Description

In _swapBaseToBase, a base-to-base swap is executed by dividing the max spread of the two pairs, base-to-quote and quote-to-second-base, by two:

function _swapBaseToBase(
    address baseToken1,
    address baseToken2,
    uint256 base1Amount,
    uint256 minBase2Amount,
    address to,
    address rebateTo
) private nonReentrant whenNotPaused returns (uint256 base2Amount) {
    // [...]

    uint64 spread = _maxUInt64(state1.spread, state2.spread) / 2;
    uint16 feeRate = _maxUInt16(tokenInfos[baseToken1].feeRate, tokenInfos[baseToken2].feeRate);

    state1.spread = spread;
    state2.spread = spread;

    uint256 newBase1Price;
    (quoteAmount, newBase1Price) = _calcQuoteAmountSellBase(baseToken1, base1Amount, state1);

    // [...]

    uint256 newBase2Price;
    (base2Amount, newBase2Price) = _calcBaseAmountSellQuote(baseToken2, quoteAmount, state2);

    // [...]
}

Note that the swap-fee logic has been omitted from the above snippet to simplify the math below. Also, note that _sellBase and _sellQuote directly pass the state into the _calc* functions.

Assume that a user is swapping token A for token Q, which is the quote token, and then subsequently token Q for token B. Leaving out constant factors like decimals, let $n_A$ be the notional amount of A, defined as $\Delta A\ast p_A$ where $\Delta A$ is the amount of token A swapped and $p_A$ is the price.

With this, according to _calcQuoteAmountSellBase, for the sell base we have $\gamma =\Delta A\ast p_A\ast k_A=n_A\ast k_A$ and then $\Delta Q=\Delta A\ast p_A\ast (1-\gamma -s_A)=n_A(1-n_A{k}_A-s_A)$. Note that this is a quadratic polynomial in $n_A$ where the only occurrence of the spread is $-s_A$ as a constant linear term in the coefficient for the first degree $n_A$.

Similarly, according to _calcBaseAmountSellQuote, we have (note, this is a different $\gamma$) $\gamma =\Delta Q\ast k_B$ and then $\Delta B=\Delta Q\ast (1-\gamma -s_B)/p_B$. So, again substituting, we have $n_B=\Delta Q\ast (1-\Delta Q\ast k_B-s_B)$.

Now, if we substitute in $\Delta Q$, the result becomes a fourth-degree polynomial in $n_A$.

Now on the other hand, if they executed _swapBaseToBase instead of base-to-quote and then quote-to-base, then according to the above code snippet, the calculation will be the same except $s_A^{\prime }=s_B^{\prime }=\text{max}(s_A,s_B)/2$ — let this just be $s^{\prime }$. This means that the same expansion will follow in the calculation of $n_B^{\prime }$ in terms of $n_A$, and so the difference $n_B^{\prime }-n_B$ will cancel out all the terms that do not depend on any $s$.

So, with that simplification, the terms of $n_B$ that include some $s$ will sum to $\Delta Q\ast (1-\Delta Q\ast k_B-s_B)=(1-s_B)\Delta Q-k_B(\Delta Q{)}^2=(1-s_B)(1-s_A)n_A-(1-s_B)k_A{n}_A-k_B(1-s_A{)}^2{n}_A^2$.

So, whether the base-to-base return is better than the base-to-quote-to-base case clearly depends on the exact quantities involved, and there are examples for both. As a quick example, if we assume the spread for token B were zero, then

• The first term is always higher, because $(1-s_A/2{)}^2>(1-s_A)$.

• The second term cancels out because there is no dependence on $s_A$.

• The third term is always lower, because $-k_B(1-s_A/2{)}^2<-k_B(1-s_A{)}^2$ (note that the domain of $s_A$ is from zero to one, and the breakeven points are the roots at zero and 1.33).

This means that which is bigger depends entirely on whether $k_B$ is larger or smaller, because it will control whether the first or third term dominates and the liquidity coefficient is a completely free parameter.

Impact

Due to nonlinearity, the approximation spread = _maxUInt64(state1.spread, state2.spread) / 2 may produce a large error if there is a large difference in oracle parameters, especially the liquidity coefficient k, such that users may get better execution doing a two-stage swap over a direct swap — whether that be base-to-base-to-quote instead of base-to-quote, or a base-to-quote-to-base instead of a base-to-base depending on the direction of the discrepancy.

Recommendations

We recommend not improving the spread in the base-to-base case and using the original spreads such that the only savings of base-to-base compared to base-to-quote-to-base is that the swap fee is not charged twice.

Remediation

This issue has been acknowledged by WOOFI, and a fix was implemented in commit 879afe26↗.