Consider a probability distribution represented by $F(x) = 1- \frac{1}{1+x}$, leading to a virtual valuation function $ \varphi(v) = -1 $. This results in all players having negative virtual valuations, indicating an irregular distribution.
We also know that the expected revenue is equal to the expected virtual welfare/surplus. $$ \mathbb{E}{[\sum_{i=1}^{n} p_i(v)]} = \mathbb{E}{[\sum_{i=1}^{n} \varphi(v_i)x_i(v)]} $$
In our scenario, this equality clearly does not hold. If it did, it would imply a negative revenue for the auction, suggesting that simply refraining from selling the item could prevent this undesirable outcome.
I want to show that, in our specific scenario where virtual valuations are consistently negative, we can invalidate the equation by highlighting errors in its proof. I am referring to this proof on pages 4-5.
I suspect issues might lie in Myerson's lemma, like the absence of a monotone allocation rule, and in Fubini's theorem, perhaps due to constraints in reversing the order of integration. However, I couldn't complete the analysis. Any help would be welcomed.