One-dimensional vs. Multi-dimensional Pricing in Blockchain Protocols

Aggelos Kiayias; Elias Koutsoupias; Giorgos Panagiotakos; Kyriaki Zioga

REVIEW 5 major objections 6 minor 13 references

One-dimensional vs. Multi-dimensional Pricing in Blockchain Protocols

T0 review · 5 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that multi-dimensional transaction pricing beats one-dimensional pricing for welfare when prices are stable, but stabilizes more slowly and is computationally harder after shocks.

desk verdict A timely fee-market comparison with one solid hardness result, but the two headline theorems have load-bearing proof errors, so the central trade-off is not established. read the letter →

arxiv 2506.13271 v1 pith:V3EH6FWL submitted 2025-06-16 cs.GT

Aggelos Kiayias , Elias Koutsoupias , Giorgos Panagiotakos , Kyriaki Zioga This is my paper · ORCID

classification cs.GT

keywords blockchainfeestransactionfeemechanismsmulti-dimensionalpricingEIP-1559welfarestabilizationtimemultidimensionalknapsackresource

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Blockchains today mostly price transactions through a single gas fee, even though transactions consume several distinct resources. This paper compares that one-dimensional scheme with multi-dimensional pricing, which charges each resource separately. Under stable conditions, with truthfully reported values and identical mempools, the paper proves the multi-dimensional mechanism generates strictly higher welfare. Outside the stable state, however, the one-dimensional mechanism converges to equilibrium faster and keeps revenue maximization easy for block producers, while the multi-dimensional problem becomes computationally intractable under congestion. The paper identifies a trade-off: multi-dimensional pricing is better in equilibrium but pays a transition cost.

What carries the argument

The central objects are the two mechanisms: $M_1$, the EIP-1559-style scheme with a single gas base fee, and $M_m$, the scheme with an independent base fee per resource. The welfare proof works by partitioning the mempool into the set $S$ included by both mechanisms, the set $B_m\setminus S$ included only by $M_m$, and the set $B_1\setminus S$ included only by $M_1$, then bounding each side's value by the multi-dimensional base-fee payments. The stabilization-time result models each resource's stabilization time $Z_i$ as independent and uses the fact that the overall stabilization time is at least the maximum of the individual times. The congestion result reduces revenue maximization to multidimensional knapsack by choosing transaction values so that tips, under a monotone and invertible tipping function, reproduce arbitrary knapsack values.

What would settle it

Examine a two-resource mempool with stable base fees (1,1), targets (1,1), and transactions A=(1,0) with value 3, B=(0,1) with value 2.5, and C=(0.5,0.5) with value 2. Multi-dimensional pricing selects A and B and excludes C even though C's value exceeds its base fee, so the assumption that all fee-paying transactions are included already fails; comparing the same mempool under a stable one-dimensional mechanism would settle whether the welfare claim itself survives discrete capacity constraints.

Watch

Extended reading notes

Core claim

The paper's central claim is a welfare comparison between the one-dimensional mechanism $M_1$ and the multi-dimensional mechanism $M_m$. Theorem 1 states that, at stable prices, with the same mempool, truthful users, and the safety assumption, the welfare of a block produced by $M_m$ is always larger than that of a block produced by $M_1$. The proof splits transactions into those included only by $M_m$, which it lower-bounds by the sum of the base fees they pay, and those included only by $M_1$, which it upper-bounds by the same fee expression, then uses the safety assumption to show the resource consumption of the latter set is dominated. Theorem 2 quantifies the transient cost: the expected stabilization time of $M_m$ grows with the number of dimensions, with a lower bound that can approach roughly three times that of $M_1$ for heavy-tailed shocks. Theorem 4 shows that during congestion, the revenue maximization problem under $M_m$ is at least as hard as multidimensional knapsack, so it admits no fast approximation under the Gap-ETH assumption, whereas the one-dimensional version is solvable in linear time.

Load-bearing premise

The welfare theorem assumes that in a stable state, every transaction that is willing to pay the base fee actually gets into the block; if a block fills up before some fee-paying transactions are admitted, the proof's key bound no longer follows.

Editorial extensions

If this is right

Under stable prices, switching from one-dimensional to multi-dimensional fee setting raises the total welfare of each block, assuming the stable-state conditions hold.
After a demand shock, the expected time for all prices to stabilize in a multi-dimensional system grows with the number of priced dimensions, and for heavy-tailed shocks can be about three times the one-dimensional stabilization time.
During congestion, a rational block producer faces a revenue-maximization problem that is computationally intractable in multiple dimensions, while the one-dimensional version remains linear-time solvable.
Two proposed mitigations—slowly changing gas weights and a small number of synthetic dimensions—aim to keep transition costs low while capturing some of the steady-state welfare gains.
The welfare gap between the two mechanisms cannot be bounded in general, since a single transaction could consume all resource targets and be stable only under multi-dimensional pricing.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The welfare theorem implicitly assumes a continuous market in which every transaction whose value exceeds its base fee can be admitted; in a discrete mempool with a full block, some fee-paying transactions are necessarily excluded, so the theorem's domain is narrower than a literal reading.
The stabilization-time bound suggests that even a small number of dimensions materially slows convergence for heavy-tailed demand shocks, so hybrid mechanisms with two or three synthetic dimensions face a measurable trade-off between welfare and transition speed.
The welfare comparison counts only the value of included transactions, not the welfare of excluded users or the fees burned, so a fuller accounting of user surplus and block-producer revenue might alter the ranking in some regimes.
The computational hardness result indicates that practical multi-dimensional fee markets would need off-chain or heuristic price discovery, since exact revenue maximization becomes infeasible as the number of resources grows.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Hi [Colleague],

You asked about arXiv:2506.13271. Short version: the paper is worth a look if you care about fee market design, but the two headline claims don't hold up as written. The computational hardness result (Theorem 4) is a legitimate reduction from multidimensional knapsack to revenue-maximizing tip selection under monotone tipping strategies. That part is clean and is a real contribution, though concurrent work by Lavee et al. already flags similar computational difficulties.

The model is well laid out, and the paper does a good job positioning itself relative to Roughgarden's TFM framework, EIP-1559, and EIP-4844. The proposed high-level fixes (slowly changing gas weights, limited synthetic dimensions) are sensible directions, even if they are not formalized.

The problems are in Theorems 1 and 2. Theorem 1's proof asserts that at stability M_m includes every transaction that can pay its base fee. That is not entailed by the model. Stability only pins down aggregate consumption at the target; it does not say the block is empty enough to admit all positive-tip transactions. A block can be at target and still exclude a transaction with value above the base fee because higher-tip transactions fill it. Without an explicit market-clearing assumption, the bound in inequality (8) is unjustified. This is load-bearing: the whole welfare comparison rests on it.

Theorem 2 is worse. The proof uses E[Z_m | Z_m < k] >= E[Z_{m-1}], which is the wrong direction: conditioning on the maximum being below k cannot raise the conditional expectation. The probability bound P[Z_m < k] >= 1 - p - delta also looks suspect; with independence, the probability that all Z_i are below k is a product, which is generally smaller, not larger, than 1-p. So the stabilization-time ratio bound is unsupported.

There's also a minor remark claiming one-dimensional knapsack can be solved in linear time; the cited reference gives an FPTAS, not an exact linear algorithm, and 1D knapsack is NP-hard.

Net: the paper asks a live question, the model is clean, and the hardness result stands. But as it stands, the main advertised trade-off between equilibrium welfare and transient convergence is not proven. I'd send this to peer review only if the venue expects major revision; the referees will need to see Theorems 1 and 2 either repaired with additional assumptions or substantially reframed. If the authors can add a market-clearing assumption for Theorem 1 and fix the probabilistic argument in Theorem 2, there is a useful paper here. Otherwise this becomes a mostly-negative result with a solid hardness core.

Recommendation: send to referees, not desk reject, but be prepared for a reject-and-resubmit outcome.

Referee Report

5 major / 6 minor

Summary. The paper studies transaction fee mechanisms for blockchains, comparing a one-dimensional mechanism M_1 (like EIP-1559, pricing a single synthetic gas) with a multi-dimensional mechanism M_m that prices each of m resources independently. The main claims are: (i) at stable prices, M_m generates strictly higher welfare than M_1 (Theorem 1); (ii) after a demand shock, M_1 stabilizes faster, quantified by a lower bound on the ratio of expected stabilization times (Theorem 2); and (iii) during congestion, the block producer's revenue-maximization problem is as hard as multidimensional knapsack for M_m, while remaining tractable for M_1 (Theorem 4). The paper also proposes two practical alternatives: slowly-changing gas weights and a small number of synthetic dimensions. The proofs of Theorems 1 and 2 contain technical gaps that need to be addressed before the paper's central trade-off can be accepted.

Significance. The question addressed is timely and relevant, especially with the introduction of multiple resource types in systems such as EIP-4844. If the results were correct, the welfare-superiority of multi-dimensional pricing at equilibrium, combined with a formal stabilization-time separation and a fine-grained hardness result, would be a valuable design insight for blockchain fee markets. The hardness reduction in Theorem 4 appears sound and is a concrete contribution that can stand on its own. However, the two analytical theorems that carry the main narrative—welfare superiority and faster stabilization—have serious proof issues. These issues are not merely cosmetic; they concern load-bearing steps. The paper would need a substantive revision to make the central claims rigorous, including added assumptions and corrected derivations for Theorems 1 and 2.

major comments (5)

[Section 3, proof of Theorem 1, Eq. (8)] The proof asserts that 'all transactions that can pay the base fees are included by M_m, as the system is assumed to be in stability, and thus there is available space to include all transactions paying the required fee.' This does not follow from the model. Stability only implies that the aggregate resource consumption of the block equals the target vector T_i; the allocation rule maximizes tips subject to upper bounds G_i, so a positive-tip transaction can be excluded if the block is already filled with higher-tip transactions or if its consumption vector exceeds the residual capacity in some dimension. Consequently, the strict upper bound v_j < Σ_i r_i c_i for every j ∈ B_1 \ S is not justified as written. A repair is available: from inequality (7), every transaction in B_1 \ S has per-resource consumption at most T'_i ≤ T_i, so adding it to B_m would respect the bounds G_i = 2T_i. The authors should use this fact to argue that if v_j ≥ Σ_i r_i c_i, then B_m was not tip-maximizing. Without either this argument or an explicit additional assumption, the main welfare theorem is not proven.
[Section 4, proof of Theorem 2] The step 'E[Z_m | Z_m < k] is greater than or equal to E[Z_{m-1}]' is false in general. Conditioning on the event that the maximum of m variables is below k truncates all variables, and the conditional expectation of the maximum can be strictly below the unconditional expectation of Z_{m-1}. For example, if Z_i takes value 0 with probability 1/2 and value 10 with probability 1/2, then for k = 6 the event Z_m < k forces all Z_i = 0, so E[Z_m | Z_m < k] = 0 while E[Z_{m-1}] = 5. The recursive inequality derived from this step is therefore not established. The authors should replace this step with a valid argument; note that the trivial bound E[Z_m] ≥ k P(Z_m ≥ k) already yields a related lower bound, but it does not directly give the displayed formula.
[Section 4, before Theorem 2] The proof claims that Assumption 2 implies the Z_i are mutually independent. This does not follow. Assumption 2 only states that each D_i depends on the corresponding base fee r_i; it does not rule out common factors in the mempool distribution that induce correlation among the D_i and hence among the Z_i. If the Z_i are perfectly correlated (e.g., a common shock affects all resources identically), then Z_m = Z and Ratio_m = 1, which contradicts the claimed lower bound for m > 1 and c > 0. An explicit independence assumption on the Z_i (or on the underlying demand shocks) must be added, or the theorem must be reformulated to handle correlation.
[Section 4, Theorem 2 statement vs. proof] The theorem statement includes a term '-O(δ/p)' in the lower bound, but the proof concludes with the expression (1+c)(p−δ)(1−(1−p−δ)^{m−1})/(p+δ). These two expressions are not algebraically identical, and the O(δ/p) term is not defined or derived. The authors should either prove the stated bound or align the theorem statement with the actual derivation.
[Section 6, Lemma 1] Lemma 1 states that the maximum total consumption of resource k in M_1 is min_i{w_i G_i}/w_k. This is incorrect. The gas bound G is a fixed protocol parameter, not a variable to be maximized over safety limits. A block can devote its entire gas budget to resource k, giving maximum consumption G/w_k, and this is consistent with the safety constraints because G ≤ w_k G_k. The proof incorrectly treats the maximum possible value of G as min_i{w_i G_i}. This error does not affect the main theorems, but it undermines the discussion of the proposed slowly-changing-weights mechanism.

minor comments (6)

[Section 5.2, proof of Theorem 4] The reduction constructs an RM instance but does not specify the price vector r. For the reduction to be well-defined, the authors should set r to a fixed vector (for example, r_i = 0 for all i) and state this explicitly.
[Section 2.1, Definition 1] The phrase 'mempool in k' appears to be a typo; it should read 'mempool in_k' or 'the mempool at round k'.
[Section 3, proof of Theorem 1] The sentence 'for at most one resource k the total consumption can be equal to T_k' is correct but terse; the argument can be made clearer by explicitly noting that if two resources hit their targets, the weighted gas sum would exceed the single-dimensional target T.
[Section 4, Figure 1] The four panels in Figure 1 have no axis labels and do not explain how the 'best-case' assumption for M_m is implemented. A reader cannot reproduce the plots from the information given.
[Section 1, Introduction] The reference to EIP-4844 as an example of multidimensional pricing is somewhat imprecise; EIP-4844 introduces a separate blob gas market, which is a step toward multidimensionality but is not a full multidimensional pricing mechanism. The relationship could be clarified.
[Section 3, footnote 3] The claim that truthful bidding is a symmetric ex-post Nash equilibrium for M_m is asserted without proof or citation. Since M_m differs from M_1 in the payment rule, a brief justification or reference is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: all three main results are derived from explicit model assumptions and external hardness theorems; the gap in Theorem 1 is an unproven premise, not a self-referential derivation.

full rationale

No load-bearing circular step is present. Theorem 1 is a formal welfare comparison based on the defined allocation, payment, and burning rules of M_1 and M_m; the proof does contain an unsupported assertion, namely "all transactions that can pay the base fees are included by M_m, as the system is assumed to be in stability," which does not follow from stability or from tip-maximization. However, this is a proof gap and a correctness concern, not an instance of the conclusion being equivalent to an input by construction: the paper does not define stability as 'all value-positive transactions are included,' and the assertion is not presented as a fitted parameter later renamed a prediction. Theorem 2 rests on explicitly stated Assumptions 2 and 3, and the expectation-ratio bound is derived from them through standard total-expectation reasoning rather than by assuming the theorem's conclusion. Theorem 4 is a reduction from RM to the known Gap-ETH hardness of multidimensional knapsack, citing Doron-Arad et al.; the reduction and the scaling claim are standard and do not import the result from the authors' own prior work. The only self-referential elements are ordinary citations to the authors' related work on multidimensional fee markets, and none of those citations carries the load of the proofs. Accordingly, the skeptical objection to Theorem 1 should be weighed as a possible proof gap, but it does not make the paper's derivation circular.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper introduces no fitted parameters. It relies on standard mechanism definitions from Roughgarden [13], EIP-1559 price update formulas, and several stated and unstated modeling assumptions. The most consequential are the unstated 'all fee-paying transactions are included' assertion in Theorem 1 and Assumptions 2 and 3 in Theorem 2.

assumptions (7)

domain assumption Stability implies block consumption equals target: for M1, sum gas = T; for Mm, each resource consumption = T_i (from price update equations 1 and 3).
Used in Theorem 1 proof to compute residual targets T'_i; also in Theorem 2 setup.
domain assumption Safety assumption: maximum consumption of resource j in a block is at most G_j (Assumption 1).
Used in Theorem 1 proof inequality 6 and in constraints.
domain assumption Users are truthful: bid equals value (stated before Theorem 1).
Needed to relate welfare to base fees in Theorem 1 proof.
ad hoc to paper In the stable state, M_m includes all transactions that can pay the base fees (asserted in Theorem 1 proof, not a stated assumption).
This is needed for the upper bound on B1\S welfare; it does not follow from stability or tip maximization when the block is exactly at target.
ad hoc to paper Assumption 2: total consumption D_i depends only on base fee r_i.
Decouples resources so that Z_i are independent in Theorem 2; not implied by the M_m allocation rule.
ad hoc to paper Assumption 3: statistical distance between Z and each Z_i is at most delta.
Used to bound P[Z_i >= k] by p +- delta; its validity for EIP-1559-like price updates is asserted, not derived.
domain assumption Gap-ETH hardness assumption for multidimensional knapsack (Theorem 3, cited from [9]).
Underlies Theorem 4 hardness lower bound.

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Cite this review

Pith. "Pith review of One-dimensional vs. Multi-dimensional Pricing in Blockchain Protocols." pith.science (2026). https://pith.science/paper/V3EH6FWL

@misc{pith2026250613271,
  author       = {Pith},
  title        = {Pith review of: One-dimensional vs. Multi-dimensional Pricing in Blockchain Protocols},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V3EH6FWL}},
  note         = {Machine review of arXiv:2506.13271}
}

read the original abstract

Blockchain transactions consume diverse resources, foremost among them storage, but also computation, communication, and others. Efficiently charging for these resources is crucial for effective system resource allocation and long-term economic viability. The prevailing approach, one-dimensional pricing, sets a single price for a linear combination of resources. However, this often leads to under-utilization when resource capacities are limited. Multi-dimensional pricing, which independently prices each resource, offers an alternative but presents challenges in price discovery. This work focuses on the welfare achieved by these two schemes. We prove that multi-dimensional pricing is superior under stable blockchain conditions. Conversely, we show that one-dimensional pricing outperforms its multi-dimensional counterpart in transient states, exhibiting faster convergence and greater computational tractability. These results highlight a critical trade-off: while multi-dimensional pricing offers efficiency gains at equilibrium, its implementation incurs costs associated with system transitions. Our findings underscore the necessity for a deeper understanding of these transient effects before widespread adoption. Finally, we propose mechanisms that aim to mitigate some of these issues, paving the way for future research.

Figures

Figures reproduced from arXiv: 2506.13271 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 11 canonical work pages

[1]

arXiv preprint arXiv:2402.08661 (2024)

Angeris, G., Diamandis, T., Moallemi, C.: Multidimensional blockchain fees are (essentially) optimal. arXiv preprint arXiv:2402.08661 (2024)

arXiv 2024
[2]

Buterin, V.: Multidimensional gas pricing (2024),https://vitalik.eth.limo/general/2024/05/09/ multidim.html

work page 2024
[3]

Buterin, V., Conner, E., Dudley, R., Slipper, M., Norden, I.: Ethereum improvement proposal 1559: Fee market change for eth 1.0 chain (2019),https://github.com/ethereum/EIPs/blob/master/EIPS/ eip-1559.md

work page 2019
[4]

Buterin, V., Feist, D., Loerakker, D., Kadianakis, G., Garnett, M., Taiwo, M., Dietrichs, A.: Multidi- mensional gas pricing (2024),https://github.com/ethereum/EIPs/blob/master/EIPS/eip-4844.md

work page 2024
[5]

European Journal of Operational Research123(2), 333–345 (2000)

Caprara, A., Kellerer, H., Pferschy, U., Pisinger, D.: Approximation algorithms for knapsack problems with cardinality constraints. European Journal of Operational Research123(2), 333–345 (2000)

work page 2000
[6]

In: Proceedings of the 56th Annual ACM Symposium on Theory of Computing

Chen, L., Lian, J., Mao, Y., Zhang, G.: A nearly quadratic-time fptas for knapsack. In: Proceedings of the 56th Annual ACM Symposium on Theory of Computing. pp. 283–294 (2024)

work page 2024
[7]

resource space

Crapis, D., Moallemi, C.C., Wang, S.: Optimal dynamic fees for blockchain resources. In: International Conference on Financial Cryptography and Data Security. pp. 271–291. Springer (2024) 9Interestingly, in concurrent work [11], limiting the number of dimensions is also explored as a possible solution to a related problem, i.e., that of limited “resource ...

work page 2024
[8]

In: 5th Conference on Advances in Financial Technologies (AFT 2023)

Diamandis, T., Evans, A., Chitra, T., Angeris, G.: Designing multidimensional blockchain fee markets. In: 5th Conference on Advances in Financial Technologies (AFT 2023). pp. 4–1. Schloss Dagstuhl– Leibniz-Zentrum f¨ ur Informatik (2023)

work page 2023

Show all 13 references

[9]

arXiv preprint arXiv:2407.10146 (2024)

Doron-Arad, I., Kulik, A., Manurangsi, P.: Fine grained lower bounds for multidimensional knapsack. arXiv preprint arXiv:2407.10146 (2024)

2024 arXiv
[10]

arXiv preprint arXiv:2502.12966 (2025)

Heimbach, L., Milionis, J.: The early days of the ethereum blob fee market and lessons learnt. arXiv preprint arXiv:2502.12966 (2025)

2025 arXiv
[11]

Lavee, N., Nisan, N., Pai, M., Resnick, M.: Does your blockchain need multidimensional transaction fees? arXiv preprint arXiv:2504.15438 (2025)

2025 arXiv
[12]

Rebuffo, G.: Cap on the maximum gas usage of a transaction (2024),https://github.com/ethereum/ EIPs/blob/master/EIPS/eip-7825.md

2024
[13]

Journal of the ACM71(4), 1–25 (2024) 15

Roughgarden, T.: Transaction fee mechanism design. Journal of the ACM71(4), 1–25 (2024) 15

2024

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

arXiv preprint arXiv:2402.08661 (2024)

Angeris, G., Diamandis, T., Moallemi, C.: Multidimensional blockchain fees are (essentially) optimal. arXiv preprint arXiv:2402.08661 (2024)

arXiv 2024

[2] [2]

Buterin, V.: Multidimensional gas pricing (2024),https://vitalik.eth.limo/general/2024/05/09/ multidim.html

work page 2024

[3] [3]

Buterin, V., Conner, E., Dudley, R., Slipper, M., Norden, I.: Ethereum improvement proposal 1559: Fee market change for eth 1.0 chain (2019),https://github.com/ethereum/EIPs/blob/master/EIPS/ eip-1559.md

work page 2019

[4] [4]

Buterin, V., Feist, D., Loerakker, D., Kadianakis, G., Garnett, M., Taiwo, M., Dietrichs, A.: Multidi- mensional gas pricing (2024),https://github.com/ethereum/EIPs/blob/master/EIPS/eip-4844.md

work page 2024

[5] [5]

European Journal of Operational Research123(2), 333–345 (2000)

Caprara, A., Kellerer, H., Pferschy, U., Pisinger, D.: Approximation algorithms for knapsack problems with cardinality constraints. European Journal of Operational Research123(2), 333–345 (2000)

work page 2000

[6] [6]

In: Proceedings of the 56th Annual ACM Symposium on Theory of Computing

Chen, L., Lian, J., Mao, Y., Zhang, G.: A nearly quadratic-time fptas for knapsack. In: Proceedings of the 56th Annual ACM Symposium on Theory of Computing. pp. 283–294 (2024)

work page 2024

[7] [7]

resource space

Crapis, D., Moallemi, C.C., Wang, S.: Optimal dynamic fees for blockchain resources. In: International Conference on Financial Cryptography and Data Security. pp. 271–291. Springer (2024) 9Interestingly, in concurrent work [11], limiting the number of dimensions is also explored as a possible solution to a related problem, i.e., that of limited “resource ...

work page 2024

[8] [8]

In: 5th Conference on Advances in Financial Technologies (AFT 2023)

Diamandis, T., Evans, A., Chitra, T., Angeris, G.: Designing multidimensional blockchain fee markets. In: 5th Conference on Advances in Financial Technologies (AFT 2023). pp. 4–1. Schloss Dagstuhl– Leibniz-Zentrum f¨ ur Informatik (2023)

work page 2023