  We introduce a new hierarchy over monotone set functions, that we refer to as
$\mathcal{MPH}$ (Maximum over Positive Hypergraphs). Levels of the hierarchy
correspond to the degree of complementarity in a given function. The highest
level of the hierarchy, $\mathcal{MPH}$-$m$ (where $m$ is the total number of
items) captures all monotone functions. The lowest level, $\mathcal{MPH}$-$1$,
captures all monotone submodular functions, and more generally, the class of
functions known as $\mathcal{XOS}$. Every monotone function that has a positive
hypergraph representation of rank $k$ (in the sense defined by Abraham,
Babaioff, Dughmi and Roughgarden [EC 2012]) is in $\mathcal{MPH}$-$k$. Every
monotone function that has supermodular degree $k$ (in the sense defined by
Feige and Izsak [ITCS 2013]) is in $\mathcal{MPH}$-$(k+1)$. In both cases, the
converse direction does not hold, even in an approximate sense. We present
additional results that demonstrate the expressiveness power of
$\mathcal{MPH}$-$k$.
  One can obtain good approximation ratios for some natural optimization
problems, provided that functions are required to lie in low levels of the
$\mathcal{MPH}$ hierarchy. We present two such applications. One shows that the
maximum welfare problem can be approximated within a ratio of $k+1$ if all
players hold valuation functions in $\mathcal{MPH}$-$k$. The other is an upper
bound of $2k$ on the price of anarchy of simultaneous first price auctions.
  Being in $\mathcal{MPH}$-$k$ can be shown to involve two requirements -- one
is monotonicity and the other is a certain requirement that we refer to as
$\mathcal{PLE}$ (Positive Lower Envelope). Removing the monotonicity
requirement, one obtains the $\mathcal{PLE}$ hierarchy over all non-negative
set functions (whether monotone or not), which can be fertile ground for
further research.
