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"

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?

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?