The Math Behind Blockchain Scheduling and Transaction Fee Mechanisms

Written by escholar | Published 2025/10/14
Tech Story Tags: blockchain-economics | proof-of-stake-mining | transaction-fee-mechanisms | online-buffer-management | non-myopic-miners | blockchain-auction-design | blockchain-auction-theory | deadline-aware-blockchain

TLDRThis appendix details the mathematical proofs and performance analysis underlying a blockchain transaction fee mechanism. It compares the expected outcomes of algorithmic (ALG) versus adversarial (ADV) scheduling, defines key parameters like miner power, immediacy ratio, and discount factors, and provides a glossary of symbols and acronyms used throughout the paper. The section reinforces the theoretical foundation for non-myopic allocation rules and miner revenue optimization under Proof-of-Work (PoW) and Proof-of-Stake (PoS) models.via the TL;DR App

Abstract and 1. Introduction

1.1 Our Approach

1.2 Our Results & Roadmap

1.3 Related Work

  1. Model and Warmup and 2.1 Blockchain Model

    2.2 The Miner

    2.3 Game Model

    2.4 Warm Up: The Greedy Allocation Function

  2. The Deterministic Case and 3.1 Deterministic Upper Bound

    3.2 The Immediacy-Biased Class Of Allocation Function

  3. The Randomized Case

  4. Discussion and References

A: Missing Proofs for Sections 2, 3

We now note that a few facts that hold true for any n when x1 ≥ ℓ + ϵ:

We separate to several subcases:

B Missing Proofs for Section 4

We now compare ALG and ADV ’s performance in different steps along the adversary schedule, separating the steps before n and the last two steps.

Step i < n.

ALG expected performance:

Notice that this amortization of considering the i + 1 is only relevant for ADV, as ALG in such case necessarily has no transactions remaining to choose from at step i + 1.

where the last transition is since for any 0 ≤ λ ≤ 1, the expression

We now move on to analyze steps n, n + 1.

ALG expected performance at step n, n + 1:

As the base case, consider k = n. Then,

For the inductive step,

We thus need to show that

With this potential function, we can thus write at step i,

C Glossary

A summary of all symbols and acronyms used in the paper.

C.1 Symbols

ψ Transaction schedule function.

x Allocation function.

B Predefined maximal block-size, in bytes.

λ Miner discount factor.

ϕ Transaction fee of some transaction, in tokens.

T Miner planning horizon.

ℓ Immediacy ratio for our non-myopic allocation rule.

µ TTL of past transactions.

α Miner’s relative mining power, as a fraction. u Miner revenue.

t TTL of a transaction.

tx A transaction.

C.2 Acronyms

mempool memory pool

PoS Proof-of-Stake

PoW Proof-of-Work

QoS quality of service

TFM transaction fee mechanism

TTL time to live

Authors:

(1) Yotam Gafni, Weizmann Institute (yotam.gafni@gmail.com);

(2) Aviv Yaish, The Hebrew University, Jerusalem (aviv.yaish@mail.huji.ac.il).


This paper is available on arxiv under CC BY 4.0 DEED license.

[2] The argument of [CCFJST06] is done by showing conditions that hold for any fixed x ∈ [−1, 0], and so they hold for any fixed x ∈ [−λ, 0] as well.


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Published by HackerNoon on 2025/10/14