Understanding Incentive Compatibility of pooled Bitcoin Mining paper
Let's assume $n=3$ and at round $t$, miner_1 reported 2 shares, miner_2 reported 3 shares, and miner_3 reported 5 shares. And we'll further assume that in rounds from 0 to $t-1$, none of them reported any shares.

$R$ is the reward function that takes the history of the shares reported by all miners and returns a n-length vector, $a$, consisting of real numbers between 0 and 1.
Where $a_i$ corresponds to the fraction of total reward earned by miner $i$. So $ \sum_i a_i =1 $. $R_i$ corresponds to the reward earned by $i^{th}$ miner. So $R$ returns $&lt;.2, .3, .5&gt;$. When $i=1$, $R_i$ returns $.2$

$b_i$ (normal font) is the total number of shares reported by miner $i$. And $\textbf{b}$ (bold font) is a vector of length $n$, where each element corresponds to the shares reported by each miner. So $\textbf{b} = &lt;2, 3, 5&gt;$. When $i=1$, $b_i = 2$. And vector at round $t$, vector $\textbf{b}$ is denoted as $\textbf{b}_t$. So $\textbf{b}_1$ corresponds to the vector of shares reported at round 1 and $b_1$ corresponds to the shares computed by miner $i$. 

Say at round $t+1$, $\textbf{b} = &lt;1, 2, 7&gt;$. So total shares reported in round $t$ and $t+1$ is component wise sum of $\textbf{b}_t$ and $\textbf{b}_{t+1}$.

Hope this helps.