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Diffstat (limited to 'final/main.tex')
| -rw-r--r-- | final/main.tex | 14 |
1 files changed, 7 insertions, 7 deletions
diff --git a/final/main.tex b/final/main.tex index 3345ee6..bf8e6c8 100644 --- a/final/main.tex +++ b/final/main.tex @@ -3,12 +3,11 @@ \usepackage[english]{babel} \usepackage{paralist} \usepackage[utf8x]{inputenc} -\usepackage[pagebackref=true,breaklinks=true,colorlinks=true,citecolor=blue]{hyperref} +\usepackage[pagebackref=false,breaklinks=true,colorlinks=true,citecolor=blue]{hyperref} \usepackage[capitalize, noabbrev]{cleveref} \usepackage[square,sort]{natbib} - % these are compressed lists to help fit into a 1 page limit \newenvironment{enumerate*}% {\vspace{-2ex} \begin{enumerate} % @@ -43,6 +42,7 @@ \newcommand{\inprod}[1]{\left\langle #1 \right\rangle} \newcommand{\R}{\mathbb{R}} +\newcommand{\M}{\mathfrak{M}} \newcommand{\eqdef}{\mathbin{\stackrel{\rm def}{=}}} \newcommand{\llbracket}{[\![} @@ -201,7 +201,7 @@ where we used the notation $(x)^+\eqdef\max(x, 0)$, so that the above inequality could also be written as $$\sum_{i: t_i\geq \hat{v}_i} (t_i- \hat{v}_i)\geq \hat{v}_0.$$ -We can now formally describe the candidate mechanism. +We can now formally describe the candidate mechanism $\M$. \begin{center} \fbox{ @@ -248,8 +248,8 @@ two-part tariff mechanism with ex-ante allocation constraint $\hat{x}$: \end{split} \end{displaymath} The question we introduced in Section~\ref{sec:intro} can then be formulated for -the two-part tariff mechanism: \emph{is $\TPRev(\hat{x}, F)$ a constant -approximation to $\Rev(\hat{x}, F)$?} +the two-part tariff mechanism: $$\text{\emph{is $\TPRev(\hat{x}, F)$ a constant +approximation to $\Rev(\hat{x}, F)$?}}$$ The following simple Lemma shows that at least in the unconstrained case, the answer to the previous question is positive. In fact, the proof shows that the @@ -288,12 +288,12 @@ price. Then we have: \subsection{$p$-exclusivity} -As noted by Yao, the above mechanism has the additional property of being +As noted by \citep{yao}, the above mechanism has the additional property of being $p$-exclusive, where $p$-exclusivity 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 Yao was crucial in his reduction +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 from the $k$-item n-buyer setting to the $k$-item single buyer setting. $p$-exclusivity can easily be enforced in the optimization we formulated in Section~\ref{sec:intro}, by adding the following non-linear constraints: |