Do shared MEV auctions actually increase revenue?

It’s sometimes taken as an article of faith that selling access to MEV on multiple chains will yield more revenue than selling it separately on each chain. The argument for this is that controlling MEV extraction on multiple chains is more valuable because of opportunities to capture cross-chain MEV are available to someone who knows they control MEV across multiple chains.

A simple model shows that this isn’t always the case. Sometimes selling MEV separately gives more revenue.

Let’s start by analyzing separate auctions for each chain. We’ll assume a chain uses a sealed-bid, second-price auction to allocate MEV control over some time period. So we know that the revenue will equal the second-highest valuation of any bidder.

We’ll assume that players i's valuation for MEV on chain A will be

V_{A,i} = V_A^*+A_i

where V_A^* is the “true valuation” which is unknown to the players and A_i is a player-specific estimation error that is random and independent for each player, drawn from a normal distribution \mathcal{N}(0, \sigma^2).

If we auction the MEV rights on chain A among n bidders, the resulting revenue will be

R_A = V_A^* + \sigma\cdot\alpha(n)

where \alpha(n) is defined to be the expected value of the second-largest of n samples from the standard normal distribution \mathcal{N}(0, 1). Note that \alpha(n) is increasing in n and is positive for n > 3.

If we have two such chains, A and B, with different “true” values V_A^* and V_B^* but (for simplicity) the same \sigma, then the expected total revenue from auctioning the two chains’ value separately is

R_\mathrm{sep} = V_A^* + V_B^* + 2\sigma\cdot\alpha(n)

What if we auction the rights to both chains together, as a single unit? Then the valuation for party i is

V_{AB, i} = V_A^* + V_B^* + C_i + M

where C_i = A_i+B_i is random with distribution \mathcal{N}(0, 2\sigma^2), and M is the extra value due to cross-chain MEV.

The revenue is then the expected second-highest value, which is

R_\mathrm{joint} = V_A^*+V_B^*+M+\sqrt{2}\sigma\cdot \alpha(N)

So we can see that

R_\mathrm{joint} - R_\mathrm{sep} = M-(2-\sqrt{2})\sigma\cdot\alpha(N)

If we generalize this to K chains, we get that

R_\mathrm{joint}-R_\mathrm{sep} = M-(K-\sqrt{K})\sigma\cdot\alpha(N)

So we can see that even if there is some cross-chain MEV, a separate auction might still be better, because it assigns each chain to the party who values it most–which may not be possible in a joint auction.

At the very least, we can see that a joint auction is not a no-brainer. We would want to do some measurement, or adopt an auction structure that can accurately determine when to sell jointly versus separately.

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Why not sell the blocks through a combinatorial auction?

A combinatorial auction is a reasonable alternative, especially if there are only two chains. If the number of chains gets large, a fully general combinatorial auction is unwieldy and creates a complex strategy space. There are some very interesting research questions around how best to do that.

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Doesn’t this ignore the reality that today’s MEV markets (at least on Ethereum) are two-tiered? Searched submit their bids to builders (usually multiple), and builders submit their bids to block proposers.

As such you would expect most builders to value the MEV on every chain pretty similarly as they are all aware of the opportunities submitted by the searchers.

The existence of builder private order flow (as notably exists with Telegram trading bots) complicates the situation.

On the one hand, in the presence of private order flow to different builders on different chains, it leads to a situation where the MEV revenue in a shared auction can be less than in separate auctions.

On the other hand, as long as there enough competition on private order flow, this disincentives private order flow, and avoids the emergence of one dominant builder (which is bad for decentralization and censorship-resistance).

Sadly, the emergence of a dominant builder is worse in the shared auction scenario than in the separate auctions one, as the dominant builder would now control block production for every chain in the system.

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I don’t think that two-tier markets invalidate this model. In a two-tier world, builders are the bidders. Their valuations will vary, which this model would treat as part of the normally-distributed per-bidder part of the valuation.

There is definitely good reason to worry about centralization in the builder model, whether on a single chain or cross-chain.

I think the simple model is great as a starting point and baseline measurement. The choice of normal distribution is also clever because you can easily get a nice closed form solution that you can compare and simulate easily.

In my recent article, I took inspiration from @edfelten’s post and expanded some thoughts on how to model in scenarios where bidders have different risk preferences (risk aversion), dynamic auctions models like sequential auctions instead of simultaneous and relaxed combinatorial auctions. You can read it here: Cross-Chain MEV Auctions - Modeling Proposer Revenue Maximization - Dare to Know

Any comments, corrections and feedback will help me improve these thoughts.

Thanks! That looks very interesting. I look forward to reading it carefully.

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