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### Token Dilution#

Similarly we can look at the expected Staked Dilution (i.e. Adjusted Staking Yield) and Un-staked Dilution as previously defined. Again, dilution in this context is defined as the change in fractional representation (i.e. ownership) of a set of tokens within a larger set. In this sense, dilution can be a positive value: an increase in fractional ownership (staked dilution / Adjusted Staking Yield), or a negative value: a decrease in fractional ownership (un-staked dilution).

We are interested in the relative change in ownership of staked vs un-staked tokens as the overall token pool increases with inflation issuance. As discussed, this issuance is distributed only to staked token holders, increasing the staked token fractional representation of the Total Current Supply.

Continuing with the same Inflation Schedule parameters as above, we see the fraction of staked supply grow as shown below. Due to this relative change in representation, the proportion of stake of any token holder will also change as a function of the Inflation Schedule and the proportion of all tokens that are staked.

Of initial interest, however, is the dilution of un-staked tokens, or $D_{us}$. In the case of un-staked tokens, token dilution is only a function of the Inflation Schedule because the amount of un-staked tokens doesn't change over time.

This can be seen by explicitly calculating un-staked dilution as $D_{us}$. The un-staked proportion of the token pool at time $t$ is $P_{us}(t_{N})$ and $I_{t}$ is the incremental inflation rate applied between any two consecutive time points. $SOL_{us}(t)$ and $SOL_{total}(t)$ is the amount of un-staked and total SOL on the network, respectively, at time $t$. Therefore $P_{us}(t) = SOL_{us}(t)/SOL_{total}(t)$.

\begin{aligned} D_{us} &= \left( \frac{P_{us}(t_{1}) - P_{us}(t_{0})}{P_{us}(t_{0})} \right)\\ &= \left( \frac{ \left( \frac{SOL_{us}(t_{2})}{SOL_{total}(t_{2})} \right) - \left( \frac{SOL_{us}(t_{1})}{SOL_{total}(t_{1})} \right)}{ \left( \frac{SOL_{us}(t_{1})}{SOL_{total}(t_{1})} \right) } \right)\\ \end{aligned}

However, because inflation issuance only increases the total amount and the un-staked supply doesn't change:

\begin{aligned} SOL_{us}(t_2) &= SOL_{us}(t_1)\\ SOL_{total}(t_2) &= SOL_{total}(t_1)\times (1 + I_{t_1})\\ \end{aligned}

So $D_{us}$ becomes:

\begin{aligned} D_{us} &= \left( \frac{ \left( \frac{SOL_{us}(t_{1})}{SOL_{total}(t_{1})\times (1 + I_{1})} \right) - \left( \frac{SOL_{us}(t_{1})}{SOL_{total}(t_{1})} \right)}{ \left( \frac{SOL_{us}(t_{1})}{SOL_{total}(t_{1})} \right) } \right)\\ D_{us} &= \frac{1}{(1 + I_{1})} - 1\\ \end{aligned}

Or generally, dilution for un-staked tokens over any time frame undergoing inflation $I$:

$D_{us} = -\frac{I}{I + 1} \\$

So as guessed, this dilution is independent of the total proportion of staked tokens and only depends on inflation rate. This can be seen with our example Inflation Schedule here: We can do a similar calculation to determine the dilution of staked token holders, or as we've defined here as the Adjusted Staked Yield, keeping in mind that dilution in this context is an increase in proportional ownership over time. We'll use the terminology Adjusted Staked Yield to avoid confusion going forward.

To see the functional form, we calculate, $Y_{adj}$, or the Adjusted Staked Yield (to be compared to D_{us} the dilution of un-staked tokens above), where $P_{s}(t)$ is the staked proportion of token pool at time $t$ and $I_{t}$ is the incremental inflation rate applied between any two consecutive time points. The definition of $Y_{adj}$ is therefore:

$Y_{adj} = \frac{P_s(t_2) - P_s(t_1)}{P_s(t_1)}\\$

As seen in the plot above, the proportion of staked tokens increases with inflation issuance. Letting $SOL_s(t)$ and $SOL_{\text{total}}(t)$ represent the amount of staked and total SOL at time $t$ respectively:

$P_s(t_2) = \frac{SOL_s(t_1) + SOL_{\text{total}}(t_1)\times I(t_1)}{SOL_{\text{total}}(t_1)\times (1 + I(t_1))}\\$

Where $SOL_{\text{total}}(t_1)\times I(t_1)$ is the additional inflation issuance added to the staked token pool. Now we can write $Y_{adj}$ in common terms $t_1 = t$:

\begin{aligned} Y_{adj} &= \frac{\frac{SOL_s(t) + SOL_{\text{total}}(t)\times I(t)}{SOL_{\text{total}}(t)\times (1 + I(t))} - \frac{SOL_s(t)}{SOL_{\text{total}}(t)} }{ \frac{SOL_s(t)}{SOL_{\text{total}}(t)} } \\ &= \frac{ SOL_{\text{total}}(t)\times (SOL_s(t) + SOL_{\text{total}}(t)\times I(t)) }{ SOL_s(t)\times SOL_{\text{total}}\times (1 + I(t)) } -1 \\ \end{aligned}

which simplifies to:

$Y_{adj} = \frac{ 1 + I(t)/P_s(t) }{ 1 + I(t) } - 1\\$

So we see that the Adjusted Staked Yield a function of the inflation rate and the percent of staked tokens on the network. We can see this plotted for various staking fractions here: It is also clear that in all cases, dilution of un-staked tokens $>$ adjusted staked yield (i.e. dilution of staked tokens). Explicitly we can look at the relative dilution of un-staked tokens to staked tokens: $D_{us}/Y_{adj}$. Here the relationship to inflation drops out and the relative dilution, i.e. the impact of staking tokens vs not staking tokens, is purely a function of the % of the total token supply staked. From above

\begin{aligned} Y_{adj} &= \frac{ 1 + I/P_s }{ 1 + I } - 1,~\text{and}\\ D_{us} &= -\frac{I}{I + 1},~\text{so} \\ \frac{D_{us}}{Y_{adj}} &= \frac{ \frac{I}{I + 1} }{ \frac{ 1 + I/P_s }{ 1 + I } - 1 } \\ \end{aligned}

which simplifies as,

\begin{aligned} \frac{D_{us}}{Y_{adj}} &= \frac{ I }{ 1 + \frac{I}{P_s} - (1 + I)}\\ &= \frac{ I }{ \frac{I}{P_s} - I}\\ \frac{D_{us}}{Y_{adj}}&= \frac{ P_s }{ 1 - P_s}\\ \end{aligned}

Where we can see a primary dependence of the relative dilution of un-staked tokens to staked tokens is on the function of the proportion of total tokens staked. As shown above, the proportion of total tokens staked changes over time (i.e. $P_s = P_s(t)$ due to the re-staking of inflation issuance thus we see relative dilution grow over time as: As might be intuitive, as the total fraction of staked tokens increases the relative dilution of un-staked tokens grows dramatically. E.g. with $80\%$ of the network tokens staked, an un-staked token holder will experience ~$400\%$ more dilution than a staked holder.

Again, this represents the change in fractional change in ownership of staked tokens and illustrates the built-in incentive for token holder to stake their tokens to earn Staked Yield and avoid Un-staked Dilution.