  We study a submodular maximization problem motivated by applications in
online retail. A platform displays a list of products to a user in response to
a search query. The user inspects the first $k$ items in the list for a $k$
chosen at random from a given distribution, and decides whether to purchase an
item from that set based on a choice model. The goal of the platform is to
maximize the engagement of the shopper defined as the probability of purchase.
This problem gives rise to a less-studied variation of submodular maximization
in which we are asked to choose an $\textit{ordering}$ of a set of elements to
maximize a linear combination of different submodular functions.
  First, using a reduction to maximizing submodular functions over matroids, we
give an optimal $\left(1-1/e\right)$-approximation for this problem. We then
consider a variant in which the platform cares not only about user engagement,
but also about diversification across various groups of users, that is,
guaranteeing a certain probability of purchase in each group. We characterize
the polytope of feasible solutions and give a bi-criteria
$((1-1/e)^2,(1-1/e)^2)$-approximation for this problem by rounding an
approximate solution of a linear programming relaxation. For rounding, we rely
on our reduction and the particular rounding techniques for matroid polytopes.
For the special case in which underlying submodular functions are coverage
functions -- which is practically relevant in online retail -- we propose an
alternative LP relaxation and a simpler randomized rounding for the problem.
This approach yields to an optimal bi-criteria $(1-1/e,1-1/e)$-approximation
algorithm for the special case of the problem with coverage functions.
