# (WIP) Notes on Everything is a Race and Nakamoto Always Wins

[🔗 Everything is a Race and Nakamoto Always Wins 🔗](https://arxiv.org/abs/2005.10484)

## Intro

### Set Up

Adversay and honest nodes, respectively `λh` and `λa` represent each set of nodes block mining rate. The game to analyze is one where:

"The adversary grows a private chain of blocks in a race to attempt to outpace the public longest chain and thereby replacing it after a block in the public chain becomes `k`-deep"

- If `λh` > `λa`, 🥳 probability of the adversary succeeding decreases exponentially with `k`
- If `λh` < `λa`, 😱 adversary succeeds with high probability no matter size of `k`

But this model is too simple. In reality an adversary will be able to communicate faster between it's own adverserial nodes than the network delay of ∆ between honest nodes. Thus we expand by,

`λa < λgrowth(λh,∆)`

where "`λgrowth(λh, ∆)` is the growth rate of the honest chain under worst-case forking"

"In a fully decentralized setting with many hon- est nodes each having small mining power, [SZ15] calculates this to be `λgrowth = λh /(1 + λh ∆)`. If we let `β` to be the adversary fraction of power, then (1) yields the following condition:
`β< 1−β` . (2) `1+(1−β)λ∆`
Here, `λ` is the total mining rate, and `λ∆` is the number of blocks mined per network delay. `1/(λ∆)` is the block speed normalized by the network delay."