One of the central challenges in online advertising is attribution, namely, assessing the contribution of individual advertiser actions including emails, display ads and search ads to eventual conversion. Several heuristics are used for attribution in practice; however, there is no formal justification for them and many of these fail even in simple canonical settings. The main contribution in this work is to develop an axiomatic framework for attribution in online advertising. In particular, we consider a Markovian model for the user journey through the conversion funnel, in which ad actions may have disparate impacts at different stages. We propose a novel attribution metric, that we refer to as counterfactual adjusted Shapley value, which inherits the desirable properties of the traditional Shapley value. Furthermore, we establish that this metric coincides with an adjusted “unique-uniform” attribution scheme. This scheme is efficiently computable and implementable and can be interpreted as a correction to the commonly used uniform attribution scheme.

We consider the assortment optimization problem when customer preferences follow a mixture of Mallows distributions. The assortment optimization problem focuses on determining the revenue/profit maximizing subset of products from a large universe of products; it is an important decision that is commonly faced by retailers in determining what to offer their customers. There are two key challenges: (a) the Mallows distribution lacks a closed-form expression (and requires summing an exponential number of terms) to compute the choice probability and, hence, the expected revenue/profit per customer; and (b) finding the best subset may require an exhaustive search. Our key contributions are an efficiently computable closed-form expression for the choice probability under the Mallows model and a compact mixed integer linear program (MIP) formulation for the assortment problem.

Assortment planning is an important problem that arises in many industries such as retailing and airlines where one of the key challenges is to identify the "right model" for the consumer preferences and substitution behavior. Distribution over preference lists or permutations is the most general framework for modeling preferences but is intractable in general. In this paper, we present a parsimonious distribution over permutations model that is induced by random swaps from a central preference list (which we also refer to as the prototype list. In particular, a random list from this distribution can be sampled as follows: sample N from a given distribution on the number of swaps and starting from the initial prototype list, perform N random swaps. A random swap operation consists of selecting a random pair of items in the current list and swapping their positions. We consider two types of random swaps: i) swapping an arbitrary pair of items, and $ii)$ swapping an adjacent pair of items. More precisely, a pair is picked uniformly at random out of all n+1\choose 2 possible pairs in i), and out all all $n$ pairs of adjacent items in ii). If the distribution over the number of swaps has sufficient large support, the distribution over permutations has non-zero probability on all preference lists. This model is motivated by practical applications where consumers preferences generally have many common items appear according to the same relative order and differ only in a small number of items. The prototype list used to generate a random preference list can intuitively be thought of as the mode of the distribution implied by the random swap model. This model also captures the well known Mallows distribution over permutation that is specified by a model permutation and a concentration parameter that determines how the probability of permutations decrease as a function of the distance from the modal permutation. Therefore, this is a fairly general model for consumer preferences. This model is motivated by practical applications where consumer preference are more or less similar over most items and differ in the relative order of only a few items.