Small changes

1 files changed, 4 insertions, 4 deletions

diff --git a/final/main.tex b/final/main.tex
index 1a101f5..dae0503 100644
--- a/final/main.tex
+++ b/final/main.tex
@@ -169,7 +169,7 @@ $\gamma$ is a constant which is at least $\frac{1}{2}$.
 Given the simplicity of posted-price mechanisms and the fact that they are
 optimal in the single-agent single-item setting (for a regular distribution
 $F$), it would be desirable to obtain a mechanism as close as possible to posted-price.
-Unfortunately, Hart and Nisan \citep{hart-nisan} showed that even in the
+Unfortunately, \citep{hart-nisan} showed that even in the
 unconstrained, regular case, no posted-price mechanism for the single-agent
 problem has an approximation ratio better than $\Omega(\log n)$.
 
@@ -223,7 +223,7 @@ 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
-can easily be enforced in the problem we formulated in Section~\ref{sec:intro},
+can easily be enforced in the optimization we formulated in Section~\ref{sec:intro},
 by adding the following non-linear constraints:
 \begin{displaymath}
     x_i(t_i - p_i)\geq 0,\quad \forall i\in[m] 
@@ -234,8 +234,8 @@ Lemma.
 
 \begin{lemma}
     If a mechanism is $p$-exclusive for some vector $p=(p_1,\dots, p_m)$, then
-    it satisfies the ex-ante allocation constraint defined by $\hat{x}
-    = \big(F^{-1}(1-p_1), \dots, F^{-1}(1-p_m)\big)$.
+    it satisfies the ex-ante allocation constraint defined by $$\hat{x}
+    = \big(F^{-1}(1-p_1), \dots, F^{-1}(1-p_m)\big).$$
 \end{lemma}