Owner: @David Rusu

An updated version of this document has been moved to overleaf:

https://www.overleaf.com/1481687363pbpfjjvjgpgb#63be11

Summary

Summary is we got bounds on the expected number of leaders per slot:

• Expected number of leaders per slot is written as $\mathbb{E}\left[\sum_i^N\phi_f(\alpha_i)]\right]$
• The bounds we got were:

$$ f\leq\mathbb{E}\left[\sum_i^N\phi_f(\alpha_i)]\right] \leq N\cdot\phi_f\left(\frac{1}{N}\right)\leq-\log(1-f) $$

where

• $f\in[0,1]$ is the “active slots coefficient”
• $N$ is number of nodes
• $\alpha_i$ is the relative stake of node $i$.
• $\phi_f$ is the praos lottery function: $\phi_f(\alpha)=1-(1-f)^\alpha$

Plotting the bounds as we vary $f$, we see that the tighter upper bound $N\phi_f(1/N)$ which depends on number of nodes very quickly converges to the maximal upper bound $-log(1-f)$. Therefore we can work with the upper bound and drop the $N$ parameter.