Leader election process

• We consider $N$ nodes participating in the leader election process.
• A node $i$ has (relative) stake $\alpha_i\in(0,1)$ such that $\sum_{i=1}^N\alpha_i=1$.
• For node $i$ the probability of winning is given by the lottery function $\phi(\alpha_i)\in[0,1]$.
• We assume that all $N$ nodes participate in the leader election process at specific times. We assume (without loss of generality) that these times are labelled by the set $[T]$.
• We model outcome of election process for node i at time t by the binary variable $s_i(t)\in\{0,1\}$, where $0$ and $1$ corresponds to, respectively, loosing and wining.
• For the outcome of leader election process $\mathbf{s}(t)=(s_1(t),\ldots,s_N(t))$ at the time $t$, the probability of outcomes $\left(\mathbf{s}(1),\ldots,\mathbf{s}(T)\right)$ at times $t\in[T]$ is given by

$$ \mathrm{P}[\mathbf{s}(1),\ldots,\mathbf{s}(T)\vert \phi(\alpha)]=\prod_{t=1}^T\prod_{i=1}^N \left[\phi(\alpha_i)\,\delta_{1;s_i(t)}+\{1-\phi(\alpha_i)\}\,\delta_{0;s_i(t)}\right], $$

where we defined the vector $\phi(\alpha)=(\phi(\alpha_1),\ldots,\phi(\alpha_N))$.

• The probability of the event $\sum_{i=1}^N s_i(t)\geq1$, i.e. there is at least one winner at time $t$, is given by

$$ 1-\prod_{i=1}^N \left[1-\phi(\alpha_i)\right]. $$

• We note that above is the average of the density $\sum_{t=1}^T \mathbf{1}\left[\sum_{i=1}^N s_i(t)\geq1\right]/T$. Here the sum $\sum_{t=1}^T \mathbf{1}\left[\sum_{i=1}^N s_i(t)\geq1\right]$ counts number of elections with at least one winner.
• Let us assume that the lottery function $\phi(x)$ is such that

$$ \boxed{f=1-\prod_{i=1}^N \left[1-\phi(\alpha_i)\right]}, $$

i.e. we fix the average number of elections with at least one winner to $fT$.

• We note that above is equivalent to

$$ 1-f=\prod_{i=1}^N \left[1-\phi(\alpha_i)\right] $$

• Furthermore, we can write $\phi(x)=1-\tilde{\phi}(x)$ and using the latter in above gives us the equality

$$ 1-f=\prod_{i=1}^N\tilde{\phi}(\alpha_i) $$

• Now we know that $\sum_{i=1}^N\alpha_i=1$ and above suggests that the product $\prod_{i=1}^N\tilde{\phi}(\alpha_i)$ is independent of $\alpha_i$ variables. The latter suggests to try $\tilde{\phi}(x) =\mathrm{e}^{Bx}$ which gives us