Understanding Incentive Compatibility of pooled Bitcoin Mining paper I'm trying to understand the paper Incentive Compatibility of
Bitcoin Mining Pool Reward Functions (Schrijvers, Bonneau, Doneh and Roughgarden, in Financial Cryptography and Data
Security&nbsp;&ndash; FC 2016 Workshops, BITCOIN, 2016; PDF).

In page 3 Section 2.1, they say pool operator does not know actual $\alpha_i$ mining power of player $i$. To estimate $\alpha_i$ depends on the reported shares and solutions.

A reward function $ R\colon H \mapsto [0,1]^n$ is a function from a history transcript to an allocation $ \{a_i\}_{i=0} ^ {n} $ with $ \sum_i a_i =1 $. 
I don't understand this? What allocation does the authors mean?

What I somewhat understand from next para is $ H(k) = b = (y_1(k), \cdots, y_i(k) ) $ where $y_i(k)$ is no of shares reported by player $i$ in round $k$.
This I am basing on $H$ contains for each miner $i$ the total no of shares $b
_i$ reported in that round. History transcript is given by a vector $b \in N^n $.

Also what do they mean when they say " We use vector notation for $b$, so $b_1 + b_2 $ means component wise addition of these, and  $\|b\|_1 = \sum_{i=1}^n b_i $ is the sum of components of $b$"

Could someone explain with examples or in a more concrete way ?
Also any suggestions for understanding this paper would be much appreciated. Thanks.
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.