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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 – 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.

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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 $<.2, .3, .5>$. 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 $\textbf{b} = <2, 3, 5>$. 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 $t+1$, $\textbf{b} = <1, 2, 7>$. 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.

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