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\begin{document}

\section{Problem}

\begin{itemize}
\item  database $D = (X_i)_{1\leq i\leq n}$
\item $(X_i)_{1\leq i\leq n}\in\big(\mathbf{R^d}\big)^n$ sampled in an i.i.d
    fashion from the probability distribution $\mu$
\end{itemize}

We assume that there is a common variable $P$ which is the underlying
phenomenon that we are interesting in, and all the variables in the database,
are noisy realisations of this phenomenon:
\begin{displaymath}
    X_i = P + N_i
\end{displaymath}
where the $N_i$ are pairwise independent and also independent from $H$.


We are interested in defining the $\emph{value}$ of a subset $S$ of the
database $D$. The approaches to such a definition can be classified in two main
families:
\begin{itemize}
    \item \emph{Information theoretic:} We use the entropy of $P$ as a measure
        of the uncertainty. Then the value of a subset $S$ is the decrease of
        entropy induced by revealing it:
        \begin{displaymath}
            V(S) = H(P) - H(P\,|\,S) 
        \end{displaymath}
    \item \emph{Statistical:} in this case you are trying to estimate the
        phenomenon. The variance of the estimation is used as a measure of
        the uncertainty. The value of a set $S$ can be defined as the average
        reduction of the variance over all the points in the database once you
        observe $S$:
        \begin{displaymath}
            V(S) = \frac{1}{|D|}\sum_{x\in D} \sigma_x^2 - \sigma_{x|A}^2
        \end{displaymath}
\end{itemize}

Once you have defined a notion of value, there are several problems you can
address:
\begin{enumerate}
    \item \emph{budget auctions:} the points in the database are associated
        with users. Each user $i$ has a cost $c_i$ for revealing his data. You
        are given a budget $B$ and you want to select a subset $S$ of users and
        a payment scheme which is truthful. The value of the subset
        $S$ should be as big as possible with the constraint that the payments
        should stay within budget.
    \item \emph{value sharing:} the goal is to find a value sharing method to
        split the whole value of a subset $S$ among its participants.
    \item \emph{k-best auctions:} this one is very similar to \emph{budget
        auctions}, but instead of having a budget constraint, the constraint is
        on the number of people in the subset $S$.
\end{enumerate}

These problems have already been studied, and optimal or near-optimal solution
are already known when the value function is submodular. Fortunately, it can be
shown that the information theoretic value defined above is submodular.

\section{Submodularity}

In this section, we will consider that we are given a \emph{universe} set $U$.
A set function $f$ is a function defined on the power set of $U$, $\mathfrak{P}(U)$.

A set function $f$ defined on $\mathfrak{P}(U)$ will be said \emph{increasing} if
it is increasing with regards to inclusion, that is:

\begin{displaymath}
  \forall\,S\subseteq T,\quad f(S)\leq f(T)
\end{displaymath}

A \emph{decreasing} function on $\mathfrak{P}(U)$ is defined similarly.

A set function $f$ defined on $\mathfrak{P}(U)$ is said to be
\emph{submodular} if it verifies the diminishing returns property, that is,
the marginal increments when adding a point to a set, is a set decreasing
function. More formally, for any point $x$ in $U$, we can define the marginal
increment of $f$ regarding $x$, it is the set function defined as:
\begin{displaymath}
    \Delta_x f(S) = f(S\cup\{x\}) - f(S)
\end{displaymath}
Then, $f$ is \emph{submodular} iff. for all $x$ in $U$, $\Delta_x f$ is a set
decreasing function.

Similarly, a \emph{supermodular} is a function whose marginal increments are set
increasing functions.
\begin{prop}
  Let $R:\mathbf{R}\rightarrow \mathbf{R}$ be a decreasing concave function and
  $f:\mathfrak{P}(U)\rightarrow\mathbf{R}$ be a decreasing submodular function,
  then the composed function $R\circ f$ is increasing and supermodular.
\end{prop}

\begin{proof}
  The increasingness of $R\circ f$ follows immediately from the decreasingness
  of $R$ and $f$.
  
  For the supermodularity, let $S$ and $T$ be two sets such that $S\subseteq
  T$. By decreasingness of $f$, we have:
  \begin{displaymath}
    \forall\,V,\quad f(T)\leq f(S)\quad\mathrm{and}\quad f(T\cup V)\leq f(S\cup V)
  \end{displaymath}
  
  Thus, by concavity of $R$:
  \begin{displaymath}\label{eq:base}
    \forall\,V,\quad\frac{R\big(f(S)\big)-R\big(f(S\cup V)\big)}{f(S)-f(S\cup V)}
    \leq\frac{R\big(f(T)\big)-R\big(f(T\cup V)\big)}{f(T)-f(T\cup V)}
  \end{displaymath}
  
  $f$ is decreasing, so multiplying this last inequality by
  $f(S)-f(S\cup V)$ and $f(T)-f(T\cup V)$ yields:
  \begin{multline}
    \forall V,\quad\Big(R\big(f(S)\big)-R\big(f(S\cup V)\big)\Big)\big(f(T)-f(T\cup V)\big)\\
    \leq \Big(R\big(f(T)\big)-R\big(f(T\cup V)\big)\Big)\big(f(S)-f(S\cup V)\big)
  \end{multline}

  $f$ is submodular, so:
  \begin{displaymath}
    f(T\cup V)-f(T)\leq f(S\cup V) - f(S)
  \end{displaymath}
  
  $R\circ f$ is increasing, so:
  \begin{displaymath}
    R\big(f(S)\big)-R\big(f(S\cup V)\big)\leq 0
  \end{displaymath}

  By combining the two previous inequalities, we get:
    \begin{multline*}
    \forall V,\quad\Big(R\big(f(S)\big)-R\big(f(S\cup V)\big)\Big)\big(f(S)-f(S\cup V)\big)\\
    \leq \Big(R\big(f(S)\big)-R\big(f(S\cup V)\big)\Big)\big(f(T)-f(T\cup V)\big)
  \end{multline*}

  Injecting this last inequality into \eqref{eq:base} gives:
  \begin{multline*}
    \forall V,\quad\Big(R\big(f(S)\big)-R\big(f(S\cup V)\big)\Big)\big(f(S)-f(S\cup V)\big)\\
    \leq \Big(R\big(f(T)\big)-R\big(f(T\cup V)\big)\Big)\big(f(S)-f(S\cup V)\big)
  \end{multline*}

  Dividing left and right by $f(S)-f(S\cup V)$ yields:
  \begin{displaymath}
    \forall V,\quad R\big(f(S)\big)-R\big(f(S\cup V)\big)
    \leq R\big(f(T)\big)-R\big(f(T\cup V)\big)
  \end{displaymath}
  which is exactly the supermodularity of $R\circ f$.
\end{proof}

\end{document}
