edits and fixes to examples

Author: YaserJazouane

pages/home/oracle-integrity-staking/examples.mdx

@@ -8,72 +8,72 @@ NOTE: All the symbols used in the examples are explained in the [Mathematical Re
 
 ## Example 1: Only Publisher Stake
 
-This example take the case where only the publisher has staked PYTH tokens.
+This example take the case one pool assigned where the publisher assigned has staked PYTH tokens, while the pool has no tokens delegated to it.
 
 $$
 \begin{aligned}
 {S^p_p} &= 100 \\
 {S^d_p} &= 0 \\
 {S^p} &= {S^p_p} + {S^d_p} = 100 + 0 = 100 \\
 {C}_p &= 500 \\
-\text{Total Amount eligible for Reward} \quad{R_p} &= min({S}_p, {C}_p) = min(500, 100) = 100 \\
-\text{Reward Rate} \quad{r} &= 10\% \\
+\text{Total Amount eligible for Reward} \quad{E_p} &= min({S}_p, {C}_p) = min(500, 100) = 100 \\
+\text{Reward Rate (Yearly)} \quad{r} &= 10\% \\
 
-\text{Total Reward} \quad{R_p} &= {r} \times {E_p} = 10\% \times 100 = 10 \\
+\text{Total Reward for one year} \quad{R_p} &= {r} \times {E_p} = 10\% \times 100 = 10 \\
 \text{Publisher Reward} \quad{R^p_p} &= {r} \times min({S^p_p}, {C}_p) = 10\% \times 100 = 10 \\
 \text{Delegator Reward} \quad{R^d_p} &= {R_p} - {R^p_p} = 10 - 10 = 0 \\
 \end{aligned}
 $$
 
-
 # Example 2: Publisher and Delegator Stake
 
-This example take the case where both the publisher and the delegator have staked PYTH tokens.
+This example take the case where the pool has stake from both the publisher and the delegator.
 
 $$
 \begin{aligned}
 {S^p_p} &= 100 \\
 {S^d_p} &= 100 \\
-{S^p} &= {S^p_p} + {S^d_p} = 100 + 100 = 200 \\
+{S_p} &= {S^p_p} + {S^d_p} = 100 + 100 = 200 \\
 {C}_p &= 500 \\
-\text{Total Amount eligible for Reward} \quad{R_p} &= min({S}_p, {C}_p) = min(500, 200) = 200 \\
-\text{Reward Rate} \quad{r} &= 10\% \\
-\text{Total Reward} \quad{R_p} &= {r} \times {E_p} = 10\% \times 200 = 20 \\
+\text{Total Amount eligible for Reward} \quad{E_p} &= min({S}_p, {C}_p) = min(500, 200) = 200 \\
+\text{Reward Rate (Yearly)} \quad{r} &= 10\% \\
+
+\text{Total Reward for one year} \quad{R_p} &= {r} \times {E_p} = 10\% \times 200 = 20 \\
 \text{Publisher Reward} \quad{R^p_p} &= {r} \times min({S^p_p}, {C}_p) = 10\% \times 100 = 10 \\
 \text{Delegator Reward} \quad{R^d_p} &= {R_p} - {R^p_p} = 20 - 10 = 10 \\
 \end{aligned}
 $$
 
 # Example 3: Publisher and Delegator Stake more than the Cap
 
-This example take the case where the combined stake of both the publisher and the delegator is more than the cap.
+This example take the case where the combined stake of both the publisher and the delegator exceeds the cap.
 
 $$
 \begin{aligned}
 {S^p_p} &= 300 \\
 {S^d_p} &= 300 \\
-{S^p} &= {S^p_p} + {S^d_p} = 300 + 300 = 600 \\
+{S_p} &= {S^p_p} + {S^d_p} = 300 + 300 = 600 \\
 {C}_p &= 500 \\
-\text{Total Amount eligible for Reward} \quad{R_p} &= min({S}_p, {C}_p) = min(500, 600) = 500 \\
-\text{Reward Rate} \quad{r} &= 10\% \\
-\text{Total Reward} \quad{R_p} &= {r} \times {E_p} = 10\% \times 500 = 50 \\
+\text{Total Amount eligible for Reward} \quad{E_p} &= min({S}_p, {C}_p) = min(500, 600) = 500 \\
+\text{Reward Rate (Yearly)} \quad{r} &= 10\% \\
+
+\text{Total Reward for one year} \quad{R_p} &= {r} \times {E_p} = 10\% \times 500 = 50 \\
 \text{Publisher Reward} \quad{R^p_p} &= {r} \times min({S^p_p}, {C}_p) = 10\% \times 300 = 30 \\
 \text{Delegator Reward} \quad{R^d_p} &= {R_p} - {R^p_p} = 50 - 30 = 20 \\
 \end{aligned}
 $$
 
-
 # Example 4: Introducing Delegator Fees
 
-This example demonstrates how delegator fees affect the reward distribution between publishers and delegators.
+This example demonstrates how the delegation fee affect the reward distribution between the publisher and the delegator.
 
 $$
 \begin{aligned}
 \quad{S^p_p} &= 200 \\
 \quad{S^d_p} &= 300 \\
-\quad{S^p} &= {S^p_p} + {S^d_p} = 200 + 300 = 500 \\
+\quad{S_p} &= {S^p_p} + {S^d_p} = 200 + 300 = 500 \\
 \quad{C}_p &= 500 \\
-\quad{R_p} &= min({S}_p, {C}_p) = min(500, 500) = 500 \\
+\quad{E_p} &= min({S}_p, {C}_p) = min(500, 500) = 500 \\
 \quad{r} &= 10\% \\
 \quad{f} &= 2\% \\
 
@@ -86,59 +86,26 @@ $$
 \end{aligned}
 $$
 
-In this example, the delegator pays a 2% fee on their rewards to the publisher. This fee is deducted from the delegator's reward and added to the publisher's reward.
-
+In the example, the delegator pays a 2\% fee on their rewards to the publisher. This fee is deducted from the delegator's reward and added to the publisher's reward.
 
 # Example 5: Slashing event on the pool
 
-This example demonstrates the impact of a slashing event on the staked PYTH tokens.
+This example demonstrates the impact of a slashing event on the staked PYTH tokens and rewards distributed to both the publisher and the delegator.
 
 $$
 \begin{aligned}
-\quad{S^p_p} &= 200 \\
-\quad{S^d_p} &= 300 \\
-\quad{S^p} &= {S^p_p} + {S^d_p} = 200 + 300 = 500 \\
-\quad{C}_p &= 500 \\
-\quad{R_p} &= min({S}_p, {C}_p) = min(500, 500) = 500 \\
-\quad{r} &= 10\% \\
-\quad{f} &= 2\% \\
-
-\quad{R_p} &= {r} \times {R_p} = 10\% \times 500 = 50 \\
-\quad{R^p_p} &= {r} \times min({S^p_p}, {C}_p) = 10\% \times 200 = 20 \\
-\quad{R^d_p} &= {R_p} - {R^p_p} = 50 - 20 = 30 \\
-\text{Fee paid by Delegator} \quad{F^d_p} &= {f} \times {R^d_p} = 2\% \times 30 = 0.6 \\
-\text{Final Delegator Reward} \quad{R^d_p} &= {R^d_p} - {F^d_p} = 30 - 0.6 = 29.4 \\
-\text{Total Publisher Reward} \quad{R^p_p} &= {R^p_p} + {F^d_p} = 20 + 0.6 = 20.6 \\
-\text{Slashed percentage}\quad{w} &=50\% \\
-\end{aligned}
-$$
+\quad{S^p_p} &= 300 \\
+\quad{S^d_p} &= 200 \\
+\quad{S_p} &= {S^p_p} + {S^d_p} = 300 + 200 = 500 \\
 
+\text{Maximum slashing rate}\quad{z} &= 5\% \\
 
-$$
-\begin{equation}
-\begin{split} \text{Net Publisher Stake} \quad{\Pi^p_p} 
-& = ( R^p_p + f \cdot R^d_p ) - w \cdot S^p_p \\
-& = (20 + 0.6) - 0.5 \cdot 200 \\
-& = 20.6 - 100 \\
-& = -79.4 
-\end{split}
-\end{equation}
-$$
+\text{Publisher Stake post slashing}\quad{S^p_p_{ps}} &= (1 - 5\%) \times 300 = 285 \\
+\text{Delegator Stake post slashing}\quad{S^d_p_{ps}} &= (1 - 5\%) \times 200 = 190 \\
 
-$$
-\begin{equation}
-\begin{split} \text{Net Deligator Stake} \quad{\Pi^d_p} 
-& = R^d_p - ( f \cdot R^d_p + w \cdot S^d_p ) \\
-& = 30 - ( 0.6 + 0.5 \cdot 300 ) \\
-& = 30 - 150.6 \\
-& = -120.6 
-\end{split}
-\end{equation}
+\end{aligned}
 $$
 
-In this example, the publisher's stake is slashed by 50%. This means that the publisher's stake is reduced by 79.4 PYTH tokens and the delegator's stake is reduced by 120.6 PYTH tokens.
-
-
+In this example, the stake is uniformly slashed by 5\%, affecting both the publisher and the delegator. Slashing impact the total stake into the pool, regardless of the Cap.
 
 {/* <StakingCapBar fillPercentage={50} secondFillPercentage={20} labelText="100" /> */}
-

pages/home/oracle-integrity-staking/mathematical-representation.mdx

@@ -1,57 +1,51 @@
 # Mathematical Representation
 
-This section outlines the mathematical representation of the Oracle Integrity Staking (OIS) protocol. 
+This section outlines the mathematical representation of the Oracle Integrity Staking (OIS) protocol.
 
-As explained in the [implementation](./implementation.mdx) section, every publisher is assigned a staking pool where they can self-stake and to which other stakers can delegate. 
+As explained in the [implementation](./implementation.mdx) section, every publisher is assigned a staking pool where they can self-stake and to which other stakers can delegate.
 
 ## Pool Cap
 
 The **pool cap** is calculated as follows:
 
-
 $$
 \large{\text{Pool Cap}: {\bold{C_p}} = M \cdot \sum_{s \in \text{Symbols\_p}} \frac{1}{\max(n_s, Z)}}
 $$
 
 Where:
 
 - $M$ is a constant parameter representing the target stake per symbol.
-- $\text{Symbols\_p}$ is the number of symbols published by the publisher $p$.
+- $\text{Symbols\_p}$ is the set of symbols published by the publisher $p$.
 - $n_s$ be the number of publishers for symbol $s$.
 - $Z$ is a constant parameter to control cap contribution from symbols with a low number of publishers.
 
-This formula actually ensures that symbols with a lower number of publishers contribute more to the overall cap, while symbols with a higher number of publishers contribute less. This is because the contribution of each symbol is inversely proportional to the number of publishers (or Z, whichever is larger).
-
+This formula ensures that symbols with a lower number of publishers contribute more to the overall cap, while symbols with a higher number of publishers contribute less. This is because the contribution of each symbol is inversely proportional to the number of publishers (or Z, whichever is larger).
 
 ## Reward
 
-The reward $R_p$ paid to each pool is calculated as follows:
+The reward $R_p$ distributed to each pool is calculated as follows:
 
 $$
 \large{R_p = y \cdot \min(S_p, C_p)}
 $$
 
 Where:
 
-- $y$ is the yearly cap to the rate of rewards for any pool.
+- $y$ is the cap to the rate of rewards for any pool
 - $S_p$ be the stake assigned to the publisher p pool , made of self-staked amount $S^{p}_{p}$ and delegated stake $S^{d}_{p}$ , or $S_{p} = S^{p}_{p} + S^{d}_{p}$.
 - $C_p$ be the stake cap for the pool assigned to publisher p.
 
-The reward is capped at the pool cap, $C_p$, to ensure that publishers and delegators are not over-rewarded. (I don't this this is necessary to mention, but it is a good sanity check)
-
-
-The total amount of rewards paid to all pools is bound the the same cap relative to the amount of rewards available to the OIS protocol.
+The total amount of rewards distributed to all pools is bound the the same cap relative to the amount of rewards available to the OIS protocol.
 
 $$
 \large{\sum_{p \in \text{Publishers}} R_p \leq y \cdot \min(NumSymbols \cdot M, \sum_{p=1}^{P} S_p)}
 $$
 
-Where: 
+Where:
 
 - $NumSymbols$ is the total number of symbols in the system.
 - $P$ is the total number of publishers in the system.
 
-
 Whereas the reward component relative to the amount self-staked by the publisher $p$ is defined as:
 
 $$
@@ -62,10 +56,9 @@ Where:
 
 - $R^d_p$ is the reward component relative to the amount delegated to the publisher $p$.
 
-
 ## Slashing
 
-Slashing is an important aspect of the OIS protocol to ensure the integrity of the system. 
+Slashing is an important aspect of the OIS protocol to ensure the integrity of the system.
 
 The slashed amount for each pool is calculated as follows:
 
@@ -81,10 +74,7 @@ Where:
 
 Here $SL_p$ is uniformly allocated to both the self-staking publisher and delegators in the pool, pro-rata to their respective stake.
 
-
-
-
-Subsequently, the rewards received  by a publisher and delegators into a pool, net of any slashed amounts can be expressed as below:
+Subsequently, the rewards received by a publisher and delegators into a pool, net of any slashed amounts can be expressed as below:
 
 $$
 \large{\Pi^p_p = ( R^p_p + f \cdot R^d_p ) - w \cdot S^p_p}
@@ -94,11 +84,11 @@ $$
 \large{\Pi^d_p = R^d_p - ( f \cdot R^d_p + w \cdot S^d_p )}
 $$
 
-Where: 
+Where:
 
 - $\Pi^p_p$ is the net reward received by the publisher $p$ after slashing.
 - $\Pi^d_p$ is the net reward received by the delegators after slashing.
-- $f$ is the delegation fee charged by the publisher.
+- $f$ is the delegation fee charged by the pool.
 - $w$ is the slashing rate.
 - $S^p_p$ is the amount self-staked by the publisher $p$.
 - $S^d_p$ is the amount delegated to the publisher $p$.