The LVR gap for random block times: a quick and useful approximation

Research
1

Thank you to @dataalways for promting me on LVR and slot times.

In a recent arxiv paper the difference in Loss Versus Rebalancing (LVR) under constant slot times and random slot times was approximated. They differ (up to correction by higher order terms) by a factor \tfrac{\sqrt{\pi}}{|\zeta(1/2)|}C_{\mu} which in general depends on the whole distribution of the random slot times. While this constant can be derived, in principle, for the slot time distribution one is interested in, either analytically (if feasible) or through Monte Carlo simulation, it would be nice to have something a bit easier, for us smol brain researchers that prefer to do mean-variance analysis. This has two advantages: 1) we just need to plug in two numbers (or one number after we have normalized average slot times) and 2) we can very easily do comperative statics.

Let’s therefore do an approximation of C_\mu. I normalize average slot times to 1 and want to approximate C_\mu by an expression that should only depends on the variance \sigma_U^2 of block times, but not on higher moments. Let U_1,U_2,\ldots be the slot times, and let us denote the average slot time for the first n slots by \bar{U}_n:=\tfrac{1}{n}\sum_{i=1}^nU_i.
We have

C_\mu=-\tfrac{1}{2\sqrt{\pi}}\sum_{n=1}^\infty\tfrac{1}{\sqrt{n}}\left(\sqrt{\bar{U}_n}-1\right).

By the Strong LLN, we have

\lim_{n\to\infty}\bar{U}_n=1\quad a.s.

Taylor expansion around the expected slot time, gives an expression for the sqrt of the average slot time for the first n slots:

\sqrt{\bar{U}_n}\approx1+\tfrac{1}{2}(\bar{U}_n-1)-\tfrac{1}{8}(\bar{U}_n-1)^2+\ldots

Taking expectations:

\mathbb{E}\sqrt{\bar{U}_n}\approx1-\tfrac{1}{8}Var(\bar{U}_n)+\ldots=1-\tfrac{1}{8n}\sigma_U^2+\ldots

Thus

C_{\mu}\approx \tfrac{1}{2\sqrt{\pi}}\sum \tfrac{1}{\sqrt{n}}\tfrac{1}{8n}\sigma_U^2=\tfrac{1}{2\sqrt{\pi}}\tfrac{\sigma_U^2}{8}\zeta(3/2)\approx 0.092\sigma_U^2

where \zeta is the Riemann zeta function so that \zeta(3/2)\approx 2.612.

So far so great, but what does it tell us? Well that the factor scales with variance. For the Poisson distribution, for example, variance is equal to the mean. Thus, the gap C_\mu\approx 9.2\% is substantial.[1] Let’s look, however, (motivated by this interesting tweet) at the empirical distribution of Ethereum slot times (which is the result of a mixture of latency effects and timing games): average slot times are 12 seconds with std of \sigma_U\approx 0.58\,\text{seconds} (estimated from data provided by @dataalways, any errors in this estimation are due to me). Normalizing to slot time, we have \sigma_U=0.04833 . Thus C_{\mu}\approx 0.0213\%. It’s a matter of interpretation of course, but I would call 2 basis points nothing. So yes, timing games increase LVR, but, in theory, only by a small, small negligible factor.

Let’s look at the bigger picture: do random slot times increase LVR substantially? They do for the Poisson as documented in the paper. But the Poisson has high variance relative to its mean. So if you want to have one take away from the math above:

:robot: Random Slot Times only influence LVR in a meaningful way if variance is within the same order of magnitude as average slot times.

  1. My approximation slightly overestimates the gap, but not by too much. The more precise gap from the paper is \tfrac{1}{2}-\tfrac{|\zeta(1/2)|}{2\sqrt{\pi}}\approx 8.8\%. ↩︎

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