# Ethereum Block Times, MEV, and LP returns

## An interesting discussion is brewing on Twitter on how the changes in Ethereum block times after the Merge are going to affect the MEV. We could not help but chime in.

The world is now days away from the Merge, and this DeFi blog has remained as-of-yet silent on the event.

Two circumstances are giving us a reason to break the silence. First, a few days ago, we attended the Science of Blockchain Conference at Stanford, where Tim Roughgarden presented a paper (hereafter MMRZ) on the losses of automated market makers to arbitrage traders. Second, we came across an interesting Twitter thread on how the changes in Ethereum block times after the Merge will affect MEV extraction.

Roughly speaking, the Merge is expected to have two effects on block times. First, the average block time is expected to decline from approximately 13 to 12 seconds. Second, there will be considerably less variation around the new mean. How will these changes affect the profits of arbitrageurs and LPs?

In this post, we will use some of the intuition from the paper presented at Stanford to answer this question for one source of MEV — arbitrage between centralized and decentralized exchanges.

# CEX-DEX arbitrage and loss versus rebalancing

One can think of the prices on decentralized exchanges that are run on the Ethereum network as being stuck between the blocks. For example, if the last block was mined six seconds ago, then the ETH-USDT price on Uniswap reflects only the information that was available to the market at that time. In contrast, prices on centralized exchanges update much more quickly. This creates an opportunity for arbitrageurs. If the ETH price on centralized exchanges went up during the last six seconds, an arbitrageur can buy ETH on Uniswap at the beginning of the next Ethereum block and sell it for a profit on a centralized exchange. The arbitrageur’s profit is a loss for Uniswap’s liquidity providers since they essentially end up selling ETH at a stale price, which is too low according to the current market conditions.

The MMRZ paper linked above calls the aggregate profit of arbitrageurs earned this way over some period of time *loss versus rebalancing*, or LVR. Our first question then is: How is LVR affected by (i) the average block time and (ii) the block time uncertainty?

To keep matters simple, we will make several assumptions. We focus on a single Uniswap v2 position, say in the ETH-USDT pair. Assume that the price of the risky asset (ETH) follows a martingale geometric Brownian motion. The LP does not add or remove liquidity for some period of time. And we will initially ignore the swap fees but will revisit this point later.

Will the arbitrageurs extract more value from the LP with shorter and more certain block times? Intuitively, shorter block times lead to smaller arbitrage opportunities since the prices on centralized and decentralized exchanges have less time to diverge. On the other hand, the number of arbitrage opportunities increases with shorter block times since more blocks are produced in any fixed period of time.

There turns out to be a strong argument that the *expected* LVR should not depend on the distribution of block times at all. The key to this argument is the following observation. (While it is not directly stated in MMRZ, it follows quickly from their Corollary 2.)

The expected value of LVR over some period of time is equal to the expected impermanent loss (IL) over the same period.

Note that the *realized* LVR and IL are generally two different quantities. LVR measures arbitrageurs’ profits and LPs’ losses on a transaction-by-transaction basis. This number monotonically increases over time. IL measures the overall loss of LPs from the opening to closing of their liquidity position. In contrast to LVR, IL can fully reverse if the asset prices return their initial values. Why are the two quantities equal in expectation?

Consider the following two strategies for an LP who has $1,000 worth of ETH and a large amount of USDT.

- LP opens a liquidity position on Uniswap v2, contributing $1,000 worth of ETH and $1,000 in USDT. LP stays put for one month and then closes the position. If the amount of ETH is different from what it was at the beginning, the LP buys or sells the difference on a centralized exchange.
- LP opens the same liquidity position on Uniswap v2. Every time a swap takes place, LP calculates the amount of ETH bought (or sold) by the liquidity position and makes an offsetting ETH sale (purchase) on a centralized exchange.

For both strategies, the LP has exactly the same amount of ETH at the end of the month as at the beginning. However, the USDT amounts will be different. For Strategy 1, the amount of USDT declines from the beginning to end of the month by IL. For Strategy 2, it declines by LVR.

Note that the two strategies are different only in transactions on *centralized* exchanges. Both strategies end up buying or selling the same net amount of ETH at market prices, with the only difference being in the timing of the transactions. If the expected payoffs of these strategies were different, then this would imply that the LP can, in expectation, make money by timing transactions on centralized exchanges. But this is impossible because we assumed that the market price of ETH follows a martingale process, and one cannot generate positive expected returns by timing a martingale. (For those interested, the formal result behind this point is the Doob’s optional sampling theorem.)

Where does this argument leave us with our question? If the expected LVR and IL are one and the same, then to have an impact on the former, the distribution of block times would have to have an impact on the latter. Yet IL only depends on the beginning and ending ETH prices. It does not matter how exactly the arbitrageurs extract value, the total amount of IL over a sufficiently long period of time will be the same in expectation. The observation above then implies that the expected arbitrageurs’ profit is also unaffected by block times.

Our story is not yet complete for two reasons. First, it is conceivable that while the *expected* LVR does not depend on block times, its *distribution* does. This is indeed the case. This point can be illustrated with some math, but we will opt out for a picture instead.

The figure above plots the distributions of simulated daily LVR for three block times — 6, 12, and 24 seconds. For daily price volatility, we use the same value as MMRZ, 5%. As a consequence the mean of all three distributions is consistent with their paper — 3.125 basis points. Note that while the mean LVR does not change with decreasing block times, the distribution of LVR becomes more concentrated around the mean. In other words, the profits of arbitrageurs and the losses of LPs become more certain with shorter block times.

There is also a second reason why our analysis is not yet complete. So far we have focused only on the losses of LPs to informed trading. But what about their revenues — the swap fees that LPs earn for providing liquidity?

# Block times and swap fees

Holding the swap fee fixed, LP revenues are increasing in the trading volume. Our next figure depicts the distributions of the daily trading volume by arbitrageurs for the three block times — 6, 12, and 24 seconds. (Caveat emptor: In simulating these trading volumes, we ignored the impact of swap fees. Arbitrageurs trade less when they have to pay a fee because the CEX and DEX prices need to diverge sufficiently far to cover all transaction costs. Since our goal is to study the impact of block times *holding fees fixed*, we will focus on zero-fee trading volumes.)

In Figure 1, the three different block times looked very similar in terms of the resulting expected LVR. Figure 2, however, shows that the same expected LVR is generated in very different ways. Arbitrageurs trade considerably more with shorter block times to achieve the same expected gains. This is good news for LPs since they can collect a higher fee revenue incurring the same expected loss.

Shorter block times lead to larger trading volumes and LP revenues from swap fees.

To see the intuition behind this result consider two situations: (1) arbitrageurs can only trade at 12 pm each day, and (2) arbitrageurs can trade very often. In scenario (1), the volume of trade will be determined by the net change in price from the afternoon of one day to the next. For instance, if these two prices are equal, then there will be no arbitrage trading even though the price may have fluctuated a lot during the day. In contrast, in scenario (2), there will almost always be some trading, and its exact volume will be determined by the length of path that the market price takes during the day.

This argument shows that shorter block times can ameliorate the problem of information asymmetry. Of course, the problem cannot be resolved fully: even after accounting for the swap fees earned, LPs lose, on average, with every arbitrage swap. Moreover, for a fixed swap fee, the benefits to reducing block time are limited. Producing the blocks more often than the time it takes for CEX and DEX prices to deviate by the swap fee will not significantly increase the arbitrage trading volume.

It turns out that *more consistent* block times also lead to larger trading volumes. In the figures below, we consider two scenarios — blocks are produced exactly 12 seconds apart or each following block is produced at random, in 6 or 18 seconds, with equal probabilities. The average block time is 12 seconds in both cases. First, consider the distributions of LVR:

These distributions are very close to each other. The distribution of LVR is mostly determined by the average block time and not the uncertainty around it.

Now compare the distribution of daily trading volumes by arbitrageurs:

Even though the average block times are the same, the arbitrage trading volume is smaller when block times are more uncertain. Why? The size of the arbitrage swap available after some time Δ*t *is proportional to the absolute value of the difference between CEX and DEX prices (at least for sufficiently small Δ*t*). For traditional price processes. such as a geometric Brownian motion, this deviation has the order of sqrt(Δ*t*). Square root is a concave function, so by Jensen’s inequality the expected value of sqrt(Δ*t*) decreases as Δ*t *gets more volatile.

Why doesn’t the same argument apply to arbitrage pre-fee profits? This is because the *profit* from an arbitrage transaction is roughly proportional to Δ*t, *not sqrt(Δ*t*). The profit is given by the product of amount swapped and the difference between CEX and DEX prices, and both of these factors are proportional to sqrt(Δ*t*).

# Conclusion

LPs benefit from both shorter and less volatile block times. In expectation, their losses due to the arbitrageurs’ informational advantage are the same. But with shorter and more certain block times, arbitrageurs trade more to generate the same expected pre-fee profit. This is beneficial to LPs since they earn more in swap fees.

Thoughts/ideas/suggestions? Leave a comment here or reach us at nezlobinalexander@gmail.com. You may also want to check out our series on order flow toxicity or our low-toxicity AMM algorithm.