appendix and biblio fonts

1 files changed, 9 insertions, 8 deletions

diff --git a/appendix.tex b/appendix.tex
index 2010227..8e30586 100644
--- a/appendix.tex
+++ b/appendix.tex
@@ -1,5 +1,5 @@
 \begin{lemma}\label{lemma:monotone}
-The mechanism is monotone.
+The mechanism is monotone and budget feasible.
 \end{lemma}
 \begin{proof}
     Consider an agent $i$ with cost $c_i$ that is
@@ -25,13 +25,14 @@ The mechanism is monotone.
     by the monotonicity and submodularity  of $V$. Hence  $i\in S_G'$. As $L(\xi)$,  is the optimal value of \eqref{relax} 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 $s_i(c_i',c_{-i}) =1$.
     Suppose now that when $i$ reports $c_i$, $L(\xi) < C V(i^*)$. Then $s_i(c_i,c_{-i})=1$ iff $i = i^*$. 
     Reporting $c_{i^*}'\leq c_{i^*}$ does not change $V(i^*)$ nor
-    $L(\xi) \leq C V(i^*)$; thus $s_{i^*}(c_{i^*}',c_{-i^*})=1$. 
-\end{proof}
-\begin{lemma}\label{lemma:budget-feasibility}
-The mechanism is budget feasible.
-\end{lemma}
-\begin{proof}
-Suppose that $L(\xi) < C V(i^*)$. Then the mechanism selects $i^*$. Since the bid of $i^*$ does not affect the above condition, the threshold payment of $i^*$ is $B$ and the mechanism is budget feasible.
+    $L(\xi) \leq C V(i^*)$; thus $s_{i^*}(c_{i^*}',c_{-i^*})=1$, so the mechanism is monotone.
+%\end{proof}
+%\begin{lemma}\label{lemma:budget-feasibility}
+%The mechanism is budget feasible.
+%\end{lemma}
+%\begin{proof}
+
+To show budget feasibility, suppose that $L(\xi) < C V(i^*)$. Then the mechanism selects $i^*$. Since the bid of $i^*$ does not affect the above condition, the threshold payment of $i^*$ is $B$ and the mechanism is budget feasible.
 Suppose  that $L(\xi) \geq C V(i^*)$. 
 Denote by $S_G$ the set selected by the greedy algorithm, and for $i\in S_G$,  denote by
 $S_i$ the subset of the solution set that was selected by the greedy algorithm just prior to the addition of $i$---both sets determined for the present cost vector $c$.