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

\maketitle

\section{Introduction to the Problem}

\blfootnote{We are grateful to Jason Hartline who provided the original idea
for this project.}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.

If the distribution $F$ is correlated, a result from \cite{hart} shows that
the gap between the revenue of auctions with finite menu size and
a revenue-maximizing auction can be arbitrarily large.

When $F$ is a product distribution however, there is a hope to obtain simple
auctions with constant approximation ratios to the optimal revenue. In fact,
a result from \cite{babaioff} shows 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
as a single item achieves a 6-approximation to the optimal revenue.

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
$\hat{x}$.

We use the notation from \cite{alaei} where $R(\hat{x})$ denotes the revenue
obtained by the optimal mechanism solution to Problem~\ref{eq:opt} with ex-ante
allocation constraint $\hat{x}$ given by \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 $R(\hat{x})$?}

Note that 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 to $R((1,...,1))$.

\section{Importance of the Problem and Possible Approach}

\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
$\frac{1}{2}$.


In the general case, a candidate simple mechanism suggested by Jason Hartline
is the following: the buyer is charged a price $p_0$ to participate in the
mechanism, and then is offered each item separately with posted prices
$p_1,\ldots, p_m$. This is essentially the concept of a two-part tariff, as
discussed in \cite{armstrong}. Our problem is then to understand how to set
$p_0$ and the posted prices $p_1,\ldots, p_m$ such that in expectation over the
agent's type, item $i$ is sold with probability $x_i\leq \hat{x}_i$.

\section{Roadmap}

Here are the steps we plan to execute before the completion of our project.

\begin{enumerate}
    \item work out simple examples for which we will be able to do exact
        computation on the proposed two-part mechanism. The goal is to obtain
        a ``proof-of-concept'' that this mechanism can yield, at least in some
        cases, a constant approximation to the optimal revenue.
    \item prove (or disprove) that this two-part mechanism achieves
        a constant-approximation for any product distribution $F$.
    \item by then, we should have a better understanding of two-part tariff
        ideas. Use this understanding to re-interpret the results from
        \cite{yao} as two-part tariff mechanisms. Indeed, it seems that many
        mechanisms in this paper have a ``two-part'' structure (entrance fee
        plus separate prices) and could benefit from this re-interpretation.
\end{enumerate}

In terms of timeline, we estimate that these steps will take us until beginning
of January 2015.

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