Sequential Multi-Purchase Choice Model

Updated 9 July 2026

Sequential multi-purchase choice models are discrete-choice frameworks that capture customer purchases as a sequential process with RUM-based rankings and Markovian transitions encoding asymmetric complementarity.
The methodology combines within-category substitution with cross-category effects via a DAG structure, enabling tractable assortment optimization and efficient EM-based parameter estimation.
Empirical results demonstrate improved predictive accuracy and revenue gains over traditional models, offering actionable insights for category management and diagnostic assessments.

Searching arXiv for papers on sequential multi-purchase choice models and closely related frameworks. Sequential multi-purchase choice models are discrete-choice frameworks in which a consumer may purchase several items during a single shopping episode rather than making at most one choice. In the cross-category setting studied in "A Markovian Approach for Cross-Category Complementarity in Choice Modeling" (Housni et al., 25 Aug 2025), the central problem is cross-category assortment optimization: retailers jointly determine assortments across categories to maximize expected revenue when consumer behavior exhibits both substitution within categories and complementarity across categories. The model is sequential because choice unfolds category by category, and it is multi-purchase because a customer may select one product in a category, proceed to downstream categories, and continue purchasing. Its distinguishing feature is a Markovian transition structure that captures asymmetric complementarity across categories while permitting general Random Utility Maximization (RUM)-based choice models within each category.

1. Decision process and formal structure

The Markovian cross-category formulation places product categories on a directed acyclic graph (DAG). A customer arrives facing several categories arranged in that DAG. In each category $U$ , she may either make a no-purchase choice or select one product, then proceed to its child categories. Within category $U$ , she has a fixed RUM-type ranking over all items, including the no-purchase option. After selecting product $i$ in category $U$ , she forms an initial attraction to some product $j$ in each child category $W$ ; the cross-category transition probabilities $\lambda^i_j$ encode the dependence of downstream attraction on the upstream purchase. The process repeats until she either chooses the no-purchase option or exits (Housni et al., 25 Aug 2025).

For two categories $A\to B$ , let $N_A$ and $N_B$ denote the product sets, each augmented by no-purchase options $U$ 0. Within-category choice probabilities are written as

$U$ 1

for assortments $U$ 2 and $U$ 3. Complementarity is represented by transition vectors

$U$ 4

Conditioned on choosing $U$ 5, the customer is initially attracted to $U$ 6 with probability $U$ 7. If $U$ 8 is offered, she buys it; if $U$ 9, she substitutes among the offered set according to her original within-category RUM ranking, conditional on $i$ 0 having been her first preference. The resulting joint probability is

$i$ 1

This construction makes the cross-category effect directional and product specific. The paper explicitly interprets $i$ 2 as capturing asymmetric complementarity. A plausible implication is that the framework can encode situations in which purchasing one upstream item increases attraction toward one downstream item but not its close substitute.

2. Within-category substitution and Markovian complementarity

A defining aspect of the model is the separation between within-category substitution and cross-category interaction. Any RUM, equivalently any ranking-based choice model, may be used within each category. Two tractable special cases are emphasized: multinomial logit (MNL) and the Markov-chain (MC) choice model (Housni et al., 25 Aug 2025).

Under MNL,

$i$ 3

By independence of irrelevant alternatives, $i$ 4 for all unavailable $i$ 5. This yields the closed-form expression

$i$ 6

Under the MC model, each category is defined by an initial arrival distribution $i$ 7 and a transition matrix $i$ 8. Choice is a random walk on states until hitting $i$ 9. Conditional on a purchase $U$ 0 in category $U$ 1, category $U$ 2 itself becomes an MC model with initial arrival $U$ 3 and the same transition matrix $U$ 4. This exact compatibility between the cross-category transition mechanism and within-category Markov substitution is what later enables polynomial-time optimization.

The model therefore occupies an intermediate position between purely independent category models and fully general joint-basket models. It retains within-category flexibility through general RUM while using a structured transition layer to encode complementarity. The paper states that most prior work either overlooks complementarity or proposes models that lead to intractable optimization problems despite being based on MNL; the Markovian construction is intended precisely to address that tension.

3. Estimation by expectation-maximization

For estimation, the observed data consist of transactions

$U$ 5

In the MNL-MNL special case, the observed-data log-likelihood is not jointly concave in $U$ 6. To address this, the paper introduces latent indicators

$U$ 7

and constructs a complete-data log-likelihood that decomposes into three terms: an MNL- $U$ 8 term, a $U$ 9 term, and a substitution term (Housni et al., 25 Aug 2025).

The EM iteration has the usual two stages. In the E-step, one computes the posterior expectations of the latent attraction indicators under the current parameter iterate. In the M-step, the expected complete-data log-likelihood separates into three concave subproblems:

update the MNL- $j$ 0 weights $j$ 1 by standard MNL maximum-likelihood estimation;
update each transition vector $j$ 2 by normalizing expected counts;
update $j$ 3 through a concave maximization over the substitution term.

Theorem 4 guarantees that each EM iteration increases the observed log-likelihood and converges to a stationary point. This gives the model an estimability result that is explicit rather than heuristic.

The estimation strategy is notable because the complementarity parameters are not treated as opaque interaction coefficients. Instead, they are tied to latent initial attractions, which preserves an interpretable sequential semantics. This suggests a decomposition between upstream-induced attraction and downstream substitution behavior that is useful both statistically and managerially.

4. Assortment optimization and computational complexity

The cross-category revenue objective for two categories is

$j$ 4

The paper states that this problem is, in general, NP-hard under MNL-based complementarity. However, when both $j$ 5 and $j$ 6 are MC models, the two-category problem decouples and admits an exact polynomial-time solution (Housni et al., 25 Aug 2025).

Algorithm 1 proceeds in three steps. First, compute an unconstrained invariant optimal assortment $j$ 7 for category $j$ 8 under any initial distribution via known polynomial-time MC methods. Second, for each product $j$ 9, compute downstream revenue

$W$ 0

and define adjusted prices $W$ 1. Third, solve the unconstrained MC assortment problem for category $W$ 2 with prices $W$ 3. Theorem 3 states that this algorithm runs in polynomial time and yields a global optimum.

The proof sketch rests on three claims stated in the paper: the optimal $W$ 4-assortment is invariant to initial arrivals; with $W$ 5 fixed, the $W$ 6-problem reduces to a single-category MC assortment problem with adjusted prices; and this invariance removes circular dependence. For a general DAG, the extension is by backward induction in reverse topological order: each category solves an unconstrained MC assortment problem with adjusted prices that include the expected revenues of its children.

The complexity picture is explicitly stratified. The paper reports a polynomial-time exact algorithm for unconstrained MC assortment optimization for any number of categories, a constant-factor approximation via backward induction when cardinality constraints apply only to root categories, and provable inapproximability under ETH when cardinality constraints apply to all categories. A common misconception is that introducing complementarity necessarily destroys tractability altogether; the results here are more specific. Tractability is recovered under MC within-category substitution, but not under all model classes.

5. Complementarity diagnostics and empirical findings

To screen for cross-category complementarity, the paper introduces the empirical metric

$W$ 7

where $W$ 8 is the count of transactions in which $W$ 9 was chosen in $\lambda^i_j$ 0, $\lambda^i_j$ 1 is the empirical conditional probability of $\lambda^i_j$ 2 in $\lambda^i_j$ 3 given $\lambda^i_j$ 4, and $\lambda^i_j$ 5 is the unconditional probability. The paper states that $\lambda^i_j$ 6, that zero indicates independence, and that it rises monotonically with the synthetic complementarity parameter $\lambda^i_j$ 7. At the product level, the Specific Complementarity Score is

$\lambda^i_j$ 8

which quantifies the lift in attraction to $\lambda^i_j$ 9 from choosing $A\to B$ 0 in $A\to B$ 1 (Housni et al., 25 Aug 2025).

The numerical evidence is reported for both synthetic and real data. On synthetic data with 10k transactions and complementarity parameter $A\to B$ 2 varying from 0 to 5, MarkovMNL outperforms independent MNL (IndMNL) and the two-stage MNL of Ke et al. (MultiMNL) in training and test log-likelihood, with gains up to $A\to B$ 3; in top-3 hit rates, with gains up to $A\to B$ 4 percentage points; in rank accuracy, up to $A\to B$ 5; and in expected revenue from optimized assortments, with $A\to B$ 6 to $A\to B$ 7 higher revenue when $A\to B$ 8 is large.

On a large-scale transaction-level dataset from a major US grocery store containing 6.2M transactions, the paper reports several concrete findings. CM scores identify high-signal category pairs such as Cake Mix $A\to B$ 9 Frosting with $N_A$ 0. On Cake Mix-Frosting, MarkovMNL improves test log-likelihood by $N_A$ 1, top-3 hit rate by $N_A$ 2 percentage points, and Effective Hit Rate (E-HR) by $N_A$ 3 percentage points relative to IndMNL; MultiMNL is comparable on fit but NP-hard to optimize. Product-level $N_A$ 4 reveals strong brand-matching lifts, including Betty Crocker $N_A$ 5 Crocker frosting and Pillsbury $N_A$ 6 Pillsbury frosting. For Meats $N_A$ 7 Buns with $N_A$ 8, the paper reports franks $N_A$ 9 hot-dog buns with $N_B$ 0, patties $N_B$ 1 burger buns with $N_B$ 2, and anti-complementarity for patties versus hot-dog buns with $N_B$ 3.

These results are used in the paper to argue that the model improves predictive accuracy, model fit, and expected revenue in settings with complementarity, while also revealing intuitive market structure such as brand-loyal cross-category purchasing. A plausible implication is that the model is not only a predictive device but also a diagnostic instrument for category management and promotion design.

The Markovian cross-category model belongs to a broader family of sequential models in which basket formation, ranking, or repeated decisions unfold over time. Several adjacent formulations clarify which parts of the problem are shared and which are distinct.

Model	Sequential mechanism	Main emphasis
Markovian cross-category model	DAG over categories with transition probabilities $N_B$ 4	Asymmetric complementarity and assortment optimization
SHOPPER	Repeated softmax picks until a special checkout item	Substitutes, complements, price elasticities, seasonality
Multiple-purchase ranking with budget	Ordered list with random attention span and purchase budget	Revenue-maximizing ranking and online learning
Ranked-list column generation	Top- $N_B$ 5 selection from a type-specific ranking	Nonparametric estimation and downstream assortment optimization

SHOPPER models a shopping trip as a sequential probabilistic process in which, at each step, the next item is chosen by a softmax over utilities that combine item popularity, user taste, price sensitivity, seasonality, and item-to-item interaction terms. Its interaction term $N_B$ 6 is interpreted so that positive values indicate complements and negative values indicate substitutes. The model includes a special checkout item to terminate the sequence, uses variational inference based on the ELBO, and supports counterfactual price-intervention queries through posterior predictive integration (Ruiz et al., 2017).

"Product Ranking for Revenue Maximization with Multiple Purchases" studies a different sequential mechanism: an online retailer presents an ordered list, the consumer has a geometric attention span $N_B$ 7 and a geometric purchase budget $N_B$ 8, and expected revenue admits a closed form. When parameters are known, the optimal ranking is obtained by sorting in descending order of the index

$N_B$ 9

When parameters are unknown, the paper develops MPB-UCB algorithms with $U$ 00 regret in non-contextual and contextual settings (Xu et al., 2022).

"Modern column generation for estimating single- and multi-purchase ranked list choice models" takes a nonparametric perspective. Each consumer type is defined by a strict ranking over products and a desired purchase count $U$ 01; given an assortment $U$ 02, the type purchases the top $U$ 03 available items. Estimation is formulated as a large linear program under either maximum likelihood or minimum $U$ 04-error, and solved by column generation. Its key technical contribution is a dynamic-programming pricing subproblem with dominance pruning, completion bounds, unreachable-products pruning, and heuristic pricing; the estimated model can then be embedded in assortment optimization (Costa et al., 7 May 2026).

A more distant but conceptually related line studies sequential consumer choice with cross-context learning rather than basket formation. "Boundedly Rational Meta-Learning in Sequential Consumer Choice" formalizes repeated choices across routes as a hierarchical Beta-Bernoulli bandit, with full Bayesian MetaDP and boundedly rational BRMDP $U$ 05 policies; BRMDP $U$ 06 is reported to fit trial-by-trial human choices better than both no transfer and fully integrated Bayesian transfer (Khosravi et al., 15 May 2026). This suggests that sequential consumer-choice modeling extends beyond basket assembly to dynamic learning and transfer across contexts.

Across these papers, the common structure is sequential dependence: earlier events alter later choice probabilities, feasible actions, or beliefs. The principal modeling differences concern what carries the dependence—cross-category attraction, item-item interaction, attention and budget constraints, ranked preferences, or hyper-posterior learning—and whether the downstream task is prediction, counterfactual analysis, assortment optimization, ranking, or online learning.