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\author{Thibaut Horel \and Paul Tylkin}
\title{Economics 2099 Project}

\begin{document}

\maketitle

\section{Introduction}

We consider the problem of selling $m$ heterogeneous items to a single agent
with additive utility. The type $t$ of the agent is drawn from a distribution
$F$ over $\R_+^m$. For an allocation $x = (x_1,\ldots,x_m)$ where $x_i$,
$i\in[m]$, denotes the probability of being allocated item $i$ and payment $p$,
the utility of an agent with type $t=(t_1,\ldots, t_m)$ is defined by:
\begin{displaymath}
    u(t, x, p) = \left(\sum_{i=1}^m x_i t_i\right) - p
\end{displaymath}

By the taxation principle, a mechanism for this setting can be described by
a pair $x(\cdot), p(\cdot)$ where the allocation rule $x$ and the payment rule
$p$ are indexed by the agent's type. For an agent with type $t$, $\big(x(t),
p(t)\big)$ is her preferred menu option.

The revenue-optimal mechanism for this single-agent problem can be formally
obtained as the solution to the following optimization problem:
\begin{equation}
    \label{eq:opt}
    \begin{split}
        \max_{(x, p)} & \E_{t\sim F}\big[p(t)\big]\\
        \text{s.t.} &\, u(t, x(t), p(t)) \geq u(t', x(t'), p(t')),\quad
        \forall t, t'\in\R_+^m\\
        &\, u(t, x(t), p(t)) \geq 0,\quad\forall t\in\R_+^m\\
        &\,x_i(t)\leq 1,\quad \forall t\in\R_+^m,\,\forall i\in[m]
\end{split}
\end{equation}
where the first constraint expresses incentive compatibility and the second
constraint expresses individual rationality.

Unfortunately, it has been shown in \cite{hart} that even in this simple case
and as soon as $m\geq 2$, the optimal mechanism can have a menu size which is
exponential in $m$. Therefore it is important to understand to which extent
simple mechanisms can approximate the revenue-optimal mechanism.

A remarkable result of \cite{babaioff} proves that a simple mechanism provides
a constant approximation to Problem~\ref{eq:opt}. More precisely, it is shown
that the maximum of \emph{item pricing} where each item is priced separately at
its monopoly reserve price and \emph{bundle pricing} where the grand bundle of
all items is priced and sold together achieves a 6-approximation to the optimal
mechanism.

We propose to extend this result by considering a more general version of
Problem~\ref{eq:opt} where there is an additional \emph{ex-ante allocation
constraint}. Formally, we are given a vector $\hat{x}
= (\hat{x}_1,\ldots,\hat{x}_m)$ and we add the following constraint to
Problem~\ref{eq:opt}:
\begin{equation}
    \label{eq:ea}
    \E_{t\sim F}\big[x_i(t)] \leq \hat{x}_i,\quad\forall i\in[m]
\end{equation}
which expresses the assumption that the ex-ante allocation rule is upper-bounded by
$\hat{x}$.

To provide some more intuition about this, we assume that the buyer is charged a price $p_0$ to participate in the mechanism, 
and then is offered a menu of goods with prices $p_1,...,p_m$. The buyer's utility over the goods is additive, as above. However, there is an ex-ante constraint of being allocated a given good $i$, given by $\hat{x}_i$. For each good, if the buyer is allocated the good, which he is with probability $x_i \leq \hat{x}_i$, then he pays $p_i$; otherwise, he pays nothing. This is essentially the concept of a two-part tariff, as discussed in \cite{armstrong}.

Let us denote by $R(\hat{x})$ the revenue obtained by the optimal mechanism
solution to Problem~\ref{eq:opt} with ex-ante allocation constraint
\eqref{eq:ea}.

\paragraph{Problem.} \emph{Given an ex-ante allocation constraint $\hat{x}$, is
there a simple mechanism whose ex-ante allocation rule is upper-bounded by
$\hat{x}$ and whose revenue is a constant approximation to $R(\hat{x})$?}

\paragraph{} One reason why one might want to consider this problem is that it
can be used as a building block for more general mechanisms. Indeed, in
\cite{alaei}, Alaei shows that under fairly general assumptions, given an
$\alpha$-approximate mechanism for the single-agent, ex-ante allocation
constrained problem it is possible to construct in polynomial time a mechanism
for the multi-agent problem which is a $\gamma\cdot\alpha$ approximation to the
revenue-optimal mechanism where $\gamma$ is a constant which is at least
$\frac{1}{2}$.

It is interesting to consider two specific cases for which we already have an
answer to our problem:
\begin{itemize}
    \item when $\hat{x} = (1,\ldots,1)$, \eqref{eq:ea} does not further
        constrain Problem~\ref{eq:opt}, i.e. the constraint is trivially satisfied. 
        In this case, as discussed above, the
        mechanism described in \cite{babaioff} provides a 6-approximation.
    \item when $\hat{x} = (\frac{1}{m},\ldots,\frac{1}{m})$, by summing the
        constraint \eqref{eq:ea} for all $i\in[m]$, we see that the ex-ante
        allocation constraint implies that in expectation, no more than
        one-item is sold to the agent. This is exactly the unit-demand case for
        which TODO:cite provides a 2 approximation.
\end{itemize}

\section{Related Work}

In \cite{babaioff}, the authors describe a setting with a monopolist seller,
offering $n$ heterogeneous goods, and a single buyer. 
\cite{hart}
\cite{hartline}
\cite{yao} 
\cite{armstrong}
\cite{alaei}

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