Tiny changes

1 files changed, 3 insertions, 1 deletions

diff --git a/final/main.tex b/final/main.tex
index 3ae2d0e..8f07ed1 100644
--- a/final/main.tex
+++ b/final/main.tex
@@ -447,7 +447,9 @@ While this mechanism's revenue does indeed provide a constant approximation to t
 
 \section{Conclusion}
 
-We have discussed the problem of selling $m$ heterogeneous items, with ex-ante allocation constraint $\hat{x}$, to a single buyer with additive utility, having type drawn from the distribution $F$. Following a suggestion by J.D. Hartline, we have proposed a mechanism $\M$ based on a two-part tariff approach (in which the price charged to the buyer consists of an entrance fee plus posted prices for the items) and then drew a connection with a technique from \citep{yao}, in which we relate this mechanism to a $\beta$-exclusive mechanism. We also surveyed the $n$-to-1 buyer reduction techniques within \citep{alaei} and \citep{yao}, unifying the exposition and terminology. Our proposed mechanism $\M$ can be viewed as a generalization of the work of \citep{babaioff}, but with the addition of allocation constraints. We showed, via a counterexample, that the approximation ratio of the revenue for the optimal $\beta$-exclusive mechanisms to the optimal revenue of the two-party tariff mechanism $\M$ is unbounded. Whether the optimal revenue of the two-party tariff mechanism is a constant approximation to the revenue of the optimal mechanism with allocation constraints remains an open and interesting path for future research.
+We have discussed the problem of selling $m$ heterogeneous items, with ex-ante allocation constraint $\hat{x}$, to a single buyer with additive utility, having type drawn from the distribution $F$. Following a suggestion by J.D. Hartline, we have proposed a mechanism $\M$ based on a two-part tariff approach (in which the price charged to the buyer consists of an entrance fee plus posted prices for the items) and then drew a connection with a technique from \citep{yao}, in which we relate this mechanism to a $\beta$-exclusive mechanism. We also surveyed the $n$-to-1 buyer reduction techniques within \citep{alaei} and \citep{yao}, unifying the exposition and terminology. 
+
+Our proposed mechanism $\M$ can be viewed as a generalization of the work of \citep{babaioff}, but with the addition of allocation constraints. We showed, via a counterexample, that the approximation ratio of the revenue for the optimal $\beta$-exclusive mechanisms to the optimal revenue of the two-party tariff mechanism $\M$ is unbounded. Whether the optimal revenue of the two-party tariff mechanism is a constant approximation to the revenue of the optimal mechanism with allocation constraints remains an open and interesting path for future research.
 
 \bibliographystyle{abbrvnat}
 \bibliography{main}