Fixed-Point Arithmetic for Rate Computation

To compute base reward rate, base exchange rate, validator-specific exchange rates, and total validator voting power, we need to carefully perform arithmetic to avoid issues with precision and rounding. We use explicitly specified fixed-precision arithmetic for this, with a precision of 8 digits. This allows outputs to fit in a u64, with all products fitting in the output and plenty of headroom for additions.

All integer values should be interpreted as unsigned 64-bit integers, with the exception of the validator’s commission rate, which is a u16 specified in terms of basis points (one one-hundredth of a percent, or in other words, an implicit denominator of $10^4$). All integer values, with the exception of the validator’s commission rate, have an implicit denominator of $10^8$.

Throughout this spec representations are referred to as $\mathtt{x}$, where $\mathtt{x}=x\cdot 10^8$, and the value represented by representations is $x=\mathtt{x}\cdot 10^{-8}$.

As an example, let’s take a starting value $\mathtt{x}$ represented in our scheme (so, $x\cdot 10^8$) and compute its product with $y$, also represented by our fixed point scheme, so $y\cdot 10^8$. The product $\mathtt{xy}$ is computed as

$$\lfloor \frac{(x10^8)(y10^8)}{10^8}\rfloor =\lfloor \frac{\mathtt{xy}}{10^8}\rfloor$$

Since both $x$ and $y$ are both representations and both have a factor of $10^8$, computing their product includes an extra factor of 10^8 which we divide out. All representations are u64 in our scheme, and any fixed-point number or product of fixed point numbers with 8 digits fits well within 64 bits.

Summary of Notation

• ${\mathtt{r}}_e$: the fixed-point representation of the base rate at epoch $e$.
• $\mathtt{psi}(e)$: the fixed-point representation of the base exchange rate at epoch $e$.
• $s_{i,e}$: the funding rate of the validator’s funding stream at index $i$ and epoch $e$, in basis points.
• $c_{v,e}$: the sum of the validator’s commission rates, in basis points.
• ${\mathtt{r}}_{v,e}$: the fixed-point representation of the validator-specific reward rate for validator $v$ at epoch $e$.
• ${\mathtt{psi}}_v(e)$: the fixed-point representation of the validator-specific exchange rate for validator $v$ at epoch $e$.
• ${\mathtt{y}}_v$: the sum of the tokens in the validator’s delegation pool.
• ${\mathtt{iota}}_v(e)$: the validator’s voting power for validator $v$ at epoch $e$.

Base Reward Rate

The base reward is an input to the protocol, and the exact details of how this base rate $r_e$ is determined is not yet decided. For now, we can assume it is derived from the block header.

Base Exchange Rate

The base exchange rate, $\psi (e)$, can be safely computed as follows:

$$\mathtt{psi}(e)=\lfloor \frac{\mathtt{psi}(e-1)\cdot (10^8+{\mathtt{r}}_e)}{10^8}\rfloor$$

Commission Rate from Funding Streams

To compute the validator’s commission rate from its set of funding streams, compute $$c_{v,e}=\sum _i{s}_{i,e}$$ where $s_{i,e}$ is the rate of validator $v$’s $i$-th funding stream at epoch $e$.

Validator Reward Rate

To compute the validator’s base reward rate, we compute the following:

$${\mathtt{r}}_{v,e}=\lfloor \frac{(10^8-(c_{v,e}\cdot 10^4)){\mathtt{r}}_e}{10^8}\rfloor$$

Validator Exchange Rate

To compute the validator’s exchange rate, we use the same formula as for the base exchange rate, but substitute the validator-specific reward rate in place of the base reward rate:

$${\mathtt{psi}}_v(e)=\lfloor \frac{{\mathtt{psi}}_v(e-1)\cdot (10^8+{\mathtt{r}}_{v,e})}{10^8}\rfloor$$

Validator Voting Power

Finally, to compute the validator’s voting power, take:

$${\mathtt{iota}}_v(e)=\lfloor {\mathtt{y}}_v\cdot \frac{{\mathtt{psi}}_v(e)}{\mathtt{psi}(e)}\rfloor$$