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\jairheading{19}{2003}{209-242}{12/02}{9/03}
\ShortHeadings{Decision-Theoretic Bidding with Learned
  Density Models}
{Stone, Schapire, Littman, Csirik, \& McAllester}


\firstpageno{209}








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    \item Compute $G_w = \mbox{opt}(H_w,Y)$, the optimal set of
      purchases if the agent wins a unit in auction $a_i$.  Note that no
      additional units of the good will be purchased, since the
      $i$-th component of $Y$\  is $\infty$.
    \item Compute $G = \mbox{opt}(H,Y)$, the optimal set of purchases
      if the agent never acquires any additional units in the auction
      $a_i$\  and prices are set to $Y$.
    \item $\mbox{diff} =  [v(G_w+H) - G_w \cdot Y] - [v(G+H) - G \cdot Y]$
    \item $\mbox{total-diff} =  \mbox{total-diff} + \mbox{diff}$
    \item $\mbox{counter} = \mbox{counter}+1$
    \end{itemize}
  \item $r = \mbox{total-diff} / \mbox{counter}$
  \item Bid $r$\  in auction $a_i$.
  \end{itemize}
\end{itemize}
\vspace{\hrulegap}
\hrule
}

\end{table}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1654}%
$a_0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1656}%
$a_1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1658}%
$H=(0,0), H_w=(1,0), Y=(\infty,70)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1660}%
$G_w = \mbox{opt}(H_w,Y)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1662}%
$100 - 70 = 30$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1664}%
$10 - 0 = 10$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1666}%
$G_w = (0,1)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1668}%
$[v(G_w+H) - G_w \cdot Y]  = v(1,1) - (0,1) \cdot (\infty,70) = 100 - 70$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1670}%
$G = (0,0)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1672}%
$50 - 70 = -20$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1674}%
$0 - 0 = 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1676}%
$[v(G+H) - G \cdot Y] = 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1678}%
$\mbox{val} = 30 - 0 = 30$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1680}%
$\mbox{val} = 60 - 10 = 50$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1682}%
$\mbox{val} = 10 - 0 = 10$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1684}%
$.5\times 30 + .25\times 50 +
.25\times 10 = \$30$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1686}%
$1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1688}%
$4$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1690}%
$2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1692}%
$5$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1694}%
$-\$10$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1696}%
$\$x$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1698}%
$x$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1700}%
$y$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1702}%
$y \in [10,90]$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1704}%
$\$150$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1706}%
$\$800$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1712}%
$\$300$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1732}%
$\$50$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1736}%
$\$0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1738}%
$\$200$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1740}%
$d$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1742}%
$AD \leq d < DD$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1371}%
\begin{displaymath}1000 - \emph{travelPenalty} + \emph{hotelBonus} + \emph{funBonus}\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1744}%
$100(|AD - {\it IAD}| + |DD - {\it IDD}|)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1746}%
$\perp $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlfigureA{table392}%
\begin{table}
% latex2html id marker 392
{\sloppy\small
\hrule
\vspace{\hrulegap}
\begin{tabbing}
{\bf Input}:~\=$(x_1,y_1),\ldots,(x_m,y_m)$\  where $x_i\in X$, $y_i\in\mbox{\msym R}$\\
       \>positive integers $k$\  and $T$
\end{tabbing}
\par
\medskip
\par
{\bf Compute breakpoints}: $b_0 < b_1 < \cdots < b_{k+1}$\  where
\begin{s_itemize}
\item
$  b_0 =  \min_i y_i$
\item
$b_{k+1} =  \max_i y_i$
\item
$b_1, \ldots, b_k$\  chosen to minimize $\sum_{j=0}^k q_j \ln q_j$
where $q_0,\ldots,q_k$\  are fraction of $y_i$'s in\\
$[b_0,b_1),[b_1,b_2),\ldots,[b_k,b_{k+1}]$\  (using dynamic programing)
\end{s_itemize}
\par
\medskip
\par
{\bf Boosting}:
\begin{s_itemize}
\item
for $t=1,\ldots T$:
\begin{s_itemize3}
\item
compute weights
$\displaystyle
 W_t(i,j) = \frac{1}{1+e^{s_j(y_i) f_t(x_i,j)}}
$\\
where $s_j(y)$\  is as in Eq.~(\ref{eqn:1})
\item
use $W_t$\  to obtain base function $h_t:X\times\{1,\ldots,k\}\rightarrow\mbox{\msym R}$
minimizing\\
$\displaystyle
  \sum_{i=1}^m \sum_{j=1}^k W_t(i,j) e^{- s_j(y_i) h_t(x_i, j)}
$
over all decision rules $h_t$\  considered.  The decision rules can take any form.  In
our work, we use ``decision stumps,'' or simple thresholds on one of
the features.
\end{s_itemize3}
\end{s_itemize}
\par
\medskip
\par
{\bf Output sampling rule}:
\begin{s_itemize}
\item
let $\displaystyle f = \sum_{t=1}^T h_t$
\item
let $f' = (\overline{f}+ \underline{f}) / 2$\  where
\begin{eqnarray*}
\overline{f}(x,j) &=& \max\{f(x, j') : j \leq j' \leq k\} \\
\underline{f}(x,j) &=& \min\{f(x, j') : 1 \leq j' \leq j\}
\end{eqnarray*}
\item
to sample, given $x\in X$
\begin{s_itemize3}
\item
let $\displaystyle p_j = \frac{1}{1+e^{-f'(x,j)}}$
\item
let $p_0 = 1, p_{k+1} = 0$
\item
choose $j\in \{0,\ldots,k\}$\  randomly with probability $p_j - p_{j+1}$
\item
choose $y$\  uniformly at random from $[b_j, b_{j+1}]$
\item
output $y$
\end{s_itemize3}
\end{s_itemize}
\vspace{\hrulegap}
\hrule
}

\end{table}%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1812}%
$(x_1,y_1),\ldots,(x_m,y_m)$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1814}%
$x_i$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1820}%
$\mbox{\msym R}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1826}%
$X\subseteq (\mbox{\msym R}\cup \{\perp \})^n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1836}%
$(x,y)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1372}%
\begin{displaymath}
   [b_0,b_1), [b_1,b_2), \ldots, [b_k,b_{k+1}]
\end{displaymath}%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1840}%
$b_0 < b_1 < \cdots < b_k \leq b_{k+1}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1842}%
$k=50$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1844}%
$k$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1848}%
$b_0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1850}%
$b_{k+1}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1854}%
$b_1,\ldots,b_k$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1858}%
$b_j$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1860}%
$j=1,\ldots,k$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1868}%
$p_j$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1874}%
$[b_j,b_{j+1})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1876}%
$p_{j+1}-p_j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1878}%
$y\geq b_j$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1880}%
$f:X\times \{1,\ldots,k\} \rightarrow \mbox{\msym R}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath439}%
\begin{displaymath} 
   \frac{1}{1 + \exp(-f(x,j))}
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{\newpage\clearpage
\lthtmldisplayA{displaymath1373}%
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\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath445}%
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                      \end{array}
              \right.
\end{displaymath}%
\lthtmldisplayZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1894}%
$(x_i,y_i)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1898}%
$f$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmldisplayA{displaymath1374}%
\begin{displaymath}
\sum_{i=1}^m \sum_{j=1}^k  \ln\left({1 + e^{- s_j(y_i) f(x_i, j)}}\right).
\end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1902}%
$t$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1904}%
$h_t: X \times \{1,\ldots,k\}\rightarrow \mbox{\msym R}$%
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{\newpage\clearpage
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{\newpage\clearpage
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$h_t$%
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{\newpage\clearpage
\lthtmldisplayA{displaymath1376}%
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmldisplayA{displaymath1377}%
\begin{displaymath}
   h(x,j) = 
         \left\{ \begin{array}{ll}
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                        C_j & \mbox{if $\phi(x) = \perp $}
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\lthtmldisplayZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1920}%
$\phi(\cdot)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1922}%
$\theta$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1924}%
$A_j$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1926}%
$B_j$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1928}%
$C_j$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1932}%
$\phi$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1936}%
$h(x,\cdot)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1938}%
$\phi(x)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1962}%
$f_t$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1964}%
$T=300$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1966}%
$f=f_{T+1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1974}%
$p_0 = 1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1976}%
$p_{k+1}=0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1978}%
$p_1\geq p_2\geq \cdots \geq p_k$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1980}%
$f(x,1)\geq f(x,2)\geq \cdots \geq f(x,k)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1986}%
$f'$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1988}%
$j$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1990}%
$f' = (\overline{f}+ \underline{f}) / 2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1992}%
$\overline{f}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1994}%
$\underline{f}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1996}%
$g$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2012}%
$p_j - p_{j+1}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmldisplayA{displaymath1378}%
\begin{displaymath} \sum_{j=0}^k  (p_j - p_{j+1}) \left({\frac{b_{j+1} + b_j}{2}}\right). \end{displaymath}%
\lthtmldisplayZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2014}%
$G^*$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2022}%
$v(G^*)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
{\newpage\clearpage
\lthtmlfigureA{table500}%
\begin{table}{\small
\begin{center}
\hrule
\vspace{\hrulegap}
\begin{description}
\item[When the first flight quotes are posted:] \
\begin{noinds_itemize}
\item Compute {$G^*$}\ with current holdings and 
\emph{expected prices}
\item Buy the flights in {$G^*$}\ for which \emph{expected cost of
postponing commitment} exceeds the \emph{expected benefit of
postponing commitment}
\end{noinds_itemize}
\par
\item[Starting 1 minute before each hotel close:] \
\begin{noinds_itemize}
\item Compute {$G^*$}\ with current holdings
and \emph{expected prices}
\item Buy the flights in {$G^*$}\ for which \emph{expected cost of
postponing commitment} exceeds \emph{expected benefit of postponing
commitment} (30 seconds)
\item Bid \emph{hotel room expected marginal values} given holdings,
  new flights, and \emph{expected hotel purchases} (30 seconds)
\end{noinds_itemize}
\par
\item[Last minute:] Buy remaining flights as needed by {$G^*$}
\par
\item[In parallel (continuously):] Buy/sell entertainment tickets
based on their \emph{expected values}
\end{description}
\vspace{\hrulegap}
\hrule
\end{center}
}

\end{table}%
\lthtmlfigureZ
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{\newpage\clearpage
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$16$%
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{\newpage\clearpage
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$4pc^2$%
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$14$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2088}%
$18$%
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\begin{table}{\small
\begin{center}
\hrule
\vspace{\hrulegap}
\begin{itemize}
\item For each hotel (in order of increasing expected price):
\item Repeat until time bound
  \begin{s_enumerate}
  \item Generate a random hotel closing order (only over open hotels)
  \item \emph{Sample} closing prices from predicted hotel price distributions 
  \item Given these closing prices, compute $V_0, V_1, \ldots V_n$ 
    \begin{s_itemize3}
    \item $V_i \equiv$\  {$v(G^*)$}\ if owning $i$\  of the hotel 
    \item Estimate {$v(G^*)$}\ with LP relaxation
    \item Assume that no additional hotel rooms of this type can be bought
    \item For other hotels, assume outstanding bids above sampled price
      are already owned ({\it i.e.}, they cannot be withdrawn).
    \item Note that $V_0 \leq V_1 \leq \ldots \leq V_n$:  the values
      are monotonically increasing since having more goods cannot be
      worse in terms of possible allocations.
    \end{s_itemize3}
  \end{s_enumerate}
\item The value of the $i$th copy of the room is the mean of $V_i - V_{i-1}$\  over
all the samples.
\item Note further that $V_1 - V_0 \geq V_2 - V_1 \ldots \geq V_n -
  V_{n-1}$:  the value differences are monotonically decreasing since
  each additional room will be assigned to the client who can derive
  the most value from it.
\item Bid for one room at the value of the $i$th copy of the room for
  all $i$\  such that the value is at least as much as the current
  price.  Due to the monotonicity noted in the step above,
  no matter what the closing price, the desired number of rooms at
  that price will be purchased.
\end{itemize}
\vspace{\hrulegap}
\hrule
\end{center}
}

\end{table}%
\lthtmlfigureZ
\lthtmlcheckvsize\clearpage}

\stepcounter{subsection}
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$0$%
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{\newpage\clearpage
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$Mt^2$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2134}%
$M$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2138}%
$y-10$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2144}%
$T$%
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\lthtmlinlinemathA{tex2html_wrap_inline2146}%
$m(T^2-t^2)(y-10)$%
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\lthtmlfigureA{table556}%
\begin{table}\begin{center}
\hrule
\vspace{\hrulegap}
\begin{itemize}
\item Assume we're considering buying $n$\  flights of a given type
\item Repeat until time bound
  \begin{s_enumerate}
  \item Generate a random hotel closing order (open hotels)
  \item \emph{Sample} closing prices from predicted price distributions (open hotels)
  \item Given these closing prices compute $V_0, V_1, \ldots V_n$ 
    \begin{s_itemize3}
    \item $V_i$\  = {$v(G^*)$}\ if forced to buy $i$\  of the
      flight
    \item Estimate {$v(G^*)$}\ with LP relaxation
    \item Assume more flights can be bought at the \emph{current price}
    \item Note that $V_0 \geq V_1 \geq \ldots \geq V_n$\  since it is
      never worse to retain extra flexibility.
    \end{s_itemize3}
  \end{s_enumerate}
\item The value of waiting to buy copy $i$\  is the mean of $V_i -
  V_{i-1}$\  over all the samples.  If all price samples lead to the
  conclusion that the $i$th flight should be bought, then $V_i =
  V_{i-1}$\  and there is no benefit to postponing commitment.
\end{itemize}
\vspace{\hrulegap}
\hrule
\end{center}

\end{table}%
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\stepcounter{subsection}
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\lthtmlfigureA{table577}%
\begin{table}\begin{center}
\hrule
\vspace{\hrulegap}
\begin{itemize}
\item Assume $n$\  of a given ticket type are currently owned
\item Repeat until time bound
  \begin{s_enumerate}
  \item Generate a random hotel closing order (open hotels)
  \item \emph{Sample} closing prices from predicted price distributions (open hotels)
  \item Given these closing prices compute $V_{n-1}, V_n, V_{n+1}$ 
    \begin{s_itemize3}
    \item $V_i$\  = {$v(G^*)$}\ if own $i$\  of the ticket
    \item Estimate {$v(G^*)$}\ with LP relaxation
    \item Assume no other tickets can be bought or sold
    \item Note that $V_{n-1} \leq V_n \leq V_{n+1}$\  since it is never
      worse to own extra tickets.
    \end{s_itemize3}
  \end{s_enumerate}
\par
\item The value of buying a ticket is the mean of $V_{n+1} - V_{n}$
  over all the samples;  the value of selling is the mean of $V_{n} -
  V_{n-1}$.
\par
\item Since tickets are considered sequentially, if the determined buy
or sell bid leads to a price that would clear according to the current
quotes, assume the transaction goes through before computing the
values of buying and selling other ticket types.
\end{itemize}
\vspace{\hrulegap}
\hrule
\end{center}

\end{table}%
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$ev$%
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%
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2278}%
${    -91.1}\pm {   17.6\ \ (7)}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2280}%
${    -33.7}\pm {   52.4\ \ (6)}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2282}%
${   -157.1}\pm {   54.8\ \ (8)}$%
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{\newpage\clearpage
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${   -198.8}\pm {   26.0\ \ (8)}$%
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{\newpage\clearpage
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$p_t$%
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{\newpage\clearpage
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$\Delta_t$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1368}%
\( t \)%
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$p_t + \Delta_t$%
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\end{document}
