From d5f4afbbf188d745439e0e15b1857fb696477d70 Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Mon, 24 Jun 2013 23:01:34 +0200
Subject: Unifying pass over the whole paper

---
 appendix.tex | 11 ++++++-----
 1 file changed, 6 insertions(+), 5 deletions(-)

(limited to 'appendix.tex')

diff --git a/appendix.tex b/appendix.tex
index 53bc328..2c83e04 100644
--- a/appendix.tex
+++ b/appendix.tex
@@ -1,12 +1,12 @@
 \begin{lemma}\label{lemma:monotone}
-Our mechanism for \EDP{} is monotone and budget feasible.
+Our mechanism for \EDP{} is $\delta$-monotone and budget feasible.
 \end{lemma}
 \begin{proof}
     Consider an agent $i$ with cost $c_i$ that is
     selected by the mechanism, and suppose that she reports
-    a cost $c_i'\leq c_i$ while all  other costs stay the same.
+    a cost $c_i'\leq c_i-\delta$ while all  other costs stay the same.
     Suppose that when $i$ reports $c_i$, $OPT'_{-i^*} \geq C V(i^*)$; then, as $s_i(c_i,c_{-i})=1$, $i\in S_G$.
-     By reporting a cost $c_i'\leq c_i$, $i$ may be selected at an earlier iteration of the greedy algorithm.
+     By reporting cost $c_i'$, $i$ may be selected at an earlier iteration of the greedy algorithm.
     %using the submodularity of $V$, we see that $i$ will satisfy the greedy
     %selection rule:
     %\begin{displaymath}
@@ -22,8 +22,9 @@ Our mechanism for \EDP{} is monotone and budget feasible.
         \frac{B}{2}\frac{V(S_i\cup\{i\})-V(S_i)}{V(S_i\cup\{i\})}
          \leq \frac{B}{2}\frac{V(S_i'\cup\{i\})-V(S_i')}{V(S_i'\cup\{i\})}
     \end{align*}
-    by the monotonicity and submodularity  of $V$. Hence  $i\in S_G'$. As
-    $OPT'_{-i^*}$,  is the optimal value of \eqref{eq:primal} under relaxation $L$ when $i^*$ is excluded from $\mathcal{N}$, reducing the costs can only increase this value, so under $c'_i\leq c_i$ the greedy set is still allocated and $s_i(c_i',c_{-i}) =1$.
+    by the monotonicity and submodularity  of $V$. Hence  $i\in S_G'$. By
+    $\delta$-decreasingness of
+    $OPT'_{-i^*}$, under $c'_i\leq c_i-\delta$ the greedy set is still allocated and $s_i(c_i',c_{-i}) =1$.
     Suppose now that when $i$ reports $c_i$, $OPT'_{-i^*} < C V(i^*)$. Then $s_i(c_i,c_{-i})=1$ iff $i = i^*$. 
     Reporting $c_{i^*}'\leq c_{i^*}$ does not change $V(i^*)$ nor
     $OPT'_{-i^*} \leq C V(i^*)$; thus $s_{i^*}(c_{i^*}',c_{-i^*})=1$, so the mechanism is monotone.
-- 
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