xy = k from first principles

Introduction

Uniswap V2 is a protocol that lets you exchange tokens.

Anyone can permissionlessly create a pair of two tokens ($\text{Token}_{X}$ and $\text{Token}_{Y}$).

This pair holds balances of the two tokens, known as reserves ($x$ and $y$). Anyone can permissionlessly add tokens to the reserves, making them available for token swaps.

A token swap involves adding tokens to one reserve $(x + \Delta x)$, and removing tokens from the other $(y - \Delta y)$.

In the Uniswap V2 protocol, each swap must satisfy the following condition known as the Constant Product Invariant, else the swap transaction is reverted:

$(x + \Delta x) \times (y - \Delta y) = x \times y$

The same is sometimes formulated more popularly as:

$x \times y = k$

which represents the product of current token reserves, and the state of reserves resulting from a swap having to maintain it:

$(x + \Delta x) \times (y - \Delta y) = k$

This simple looking rule creates the intuitive user experience of exchanging tokens at rates that largely match user expectations (e.g. "selling $1 \hspace{1mm} \text{ETH}$ for $2500 \hspace{1mm} \text{USDC}$") based on exchange rates found elsewhere in non-smart-contract systems like centralized exchanges or trading apps.

There are plenty of excellent articles out there that explain how Uniswap V2 works, by taking the $x \times y = k$ formula as a given, and then showing e.g. the math for how one can derive the formula for calculating token prices, or the amount of $\text{Token}_{Y}$ received in exchange for $\text{Token}_{X}$, etc.

These articles are valuable, but this is not one of them.

The goal of this article is to explain why the rule "$x \times y = k$" is not an arbitrary design choice, but rather, concretely arises from how we generally, rationally price two assets in terms of each other, whether it be pricing $\text{Token}_{Y}$ in terms of $\text{Token}_{X}$, or apples in terms of oranges.

Pricing tokens in terms of each other

To make two tokens $X$ and $Y$ tradable and let users exchange token $X$ for $Y$ and vice-versa, we need supplies of the two tokens that can be traded over.

So, let's start with a pair of tokens $X$ and $Y$, with reserves (tradable balances) $x$ and $y$ respectively. What should be the exchange rate between the two tokens?

The U.S. dollar is a highly common denomination of monetary value. Going forward, we'll use "monetary value" and "USD value" interchangeably.

When we swap one token for another, we're not trying to increase our wealth. We want to keep our wealth the same while converting the currency it is denominated in.

So, in exchange for adding some USD worth $\Delta x$ in the reserve $x$, we should be getting out the same USD worth of $\Delta y$ out of the reserve $y$.

Therefore, the reserves $x$ and $y$ should be such that:

$\text{USD\_Value}(x) = \text{USD\_Value}(y)$

And so, the ratio $\frac{y}{x}$ represents not just "the number of $Y$ tokens per one $X$ token", but also "the number of $Y$ tokens having the same monetary worth as one $X$ token".

So, for a given incoming $\Delta x$, should we calculate the $\Delta y$ to be given out as the following?

$\Delta y = -\frac{y}{x} \times \Delta x$

Note, we denominate $\Delta y$ with a minus sign since the change in $y$ reserves is a decrease.

This fraction $\frac{y}{x}$ is also referred to as the spot price of token $X$ in terms of $Y$ tokens. The converse $\frac{x}{y}$ is the spot price of token $Y$ in terms of token $X$.

Issues with spot price as the price mechanism

We have the pair of tokens $X = \text{USDC}$ and $Y = \text{ETH}$ with reserves and $x = 10,000,000$ $\text{USDC}$ and $y = 5,000$ $\text{ETH}$.

A trader wants to swap in $2000 \hspace{1mm} \text{USDC}$ ($\Delta x$) to purchase $\text{ETH}$ ($\Delta y$). The trader receives the following in return:

$\Delta y = -\frac{y}{x} \times \Delta x = -\frac{5000}{10,000,000} \times 2000 = -1 \hspace{1mm} \text{ETH}$

The new state of the pair's reserves is $x = 10,002,000$ and $y = 4,999$.

But notice how much $\text{USDC}$ the trader would receive if they trade this $1 \hspace{1mm} \text{ETH}$ back in:

$\Delta x = -\frac{x}{y} \times \Delta y = -\frac{10,002,000}{4,999} \times 1 = -2000.8 \hspace{1mm} \text{USDC}$

Interesting! The trader decides to trade their newly gained $1$ $\text{ETH}$ back for $2000.8$ $\text{USDC}$. The new reserves are $x = 9,999,999.2$ and $y = 5000$

The trader has effectively drained the pair's reserves for $0.8$ $\text{USDC}$, which is unacceptable.

So, what's going wrong? Let's make the problem clearer.

In the initial state, swapping in $\Delta x = 1$ $\text{USDC}$ gets out:

$\Delta y = -\frac{y}{x} \times \Delta x = -\frac{5000}{10,000,000} \times 1 = -0.0005 \hspace{2mm} \text{ETH}$

The updated reserves are:

$x_{new} = 10,000,001 \hspace{2mm} \text{USDC} \hspace{4mm} y_{new} = 4999.9995 \hspace{2mm} \text{ETH}$

Now, swapping in $\Delta x = 1$ $\text{USDC}$ again gets out:

$\Delta y = -\frac{y}{x} \times \Delta x = -\frac{4999.9995}{10,000,001} \times 1 = -0.00049999995 \hspace{2mm} \text{ETH}$

Notice how swapping in the first $1 \hspace{1mm} \text{USDC}$ yielded more $\text{ETH}$ compared to the next $1 \hspace{1mm} \text{USDC}$. This trend continues as we swap successive $1 \hspace{1mm} \text{USDC}$.

Let's now see what would happen if we'd have swapped $\Delta x = 2 \hspace{1mm} \text{USDC}$ at once:

$\Delta y = -\frac{y}{x} \times \Delta x = -\frac{5000}{10,000,000} \times 2 = -0.001 \hspace{2mm} \text{ETH}$

We got out more $\text{ETH}$ by simply swapping in $2 \hspace{1mm} \text{USDC}$ at once instead of two successive $1 \hspace{1mm} \text{USDC}$ swaps.

Each swap in of the smallest possible $\Delta x$ in exchange for getting out the corresponding $\Delta y$ changes the reserves; it increases $x$ and decreases $y$, which makes $\frac{y}{x}$ smaller.

Therefore, each successive trade of the smallest possible $\Delta x$ should be evaluated at the reserve ratio $\frac{y}{x}$ based on the changed reserves resulting from the prior trade. As we've seen, the reserve ratio continues to decrease on each successive trade. You get out successively less $\Delta y$ for every marginal $\Delta x$ swapped in for it.

And so, in our example of swapping in $2000 \hspace{1mm} \text{USDC}$, we end up giving out more $\text{ETH}$ than we should have because we calculate $\Delta y$ for the entire $\Delta x = 2000$ amount using the ratio of the reserves right at the start of the trade, where the ratio is at its highest!

The right calculation for $\Delta y$ would, instead, evaluate the formula

$\Delta y = -\frac{y}{x} \times \Delta x$

at infinitely many tiny steps. Each step takes in an infinitely tiny amount of $\Delta x$ and calculates $\Delta y$ at the current reserve ratio. After each step, both reserves are updated. The next $\Delta y$ is calculated using the updated reserves. The total $\Delta y$ to give out is the sum of these infinitely many $\Delta y$ outputs.

Deriving the Constant Product Invariant

The reserves $x$ and $y$ are not independent values. The whole point of the reserves being tradable is that adding some $\Delta x$ into the $x$ reserve yields some $\Delta y$ out of the $y$ reserve.

Therefore, $y$ is a function of $x$:

$y = f(x)$

Specifically, we care about a function $f$ that relates the $x$ and $y$ reserves in the way that the previous section just established: the change in $y$ from a given change in $x$ should be based on a continuously adjusting reserve ratio.

We don't know what this $f$ looks like, exactly. In fact, our goal is to find $f$ so that, given an increase $\Delta x$ in the $x$ reserve, we can calculate the correct corresponding $\Delta y$ to give out from the $y$ reserve:

$\Delta y = f(x + \Delta x) - f(x)$

Note that for an arbitrarily small $\Delta x$, the calculation $f(x + \Delta x) - f(x)$ simply corresponds to $\Delta f$ i.e. the change in $f$ resulting from an arbitrarily small change in $x$.

So, for a given set of current reserves of $x$ and $y$, the formula:

$\Delta f = \Delta y = -\frac{y}{x} \times \Delta x$

gives us the change in $f$ for an arbitrarily small change in $x$.

Therefore, though we don't know what the function $f$ looks like, we do know what the derivative of $f$ i.e. the function $f'$ ("f prime") looks like!

$f'(x) = -\frac{y}{x}$

This derivative $f'(x)$ tells us the rate at which $f(x)$ changes in a given current state of reserves $x$ and $y$ (which itself is a function of $x$), and an arbitrarily small change $\Delta x$ in the $x$ reserve.

We can now recover $f$ from $f'$ via integration, and then we'll have a concrete way of calculating the right $\Delta y$ amount that should be taken out from the $y$ reserve for a given $\Delta x$ increase in the $x$ reserve.

Given $y = f(x)$, we have:

$f'(x) = \frac{d}{dx} f(x) = \frac{dy}{dx} = -\frac{y}{x}$

since $\frac{dy}{dx}$ refers to the rate at which $y$ changes for an arbitrarily small change in $x$.

$\frac{1}{y} \times \frac{dy}{dx} = -\frac{1}{x}$

Now, integrate both sides with respect to $x$ $\int \frac{1}{y} \cdot \frac{dy}{dx} \hspace{1mm} dx= \int \frac{-1}{x} \hspace{1mm} dx$

The right hand side evaluates as

$\int \frac{-1}{x} \hspace{1mm} dx = -\ln(x) + C$

For the left hand side, note $\frac{1}{y} \cdot \frac{dy}{dx}$ is the derivative of $\ln(y)$ when $y$ is a function of $x$. Therefore:

$\int \frac{1}{y} \cdot \frac{dy}{dx} \hspace{1mm} dx = \int \frac{d}{dx} \ln(y) \hspace{1mm} dx = \ln(y)$

Putting both sides together:

$\ln(y) = -\ln(x) + C$

$\ln(y) + \ln(x) = C$

$\ln(x \times y) = C$

$x \times y = k \hspace{4mm} \equiv \hspace{4mm} f(x) = y = \frac{k}{x}$

That looks familiar! We've derived the famous Constant Product Invariant. Now, what does it mean?

We've been trying to find a way to calculate the "right" $\Delta y$ to give out of the $y$ reserve in exchange for an increase $\Delta x$ in the $x$ reserve.

We established earlier that the "right" $\Delta y$ means that for an incoming $\Delta x$, we calculate the corresponding $\Delta y$ as the sum of the outputs of infinitely many runs of the following:

$\Delta y_n = -\frac{y}{x} \times \Delta x_n$

where each run $n$ takes in an infinitely tiny amount of $\Delta x$ and calculates the corresponding $\Delta y$ output at the reserve ratio resulting from the previous run $n-1$, at reserve values $x_{n-1}$ and $y_{n-1}$.

However, performing this operation is not straightforward, since the $\Delta x_n$ in each run of the formula is arbitrarily small and does not have a definite numerical value. And so, we can't put a definite numerical value to corresponding $\Delta y_n$ output, and just sum up all the infinite $\Delta y$ outputs.

But then, we established that the $y$ reserve is a function of the $x$ reserve, and therefore, upon finding $f$ such that $y = f(x)$, we can calculate the "right" $\Delta y$ to give out in exchange for an incoming $\Delta x$ as follows:

$\Delta y = f(x + \Delta x) - f(x)$

We then concretely derived the function $f$ as a direct consequence of our initial understanding of how $\Delta y$ for an incoming $\Delta x$ should be calculated with a constantly adjusting reserve ratio in mind.

The function $f$ we found that dictates what the $y$ reserves should be for any given $x$ reserve is:

$f(x) = y = \frac{k}{x}$

where $k$ can be thought of as the product of the two reserves. This product is first established when a liquidity provider seeds a UniswapV2 pair with balances of token $X$ and $Y$, and this $k$ changes every time a liquidity provider adds to or removes liquidity (i.e. their share of the reserves) from the pair.

Therefore, when no liquidity provider is adding or removing liquidity from the pair, the function $f(x) = y = \frac{k}{x}$ describes all the values that the $x$ and $y$ reserves are allowed to take.

That is, $x$ and $y$ are allowed to change in all the ways that keeps $k$ the same i.e. constant. And apart from adding and removing liquidity by liquidity providers, $x$ and $y$ change during swaps. This is why swaps are considered valid only if:

$(x + \Delta x) \times (y - \Delta y) = x \times y = k$

As long as swaps obey this invariant, we know that the swap is taking out the right amount of $\Delta y$ in exchange for an incoming $\Delta x$, and the trade is happening, intuitively, near the spot prices of the two tokens $X$ and $Y$, comparable with non-smart-contract systems like centralized exchanges or trading apps.