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Serial Monopoly on Blockchains with Quasi-patient Users

Paolo Penna, Manvir Schneider

TL;DR

This work considers a more realistic setting with quasi-patient users, where only a fraction of pending transactions remains in the next round, and provides quantitative bounds and analytical results, showing that the bounds for $\delta=1$ are generally not tight for $\delta<1", and gives guarantees on the minimum ("admission") price for transactions.

Abstract

In the face of limited block size, miners (e.g., in Bitcoin) prioritize high-bid transactions, forming a large part of their revenue. If the block size were to expand significantly, meeting all transaction demand due to infrastructure or protocol improvements, bids could drop to zero or to a minimum fee, reducing mining incentives and potentially affecting security. To address this, Lavi et al. (2022) introduced a monopolistic pricing mechanism where miners only include transactions paying a minimum price, ensuring some revenue but resulting in an unbounded loss in welfare. Nisan (2023) expands this by modeling bidders as patient, who wait indefinitely long for lower prices, causing price fluctuations even with stable demand. In order to capture users' diminishing interest in having their transactions added to the ledger over time, we consider a more realistic setting with quasi-patient users, where only a fraction $δ\in [0,1]$ of pending transactions remains in the next round. This richer model encompasses both Lavi et al.'s impatient users ($δ=0$) and Nisan's patient users ($δ=1$) as special cases. We demonstrate that Nisan's fluctuating dynamics persist for $δ$ close to 1, while for $δ$ close to 0, the dynamics resemble the impatient case. For $δ\in (0,1)$, we establish new bounds on price dynamics, revealing unexpected effects. Unlike the fully patient case, the bounds of the dynamics for $δ<1$ depend on the demand curve and undergo a "transition phase". For some $δ$, the model mirrors the fully patient setting, and for smaller $δ' < δ$, it stabilizes at the highest monopolist price, thus collapsing to the impatient case. We provide quantitative bounds and analytical results, showing that the bounds for $δ=1$ are generally not tight for $δ<1$, and we give guarantees on the minimum ("admission") price for transactions.

Serial Monopoly on Blockchains with Quasi-patient Users

TL;DR

This work considers a more realistic setting with quasi-patient users, where only a fraction of pending transactions remains in the next round, and provides quantitative bounds and analytical results, showing that the bounds for are generally not tight for $\delta<1", and gives guarantees on the minimum ("admission") price for transactions.

Abstract

In the face of limited block size, miners (e.g., in Bitcoin) prioritize high-bid transactions, forming a large part of their revenue. If the block size were to expand significantly, meeting all transaction demand due to infrastructure or protocol improvements, bids could drop to zero or to a minimum fee, reducing mining incentives and potentially affecting security. To address this, Lavi et al. (2022) introduced a monopolistic pricing mechanism where miners only include transactions paying a minimum price, ensuring some revenue but resulting in an unbounded loss in welfare. Nisan (2023) expands this by modeling bidders as patient, who wait indefinitely long for lower prices, causing price fluctuations even with stable demand. In order to capture users' diminishing interest in having their transactions added to the ledger over time, we consider a more realistic setting with quasi-patient users, where only a fraction of pending transactions remains in the next round. This richer model encompasses both Lavi et al.'s impatient users () and Nisan's patient users () as special cases. We demonstrate that Nisan's fluctuating dynamics persist for close to 1, while for close to 0, the dynamics resemble the impatient case. For , we establish new bounds on price dynamics, revealing unexpected effects. Unlike the fully patient case, the bounds of the dynamics for depend on the demand curve and undergo a "transition phase". For some , the model mirrors the fully patient setting, and for smaller , it stabilizes at the highest monopolist price, thus collapsing to the impatient case. We provide quantitative bounds and analytical results, showing that the bounds for are generally not tight for , and we give guarantees on the minimum ("admission") price for transactions.
Paper Structure (4 sections, 1 theorem, 3 equations, 1 figure)

This paper contains 4 sections, 1 theorem, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Model

Key Result

theorem thmcountertheorem

For patient users ($\delta=1$) and for any strictly decreasing demand function $Q$ and supply $s$ the following holds: Transactions paying at least $p_{mon}$ are immediately included, and this is tight as there are infinitely steps for which paying less will delay admission to a later step.

Figures (1)

  • Figure 1: Price dynamics from Example \ref{['intro:example']} for different values of $\delta$, with 3 time steps (left) and 20 time steps (right), respectively. For $\delta$ smaller than $2\sqrt{2}-2\approx 0.828$ the prices jumps up after step $t=2$. For $\delta$ above the threshold, the price decreases after step $t=2$.

Theorems & Definitions (6)

  • remark thmcounterremark
  • remark thmcounterremark
  • definition thmcounterdefinition
  • remark thmcounterremark
  • definition thmcounterdefinition: Eventual Transaction Inclusion, (Minimum) Admission Price
  • theorem thmcountertheorem: Theorem 1 in Nisan restated