Small changes

1 files changed, 6 insertions, 6 deletions

diff --git a/final/main.tex b/final/main.tex
index a27f751..3345ee6 100644
--- a/final/main.tex
+++ b/final/main.tex
@@ -191,7 +191,7 @@ mechanism, and then is offered each item separately with posted prices
 $\hat{v}_1,\ldots, \hat{v}_m$.  This is essentially the concept of a two-part
 tariff, as discussed in \citep{armstrong}.
 
-Note that as described above, the candidate mechanism is not individually
+Note that as the candidate mechanism is described above, it is not individually
 rational because the agent gets charged $\hat{v}_0$ regardless of her type. To
 restore individual rationality, we need to have the agent pay only when
 \begin{displaymath}
@@ -274,11 +274,11 @@ price. Then we have:
     tariff mechanism simply sells each item separately in a take-it-or-leave-it
     manner. For the optimal choice of $(\hat{v}_1,\dots,\hat{v}_m)$, the
     revenue of the mechanism reaches $\SRev(F)$. Since the supremum defining
-    $\TPRev(F)$ optimizes (in particular) over all $\hat{v}$ with $\hat{v}_0
-    = 0$ we have that $\TPRev(F)\geq \SRev(F)$.
+    $\TPRev(F)$ optimizes (in particular) over all $\hat{v}$, with $\hat{v}_0
+    = 0$, we have that $\TPRev(F)\geq \SRev(F)$.
 
     When $\hat{v}_1=\dots=\hat{v}_m = 0$, the two part tariff mechanism then
-    simply sells all the item at price $\hat{v}_0$ whenever $\sum_{i=1}^m
+    simply sells all the items at price $\hat{v}_0$ whenever $\sum_{i=1}^m
     t_i\geq \hat{v}_0$, \emph{i.e} is a grand bundle pricing mechanism. For the
     optimal choice of $\hat{v}_0$, the revenue reaches $\BRev(F)$. As in the
     previous paragraph, this implies that $\TPRev(F)\geq \BRev(F)$.
@@ -289,12 +289,12 @@ price. Then we have:
 \subsection{$p$-exclusivity}
 
 As noted by Yao, the above mechanism has the additional property of being
-$p$-exclusive in the following sense: for a vector $p = (p_1,\dots,p_m)$
+$p$-exclusive, where $p$-exclusivity is defined as follows: for a vector $p = (p_1,\dots,p_m)$
 a mechanism is said to be $p$-exclusive if $x_i = 0$ whenever $p_i > t_i$. This
 is essentially saying that there is a reserve price for each item.
 
 The notion of $p$-exclusivity introduced by Yao was crucial in his reduction
-from the multiple-buyer setting to the single buyer setting. $p$-exclusivity
+from the $k$-item n-buyer setting to the $k$-item single buyer setting. $p$-exclusivity
 can easily be enforced in the optimization we formulated in Section~\ref{sec:intro},
 by adding the following non-linear constraints:
 \begin{displaymath}