• Bitcoin (proof-of-work) blockchain is not scalable
  • Limited adoption: the fraction of users who use the blockchain for payments vanishes as the number of users increases

• Permissioned blockchain is a viable alternative
  • But not for all consensus mechanisms, e.g.:
    • simple majority voting doesn't work
    • voting scaled by crypto-currency holdings does

• Blockchain as payment infrastructure 

N users need to transact & choose:

Blockchain:

"Traditional" payment system

• unmodelled
• normalized as zero reward, zero cost?

Reward (from using blockchain?)

Fee to miners

Waiting cost:

"user impatience \(c_i\)" \(\times\) "time to confirm"

• Assumed to coordinate on the longest chain, no malicious behavior

Cost of validation technology

Number of validators

Sum of user fees

• Free entry:

• Low enough impatience \(c_i\le c^* \Rightarrow \) use blockchain
• Marginal user \(c^*\) pays the highest fee and waits for:

block's arrival rate

fork/dispute resolution time

\(\infty\)

as \(N\to\infty\)

\(c^* \to 0\)  

"limited adoption"

Key Result:

• as \(N\to\infty\), the number of validators \(V\to \infty\)

• Why do forks take \(\infty\)  to resolve?

Network delay/physical system limits 

• \(\to \) with \(V=\infty\), time to agree/communicate explodes

Lemma B.1 in Appendix B ....

Number of validators V determined by free entry:

Number of users \(N \to \infty \)

• \(\Rightarrow\) expected fees \(\to \infty\)

Comment 1: need more intuition for this:

• User i pays fee \(f_i \propto (N-1) c_i^2\)
• With \(N\to\infty\), wait times explode & only super-patient users use blockchain ....
  • \(\to \)  \(c_i \approx 0 \) 
• Fraction of blockchain users vanishes ...
• What happens to fee per user?

Comment 2: 

• Confirmation times \(\to \infty\)
  \(\Rightarrow\) fees are received with infinite delay
• Technology costs are incurred in real time!(?)
• Are validators infinitely patient? No capital constraints?
  • Is there a transversality condition?

• Where does the reward come from? 
• Seemingly should depend on the value of transactions and/or malicious users? 

This paper: 2.5 models

1. Proof-of-Work -- validators' incentives unmodelled
2. Permissioned -- coordination game among validators 
   • with majority voting
   • with crypto-currency stake-weighted voting
     • \(\to\) must introduce and value cryptocurrency

No clear connection between #1 and #2

• Why move from decentralized proof-of-work to permissioned?
• Are there decentralized alternatives?
  • E.g., require minimum crypto-stake to become a validator?

Users obtain a reward from transacting on the blockchain:

• similar reward structure for permissionless PoW vs. permissioned

Where does the utility gain from blockchain use stem from?

• E.g., with bitcoin: censorship-resistance, immutability ...
• But this disappears if blockchain is permissioned

Key differences (from the user perspective) b/n the permissioned blockchain vs. traditional payment system in the model?

• Central Bank Digital Currency?