I have a confusion matrix of true and estimated $\boldsymbol{\beta}$ vectors of lasso and ridge models from a replicate of a simulation study, say. The following tables illustrate the scenario.
$$\begin{array}{c|c|c|} \text{lasso} & \#\left(\beta_{j}=0\right) & \#\left(\beta_{j}\ne0\right)\\ \hline \#\left(\hat{\beta}_{j}=0\right) & 4 & 2\\ \hline \#\left(\hat{\beta}_{j}\ne0\right) & 1 & 5 \\\hline \end{array} \begin{array}{c|c|c|} \text{ridge} & \#\left(\beta_{j}=0\right) & \#\left(\beta_{j}\ne0\right)\\ \hline \#\left(\hat{\beta}_{j}=0\right) & 0 & 0\\ \hline \#\left(\hat{\beta}_{j}\ne0\right) & 5 & 7 \\\hline \end{array}$$
The false positive rate from the lasso estimates is $2/(2+5)=2/7$.
Since the ridge can not perform variable selection, the false positive rate should be 0 all the time.
But when I study a paper, I come across a simulation result in which the median false positive rate from the ridge is reported as a value different from zero.
The paper is here (or here in arxiv).
Do I overlook something?