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diff --git a/final/main.tex b/final/main.tex index adde64f..36f0b04 100644 --- a/final/main.tex +++ b/final/main.tex @@ -305,8 +305,6 @@ 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 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 can easily be enforced in the optimization we formulated in Section~\ref{sec:intro}, @@ -324,6 +322,25 @@ Lemma. = \big(F^{-1}(1-p_1), \dots, F^{-1}(1-p_m)\big).$$ \end{lemma} +\section{$n$-to-1 Bidders Reductions} + +The goal of this section is to compare the two ways of constraining the single +buyer problem that we discussed: namely $p$-exclusivity and ex-ante allocation +constraints and draw a parallel in how these two notions are used to construct +$n$-to-1 bidders reductions in \citep{yao} and \citep{alaei} respectively. + + +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. \bibliographystyle{abbrvnat} \bibliography{main} |