The instability of Bitcoin without the block reward (Carlsten et al. 2016)

When the subsidy is gone, a block earns only the fees it collects. Fees arrive steadily, blocks at exponentially distributed intervals, so a block’s reward is proportional to the time since the previous one, and as variable as that time. Carlsten, Kalodner, Weinberg and Narayanan showed two consequences.

Right after a block the mempool is nearly empty, so the next block is worth less than the electricity to find it, and rational miners pause: a mining gap of

\[t^* = \max\left(0, \frac{c - S}{f}\right)\]

seconds, for a cost \(c\) per expected block, a subsidy \(S\) and fees arriving at \(f\) per second. And when the tip block claimed more fees than are left, a miner gains by forking it and undercutting: it claims only part of those fees, leaving the rest as a bribe for whoever builds on its block instead.

import statistics

import matplotlib.pyplot as plt

import blockchainkit as bk

Reward variance

FEE_RATE = 0.0005  # Fees per second: 0.3 BTC per 600-second block on average.
subsidized = bk.economics.simulate_block_rewards(3.125, FEE_RATE, blocks=5_000, seed=1)
fees_only = bk.economics.simulate_block_rewards(0.0, FEE_RATE, blocks=5_000, seed=1)
for label, rewards in (("with subsidy", subsidized), ("fees only", fees_only)):
    variation = statistics.pstdev(rewards) / statistics.mean(rewards)
    print(
        f"{label}: mean {statistics.mean(rewards):.3f} BTC, "
        f"coefficient of variation {variation:.2f}"
    )
assert statistics.pstdev(fees_only) / statistics.mean(fees_only) > 0.9

fig, (left, middle, right) = plt.subplots(1, 3, figsize=(14, 4.2))
left.hist(subsidized, bins=50, color="#2563eb", alpha=0.7, label="subsidy 3.125 + fees")
left.hist(fees_only, bins=50, color="#dc2626", alpha=0.7, label="fees only")
left.set(xlabel="block reward (BTC)", ylabel="blocks", yscale="log")
left.set_title("Fee-only rewards are as variable as block times")
left.legend()

with subsidy: mean 3.427 BTC, coefficient of variation 0.09
fees only: mean 0.302 BTC, coefficient of variation 0.99

The mining gap

costs = [0.1 * i for i in range(1, 11)]
for subsidy, color in ((0.0, "#dc2626"), (0.25, "#d97706"), (3.125, "#2563eb")):
    gaps = [bk.economics.mining_gap(subsidy, FEE_RATE, c) / 60 for c in costs]
    middle.plot(costs, gaps, "o-", color=color, label=f"subsidy {subsidy}")
middle.set(xlabel="mining cost per expected block (BTC)", ylabel="idle minutes after a block")
middle.set_title("Miners pause until fees accumulate")
middle.legend()
assert bk.economics.mining_gap(0.0, FEE_RATE, 0.25) == 500  # Most of a block interval.
assert bk.economics.mining_gap(3.125, FEE_RATE, 1.0) == 0

Undercutting a wealthy block

A 20% miner sees a tip that claimed tip BTC of fees while 0.05 BTC wait in the mempool. It forks the tip, keeping 80% of all the fees.

tips = [0.05 * i for i in range(1, 41)]
for subsidy, color in ((0.0, "#dc2626"), (3.125, "#2563eb")):
    gains = []
    for tip in tips:
        payoff = bk.economics.undercutting_payoffs(0.2, tip, 0.05, kept=0.8, subsidy=subsidy)
        gains.append(payoff["undercut"] / payoff["honest"] - 1)
    right.plot(tips, gains, color=color, label=f"subsidy {subsidy}")
right.axhline(0, color="black", linestyle=":")
right.set(xlabel="fees claimed by the tip (BTC)", ylabel="relative gain from undercutting")
right.set_title("Without a subsidy, forking pays handsomely")
right.legend()
fig.tight_layout()

whale = bk.economics.undercutting_payoffs(0.2, 1.0, 0.05, kept=0.8)
print("fee-only undercut of a 1 BTC block:", {k: round(v, 3) for k, v in whale.items()})
assert whale["undercut"] > 10 * whale["honest"]

plt.show()

fee-only undercut of a 1 BTC block: {'honest': 0.01, 'undercut': 0.168}

Exercise

Undercutting works only if other miners prefer the undercutting branch. For a tip of 1 BTC and 0.05 BTC pending, what is the largest fraction kept for which they do? What happens to the next miner’s incentive if everyone undercuts?