Are Liquid Restaking Tokens (LRTs) essential to restaking security and even risk mitigation, vs. being a source of systemic risk?

Surprisingly, yes! New paper w/ @malleshpai shows that smart allocation to AVSs is crucial for security against cascading failures

https://arxiv.org/abs/2408.00928

Game theory, mechanism design, blockchains, DeFi

Our paper builds off of the excellent work of Tim Roughgarden & Naveen Durvasula that formulates cascade risk in combinatorial terms and argues that overcollateralization is needed for security — but we arrived at this via a circuitous path

First: Storytime 📗

ヽ(⌐■_■)ノ♪♬ @ @gauntlet/🤖Ventures/tldr/Aera \\ ¯\_(ツ)_/¯ Struggling with the nature of randomness \\ main: @gaa \\ /red-bull /defi-research maxi

Our paper builds off of the excellent work of Tim Roughgarden & Naveen Durvasula that formulates cascade risk in combinatorial terms and argues that overcollateralization is needed for security — but we arrived at this via a circuitous path

First: Storytime 📗

https://x.com/n_durvasula/status/1819127937192308967

Jan '24: I read Appendix B of the Eigenlayer paper and found a lot of similarities between the allocation problem and an old paper of mine (+ Alex Evans) on LSTs and on-chain lending

Cascade risks seemed like a multidimensional version of our old model — I got overly optimistic about our model applying here

The old paper's insight: If rational stakers *rebalance* between LSTs and DeFi based on yield → risk of cascading slashing events 📉

So: I simulated a multidimensional version (each dim. = abs. return of each AVS) and tried to measure cascade risk (quantified by a regret measure)

Simulations show regret:
- Growing linearly in time w/o rebalancing
- Asymptotes/slow growth with rebalancing

Strongly suggests that smart rebalancing from LRTs (or large stakers) is important for reducing cascade risks

Extending our old result should easily show this right?

Wrong! Combinatorial examples in Tim + Naveen's paper show obstructions to 'naive' extensions of our old paper that don't take into account the precise structure of a restaking graph

Obstruction: "Global" nature of cascades (i.e. attack w/ Ω(n) nodes)

Wrong! Combinatorial examples in Tim + Naveen's paper show obstructions to 'naive' extensions of our old paper that don't take into account the precise structure of a restaking graph

Obstruction: "Global" nature of cascades (i.e. attack w/ Ω(n) nodes)

https://x.com/Tim_Roughgarden/status/1819086514451894638

Let's see why in pictures

Below is a restaking graph where the black dots are node operators and the boxes are the AVSs

An attack against a single node operator cascades into a series of attacks until the entire network's stake is slashed

💀💀💀 A worst-case outcome!

On the other hand, if LRTs / operators _can_ rebalance in response to slashes, we can *arrest* this attack and stop it far before it spreads to the whole network

In this figure, a node operator rebalances to form the yellow service which then leads to a graph with no attacks

This example suggests that while overcollateralization is necessary (as T&N showed), if node operators strategically rebalance (due to rewards) in response to slashing events, then one need not be as overcollateralized as in T&N's initial paper

Why does this matter?

In Appendix B of our paper, we show an example where the overcollateralization required from T&N's paper is global and potentially requires the smallest AVS (by attack profit) to attract stake equivalent to what the highest AVS has to attract for security —  super expensive!

Concretely: Consider two AVSs, one with profit from corruption of 1 ETH and the other with profit of 1,000,000 ETH

Globally overcollateralization → 1 ETH profit AVS has to attract on the order of 1,000,001 ETH

(May as well start a new PoS network vs. joining restaking network)

Our results show that one does not need to be as overcollateralized if:
- Node operators actively rebalancing (like the sims showed)
- Attackers face non-trivial profit for attacking multiple services simultaneously 

∴ Smarter LRTs + Poorer Attackers = More Capital Efficiency

So what exactly do we show?

If:
1. Incentives: Dynamic fees (or points) direct LRTs to allocate to parts of network that need more security
2. Costly Attacks: Cost of attacking n services 📈 (e.g. submodularity of profit) localizes attacks

Then:
Bounded cascade length R(G)

To achieve such security, AVSs need to provide a certain amount of rewards in ETH terms which is used to 'direct' where LRTs should allocate — how should AVSs choose their rewards? 

We provide an approximation algo for estimating optimal rewards (using a convex relaxation)

What are the main takeaways?
1. 𝐋𝐑𝐓𝐬 𝐧𝐞𝐞𝐝 𝐭𝐨 𝐚𝐥𝐥𝐨𝐜𝐚𝐭𝐞 𝐢𝐧𝐭𝐞𝐥𝐥𝐢𝐠𝐞𝐧𝐭𝐥𝐲 𝐟𝐨𝐫 𝐧𝐞𝐭𝐰𝐨𝐫𝐤-𝐰𝐢𝐝𝐞 𝐬𝐞𝐜𝐮𝐫𝐢𝐭𝐲
2. 𝐀𝐕𝐒𝐬 𝐧𝐞𝐞𝐝 𝐭𝐨 𝐜𝐡𝐨𝐨𝐬𝐞 𝐚 𝐦𝐢𝐧𝐢𝐦𝐮𝐦 𝐲𝐢𝐞𝐥𝐝 / 𝐫𝐞𝐰𝐚𝐫𝐝𝐬 𝐭𝐡𝐚𝐭 𝐚𝐯𝐨𝐢𝐝 𝐜𝐚𝐬𝐜𝐚𝐝𝐞 𝐫𝐢𝐬𝐤

3. Restaking risk combines:
a. PoS risk (e.g. combinatorial nature, rewards, attackers)
b. DeFi risk (e.g. cascades)

Analyzing risk from only one POV underestimates risk dramatically (e.g. what most people are doing)

That's why it took me so long to go from simulation to proof!

I also wanted to thank a lot of people who I ranted at about this problem over the last year — thanks!

This problem really drove me and @malleshpai crazy for so long because the simulation/intuition was "obvious" but the proof was filled with many twists, turns, and dead-ends

Some fun things about the proof:
1. Step 2 resembles the Lovasz Local Lemma (but isn't quite the same! c.f. @gaussianprocess)
2. Step 3 resembles formal verification
3. Jan Vondrák's lectures (Maryam Mirzakhani's husband) helped me figure out a lot of key steps 🐐