Make the dependency on the assumption clearer in the proof of Lemma 1

1 files changed, 6 insertions, 4 deletions

diff --git a/final/main.tex b/final/main.tex
index 1ecfbca..8ac2bbe 100644
--- a/final/main.tex
+++ b/final/main.tex
@@ -139,9 +139,10 @@ a 6-approximation to $\Rev((1,...,1),F)$.
 
 The main question underlying this work can then be formulated as:
 
-\paragraph{Question.} \emph{Given an ex-ante allocation constraint $\hat{x}$ and a type distribution $F$, is
-there a simple mechanism whose ex-ante allocation rule is upper-bounded by
-$\hat{x}$ and whose revenue is a constant approximation to $\Rev(\hat{x},F)$?}
+\paragraph{Question.} \emph{Given an ex-ante allocation constraint $\hat{x}$
+and a type distribution $F$, is there a simple mechanism whose ex-ante
+allocation rule is upper-bounded by $\hat{x}$ and whose revenue is a constant
+approximation to $\Rev(\hat{x},F)$?}
 
 \subsection{Examples and Motivations}
 
@@ -277,7 +278,8 @@ price. Then we have:
 
 \begin{proof}
     The second inequality is the main result from \citep{babaioff}; we will
-    thus only prove the first inequality. 
+    thus only prove the first inequality, which holds even without the
+    assumption that $F$ is a product distribution.
 
     In the specific case where $\hat{v} = (0,\hat{v}_1,\dots,\hat{v}_m)$,
     \emph{i.e} $\hat{v}_0 = 0$, there is no entrance fee and the two-part