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diff --git a/appendix.tex b/appendix.tex index 11607b0..b1d35d6 100644 --- a/appendix.tex +++ b/appendix.tex @@ -5,7 +5,7 @@ Our mechanism for \EDP{} is monotone and budget feasible. 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. - Suppose that when $i$ reports $c_i$, $L(\xi) \geq C V(i^*)$; then, as $s_i(c_i,c_{-i})=1$, $i\in S_G$. + 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. %using the submodularity of $V$, we see that $i$ will satisfy the greedy %selection rule: @@ -22,18 +22,18 @@ 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 $L(\xi)$, 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 $i = i^*$. + by the monotonicity and submodularity of $V$. Hence $i\in S_G'$. As $OPT'_{-i^*}$, 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 $c_i$, $OPT'_{-i^*} < C V(i^*)$. Then $s_i(c_i,c_{-i})=1$ iff $i = i^*$. Reporting $c_{i^*}'\leq c_{i^*}$ does not change $V(i^*)$ nor - $L(\xi) \leq C V(i^*)$; thus $s_{i^*}(c_{i^*}',c_{-i^*})=1$, so the mechanism is monotone. + $OPT'_{-i^*} \leq C V(i^*)$; 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 $L(\xi) \geq C V(i^*)$. +To show budget feasibility, suppose that $OPT'_{-i^*} < C V(i^*)$. 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 $OPT'_{-i^*} \geq C V(i^*)$. 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 $c$. %Chen \emph{et al.}~\cite{chen} show that, |