, subjectivity, relay availability attacks) alongside their elimination of the withholding strategy space.
Model the eclipse adversary as the binding constraint. In the low-work regime, the cost-minimizing attack is plausibly not hashpower acquisition but network isolation of the victim, which blinds the witnessing layer and thereby disarms any reactive defense. The threshold analysis should be extended with an adversary who chooses between out-working the base load and eclipsing the target, with the attack cost taken as the minimum of the two.
Specify connectivity-aware clients. Formalize the client-side trust calculation in which finality requires both sufficient work and a verified, quantified estimate of the client’s own connectedness — including concrete estimators (peer diversity across autonomous systems and jurisdictions, out-of-band tip comparison, witness attestation) and the refusal rule under which no observed chain weight is accepted when the connectivity estimate falls below threshold.
Analyze the hybrid. The most promising architecture is plausibly the combination: the paper’s base fee and fee floor sustain the visible base load and set the attack entry price, while stakeholder-funded elastic capacity, deep confirmations for high-value settlement, witnessed propagation, and connectivity-aware clients handle the tail risk. No existing work formalizes this combination; the paper’s framework is well positioned to do so.

6. Conclusion

The paper establishes, rigorously, what a fee-only Bitcoin can guarantee under the most pessimistic possible assumptions: a static, always-on miner population, no stakeholder ever lifting a finger in the network’s defense, and security purchased entirely through protocol-mediated fees. Under those assumptions the outcome is indeed dire — deviation becomes attractive at private gains of a fraction of a percent of block value — and the proposed remedies are correctly identified as what the protocol alone must do to hold the line. This reply’s contention is that this construction is a limit model — minimum security at maximum cost — and that it should be stated as such. Its assumptions are not realistic of end-game game theory, because they exclude the players whose incentives dominate the end game: the parties with existential stakes in Bitcoin’s stable operation, for whom funding an elastic, deliberately opaque surge defense is not altruism but the natural cost-minimization strategy. The fee-only catastrophe is not the equilibrium of the full game; it is the corner of the game in which every rational stakeholder inexplicably abstains.

The equilibrium above that floor has a discernible structure, and its elements form a single dependency chain whose first link is epistemic: the elastic defense only functions if attacks are seen, attacks are only seen if blocks are witnessed, and witnessing is only meaningful to a client that can verify it has not been eclipsed. Confirmation therefore becomes two conjunctive conditions — the cost of the blocks, and knowledge that the network is aware of the blocks — and the late-game security assumption must be stated in its negative form: security fails if the client is eclipsed, and the client must model that possibility explicitly in deciding whether to trust what it sees. Work, in that world, is the price of admission; witnessed common knowledge is the finality; verified connectivity is the precondition of both; and the standing security budget is sized by cost minimization, not by the worst case. The author’s threshold framework is an excellent instrument for pricing the pessimistic floor. The invitation of this reply is to name it as the floor — and to extend the framework to the game the end game will actually be.

Second Edit: Clarify that his is a limit-model under late-game game theory.

Edited to make sure the core connection between the Sybil attack and the Dynamic Mining is more clear.

Thank you J. Lee for publishing your preprint on arXiv. It is good to have proper analysis of the dynamics of the end-game system. The above is my thoughts on the matter: elaborated and little with the help of the AI to make it more accessible.

Good Regards,

Cameron.

# · Junhyuk Lee · #8 · 15 Jul 2026, 03:48 UTC · in reply to #7

Howdy, Cameron.

Thank you for your sincere comment.

I would say your critiques are fair enough and comprehensive approach.

Your suggestions are valuable and I would consider to have them in scope but I am not sure to have them right now because it could be too broad to deal with as a theoretical paper and not to be specialized. (Like your saying, it is specifically scoped within the scope of miner’s profitability and followed conservative approach for intentionally not to be too broad.)

I would consider re-scoping(change the title, state that this is conservative approach. But the reader would know this is theoretical paper so not the 3)

The reason why I did not answer to AI’s comment is that I am worries that there would be some material AI would miss if I let them summarize(I have no guarantee that my AI could summarize the comment perfectly).

It would make misunderstanding or verbose comment of mine so I refused to answer it.

Regarding your comment :

Your paper makes fundamental assumptions that are not realistic of end-game game theory

>> Could you elaborate more on end-game game theory in this context?

You assume a static model of mining. But this maximises cost and minimises security.

>> I used record of network electricity cost using CBECI’s methodology in general. Like your comment, considering variable hash rate(e.g. turn on/off rigs in Carlsten 2016) would give more flexibility to this model but I did not consider it since the elasticity of hash rate could change at any time. As a result, I could say this model is more conservative and theoretical than what you assume.

You assume that the secure is only from the difficulty of creating blocks. But this ignores the low cost of Sybil attack when the blocks are cheap to make.

>> I generally analyze the network’s security with miner’s profitability because they are the main security factor of the network. However, considering Sybil attack is realistic but I am not sure how to model the possibility of Sybil attack with proper measurement.

I know that there are general security model assuming adversaries without economic incentive(classic approach) and assuming adversaries with economic incentive(rational attackers in RPD). I chose RPD way to analyze the network’s security because it is more recent theory and reasonable.(e.g. No one would spend $3 Million to attack BTC network for fun). Having Sybil attack in the model would be more comprehensive but it could be too general and wide scope of view.

Appreciate your time,

Junhyuk.

# · Junhyuk Lee · #10 · 16 Jul 2026, 18:41 UTC · in reply to #9

Howdy Cameron,

Thank you for taking the time to write such a detailed reply. Here are my thoughts on the main points.(AI summarized my thoughts)

Stakeholders. I agree that exchanges, ETF issuers, custodians, and academia hold large stakes in Bitcoin. However, this paper is grounded in miners, which I regard as the primitive of the Bitcoin network. I cannot reliably model the behavior of the surrounding entities, and I do not know their exact incentive structures, so premature speculation could blur the direction of the paper and lower its quality. Entities other than miners and users are applications built on top of Bitcoin’s base layer, so I believe the primitive should be analyzed first.
Dark capacity. Reserve capacity could indeed be switched on to raise the hashrate when needed. But an attacker can do exactly the same, so this becomes a different game from the one analyzed here. I think it belongs to a separate game-theoretic setting rather than to this paper’s analysis.
Eclipse attacks. I find this discussion productive. If someone writes a paper on network-level attacks on Bitcoin, I believe this line of thought could provide great insight.
Common knowledge. I do not have a firm view on this part yet.
Framing. Rather than a limit case, I see this work as an analysis of the primitive element of the Bitcoin network. Only after the problem posed in this paper is resolved can the higher-layer games, such as stakeholder defense or eclipse scenarios, be meaningfully tested on top of it
Future work. If you could organize a few concrete items, I wou happy to investigate them on my own. As you noted, this paper deliberately takes a conservative approach, so I think the risks discussed abo be better analyzed in a separate, connected paper.

Thanks again for the discussion.

Regards,

Junhyuk.

# · Cameron · #11 · 17 Jul 2026, 09:10 UTC · in reply to #10

Hello Junhyuk,

I trust you are well, and thank you for your reply,

The natural first future-work item you may consider extends exactly from your “attackers can hold dark capacity too” argument:

Symmetry there is granted, a conclusion, not an assumption. However, if hidden capacity favours attack or defence can only be settled by modelling the contest: a block-level double-spend race — the next order up from the transaction-level race — which escalates into a bidding war for the capacity itself, the order above that. It is a game, and someone should model it.

The foils already exist, and they bind even if nobody does:

Budish (QJE 2024) models the double-spend contest with a fixed acceptance rule and no reactive defender. The doomsday bounds follow from that assumption.
Bonneau (FC’16) and Bonneau et al. (FC’18) model attack-capacity rental and acquisition — again with no strategic defender on the other side of the market.
Moroz, Aronoff, Narula & Parkes (2020) is the exception that proves the pattern: the moment the victim can counter-fork, deterrence emerges and the attack is not attempted.

Remove the defender’s move and you get collapse; restore it and you get deterrence. Your paper inherits the first family’s assumption. That is a legitimate scope choice — but it should be cited as one, because these foils bound your claims whether or not the game is in scope.

So, stated as a referee would: either model the contest, or cite these foils and state explicitly that the deviation results hold only under the no-reactive-defender assumption they share.

Kind Regards, Cameron