Add delayed claim analysis (#216)

* Add delayed claim analysis

* Change alignment

* Update claim-burn.md

* Update docs/claim-burn.md

Co-authored-by: Will Demaine <willdemaine@gmail.com>

---------

Co-authored-by: Will Demaine <willdemaine@gmail.com>

docs/assets/delayed_claim_er_change.py

@@ -0,0 +1,26 @@
+import matplotlib.pyplot as plt
+import numpy as np
+
+plt.rcParams['text.usetex'] = True
+
+# e / E_2
+xs = np.linspace(0, 0.4, 200)
+
+def f(x, r):
+    return  (1 - x)/ (1 - r * x)
+
+# rho_2 / rho_1
+rs = np.linspace(0.9, 1.1, 5)
+
+# rho_3 / rho_2
+ys = [f(xs, r) for r in rs]
+
+for r, y in zip(rs, ys):
+    plt.plot(xs, y, label=f"{r:.2f}")
+
+plt.grid(True)
+plt.legend(title='Fractional exchange rate change\nbetween request and claim, ' + r'$\rho_2/\rho_1$', ncol = 2)
+plt.xlabel(r'Fraction of total ETH claimed, $e/E_2$')
+plt.ylabel(r'Fractional exchange rate change after claim, $\rho_3/\rho_2$')
+
+plt.savefig('delayed_claim_er_change.png', dpi=300)

docs/claim-burn.md

@@ -4,8 +4,61 @@ In our design, mETH is sent to the contract when the user does an unstake reques
 
 As a consequence of burning at this point, it is the case that the delta for the exchange rate adjustment at claim time is slightly accelerated, and this scales with the amount of time between unstaking and claiming.
 
-We have analyzed the effect on the rate and concluded that the difference is entirely negligible in realistic cases. For example, our calculations show that to affect the exchange rate by ~0.7%, a user would have to unstake 20% of the entire TVL in the protocol and wait about 1 year before claiming it.
+We have analyzed the effect on the rate and concluded that the difference is entirely negligible in realistic cases. For example, our calculations (see below) show that to affect the exchange rate by ~1%, a user would have to unstake 20% of the entire TVL in the protocol and wait about 1 year (assuming 5% in rewards) before claiming it.
 
 Given that a user leaving an unclaimed request for a long time is extremely inefficient (they have locked money which is not earning rewards) and the TVL of the protocol is expected to be very large, we consider the likelihood of this affecting the protocol extremely small. In the normal case of a user unstaking a fraction of the TVL and claiming after a few days, the effect is almost immeasurable, so we do not consider this an issue.
 
 For reference, we made this trade-off because it simplifies other parts of the protocol mechanics.
+
+
+### Analysis
+
+Assume a user submits the request to unstake $m$ mETH for $e$ ETH at exchange rate 
+
+$$
+\begin{aligned}
+\rho_1 = \frac{E_1}{M_1} = \frac{e}{m}
+\end{aligned}
+$$
+
+where $E_1$ and $M_1$ denote the current total controlled ETH and supply of mETH.
+The user does not claim their ETH immediately but waits until the exchange rate has changed to
+
+$$
+\begin{aligned}
+\rho_2 = \frac{E_2}{M_2},
+\end{aligned}
+$$
+
+where $E_2$ and $M_2$ are defined in analogy to the above.
+Since the delayed claim removes ETH and mETH at a ratio that is different to the current exchange rate, it will change the exchange rate to 
+
+$$
+\begin{aligned}
+\rho_3 = \frac{E_2 - e}{M_2 - m}.
+\end{aligned}
+$$
+
+One can obtain a compact expression for the relative change by dividing by $\rho_2$, substituting $M_2$ and $m$ and rearranging the terms as follows
+
+$$
+\begin{aligned}
+\frac{\rho_3}{\rho_2} 
+&= \frac{1}{\rho_2} \frac{E_2 - e}{M_2 - m} \\
+&= \frac{E_2 - e}{E_2 - \frac{\rho_2}{\rho_1} e} \\
+&= \frac{1 - \frac{e}{E_2}}{1 - \frac{\rho_2}{\rho_1} \frac{e}{E_2}}
+\end{aligned}
+$$
+
+#### Numerical example
+
+Assuming that the user claims 20% of the total controlled ETH (i.e. $e/E_2 = 0.2$) after the protocol has generated 5% of returns (i.e $\rho_2 / \rho_1 = 1.05$), the exchange rate will change by 1.2%
+
+$$
+\begin{aligned}
+\frac{\rho_3}{\rho_2} 
+&= \frac{1 - 0.2}{1 - 1.05 \cdot 0.2} = 1.012 \\
+\end{aligned}
+$$
+
+![Delayed claim exchange rate change](./assets/delayed_claim_er_change.png)