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Power Adjusting Games Under Block-Dependent Pool Payout Schemes

Mustafa Doger, Sennur Ulukus

arXiv:2610.40558v1cs.CRcs.DCcs.ITmath.ITmath.PR

TL;DR

Prior analyses of advanced withholding attacks used GRP, which is not used in practice, leaving their relevance to deployed payout schemes unclear. The paper derives attack revenue ratios for practical schemes and finds that GRP overstates T-PAW gains under PROP but understates them under EXP.

  • Problem

    Prior advanced withholding analyses use GRP, which is not used in practice, leaving unclear whether their conclusions apply to deployed schemes such as round-based PROP, PPLNS, and EXP.

    Source: §I · p. 1

  • Method

    The paper derives T-PAW reward fractions under PPLNS, round-based PROP, and EXP, and introduces PAH with revenue ratios derived for PROP and EXP.

    Source: §I · pp. 1–2

  • Results

    GRP overstates T-PAW gain under PROP and understates it under EXP.

    Source: §I · p. 2

  • Takeaways & Limitations

    PROP is highly vulnerable to PAH, EXP faces a decay-rate trade-off between PAH and T-PAW, and PPLNS is immune to PAH and yields the smallest T-PAW gain among schemes considered.

    Source: §I · p. 2, §VI · p. 8

    For windows comparable to a pool cycle’s blockwork, the closed-form approximation is insufficient and Theorem 5 must be used directly.

    Source: B. Reward Fractions of the Adversary · p. 11

Abstract

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Advanced power adjusting block withholding attacks such as PAW and T-PAW have been analyzed so far under a global-reset version of the proportional scheme (GRP), which is rarely used in practice. In this paper, we analyze T-PAW, which contains FAW and PAW as special cases, under the block-dependent payout schemes deployed in practice: Pay-Per-Last-N-Shares (PPLNS), round-based Proportional (PROP) and exponential-decay (EXP). We derive the revenue ratios analytically, by renewal arguments for PPLNS and through the stationary carry process of an open round for PROP and EXP. We further introduce Power Adjusting Hopping (PAH), where an adversary of any size adjusts its power allocation towards the pool according to the blockwork age of the round without withholding any block, and derive its revenue ratios under PROP and EXP. We show that GRP misestimates the T-PAW gain in a scheme-dependent direction: it overstates the gain under PROP up to an order of magnitude and understates it under EXP for the original slush decay rate. PROP is highly vulnerable to PAH, while EXP faces a trade-off: a slow decay invites PAH and a fast decay invites T-PAW. PPLNS is the least vulnerable scheme: it is immune to PAH and it mitigates, but does not eliminate, T-PAW. All analytical results are verified by Monte Carlo simulations.

I. INTRODUCTION · II. SYSTEM MODEL

The paper analyzes withholding and hopping attacks under practical block-dependent payout schemes, finding that vulnerability depends on the scheme; it also specifies the mining model and revenue metrics used in the analysis.

I. INTRODUCTION

  • Prior analyses left open whether GRP-based withholding results generalize to deployed schemes, how EXP responds to hopping, and which of withholding or hopping is more effective across schemes.Source: §I · p. 1
  • The paper analyzes T-PAW, including FAW and PAW as special cases, under PPLNS, round-based PROP, and EXP, deriving rewards by renewal arguments or stationary carry processes.GRP follows as the zero-carry special case of PROP.Source: §I · p. 1
  • PAH lets an adversary of any size adjust pool power by blockwork age without withholding blocks; its PROP and EXP revenue ratios are derived in closed form, recovering the classical 28% gain as α → 0.Source: §I · pp. 1–2
  • GRP overstates T-PAW gains under PROP by up to an order of magnitude and understates them under EXP.Source: §I · p. 2
  • PROP is dominated by PAH, EXP faces a decay-rate trade-off between PAH and T-PAW, and PPLNS is immune to PAH and has the smallest T-PAW gain, even versus PPS.The analytical results are verified with Monte Carlo simulations.Source: §I · p. 2

II. SYSTEM MODEL

  • The model treats hashpower as independent Poisson block-finding processes, defines an adversary and target pool with α + β < 1/2, and attributes forks only to adversary actions with γ-dependent outside-miner choices.Honest miners publish immediately; the adversary splits power between pool and protocol-following solo mining.Source: §II · p. 2
  • Pool shares are lower-difficulty pPoWs, while valid-block fPoWs occur with probability dp/df; blockwork measures expected hashing needed to find one fPoW.The manager distributes block rewards under the chosen payout scheme.Source: §II · p. 2
  • The analysis measures long-run canonical-block reward shares as revenue ratios for the adversary, target-pool miners, and outside miners, with their ratios summing to one.Block redundancy counts all found blocks, including orphaned or discarded ones, per canonical block; attack gain is measured by the adversary's relative extra reward.Source: §II · pp. 2–3

A. Block-Dependent Payout Schemes … A. Classical Block Withholding (BWH)

The paper describes the practical block-dependent payout schemes PPLNS, PROP, and EXP, contrasts them with GRP, and frames classical BWH under these payout schemes.

A. Block-Dependent Payout Schemes

  • Block-dependent schemes pay miners only when the pool finds a valid block, making cashflow less stable while reducing pool reserve needs, operator risk, and fees.The paper focuses on PROP, PPLNS, and EXP; share-based schemes are unaffected by within-pool power adjustment but are vulnerable to BWH after difficulty adjustment.Source: A. Block-Dependent Payout Schemes · p. 3

1) Pay-Per-Last-N-Shares (W-PPLNS):

  • PPLNS divides each pool-block reward proportionally among the last N submitted shares, with the window parametrized by its blockwork W.Real-world PPLNS windows typically cover a small number of blockworks, including W = 2,.Source: 1) Pay-Per-Last-N-Shares (W-PPLNS) · p. 3

2) Proportional (PROP):

  • PROP distributes each round’s block reward proportionally among all shares submitted between consecutive canonical pool blocks.A miner contributing ns of the round’s nS total shares receives the fraction ns/nS of the reward.Source: 2) Proportional (PROP) · p. 3

3) Score-based: Exponential-decay (EXP):

  • EXP assigns shares time-decaying weights within a round using w(t) = e^(-λdt), then distributes the block reward according to those weights.The paper models historically deployed slush-style round-based exponential decay; the original slush method uses κ1 = 1/2 [2],.Source: 3) Score-based: Exponential-decay (EXP) · p. 3

4) Global-reset-PROP (GRP):

  • GRP resets rounds at every network block, unlike round-based PROP, and gives more weight to recent shares, thereby overstating power-adjusting gains relative to PROP –.The paper characterizes GRP as rarely, if ever, used in practice and as imposing more risk on pool miners than PROP.Source: 4) Global-reset-PROP (GRP) · p. 3

III. POWER ADJUSTING WITHHOLDING ATTACKS UNDER BLOCK-DEPENDENT PAYOUT SCHEMES

  • The attack analysis reviews BWH, FAW, PAW, and T-PAW, with revenue ratios depending on payout design through adversarial reward fractions for pool blocks.For BWH, the adversarial share of pool blockwork is constant across the four schemes because the adversary does not change its power allocation.Source: §III · p. 3, A. Classical Block Withholding (BWH) · p. 3

A. Classical Block Withholding (BWH)

  • In classical BWH, the adversary allocates fraction p of its hashpower to the target pool, submits every pool proof-of-work, withholds every full proof-of-work, and mines solo with the remainder.Because its pool power allocation stays fixed, the adversarial reward fraction is scheme-independent regardless of which past shares are paid; the adversary can optimize an objective such as revenue ratio by adjusting p.Source: A. Classical Block Withholding (BWH) · p. 3

B. Advanced Variants

This section presents T-PAW as a unified attack framework that includes FAW and PAW and models fork races explicitly. It extends the analysis beyond GRP to block-dependent payout schemes.

B. Advanced Variants

  • T-PAW spans FAW and PAW and models fork races explicitly with γ, extending analysis beyond GRP to four block-dependent payout schemes.The paper notes that prior PAW and T-PAW analyses used GRP, which overstates power-adjusting attack gains.Source: B. Advanced Variants · p. 4
  • An attack cycle starts with the adversary splitting power between solo mining and the pool, then adjusts its pool allocation when withholding a pool-forking block.If an outside miner finds a block during the withholding window, the adversary releases its withheld block, adjusts allocation again, and enters a fork race modeled with γ.Source: B. Advanced Variants · pp. 4–5
  • Setting p1 = p2 = p3 makes FAW an exact T-PAW special case, while taking T → ∞ yields PAW or FAW under that allocation constraint.The added p2 → p3 adjustment during the fork race provides flexibility beyond the earlier T-PAW formulation.Source: B. Advanced Variants · p. 5
  • Payout schemes change reward allocation between the adversary and honest pool miners, while expected block counts and outside-miner rewards remain scheme-independent.The adversary optimizes its revenue ratio over p1, p2, p3, and T; the objective is generally neither concave nor convex.Source: B. Advanced Variants · p. 5

C. Reward Fractions in Advanced Variants · 1) Reward Fractions in FAW: · 2) Reward Fractions for W-PPLNS:

Reward fractions depend on which shares each payout scheme includes and how it weights them. FAW preserves the baseline fraction across schemes, while W-PPLNS accounts for the attack cycle’s position in its payout window.

C. Reward Fractions in Advanced Variants

  • A pool block’s adversarial reward fraction is the adversarial share fraction among the shares that its payout scheme rewards.These are the last W blockworks for PPLNS, round shares for PROP, decay-weighted round shares for EXP, and shares since the network’s last block for GRP.Source: C. Reward Fractions in Advanced Variants · p. 5
  • Because the adversary submits shares with different fractions across phases, reward fractions depend on which past phases are paid and their weights.During withholding, adjustments toward the target pool affect r_s and r_u; the deviation from r_1 is more pronounced when recent shares carry more weight, as under EXP or GRP.Source: C. Reward Fractions in Advanced Variants · pp. 5–6

1) Reward Fractions in FAW:

C. Reward Fractions in Advanced Variants

  • The analysis depends on λ_1T rather than T or λ_1 individually, so an optimal T found at λ_1 = 1 can be rescaled to the actual block production rate.The adversary can first set λ_1 = 1, find the optimal T, and rescale it using the actual λ_1.Source: C. Reward Fractions in Advanced Variants · p. 6

1) Reward Fractions in FAW:

  • With unchanged pool-mining allocation, FAW’s adversarial reward fraction remains r under all four listed block-dependent payout schemes.In FAW, p_i = p for i = 1, 2, 3 and T tends to infinity.Source: 1) Reward Fractions in FAW · p. 6

2) Reward Fractions for W-PPLNS:

  • For W-PPLNS, the reward-fraction analysis uses renewal equations over the target pool’s blockwork S in an attack cycle.The result is obtained in Appendix A.Source: 2) Reward Fractions for W-PPLNS · p. 6
  • In W-PPLNS, the long-run adversarial blockwork fraction x̄ is corrected by 1/W terms for the current attack cycle at the window’s end.The cycle’s contribution uses the adversarial fraction a′_i in phase i; exact expressions are given in Theorem 5.Source: 2) Reward Fractions for W-PPLNS · p. 6

3) Reward Fractions for PROP: · 4) Reward Fractions for EXP : · 5) Reward Fractions for GRP:

PROP reward fractions are derived using a carry process and its Laplace transform, while EXP accounts for discounted carried scores; GRP is the zero-carry special case of PROP.

3) Reward Fractions for PROP:

  • Theorem 2 gives expected adversarial reward fractions under round-based PROP for x ∈ {1, s, u, m}.The derivation models blockwork from attack cycles with no canonical pool block as carry into later cycles, using the carry process’s Laplace transform.Source: 3) Reward Fractions for PROP · p. 6
  • For PROP, ωx_i denotes the expected fraction of round blockwork submitted in phase i, while rm is the adversarial fraction of a fork-race pool block when its manager is not rational.The three phases are the first phase, withholding phase, and fork-race phase; these quantities are derived in Appendix B.Source: 3) Reward Fractions for PROP · p. 6

4) Reward Fractions for EXP :

  • Under EXP, round scores are carried over from earlier attack cycles as in PROP but discounted over time by e^−λdt.The adversarial reward fractions are given in informal Theorem 3.Source: 4) Reward Fractions for EXP · p. 6
  • EXP’s A(s) is the expected value of ZAe^−sZ for the score of the corresponding paid round and is specified through a unique bounded solution to a linear fixed-point equation.The formal result is given in Theorem 6 in terms of equation (127).Source: 4) Reward Fractions for EXP · p. 6
  • Unlike PROP, EXP’s transforms are not explicit; equation (127) is solved on a grid, and rx is obtained by one-dimensional quadrature.Each operator in the fixed-point equation is a one-dimensional integral.Source: 4) Reward Fractions for EXP · p. 6

5) Reward Fractions for GRP:

  • GRP has zero carry because each network block starts a new round, making it a special case of round-based PROP.The passage notes that GRP was rigorously studied in [13] with p3 = 0 and that Theorem 2 yields the corresponding result.Source: 5) Reward Fractions for GRP · p. 6

IV. POWER ADJUSTING HOPPING ATTACKS UNDER BLOCK-DEPENDENT PAYOUT SCHEMES

PAH adjusts the adversary’s pool power allocation according to the open round’s blockwork age without withholding blocks. The section analyzes PAH under PROP and EXP for adversaries of any size, including partial power allocations.

IV. POWER ADJUSTING HOPPING ATTACKS UNDER BLOCK-DEPENDENT PAYOUT SCHEMES

  • PAH submits all pool and full-proof-of-work blocks immediately, adjusting only the fraction of adversarial hashpower allocated to the target pool.The analysis extends earlier negligible-power PAH under PROP to PROP and EXP for any adversarial power size, allowing partial allocations.Source: §IV · p. 6
  • Changing power allocation changes how quickly round blockwork age grows, but not its distribution at the pool block, which remains Exp(1).The pool block ends the round, and the ending blockwork-age distribution is the same under every policy.Source: §IV · p. 7
  • Under round-based PROP and EXP, PAH has block redundancy ratio δPAH = 1, and Theorem 4 derives its revenue ratios.When the adversary uses the same pool-power fraction in young and old rounds, the reward fractions reduce to those for honest mining.Source: §IV · p. 7
  • PROP share value varies with round blockwork age: young-round shares can be worth more than solo mining, while old-round shares are worth less.EXP reduces, but does not remove, this dependence, which PAH exploits by shifting pool power toward young rounds.Source: §IV · pp. 6–7

V. NUMERICAL RESULTS · VI. CONCLUSION

The numerical results show that payout schemes differ sharply in their vulnerability to T-PAW and PAH, and that GRP can misstate T-PAW gains. The conclusion emphasizes PAH’s threat to PROP, EXP’s decay-rate trade-off, and PPLNS’s relative resilience.

V. NUMERICAL RESULTS

  • Monte Carlo simulations verify the analytical revenue ratios, block redundancy ratios, and reward fractions, with reward fractions matching theory to the fourth digit for every scheme.The simulations explicitly model attack cycles, pool rounds, the PPLNS window, and EXP scores; optimization uses multi-start interior-point methods, with W = 2 for PPLNS and λd = 2λ1 for EXP unless stated otherwise.Source: §V · p. 8
  • For (α, β, γ) = (0.05, 0.05, 0), T-PAW gains 0.194% under PROP versus 1.698% under GRP, but 2.382% under EXP versus 1.698% under GRP.Under PROP, carried-over blockwork dilutes power adjustments; under EXP, decay emphasizes recent shares from the withholding phase. Thus GRP overstates T-PAW gains under PROP and understates them under EXP.Source: §V · p. 8
  • Under PROP, PAH gains 17.603%–27.968% while T-PAW stays below 1%; under EXP, the better attack depends on adversary and pool size, and PPLNS gives PAH no gain and at most 0.376% for T-PAW.T-PAW gains increase with γ, while PAH gains do not depend on γ. Under PROP, PAH’s gain decreases as α/β grows because the adversary dilutes young rounds with its own shares; for small adversaries, it approaches the 28% gain of a negligible miner [2],.Source: §V · p. 8
  • For α = 0.05, the PPS All-out Attack gain exceeds every attack under every block-dependent scheme except PAH under PROP; for small adversaries, PAH under PROP and EXP can far exceed PPS.For (α, β, γ) = (0.01, 0.05, 0.5), T-PAW under EXP gains 1.366% versus the PPS gain of 1.010%; PPS pays shares immediately, so PAH yields no gain, while BWH is profitable after difficulty adjustment through AoA, with gain α/(1−α) per epoch borne by the pool operator. PPLNS is least vulnerable for every (α, β, γ).Source: §V · p. 8
  • As EXP’s decay rate λd rises from λ1 to 4λ1, T-PAW gain increases from 1.438% to 3.579%, while PAH gain falls from 2.768% to 0.805% for (α, β, γ) = (0.05, 0.05, 0).Slow decay invites PAH, while fast decay invites T-PAW; as λd approaches zero, EXP reduces to PROP, which is especially vulnerable to PAH.Source: §V · p. 8
  • For (α, β, γ) = (0.05, 0.1, 0.5), increasing the PPLNS window W from 1 to 4 reduces T-PAW gain from 0.523% to 0.314%, but does not eliminate it.The power-adjustment component affects reward fractions through terms of order 1/W, whereas the fork-race component does not depend on W, so longer windows alone cannot make PPLNS fully resistant.Source: §V · p. 8

VI. CONCLUSION

  • The conclusion identifies combining withholding and hopping, and analyzing the attacks across multiple target-pool systems, as future directions.The authors suggest combined strategies may be especially useful against score-based schemes and that multiple target pools may increase rewards, especially for PAH.Source: §VI · p. 8

APPENDIX A REWARD FRACTIONS FOR W -PPLNS · 2) Withholding phase: · A. Adversarial Blockwork in the Window Before a Cycle

The appendix computes W-PPLNS reward fractions by tracking pool and adversarial blockwork within attack cycles and in the blockwork window preceding each cycle.

2) Withholding phase:

  • When withholding ends early after an outside-pool honest block, a fork race follows, with pool blockwork distributed exponentially according to hash fraction a3.The withholding phase can also last until its deadline, yielding deterministic pool blockwork.Source: 2) Withholding phase · p. 9

APPENDIX A REWARD FRACTIONS FOR W -PPLNS

2) Withholding phase:

  • Under W-PPLNS, a pool block’s reward is split according to the blockwork submitted in the preceding W units of pool blockwork.The adversarial fraction of submitted blockwork is tracked as x(σ).Source: 2) Withholding phase · p. 9
  • The cycle’s adversarial blockwork calculation splits the window contribution across the first, withholding, and fork-race phases.The phase contributions use their respective adversarial blockwork fractions.Source: 2) Withholding phase · p. 9

APPENDIX A REWARD FRACTIONS FOR W -PPLNS

A. Adversarial Blockwork in the Window Before a Cycle

  • For the preceding-window calculation, if the previous cycle’s blockwork S is at least v, the window lies within that cycle; otherwise, the calculation recurses over the remaining window v − S.Attack cycles are i.i.d., which yields the renewal equation for H(v).Source: A. Adversarial Blockwork in the Window… · pp. 9–10
  • The window analysis uses renewal function U to count cycle starts and subtracts shares outside the interval from the earliest overlapping cycle.Term (I) counts adversarial blockwork from cycles whose later boundary is inside the interval; term (II) removes the earliest cycle’s shares outside it.Source: A. Adversarial Blockwork in the Window… · p. 10

0 F(t) dt, hence · B. Reward Fractions of the Adversary

The renewal argument solves the renewal equation and establishes convergence, while Theorem 5 derives expected adversarial shares under W-PPLNS by accounting for a backward reward window across cycle phases. The closed-form approximations are negligible-error for windows of a few units of blockwork, but the theorem should be used directly when the window is comparable to a cycle’s pool blockwork.

0 F(t) dt, hence

  • The renewal equation g = z + g∗F has solution g = z∗U, and the cited renewal result gives a limit for g(v).The supplied passage truncates the displayed limit, so its full value is not reported here.Source: §0 · p. 10
  • The function ĥ is non-negative and non-increasing because a pool block at cycle position σ is counted only for v in [0, S − σ).Exchanging the order of integration is used to evaluate this term.Source: §0 · p. 10
  • Applying the renewal result with z = ĥ makes term (II) converge to E[Ψ]/m1, completing the proof when combined with term (I).Source: §0 · p. 10

B. Reward Fractions of the Adversary

  • Theorem 5 gives the expected adversarial shares under W-PPLNS for a canonical pool block.Its proof evaluates the reward window backwards from the block.Source: B. Reward Fractions of the Adversary · pp. 10–11
  • The backward W-PPLNS window counts each phase by its density times the lesser of its work and the remaining window length, with leftover length assigned to earlier cycles.For the race, withholding, and first phases, the remaining lengths are respectively W, (W − Q̂3)+, and (W − Q̂2 − Q̂3)+.Source: B. Reward Fractions of the Adversary · p. 11
  • Because a cycle ends at the first of independent exponential clocks, Q̂1, Q̂2, and Q̂3 are independent of which case occurs, allowing expectations to complete the proof.Other canonical pool blocks use the same accounting with Q̂3 = 0 and Q̂2 ∈ {0, T̂, Q̂2}.Source: B. Reward Fractions of the Adversary · p. 11
  • For windows of a few units of blockwork, the closed-form approximations neglect only negligible terms; comparable-to-cycle windows require Theorem 5 directly.The neglected contributions are the o(1) term and the event that the three phase lengths sum to more than W.Source: B. Reward Fractions of the Adversary · p. 11

APPENDIX B REWARD FRACTIONS FOR ROUND-BASED PROP … C. Transform of the Carry

This appendix derives reward fractions for round-based PROP by defining rounds around canonical pool blocks and modeling open-round blockwork as a stationary carry process. Laplace transforms of the carry and paid rounds provide the expected adversarial fractions used in the derivation.

A. Model

  • PROP splits each canonical pool-block reward by the blockwork submitted since the previous canonical pool block; only canonical pool blocks end rounds.Orphaned pool blocks and discarded withheld fPoWs do not end a round; a released fPoW ends one only if it becomes canonical.Source: A. Model · p. 11
  • Alternatively, a pool could close rounds whenever it receives an fPoW, including one that later loses a fork race; the analysis can be modified for this rule.The authors consider their canonical-block boundary definition more likely to be used in practice.Source: A. Model · p. 11

B. The Carry Process

  • Because a round can span multiple attack cycles, the pool’s reward shares are dependent across cycles, and the open round’s blockwork is tracked as a three-mode carry.The carry at the start of each cycle records pool blockwork submitted in the three modes during the still-open round.Source: B. The Carry Process · p. 11
  • The carry evolves as a Markov chain with positive recurrence and regeneration: reset cycles occur with probability η0 ≥ β > 0 and leave zero carry.A cycle resets the carry exactly when it is one of Cases 3, 4a, 4b-ii, or 4b-iii-B.Source: B. The Carry Process · p. 12

C. Transform of the Carry

  • The stationary carry’s Laplace transform is derived in Lemma 2, and the paid-round transforms are obtained using an independent stationary carry and mode-specific blockwork.These transforms support the expected adversarial fractions of paid rounds used to derive reward fractions.Source: C. Transform of the Carry · p. 12, B. The Carry Process · p. 12
  • Differentiating the transforms on the diagonal yields the expected fraction of round blockwork submitted in each mode, completing the proof of Theorem 2.The derivation applies the product rule to the paid-round transforms.Source: C. Transform of the Carry · p. 13
  • Under GRP, every block known to the manager starts a new round, so the carry is always zero and the stationary carry transform is one.Withheld fPoW blockwork remains in its round because it is not known to the manager; the race-triggering honest block starts a new round.Source: C. Transform of the Carry · p. 13

APPENDIX C REWARD FRACTIONS FOR EXP … APPENDIX D REVENUE RATIOS OF POOL HOPPING

The EXP appendix models each round’s reward using exponentially decaying share weights and derives reward fractions from the stationary carry process. The pool-hopping appendix characterizes round durations and shows that round-ending work age is exponentially distributed regardless of the power-allocation policy.

A. Model

  • Under EXP, a share submitted t time units before round end receives weight e^(-λ_dt), and the canonical pool block’s reward is split in proportion to the total weights of shares in its round.The adversarial reward fraction for a round is its adversarial score divided by its total score.Source: A. Model · p. 13

B. The Carry Process

  • The score of an open round is its carry between cycles; cycle rewards are paid using phase-specific scores, and the next carry depends on the cycle’s outcome.The carry has stationary distribution π, and the renewal-reward theorem gives each party’s revenue ratio as its expected cycle reward divided by expected canonical pool blocks.Source: B. The Carry Process · p. 14

C. Transform of the Carry

  • The stationary carry’s transforms are the unique bounded solution of the EXP update equations, yielding expected adversarial reward fractions.The equations can be iterated on a grid in s using one-dimensional integrals, followed by one-dimensional quadrature for the reward fractions.Source: C. Transform of the Carry · p. 14

APPENDIX D REVENUE RATIOS OF POOL HOPPING

  • Under PAH, rounds are i.i.d.; the pool block is adversarial with probability a′(τ), and the expected counts of outside honest and solo adversarial blocks before τ are nR and nS.The round’s survival probability is determined by the time-varying pool hash fraction and the policy’s allocation.Source: Appendix D · p. 14
  • The blockwork age Yτ is Exp(1) for every PAH policy, because pool blocks arrive at unit rate when a round is measured in blockwork age.Changing the adversary’s allocation changes how quickly blockwork age accumulates, not how much blockwork age a round lasts.Source: Appendix D · pp. 14–15

A. Proof of Theorem 4 … APPENDIX E MONTE CARLO VERIFICATIONS

The appendix validates the analytical results for PAH and T-PAW by simulation, while the preceding sections derive PAH rewards, characterize all-or-nothing hopping, and establish when PPLNS resists power adjusting. The results distinguish payout schemes and identify withholding as necessary for PPLNS gains over honest mining.

A. Proof of Theorem 4

  • With no withholding, every block is canonical, and renewal-reward calculations derive the revenue ratios from expected per-round rewards and canonical block counts.The proof computes the expected adversarial reward fraction of the pool block to complete the ratios.Source: A. Proof of Theorem 4 · p. 15

B. All-or-Nothing Hopping

  • For a negligible adversary, all-or-nothing hopping reaches a relative extra reward of about 0.2815 at y*≈0.4348; a larger adversary gains less and has a higher optimal threshold.The negligible-miner optimum satisfies e^y*E1(y*)=1; the adversary’s own shares dilute young rounds, captured by β′ in (142).Source: B. All-or-Nothing Hopping · p. 15
  • Under EXP, hopping gains depend separately on α and β: long rounds relative to 1/λ_d yield little gain, while short rounds make EXP behave like PROP.Source: B. All-or-Nothing Hopping · p. 16

C. Pool Hopping Under W-PPLNS

  • Under W-PPLNS, past-based allocation with immediate share submission yields ρ_A=α; PAW/T-PAW gains over honest mining therefore require withholding and future-block information [2], [13].Source: C. Pool Hopping Under W-PPLNS · p. 16

APPENDIX E MONTE CARLO VERIFICATIONS

  • 17.603% (theory) versus 17.601%±0.009% (simulation) for PAH under PROP at β=0.05; T-PAW theory fell within all 99% confidence intervals except PPLNS at (β, γ) = (0.1, 0).T-PAW confidence intervals were wider, and its RER values alone were not a sharp test because the gain was comparable to the interval size for PPLNS and PROP.Source: Table VIII · p. 16, Appendix E · p. 16
  • Simulations used optimal parameters at α=0.05, stationary-carry burn-in for PROP and EXP, and independent rounds for PAH.The simulations tracked pool state explicitly; for PAH, outside and solo block counts were replaced by conditional means to reduce variance without bias.Source: Appendix E · p. 16
  • Reward fractions matched theory to the fourth digit across schemes, including PROP carry (r_u = 0.1552 versus 0.1554 ± 0.0003 at β=0.05, γ=0.5).Under PPLNS, simulated r_u and r_f were at most 2 × 10^-4 below theory; simulated PAH r_h matched theory in every case, verifying Theorem 4 and the blockwork-age argument.Source: Appendix E · pp. 16–17, Table IX · p. 17

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