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Diffstat (limited to 'appendix.tex')
| -rw-r--r-- | appendix.tex | 17 |
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 $c$. |