Write the discrepancy between the block proposer(BP)’s order book and that of a node $i$ at block $j$ as the following:
$$ BP_{i}^j=MEV(O_{BP}^j,O_{i}^j) $$
Where $O_{BP}^j,O_i^j$ are the matched orders from the block proposer and node $i$ at block $j$, respectively. See Catching a bad validator: Appendix for the calculation of the MEV metric based on matched orders.
Then, the MEV Threshold can be calculated as the following:
$$ \tau_{i}^j=\mu_k^i+a\sigma_k^i $$
where $\mu_k^i$ and $\sigma_k^i$ are the sample mean and standard deviation of the sample given by:
$$ \left\{MEV(O_1,O_k), ..., MEV(O_i, O_k) ,... , MEV(O_N,O_k); i\neq k \right\} $$
$O_i$ is the matched orders of the $i$th node. We use $a = 2$.
By comparing $\tau_i^j$ and $BP_i^j$, each full node can evaluate the block proposer’s behavior. In particular, if a full node decides that the block proposer has extracted MEV if $\tau_i^j < BP_i^j$ .
For a given block proposed by a validator, we can suspect that MEV was malicious extracted if the proportion of nodes that evaluated this validator as having extracted MEV is larger than a threshold.
We write
$$ n_r = \sum_i^n \mathbb{1}{red} \newline \text{where } \mathbb{1}{red} = \begin{cases} 1 &\text{if}\quad BP_{i}^j > \tau_i^j\\ 0 &\text{else } \end{cases} $$
where $i$ indexes over $n$ nodes participating in the orderbook discrepancy measurement.
We conclude the block proposer as having extracted MEV if
$$ \frac{n_r}{n}\geq b $$