Bitcoin Security, Incentives, and Market Integrity

Abstract.
Bitcoin’s security against double‑spends depends on relative hash power, confirmation depth, and network propagation. Deviations from honest mining (e.g., selfish/stubborn mining) can be strictly profitable below 50% hash share under realistic tie‑breaking. Security budgets are ultimately an economic equilibrium: the flow paid to miners must deter stock attack payoffs. Since the April 2024 halving (block 840,000), fees’ relative role has risen, re‑exposing fee‑market fragilities. On market integrity, there is rigorous evidence of past manipulation (Mt.Gox bots), systematic wash trading on unregulated exchanges, large‑scale pump‑and‑dumps in thin venues, and contested but influential findings on stablecoin flows. Since January 2024, U.S. spot Bitcoin ETFs have changed microstructure: several studies find ETFs frequently lead price discovery, shifting the locus of “where prices are made.”

I. Mathematical foundations of Bitcoin security & incentives

I.1 Notation and set‑up

• Let the attacker’s hash share be $q\in(0,1/2)$; honest share $p=1-q$.
• A merchant waits $z$ confirmations before treating a payment as final.
• Blocks arrive as a Poisson process with rate proportional to hash share (standard in Nakamoto’s model). (Bitcoin)

I.2 Double‑spend catch‑up probability (exact vs. bound)

Exact Nakamoto–Rosenfeld formula.
Conditioned on $z$ honest blocks arriving, the attacker mines a Poisson$(\lambda)$ number of blocks with $\lambda=z\cdot q/p$. The probability the attacker ever overtakes the honest chain (i.e., goes one ahead) is:
$P_{\text{catch}}(q,z)
= 1 - \sum_{k=0}^{z} e^{-\lambda}\frac{\lambda^k}{k!}\left(1-\left(\frac{q}{p}\right)^{z-k}\right),
\qquad \lambda=\frac{zq}{p}.$
This is the formula Satoshi sketched and that Rosenfeld formalized and tabulated. (Bitcoin)

Worked examples (using the exact formula):
$q=10%$: $P_{\text{catch}}(z=6)\approx 2.428\times10^{-4}$.
$q=20%$: $P_{\text{catch}}(z=6)\approx 1.425\times10^{-2}$.
$q=30%$: $P_{\text{catch}}(z=6)\approx 1.321\times10^{-1}$.
(Computed directly from the expression above.)

Gambler’s‑ruin lower bound.
A commonly quoted bound is $P_{\text{catch}}\ge (q/p)^z$. It has the right exponential decay intuition but dramatically understates risk at realistic $q$; e.g., at $q=20%$ and $z=6$: bound $=2.44\times10^{-4}$ vs. exact $1.43\times10^{-2}$. Use it only as a bound, not as “the formula.” (Statistics LibreTexts)

Refinements.
Later work gives closed‑form expressions and clarifies that the probability to compute is “overtaking” (one ahead), not just “catching up” at parity. These refinements slightly change recommended $z$ for a given failure tolerance. (arXiv)

I.3 Propagation, forks, and why network plumbing matters

Propagation delay is a first‑order driver of stale blocks and temporary forks; large blocks propagate more slowly, which weakens effective security and interacts with strategic mining. Empirically, propagation delays cause forks; improving relay protocols reduces effective attack surfaces. 

I.4 Selfish and stubborn mining (deviation from honest protocol)

Result. Minority miners can earn more than pro‑rata by deviating. Profitability depends on the attacker’s share $\alpha$ and the tie‑breaking parameter $\gamma$ (the fraction of honest miners the attacker can sway in a race). Eyal–Sirer derive the threshold for profitability:
$\alpha;>;\frac{1-\gamma}{3-2\gamma},$
giving thresholds of 1/3 when $\gamma=0$ and 1/4 when $\gamma=1/2$. With $\gamma\to 1$, the threshold tends to 0 (any size deviator benefits). 

Beyond selfish mining. “Stubborn” strategies further expand profitable regions; the honest protocol is not optimal against these variants. (Illinois Experts)

I.5 Fees vs. subsidy: stability in the fee‑dominant regime

When the block subsidy shrinks, fee income becomes bursty. In fee‑only or fee‑dominant regimes, rational deviations such as undercutting and withholding become more attractive, degrading steady‑state security and throughput—even without majority hash power. This is a central, peer‑reviewed result. (ACM Digital Library)

Context in 2024–2025. The fourth halving occurred on April 20, 2024 at block 840,000, reducing the subsidy from 6.25 to 3.125 BTC; the fee share of miner revenue has therefore become more material post‑halving. (CoinGecko)

I.6 The economic limit: security as a flow‑vs‑stock equilibrium

Security budgets must be understood in equilibrium: to deter rewriting (k) blocks, the flow paid to miners needs to exceed the stock payoff from a successful attack of that scale. Budish formalizes this as a “three‑equation” argument; the implication is that proof‑of‑work security is expensive if it must deter large stock payoffs. (NBER)

II. Market manipulation, microstructure, and price discovery

II.1 What’s strongly evidenced vs. debated

(A) Documented manipulation episodes (Mt.Gox 2013).
Using the leaked Mt.Gox dataset, Gandal–Hamrick–Moore–Oberman show that two bot accounts (“Markus”, “Willy”) conducted suspicious trades; on days with such activity, BTC/USD rose ~4–5% on average, and prices rose on ~80% of those days—consistent with manipulation causing the late‑2013 spike. This is the cleanest single‑venue, large‑effect case. 

(B) Wash trading on unregulated exchanges.
Cong–Li–Tang–Yang develop forensic tests (Benford’s law, round‑number clustering, tail shape) across 29 exchanges and quantify wash trading averaging over 70% of reported volume on unregulated venues over their sample—distorting rankings and short‑term prices. This is large‑N, methodologically careful evidence. 

(C) Pump‑and‑dump mechanics.
Multiple studies document organized pumps (Telegram/Discord), with sharp run‑ups and quick reversals in thin markets. These are systematic and repeatable, though effects cluster in small‑cap tokens and illiquid pairs rather than BTC itself. (SSRN)

(D) Stablecoin flows and price support (contested).
Griffin–Shams (Journal of Finance, 2020) find Tether flows timed to downturns and associated with large BTC price impacts, concentrated in one entity’s activity in 2017—an influential paper often read as “support” or “manipulation.” Counter‑work by Lyons–Viswanath‑Natraj emphasizes arbitrage‑based peg dynamics rather than causal price pumping. Treat as serious but debated. (Wiley Online Library)

II.2 Where price discovery happens now (post‑ETF launch)

On Jan 10, 2024, the U.S. SEC approved the listing/trading of spot Bitcoin ETPs/ETFs, changing the microstructure and investor base. (SEC)

Early post‑launch microstructure work (minute‑level information‑share metrics) finds the largest ETFs (e.g., IBIT, FBTC, GBTC) lead price discovery over spot the majority of the time—estimates around ~85% over the 2024 sample in one peer‑reviewed study—indicating a structural shift in “where prices are made.” Treat exact percentages as sample‑ and metric‑dependent, but the qualitative shift is robust. (SpringerLink)

Industry research converges on the same direction (flows, depth, and volume concentrating in ETFs), again pointing to a re‑centered discovery locus in U.S. hours. (Kaiko Research)

III. Practical implications (what follows from I & II)

1. Confirmations should be set from the exact model, not the bound.
   For a 10% attacker, $z=6$ gives $P\approx2.4\times10^{-4}$, but at 20% it’s ~1.4%—a two‑order‑of‑magnitude jump. Merchants and custodians should calibrate (z) to tolerance and context (risk, asset size, time of day), not to folk rules. (arXiv)
2. Network engineering is part of security.
   Lower propagation delays (and relay improvements) reduce stale rates and undercut selfish/stubborn advantages; it’s not “just math”—it’s also plumbing. 
3. Fee‑dominant eras need watchfulness.
   As subsidy falls (post‑2024), fee spikes/variance can make undercutting/withholding rational. Operators should monitor fee volatility, orphan rates, and pool behavior. (ACM Digital Library)
4. Market integrity depends on venue.
   Evidence of wash trading is concentrated on unregulated venues; price discovery shifting to regulated ETFs changes manipulation surface area and surveillance capabilities. Treat volume metrics from opaque venues with skepticism; prioritize regulated feeds when building signals. 
5. Stablecoin‑flow claims deserve balance.
   The Griffin–Shams result is important, but not the last word; any analysis invoking “Tether pumps” should acknowledge the active debate. (Wiley Online Library)

References (select)

• Bitcoin & double‑spend math: S. Nakamoto, Bitcoin: A Peer‑to‑Peer Electronic Cash System, 2008. (Bitcoin)
  M. Rosenfeld, Analysis of Hashrate‑Based Double Spending, 2014. (arXiv)
  A.P. Ozisik & B.N. Levine, An Explanation of Nakamoto’s Analysis of Double‑Spend Attacks, 2017. (arXiv)
  C. Grunspan & R. Pérez‑Marco, Double Spend Races, 2018. (webusers.imj-prg.fr)
• Propagation/forks: C. Decker & R. Wattenhofer, Information Propagation in the Bitcoin Network, IEEE P2P 2013. 
• Selfish/stubborn mining: I. Eyal & E.G. Sirer, Majority Is Not Enough: Bitcoin Mining Is Vulnerable, FC 2014 (profitability threshold (\alpha>\frac{1-\gamma}{3-2\gamma}); thresholds at 1/3 and 1/4). 
  K. Nayak, S. Kumar, A. Miller, E. Shi, Stubborn Mining, 2016. (Illinois Experts)
• Fees vs. subsidy: M. Carlsten, H. Kalodner, S.M. Weinberg, A. Narayanan, On the Instability of Bitcoin Without the Block Reward, 2016. (ACM Digital Library)
• Economic limits: E. Budish, The Economic Limits of Bitcoin and the Blockchain (NBER WP 24717, 2018; updated QJE 2024). (NBER)
• 2024 halving: Coingecko Halving Tracker (date 2024‑04‑20, height 840,000, reward 3.125 BTC). (CoinGecko)
• Manipulation evidence:
  Mt.Gox: N. Gandal, J.T. Hamrick, T. Moore, T. Oberman, Price Manipulation in the Bitcoin Ecosystem, JME 2018. 
  Wash trading: L.W. Cong, X. Li, K. Tang, Y. Yang, Crypto Wash Trading, NBER WP 30783 (avg. inflation >70% on unregulated venues). 
  Pumps: J.T. Hamrick et al., An Examination of the Cryptocurrency Pump and Dump Schemes, 2019; follow‑ons incl. 2021/2024. (SSRN)
  Stablecoins: J.M. Griffin & A. Shams, Is Bitcoin Really Un‑Tethered?, JF 2020; R.K. Lyons & G. Viswanath‑Natraj, What Keeps Stablecoins Stable?, JIMF 2023. (Wiley Online Library)
• ETFs & price discovery: SEC approval statement (Jan 10, 2024). (SEC)
  A. Mohamad et al., Do Bitcoin ETFs Lead Price Discovery …? (Computational Economics, 2025) — ETFs lead ~85% of the time in sample. (SpringerLink)
  Kaiko Research, How BTC ETFs Reshaped Crypto, Jan 2025. (Kaiko Research)

Appendix: quick numeric table (exact (P_{\text{catch}}(q,z)))

(Values from the exact Poisson‑mixture expression; see §I.2. Benchmarks are consistent with Rosenfeld’s tables.) (arXiv)

AI Reasoning - ChatGPT 5Pro

• Uses the exact double‑spend model, with numbers, and distinguishes it from the gambler’s‑ruin bound. (arXiv)
• Covers selfish/stubborn mining and the (\alpha,\gamma) threshold. 
• Treats propagation as part of the threat surface. 
• Explains fee‑dominant fragility post‑halving. (ACM Digital Library)
• Anchors manipulation claims in the Mt.Gox paper and wash‑trading literature; separates contested stablecoin claims from settled facts. 
• Updates microstructure: ETFs and the changed locus of price discovery. (SEC)

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