I am having difficulty understanding the problem below:
Given CDFs $F_1, ..., F_n$, prove that the allocation rule of a second-price auction with bidder-specific reserve prices is monotone. The item is rewarded to the bidder with the highest bid that meets his reserve price. The reserve price of bidder $i$ is $r_i = 𝜙_i^{-1}(t)$ where $𝜙_i$ is the virtual valuation function
$$\phi_i(v_i) = v_i - \frac{1 - F_i(v_i)}{f_i(v_i)}$$
and $t$ is chosen such that
$$Pr\left[\max_i 𝜙_i(v_i) \ge t\right] = 1/2.$$ Since the virtual valuations are determined before the auction (if I understand it correctly), how does the reserve prices affect the monotonicity of the allocation rule? The behavior is still the same: if you keep on increasing your bid, you'll have a higher chance of getting the price. How should I go about this?