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diff --git a/final/main.tex b/final/main.tex index 5ebba6e..adde64f 100644 --- a/final/main.tex +++ b/final/main.tex @@ -299,11 +299,13 @@ price. Then we have: \subsection{$p$-exclusivity} As noted by \citep{yao}, the above mechanism $\M$ has the additional property of being -$p$-exclusive, where $p$-exclusivity is defined as follows: for a vector $p = (p_1,\dots,p_m)$ +$p$-exclusive, where $p$-exclusivity\footnote{\citep{yao} actually uses the +notation $\beta$-exclusive for the same thing, but use $p$ here to have +consistent notations.} is defined as follows: for a vector $p = (p_1,\dots,p_m)$ a mechanism is said to be $p$-exclusive if $x_i = 0$ whenever $p_i > t_i$. This is essentially saying that there is a reserve price for each item. -The notion of $p$-exclusivity introduced\footnote{\citep{yao} actually uses the notation $\beta$-exclusive for the same thing, but we thought that $p$ was a more natural choice.} by \citep{yao} was crucial in his reduction +The notion of $p$-exclusivity introduced by \citep{yao} was crucial in his reduction from the $k$-item $m$-buyer setting to the $k$-item single buyer setting. He describes a mechanism known as \emph{Best-Guess Reduction}, which conducts $m$ single-buyer $k$-item auctions, using an IR-IC $p$-exclusive mechanism, for a particular value of $p$ drawn from the joint bid distribution over all buyers conditioned on the bids of all other buyers, and then combines this with the Vickrey second-price auction, showing that this mechanism has revenue that is a constant approximation to the optimal $k$-item, $m$-buyer mechanism. He then defines another mechanism, \emph{Second-Price Bundling}, which is meant to heuristically approximate this combined mechanism, and shows that its revenue is also a constant approximation to the optimal mechanism. $p$-exclusivity |