\subsection{Related work}

Two types of mechanisms: \emph{deterministic} and \emph{randomized}. For
randomized mechanisms, people seek \emph{universally truthful} mechanisms:
mechanisms which are a randomization of truthful mechanisms.

\paragraph{Symmetric submodular functions} $V(S) = g(|S|)$ where $g$ is
a concave function. Truthful deterministic mechanism with approximation ratio
of 2. This is optimal \cite{singer-mechanisms}.

\paragraph{Knapsack} deterministic: $1+\sqrt{2}\leq \alpha \leq 2 + \sqrt{2}$,
randomized: $2\leq \alpha\leq 3$ \cite{chen}

\paragraph{Matching} deterministic: $2 \leq \alpha\leq 7.32$ \cite{singer-mechanisms}

\paragraph{Coverage} deterministic: $ ?\leq\alpha\leq 31$ \cite{singer-influence}

\paragraph{Submodular function} deterministic: $1+\sqrt{2}\leq\alpha\leq ?$,
randomized: $2\leq\alpha\leq 7.91$ \cite{chen}

For wider class of functions, \cite{singer-mechanisms} proves that even for XOS
functions, the lower bound is $\sqrt{n}$ (no constant approximation) even in the
non-strategic case). To be able to say something interesting, people extend the
computational model to the \emph{demand} query model: you have access to an
oracle which given a vector of price $[p_i]_{i\in\mathcal{N}}$ returns
$S\in\argmax_{S\subseteq\mathcal{N}} V(S) - \sum_{i\in S} p_i$

\paragraph{XOS functions} fractionally additive functions: functions which can
be written as the max of a finite set of additive functions. deterministic: 768

\paragraph{Complement-free (sub-additive) objective} derministic: $O(\log^3 n)$
\cite{dobz2011-mechanisms}. randomized: $O(\frac{\log n}{\log \log n})$
\cite{bei2012budget}. More generally \cite{bei2012budget} gives a randomized
mechanism which is $O(\mathcal{I})$ where $\mathcal{I}$ is the integrality gap
of a set cover like problem which is defined from the value function. In
particular, for XOS function, this integrality gap is $O(1)$.

\paragraph{Bayesian mechanism design} when we have a distribution over the
costs of the agents. In this case the approximation guarantee is defined in
expectations. \cite{bei2012budget} $768/512$ approximation ratio for any
subadditive function.

\paragraph{Online mechanisms} \cite{learning} in the i.i.d posted price model
for symmetric submodular functions randomized $O(1)$-competitive algorithm. For
general submodular function in the secretary posted price model randomized
$O(\log n)$-competitive algorithm. In the bidding model $O(1)$-competitive
algorithm.

\stratis{What is known about the maximization of logdet in the non-strategic case? Is it NP hard? Approximation ratios?}
\thibaut{Knapsack reduces to our problem in dimension 1, hence maximizing log
det is NP-hard. The approximation ratio is at least (1-1/e) by applying the
general approximation algorithm from Sviridenko \cite{sviridenko-submodular}.}
