In this note, we compute how much penalty per white-label validator is needed in order to disincentivize operators from using white-labeling services. Thereafter, we compute the expected profit for honest operators, in order to understand if investing in being a solo operator yields a positive expected return.
Our analysis will need to take into account the following parameters and probabilities:
$n$: total number of validators. In the case of a dishonest operator, we assume all are white-label. In the case of an honest operator, we assume none are white-label. (There is no need to model mixed strategies since these are simply the superposition of both cases)
$t$: denotes the number of months that the nodes have been active.
$r$: the expected reward per month per validator of a Lido operator, expressed in ETH. This is readily computed by dividing the yearly revenue per validator by 12:
$$ r = \frac{32 \times \text{APR } \times \text{operator reward percentage}}{12}. $$
The operator reward percentage is commonly chosen to be 5%. Also, although in practice the APR can vary in time, we shall choose its mean value to work with a constant. Consider two examples:
Then the total revenue is simply $r\cdot n \cdot t$.
$\Xi_t$: the price of Ether in USD at month $t$.
$c_{w}$ : the average ETH cost, per month per validator, of an operator using white-labeling services for $t$ months. By defining $c_{w}$ as such, the total costs are simply $c_{w} \cdot n \cdot t$. According to our analysis in Market analysis of staking providers we will use at least two different models:
Assume a fixed monthly charge in dollars per validator, denoted $\tilde c_w$. In order to price the mean cost in ETH over a period of $t$ months, we write
$$ c_{w}=\frac{\tilde c_w \cdot \Xi_0^{-1} + \tilde c_w \cdot \Xi_1^{-1}
where $M_t$ denotes the mean value of $\Xi^{-1}$ over $t$ months. This formula gives a more realistic depiction of the effect of Ether’s price on an operator who sells ETH (or stETH) month to month to pay for white-label services.
As an example, if $\tilde c_w = 5$ and $M_t = (\$ 2000/\text{ETH})^{-1}$ then $c_{w} = 0.0025 \text{ ETH}$. Once more, although $c_w$ may vary in time, we will use a mean value in practice.
A percentage of the validator’s total APR—say, 3%. Since the node operator only gets 5% of the total APR, this 3% cut corresponds to 60% of his earnings, i.e. $c_w = 0.6 \cdot r$. Note that this model is simpler since it does not depend on the price of ETH.
$c_h$: the cost for an operator (in ETH) to run his validators honestly. According to conversations with the Lido team and with the DevOps team at Nethermind, for small- to mid-scale operations (say, $n<1000$ validators) the costs are flat and can be approximated as a one-time investment in equipment given by $\tilde c_h = \$1500$. Thus, $c_h = \tilde c_h \cdot \Xi_0^{-1}$. (Note that we price the operator’s investment in ETH at time $t=0$.)
$C$: represents the dispute mechanism’s court fees. For the purposes of this note, we estimate $C = 0.237 \text{ ETH}$, as we chose in Bonding requirements for operating the courts.
$G(n)$: The penalty in ETH for an operator found guilty of an accusation of white-labeling with $n$ validators. Normally, we shall have $G(n) = R(n) + C$, since the loss to the operator will come from the accuser’s reward $R(n)$ and court fees. This penalty could be increased further if deemed necessary, however.
$I(n)$: The compensation in ETH for an operator found innocent of an accusation of white-labeling with $n$ validators. Normally, we shall have $I(n) = A(n) - C$, since the operator gets to keep the accuser’s bond $A(n)$ after subtracting court fees.
From the parameters above, the total profit for an operator using white-label services over $t$ months is simply
$$ \Pi_w^{n,t} = r \cdot n \cdot t - c_{w} \cdot n \cdot t = (r-c_{w}) \cdot n \cdot t, $$
with $c_{w}$ described by any of the two models explained above. On the other hand, the profit for an honest operator is
$$ \Pi_h^{w,t}=r\cdot n\cdot t-c_h. $$
Of course, this does not account for the possibility of a punishment $G(n)$ or reward $I(n)$ due to the dispute resolution mechanism. To account for these, we must take a probabilistic approach to both detection and conviction.