Math

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Overview

In this version of the ALTBC we incorporate liquidity additions and liquidity withdrawals. Trading works in a similar way as in the ALTBC v1.1. In order to process liquidity additions and liquidity withdrawals we will need to mint and burn NFTs that have two fields: amount and last_revenue_claim. These NFT will play the role of the LP tokens.

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Definitions

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Initialization Parameters

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Functions

We will now define some functions that will be used several times throughout the spec. $${\bf D}(x_{n+1},b_n,c_n) = \frac{1}{2} b_n x_{n+1}^2 + c_n x_{n+1}$$ $${\bf b}(x_n, C_n, V) = \frac{V}{x_n + C_n}$$ $${\bf c}(x_{n+1},b_{n+1},b_n,c_n) = \frac{1}{2} (b_n - b_{n+1}) x_{n+1} + c_n$$ $${\bf p}(x_n,b_n,c_n) = b_n x_n + c_n$$ $${\bf L}(x_n,b_n,c_n,x_{\textnormal{min}}^{(n)},C_n,V) = \frac{1}{2} x_{\textnormal{min}}^{(n)} \left( b_n x_n + 2 c_n + V \cdot \ln\left(\frac{x_{\textnormal{min}}^{(n)} + C_n}{x_n + C_n}\right) \right)$$ $${\bf h}(x_n,b_n,c_n,x_{\textnormal{min}}^{(n)},C_n,V,W_n, W_n^I, \Phi_n, Z_n) = \frac{{\bf L}(x_n,b_n,c_n,x_{\textnormal{min}}^{(n)},C_n,V) + Z_n}{W_n - W_n^I} + \Phi_n$$ In the formula for $\bf{L}$, the $\ln$ denotes the natural logarithm function.

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State variables (TBC maintained quantities)

Notational Notes: In some steps below, to simplify notation, we will use $S_n$ to denote the collection of state variables before the $n$-th transaction. If there is no ambiguity, we might write $$h_n=\textbf{h}(S_n)$$ to mean, in plain language, “we are defining $h_n$ to be the value of the function $\textbf{h}$ using the relevant state variable values from $S_n$” instead of writing out explicitly $$h_n=\textbf{h}(x_n,b_n,c_n,x_{\textnormal{min}}^{(n)},C_n,V,W_n, W_n^I, \Phi_n, Z_n)$$ For another example, we might write $$p_{n+1}=\textbf{p}(S_{n+1})$$ instead of writing $$p_{n+1}=\textbf{p}(x_{n+1},b_{n+1},c_{n+1})$$

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Notation related to the NFTs

The NFT with amount field value equal to $a$ and last_revenue_claim field value equal to $r$ will be denoted by $$\textbf{NFT}(\{\texttt{amount}\colon a,\ \texttt{last\_revenue\_claim}\colon r\})$$ In addition, the value of the last_revenue_claim field of the NFT $j$ at step $n$ will usually be denoted by $r_j^{(n)}$. Recall that these values are not tracked by the pool since they are stored in the corresponding NFT.

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Transactions

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Pool initialization

When the pool is initialized, the state variables are defined as described in the previous table. In addition, an amount $x_{\textnormal{add}}$ of tokens must be deposited into the pool. For reference in what follows, $$D_0 = {\bf D}(x_{\textnormal{min}}^{(0)},{\bf b}(0,C_0, V_0),p_{\textnormal{lower}})$$ Two NFTs must be minted and given to the pool deployer (or game owner) upon initialization of the pool. The first will have amount field value equal to $W_0 - W_0^I$ and last_revenue_claim field value equal to $\frac{D_0}{W_0 - W_0^I}$. With our notation, this NFT is $$\textbf{NFT}(\{\texttt{amount}\colon W_0 - W_0^I,\ \texttt{last\_revenue\_claim}\colon \frac{D_0}{W_0 - W_0^I}\})$$ The second, which will henceforth be referred to as the inactive-fee NFT (and which is unique in this way), will have amount field value equal to $W_0^I$ and last_revenue_claim field value equal to $\infty$. With our notation, this NFT is $$\textbf{NFT}(\{\texttt{amount}\colon W_0^I,\ \texttt{last\_revenue\_claim}\colon \infty\})$$ Note that $\infty$ can be replaced by any appropriate number or object that handles the following three actions appropriately for the inactive-fee NFT: 1. Extracting revenue: the transaction should be immediately reverted (described below). 2. Withdrawing liquidity: the transaction is allowed with some modifications (described below - for example, step 8 in Liquidity withdrawals). 3. Adding liquidity: the transaction should be immediately reverted (described below). We emphasize that a pool may have at most one inactive-fee NFT at any stage in its lifetime. The deployer must mint one at inception and if this inactive-fee NFT is liquidated by the deployer, there will never be another.

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Trades

Suppose that the $n$-th trade comes into the pool, and that the current values of the state variables is $S_n$. We perform the trade using the algorithm defined by the following steps. Step 1. We update the state variables $x_n$ and $D_n$ according to four different situations as described below.

• If the user submits an order to buy an amount $a_n > 0$ of token $X$, we check that

$$x_n + a_n \leq x_{\textnormal{max}}^{(n)}$$ If this condition does not hold, we do not perform the transaction. Otherwise, if the previous condition holds, we update the parameters $x$ and $D$ as follows. $$\begin{aligned} x_{n+1} & = x_n + a_n \\ D_{n+1} & = {\bf D}(x_{n+1},b_n,c_n) \end{aligned}$$ In this case, we define $$U_{\textnormal{in}} = \frac{D_{n+1} - D_n}{1 - \phi - \psi}$$ and an amount $U_{\textnormal{in}}$ of collateral is transferred from the user to the pool, and an amount $a_n$ of token $X$ is transferred from the pool to the user. In addition, the state variables $\Phi_n$ and $\Psi_n$ are updated as follows: $$\begin{aligned} \Phi_{n+1} &= \Phi_n + \frac{\phi \cdot U_{\textnormal{in}}}{W_n - W_n^I} \\ \Psi_{n+1} &= \Psi_n + \psi \cdot U_{\textnormal{in}} \end{aligned}$$

• If the user submits an order to sell an amount $a_n > 0$ of token $X$, we check that

$$x_n - a_n \geq x_{\textnormal{min}}^{(n)}$$ If this condition does not hold, we do not perform the transaction. Otherwise, if the previous condition holds, we update the parameters $x$ and $D$ as follows. $$\begin{aligned} x_{n+1} & = x_n - a_n \\ D_{n+1} & = {\bf D}(x_{n+1},b_n,c_n) \end{aligned}$$ The trading fee and protocol fee are defined, respectively, as $$\begin{aligned} M& = \phi \cdot (D_{n} - D_{n+1}) \\ N & = \psi\cdot(D_{n} - D_{n+1}) \end{aligned}$$ The state variables $\Phi_n$ and $\Psi_n$ are updated as follows: $$\begin{aligned} \Phi_{n+1} &= \Phi_n + \frac{M}{W_n - W_n^I} \\ \Psi_{n+1} &= \Psi_n + N \end{aligned}$$ Then, the amount $a_n$ of token $X$ is transferred from the user to the pool, and the amount $(D_{n} - D_{n+1}) - M -N$ of collateral is transferred from the pool to the user (with the latter two terms representing LP and protocol fees collected by the pool rather than transferred to the user).

• If the user submits an order to buy an amount $U_{\textnormal{out}} > 0$ of token $Y$ (collateral), we define

$$a_n = \frac{U_{\textnormal{out}}}{1 - \phi - \psi}$$ and we check that $$D_{n} - a_n \geq {\bf D}(x_{\textnormal{min}}^{(n)},b_n,c_n) \ .$$ If this condition does not hold, we do not perform the transaction. Otherwise, if the previous condition holds, we update the parameters $x$ and $D$ as follows. $$\begin{aligned} D_{n+1} & = D_n - a_n \\ x_{n+1} & = \dfrac{2 D_{n+1}}{c_n + \sqrt{c_n^2 + 2 b_n D_{n+1}}} \end{aligned}$$ In this case, an amount $x_{n} - x_{n+1}$ of token $X$ is transferred from the user to the pool, and an amount $U_{\textnormal{out}}$ of collateral is transferred from the pool to the user. In addition, the state variables $\Phi_n$ and $\Psi_n$ are updated as follows: $$\begin{aligned} \Phi_{n+1} &= \Phi_n + \frac{\phi \cdot a_n}{W_n - W_n^I} \\ \Psi_{n+1} &= \Psi_n + \psi \cdot a_n \end{aligned}$$

• If the user submits an order to sell an amount $U_{\textnormal{in}} > 0$ of token $Y$ (collateral), we define the trading fee and protocol fee, respectively, by

$$\begin{aligned} M& = \phi \cdot U_{\textnormal{in}} \\ N & = \psi \cdot U_{\textnormal{in}}\ . \end{aligned}$$ Then, we update the state variables $\Phi_n$ and $\Psi_n$ are updated as follows: $$\begin{aligned} \Phi_{n+1} &= \Phi_n + \frac{M}{W_n - W_n^I} \\ \Psi_{n+1} &= \Psi_n + N\ . \end{aligned}$$ Then, we define $$a_n = U_{\textnormal{in}} - M - N$$ and we check that $$D_{n} + a_n \leq {\bf D}(x_{\textnormal{max}}^{(n)},b_n,c_n) \ .$$ If this condition does not hold, we do not perform the transaction. Otherwise, if the previous condition holds, we update the parameters $x$ and $D$ as follows. $$\begin{aligned} D_{n+1} & = D_n + a_n \\ x_{n+1} & = \dfrac{2 D_{n+1}}{c_n + \sqrt{c_n^2 + 2 b_n D_{n+1}}} \end{aligned}$$ In this case, an amount $U_{\textnormal{in}}$ of collateral is transferred from the user to the pool, and an amount $x_{n+1} - x_n$ of token $X$ is transferred from the pool to the user. Step 2. We update the state variables $b_n$ and $c_n$ as follows $$\begin{aligned} b_{n+1} & = {\bf b}(x_{n+1}, C_n, V) \\ c_{n+1} & = {\bf c}(x_{n+1},b_{n+1},b_n,c_n)\ . \end{aligned}$$ Step 3. We update the spot price $p_n$ as $$p_{n+1} = {\bf p}(S_{n+1}) \ .$$ The remaining state variables do not have to be updated.

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The revenue parameter

In order to enable liquidity providers to collect their corresponding share of revenue and fees from the pool, we introduce the revenue parameter function, which is defined by $${\bf h}(x_n,b_n,c_n,x_{\textnormal{min}}^{(n)},C_n,V,W_n, W_n^I, \Phi_n, Z_n) = \frac{{\bf L}(x_n,b_n,c_n,x_{\textnormal{min}}^{(n)},C_n,V) + Z_n}{W_n - W_n^I} + \Phi_n$$ Now, if $x_n,b_n,c_n,x_{\textnormal{min}}^{(n)},C_n,V,W_n, W_n^I, \Phi_n, Z_n$ are the current values of the corresponding pool variables, we define the current revenue parameter (per unit of amount) as $$h_n = {\bf h}(x_n,b_n,c_n,x_{\textnormal{min}}^{(n)},C_n,V,W_n,W_n^I,\Phi_n, Z_n) \ .$$ It is important to mention that this parameter is invariant with respect to proportional liquidity deposits and liquidity withdrawals.

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Liquidity deposits

Let $S_n$ be the current values of all state variables. Suppose that a user wants to provide liquidity to the pool approving up to amounts $\tilde A \geq 0$ of token $X$ and $\tilde B \geq 0$ of collateral, with $(\tilde A, \tilde B) \neq (0,0)$. Before proceeding, check if the deposit is associated with the inactive-fee NFT. If so, revert the transaction. In all other cases, in order to process this liquidity deposit we proceed as follows. Step 1. We will determine the amounts $A$ and $B$ of the respective tokens that will actually be taken so that the liquidity deposit is proportional. Let $L_n = {\bf L}(S_n)$. If $$\tilde B \cdot \left (x_{\textnormal{max}}^{(n)}- x_n\right ) \leq \tilde A \cdot \left (D_n - L_n \right ),$$ then set $$\begin{aligned} B &= \tilde B \\ A &= \begin{cases} \frac{x_{\textnormal{max}}^{(n)}- x_n}{D_n - L_n} \cdot \tilde B, &\text{ if } D_n - L_n \neq 0 \\ \tilde A, &\text{ if } D_n - L_n = 0 \\ \end{cases} \\ q &= \begin{cases} \frac{B}{D_n - L_n}, &\text{ if } D_n - L_n \neq 0 \\ \frac{A}{x_{\textnormal{max}}^{(n)}- x_n}, &\text{ if } D_n - L_n = 0 \ . \\ \end{cases} \\ \end{aligned}$$ If $$\tilde B \cdot \left (x_{\textnormal{max}}^{(n)}- x_n\right ) > \tilde A \cdot \left (D_n - L_n \right ),$$ then set $$\begin{aligned} A &= \tilde A \\ B &= \frac{D_n - L_n}{x_{\textnormal{max}}^{(n)}- x_n} \cdot \tilde A \\ q &= \frac{A}{x_{\textnormal{max}}^{(n)}- x_n} \ . \\ \end{aligned}$$ Note that $A \leq \tilde A$ and $B \leq \tilde B$. If $A = B = 0$, the deposit should be reverted. In all other cases, move through the remaining steps with the computed values for $A$, $B$, and $q.$ Step 2. Now we update the variables $x$ and $D$ of the TBC, as follows. $$\begin{aligned} x_{n+1} &= (1+q) \cdot x_n \\ D_{n+1} &= (1+q) \cdot D_n \end{aligned}$$ Step 3. We update the parameters $x_{\textnormal{max}}$ and $x_{\textnormal{min}}$ of the pool in the following way. $$\begin{aligned} x_{\textnormal{max}}^{(n+1)} &= (1+q) \cdot x_{\textnormal{max}}^{(n)} \\ x_{\textnormal{min}}^{(n+1)} &= (1+q) \cdot x_{\textnormal{min}}^{(n)} \ . \end{aligned}$$ Step 4. We update the parameter $C$ of the pool, as follows. $$\begin{aligned} C_{n+1} &= (1+q) \cdot C_n \end{aligned}$$ Step 5. We update the parameters $b$ and $c$ of the ALTBC as follows: $$b_{n+1} = \frac{b_n}{1+q}$$ and $$c_{n+1} = c_n \ .$$ Note that $b_{n+1} = {\bf b}(x_{n+1}, C_{n+1}, V)$ and that $c_{n+1} = {\bf c}(x_{n+1},b_{n+1},\frac{b_n}{1+q},c_n)$. We also update the spot price $p_n$ as $$p_{n+1} = p_n \ .$$ Note that $p_{n+1} = {\bf p}(x_{n+1},b_{n+1},c_{n+1})$. Step 6. We update the parameters $W$ and $W^I$ of the TBC in the following way. $$\begin{aligned} W_{n+1} &= (1+q) \cdot W_n \end{aligned}$$ $$\begin{aligned} W_{n+1}^I &= W_n^I \end{aligned}$$ Step 7. We update (recall $L_n$ computed at the beginning of this section) $$Z_{n+1} = Z_n + q \cdot \frac{W_n^I}{W_n - W_n^I} \cdot \left (L_n + Z_n \right) + q \cdot Z_n,$$ we define $$w_j = q W_n,$$ and we update $$h_{n+1} = {\bf h}(S_{n+1}).$$ Step 8. We transfer the amounts $A \geq 0$ of token $X$ and $B \geq 0$ of collateral to the pool. Step 9.

• Case 1. If the liquidity provider has provided an NFT to which the new deposit should be added we proceed in the following way. Let

$$\textbf{NFT}(\{\texttt{amount}\colon \widehat{w},\ \texttt{last\_revenue\_claim}\colon \widehat{r}\})$$ be the NFT to which the new liquidity deposit should be added. We update the given NFT to $$\textbf{NFT}(\{\texttt{amount}\colon \widehat{w} + w_j,\ \texttt{last\_revenue\_claim}\colon \frac{h_{n+1} w_j + \widehat{r} \widehat{w}}{\widehat{w} + w_j} \})$$

• Case 2. If the liquidity provider has not provided an NFT to which the new deposit should be added, we mint the NFT defined by

$$\textbf{NFT}(\{\texttt{amount}\colon w_j,\ \texttt{last\_revenue\_claim}\colon h_{n+1}\})$$ and transfer it to the liquidity provider.

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Extracting revenue

We need to define a pool function that enables the game developers and liquidity providers to extract, at any moment, any excess of collateral from the pool as revenue. In order to give a liquidity provider the fair amount of collateral that his/her NFT accrued we do the following. Before proceeding, check if we are dealing with the inactive-fee NFT. If so, revert the transaction. In all other cases, we proceed as follows. Suppose that liquidity provider $j$ owns the NFT defined by $$\textbf{NFT}(\{\texttt{amount}\colon w_j,\ \texttt{last\_revenue\_claim}\colon r_j^{(n)}\}) \ .$$ Let $S_n$ be the current values of the state variables, and let $$h_n = {\bf h}(S_n)$$ be the current value of the revenue parameter. Then, the maximum amount of collateral that the liquidity provider is entitled to collect is given by the formula $$w_j \cdot (h_n - r_j^{(n)}) \ .$$ If the liquidity provider $j$ intends to extract an amount $Q$ of revenue we proceed as follows. Step 1. We check that $$Q \leq w_j \cdot (h_n - r_j^{(n)}) \ .$$ If so, we proceed. Otherwise we reject the transaction. Step 2. We update the value of the variable $r_j^{(n)}$ (of the last_revenue_claim field of the liquidity provider’s NFT) as $$r_j^{(n+1)} = r_j^{(n)} + \frac{Q}{w_j} \ .$$ This is, the NFT that the liquidity provider owns is updated to $$\textbf{NFT}(\{\texttt{amount}\colon w_j,\ \texttt{last\_revenue\_claim}\colon r_j^{(n+1)}\}) \ .$$ Step 3. We transfer an amount $Q$ of collateral to the liquidity provider’s account.

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Liquidity withdrawals

Let $S_n$ be the current values of the state variables. Suppose that liquidity provider $j$ wants to redeem liquidity corresponding to the NFT defined by $$\textbf{NFT}(\{\texttt{amount}\colon w_j,\ \texttt{last\_revenue\_claim}\colon r_j^{(n)}\}) \ .$$ Suppose, in addition, that the liquidity provider wants to redeem the part of the liquidity corresponding to the previous NFT given by an amount value of $u_j$ with $0 < u_j \leq w_j$. In order to process the liquidity withdrawal, we proceed as follows. First, determine whether the NFT in question is the inactive-fee NFT. Step 1. We define $$q = \frac{u_j}{W_n}$$ and $$h_n = {\bf h}(S_n) \ .$$ Note that if we are dealing with the inactive-fee NFT, $h_n$ does not need to be computed. Step 2. Using our computed value of $L$ from the previous step (obtained when computing $h_n$), define $$L_n = {\bf L}(S_n).$$ Then, define the amounts of token and collateral that the liquidity provider will receive by $$\begin{aligned} A_X & = q \cdot (x_{\textnormal{max}}^{(n)} - x_n ) \\ A_Y & = q \cdot (D_n - L_n) \end{aligned}$$ Step 3. We update the variables $x$ and $D$ of the TBC, as follows. $$\begin{aligned} x_{n+1} &= (1-q) \cdot x_n \\ D_{n+1} &= (1-q) \cdot D_n \end{aligned}$$ Step 4. We update the parameters $x_{\textnormal{max}}$ and $x_{\textnormal{min}}$ of the pool in the following way. $$\begin{aligned} x_{\textnormal{max}}^{(n+1)} &= (1-q) \cdot x_{\textnormal{max}}^{(n)} \\ x_{\textnormal{min}}^{(n+1)} &= (1-q) \cdot x_{\textnormal{min}}^{(n)} \ . \end{aligned}$$ Step 5. We update the parameter $C$ of the pool, as follows. $$\begin{aligned} C_{n+1} &= (1-q) \cdot C_n \end{aligned}$$ Step 6. We update the parameter $b$ of the ALTBC as follows. $$b_{n+1} = \frac{b_n}{1-q}$$ The parameter $c_n$ and the spot price $p_n$ do not need to be updated. This is, we define $$\begin{aligned} c_{n+1} & = c_n \\ p_{n+1} & = p_n \end{aligned}$$ Step 7. We update the parameter $W$ of the TBC in the following way: $$\begin{aligned} W_{n+1} &= W_n - u_j \end{aligned}$$ If we are dealing with the inactive-fee NFT, we must also update $W_I$ of the TBC in the following way: $$\begin{aligned} W_{n+1}^I &= W_n^I - u_j \end{aligned}$$ Otherwise, define $W_{n+1}^I = W_n^I.$ Finally, if $W_{n+1} - W_{n+1}^I = 0$ and $W_{n+1}^I > 0$, revert the transaction. Otherwise, proceed. Step 8. If we are dealing with the inactive-fee NFT, this step must be skipped. We give the liquidity provider $j$ the remaining revenue to which he/she is entitled (if there is any left) corresponding to the amount value $u_j$, which is given by the formula $$u_j \cdot (h_n - r_j^{(n)}) \ .$$ Step 9. The NFT used by the liquidity provider is updated to $$\textbf{NFT}(\{\texttt{amount}\colon w_j - u_j,\ \texttt{last\_revenue\_claim}\colon r_j^{(n)}\}) \ .$$ (or burned if $u_j = w_j$), and the liquidity provider receives an amount $A_X$ of token $X$ and an amount $A_Y$ of collateral. Note that if we are dealing with the inactive-fee NFT, the last_revenue_claim persists at $\infty$ (this is just noted for emphasis, not because a different action is required). Note that only the amount field of the NFT is updated. Step 10. If we are dealing with the inactive-fee NFT, we will update $Z$ as $$Z_{n+1} = Z_n + q \cdot L_n.$$ In all other cases (i.e. whenever we are dealing with an active-fee NFT), we update $Z$ as $$Z_{n+1} = Z_n - q \cdot \frac{W_n^I}{W_n - W_n^I} \cdot \left (L_n + Z_n \right) - q \cdot Z_n.$$