# Statistics of Livepeer issuance and participation rate

*Andrew W. Macpherson*
*Shtuka Research*
*2025-03-19*

**TL;DR.** To evaluate the effectiveness of the Livepeer issuance control mechanism, we gathered data on issuance, participation rate, and a market signal from 2024 and fit a power law model. The market signal and issuance together turn out to have very strong predictive power for the participation rate, while a model based on either one of the factors alone is much weaker; this lends credence to the idea that the issuance rate does significantly influence participation in practice. We conclude by using the model estimator to make predictions about the effect of capping issuance.

*The following analysis is purely informational and should not be construed as investment or policy advice. We bear no responsibility for policy decisions taken based on the analyses presented in this memo.*

## Background

The Livepeer Token issuance mechanism is governed according to a principled system of incentivising a targeted *participation rate*, that is, proportion of outstanding LPT tokens locked in a staking system.[^petkanics] Since newly minted LPT is distributed to stakers, the issuance rate directly influences the LPT-denominated value of stake positions.[^formula] Under ordinary demand conditions there is therefore a positive relationship between issuance rate and demand for staked, as opposed to liquid, LPT. With these assumptions, the idea of the issuance controller is easily understood: when participation is above, resp. below the target, reduce, resp. increase issuance to move up or down the demand curve.

[^petkanics]: https://petkanics.medium.com/inflation-and-participation-in-stake-based-token-protocols-1593688612bf
[^formula]: A simple model for the valuation of uncompounded staked LPT positions is given by the formula $V_\mathrm{stake}(x) = \sum_{n\in\mathbb{N}} \mathbb{E}[I_n/\rho_n]x - \Lambda(x)$ where $I$ is the issuance, $\rho$ is participation rate, and $\Lambda(x)$ is a liquidity premium. But we won't use this.

How well does the intuition behind this mechanism hold up in practice? At first glance, the situation doesn't look great: the participation rate has not reached the target of $50\%$ since early 2023. Correspondingly, the issuance parameter has increased every single round since then. The community has become concerned that the issuance rate may now be getting too high, potentially undermining LPT liquidity and even the attractiveness of LPT as an investment. Moreover, the persistent failure of the system to reach its target participation rate calls into question whether the controller mechanism is even working.

As a response, the community is considering measures to limit inflation.[^issuance-discussion][^issuance-lip] The Livepeer Protocol has been deployed on Arbitrum One since February 2022, so we have 3 years of data available to inform decisions on these measures. By fitting a simple statistical model, we can:

* Quantify the extent to which issuance rate actually controls demand for staked LPT;
* Make predictions about participation rate under given issuance and market conditions;
* Make predictions about conditions under which issuance limits would saturate, rendering the mechanism inert and staking demand uncontrolled.

[^issuance-discussion]: https://forum.livepeer.org/t/inflation-focused-lip-discussion-thread/2753/
[^issuance-lip]: https://github.com/dob/LIPs/blob/dob/inflation/LIPs/LIP-inflation-params.md

Since there are likely other market factors that affect the demand for staked LPT relative to liquid LPT, we should also consider introducing more factors as regressands. It turns out that controlling for issuance, the USD price of Bitcoin has a strong negative association with participation rate. Intuitively, this should not be surprising: when Bitcoin prices are surging, active trading becomes more appealing as a strategy than passive yield farming, even at the relatively high rates available for LPT stakers.

## Model

We queried historic data on Livepeer issuance $I_\bullet$ and participation rate $\rho_\bullet$ from an Arbitrum archive node. We als pulled historic BTC/USD prices from CoinGecko and low-passed them, replacing the data with its 30 day moving average $P_\bullet$. The dataset covers the calendar year of 2024 with samples taken at midnight UTC (366 data points).

Since $\rho,I,P$ are all positive variables, we'll fit a power law
$$
\rho = e^\alpha\cdot I^{\beta_I} \cdot P^{\beta_P}
$$
where $\alpha,\beta_I,\beta_P\in\mathbb{R}$ are parameters that we will learn. This is isomorphic to a linear model in $\log\rho,\log I , \log P$, which we normalise and fit using OLS regression. The results of this regression, as well as of the restricted models obtained by regressing on each of the factors alone, are summarised in the appendix.

It turns out the dollar price of Bitcoin has high explanatory power in this model: including it as a regressand pushes the adjusted $R^2$ from the $13.7\%$ of a univariate model to over $80\%$. That is, over $80\%$ of the variance of the staking rate is coming from variance in BTC price and issuance parameter.

The specific relation found was
$$
\mathbb{E}[\hat\beta] = (-1.2574, 1.3335)
$$
leading to a participation rate prediction formula
$$
\bar\rho \approx 1.33\cdot \bar I - 1.26\cdot \bar P
$$
where $\bar\rho = (\log\rho - \mu_{\log\rho}) / \sigma_{\log\rho}$ and similarly for $\bar{I}$ and $\bar{P}$.

Qualitatively, one notices that (normalised log) participation rate is roughly (to 2 s.f.) proportional to the difference between (normalised log) issuance parameter and Bitcoin price, indicating a positive dependence on issuance and negative dependence on Bitcoin price of roughly equal importance.

## Predictions

Using the model we can get a preliminary idea of what the equilibrium participation rate and staking APY might look like under different values of the issuance parameter. If the community fixes one of these as a maximum, then for sufficiently high values of $P_\mathrm{BTC}$ the controller will saturate with participation rate below $50\%$.

| parameter | annual issuance | BTCUSD | equilibrium participation rate | APY at equilibrium |
| --------- | --------------- | ------ | ------------------------------ | ------------------ |
| 900000 | 44.9% | 80K | 57% | 78.7% |
| 900000 | 44.9% | 100K | 53% | 84.6% |
| 750000 | 36.2% | 80K | 51.4% | 70.4% |
| 750000 | 36.2% | 100K | 47.9% | 75.6% |
| 250000 | 10.9% | 80K | 27.7% | 39.2% |
| 250000 | 10.9% | 100K | 25.8% | 42.1% |

## Appendix: Regression results

# ISSUANCE ONLY

R-squared (uncentered):                   0.140
Adj. R-squared (uncentered):              0.137
F-statistic:                              41.01
Prob (F-statistic):                    7.41e-10
Log-Likelihood:                         -338.00

==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
iss            0.3747      0.059      6.404      0.000       0.259       0.490
==============================================================================

# BITCOIN AND ISSUANCE

R-squared (uncentered):                   0.802
Adj. R-squared (uncentered):              0.801
F-statistic:                              507.3
Prob (F-statistic):                    9.90e-89
Log-Likelihood:                         -152.81
                                            
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
btc           -1.2574      0.043    -28.932      0.000      -1.343      -1.172
iss            1.3335      0.043     30.683      0.000       1.248       1.419
==============================================================================

# BITCOIN ONLY

R-squared (uncentered):                   0.058
Adj. R-squared (uncentered):              0.054
F-statistic:                              15.43
Prob (F-statistic):                    0.000111
Log-Likelihood:                         -349.55

==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
btc           -0.2407      0.061     -3.928      0.000      -0.361      -0.120
==============================================================================

**Participation rate data, predictions, and residuals against time (normalised)**


**Residuals plot**

The evident heteroskedasiticity looks like it could be addressed by a clustering/régime model.