The Economic Security of Exponential EIP-1559

1 Introduction

• Adjustment-rate calibration balances rapid correction against predictable prices: node-capacity constraints favor aggressive responses, while long-term state-growth limits allow gradual responses and more flexible calibration.Setting the target below the desired long-run usage bound creates slack and can allow a less aggressive adjustment rate.Source: §1 · p. 2

1.1 Our Contributions

• For each candidate target below U, the method chooses the least aggressive adjustment rate whose minimum fee revenue reaches B, ensuring average usage at least U over n blocks incurs at least B fees.The designer treats B as an amount of fee revenue considered infeasible to spend over the window.Source: §1.1 · p. 3
• For fixed parameters, the authors seek the minimum-fee distribution subject to average usage at least U, proving a simply structured optimum or approximation that can be found efficiently.The search is over a window of n blocks.Source: §1.1 · p. 3
• The single-target pure exponential mechanism has an exactly identified optimum; for mechanisms with the same maximum price increase, the two-target strategy approximates minimum revenue within 10%, and Robinhood Chain’s parameters yield 6.3%.The two-target approximation factor is at least twice as good as the maximum relative price increase in one block transition, τ.Source: §1.1 · pp. 3–4

1.2 Structure of the Paper

• The paper introduces the mechanisms in Section 2, proves exact and approximate distribution results in Sections 3–4, computes revenues in Section 5, and discusses parameter setting and tradeoffs in Section 6.Source: §1.2 · p. 4

1.3 Related Work

• Earlier work on adjustment rates assumes gas-demand elasticity, whereas this paper takes an adversarial approach to derive worst-case security bounds.Source: §1.3 · p. 4

2 Preliminaries

• The objective is to find a revenue-minimizing feasible gas-usage vector over n rounds, with each round’s usage bounded by L and average usage fixed at U.The setup assumes nU is an integer.Source: §2 · pp. 4–5

2.1 The Single Target Mechanism

• The single-target mechanism is parameterized by a gas target, adjustment rate, minimum price, and starting price; its backlog tracks excess usage relative to the target and is floored at zero.The minimum price applies exactly when the backlog is zero.Source: §2.1 · p. 5

2.2 The Two Target Mechanism

• The two-target mechanism maintains lower- and higher-target backlogs, each updated by gas usage above its corresponding target, to determine the round’s price.The mechanism also has two adjustment rates and a minimum price; the price increase per round is bounded and reaches its maximum when the round uses L gas.Source: §2.2 · p. 6

The Single Target Mechanism

• An optimal strategy is non-decreasing, with at most one round of usage strictly between 0 and T and at most one strictly between T and L.These properties hold for at least one optimal strategy, not necessarily every optimal strategy.Source: §3 · p. 7
• When usage first reaches T, the price is at its minimum or all later rounds use L; when the price first reaches its minimum, usage is at least T or only one round exceeds T.Informally, an optimal strategy can wait with zero usage until the price reaches its minimum, then use T as long as possible before raising usage toward L, with at most one transition round at either level.Source: §3 · p. 7
• Shifting gas from one round to the next can preserve or lower revenue in the stated cases: usage at or below T, or usage above T with the shifted total bounded by L or exceeding L.The propositions establish non-increasing revenue for these local transformations, which the proof uses to obtain the optimal strategy's structure.Source: §3 · pp. 8–9
• The transformations used to prove the theorem do not increase the ending price pn+1.Only the transformation used for the fifth theorem property might strictly decrease it; the others leave it unchanged.Source: §3 · p. 11
• When pstart = pmin and U ≥ T, an optimal strategy uses T as its baseline, includes a residual usage r, and uses L for q rounds when (U − T)n = (L − T)q + r.The stated conditions include q,r ≥ 0 and 0 ≤ r < L − T.Source: §3 · p. 11

4 The Two Target Mechanism

• In an example, the only optimal strategy is (10, 0, 8, 10), with ≈50.82 revenue, while the optimal non-decreasing strategy earns ≈51.37.This demonstrates that an optimal two-target strategy need not be monotonic.Source: §4 · pp. 11–12
• When L ≤ Tℓ + Th, a simply structured strategy can be identified directly; the section instead studies approximation for the general regime.The directly solvable regime is stated to differ from the one encountered in practice.Source: §4 · p. 12
• The strategy family GS has successive blocks at Tℓ, Th, and L, with at most one transition round at each threshold.The block lengths sum to n; the transition amounts satisfy 0 ≤ eℓ < Th − Tℓ and 0 ≤ eh < L − Th.Source: §4 · p. 12
• The constructed strategy delays gas usage, caps it at Th except in final rounds, and uses up to L later to fit the remaining gas.The transformation preserves feasibility and the total gas used.Source: §4 · pp. 12–13
• With practical parameters, the theorem’s constant is c ≈93.7%, so the optimal revenue is within 6.3% of the optimal simply structured strategy’s revenue.The result assumes empty initial backlogs and U ≥ Tℓ; the example uses L = 192 Mgas/s, Tℓ = 10 Mgas/s, Aℓ = 86400, Th = 60 Mgas/s, and Ah = 12.Source: §4 · pp. 12, 17

5.1 Single Target

• With an initially empty backlog and U ≥ T, the optimal single-target strategy keeps price at p_min for the first n − q rounds, and its revenue is summed geometrically.The passage describes the subsequent price progression and says the geometric sum simplifies to a closed form.Source: §5.1 · p. 18

5.2 Two Targets

• For two targets, fixing e_h and i_L determines the remaining variables, so trying n × (L − T_h) combinations finds the minimum among the structured strategies.The total-gas constraint determines e_l and i_h, after which i_l follows from i_l + i_h + i_L = n.Source: §5.2 · p. 18
• 0.1%: Searching rounded values of e_h gives a minimum revenue at most 0.1% above the minimum over simply-structured strategies.Rounding e_h upward and reducing e_l bounds the revenue increase; the passage gives b = 0.001 min(T_h, L − T_h) as an example grid spacing.Source: §5.2 · pp. 18–19
• In the running example, the upper bound i_L ≤ 58 sharply reduces the values checked by ruling out configurations whose L-round revenue exceeds a baseline.The baseline is the revenue for i_L = 0 and e_h = 0; the bound uses a conservative estimate that both backlogs are zero when the L-rounds begin.Source: §5.2 · p. 19

6 Setting Pricing Parameters

• The design fixes gas limit L and treats pmin, Th, and Ah as determined by other requirements, such as controlling usage over seconds to minutes so nodes can keep up.Th and Ah may be set directly or via Th and maximum per-round price increase τ; Ah can then be inferred once the other parameters are set from L−Tℓ.Source: §6 · p. 19
• For a long-term state-growth constraint, require average gas usage to stay at or below U over n seconds, and set Tℓ and Aℓ so usage above U requires adversarial budget at least B.The constraint is intended to limit usage over days to weeks, including to limit state growth.Source: §6 · pp. 19–20
• For each low target Tℓ in [0, U], binary search finds Aℓ satisfying Rev(Tℓ, Aℓ) ≥ B, yielding a design curve that trades higher throughput for more aggressive price increases.A larger Tℓ permits higher long-term average throughput but requires a smaller Aℓ, making the mechanism produce more aggressive price increases.Source: §6 · p. 20

7 Conclusion

• The framework guarantees a lower bound on total fees for sustaining given average gas usage over a fixed window, without assuming anything about gas demand.This lets mechanism designers make violating a usage constraint prohibitively expensive.Source: §7 · p. 20
• For the single-target mechanism, the authors characterize an exact revenue-minimizing gas distribution; for two targets with initially empty backlogs, they give a structured distribution that provably approximates the minimum.The approximation guarantee is controlled by the maximum per-round relative price increase, making fee bounds efficiently computable for parameter selection.Source: §7 · p. 20
• Moving the target farther below desired usage creates slack, allowing a less aggressive price-adjustment rate while preserving the economic security guarantee.In the two-target setting, this supports calibrating long-term constraints such as state growth alongside short-term requirements that nodes can keep up with the chain.Source: §7 · p. 21
• The framework provides a theoretical basis for parameter selection in existing mechanisms and for designing future variants.Source: §7 · p. 21

• When L ≤ Tℓ + Th, an optimal strategy is non-decreasing and has at most one round in each of the bands (0,Tℓ), (Tℓ,Th), and (Th,L).Source: A The Two Target Mechanism when the… · pp. 22–24
• For a fixed pricing state, M(g) and (M(g) − 1)/g are non-decreasing in gas usage g.Source: A The Two Target Mechanism when the… · p. 23
• For a strategy G with gj > Tℓ and gj + gj+1 − Tℓ ≤ Th in some round j, setting g′j+1 = gj + gj+1 − Tℓ yields Rev(G′) ≤ Rev(G).Source: A The Two Target Mechanism when the… · p. 24