In online advertising, marketing interventions such as coupons introduce significant confounding bias into Click-Through Rate (CTR) prediction. Observed clicks reflect a mixture of users' intrinsic preferences and the uplift induced by these interventions. This causes conventional models to miscalibrate base CTRs, which distorts downstream ranking and billing decisions. Furthermore, marketing interventions often operate as multi-valued treatments with varying magnitudes, introducing additional complexity to CTR prediction. To address these issues, we propose the \textbf{Uni}fied \textbf{M}ulti-\textbf{V}alued \textbf{T}reatment Network (UniMVT). Specifically, UniMVT disentangles confounding factors from treatment-sensitive representations, enabling a full-space counterfactual inference module to jointly reconstruct the debiased base CTR and intensity-response curves. To handle the complexity of multi-valued treatments, UniMVT employs an auxiliary intensity estimation task to capture treatment propensities and devise a unit uplift objective that normalizes the intervention effect. This ensures comparable estimation across the continuous coupon-value spectrum. UniMVT simultaneously a
We study consumption stimulus with digital coupons, which provide time-limited subsidies contingent on minimum spending. We analyze a large-scale program in China and present five main findings: (1) the program generates large short-term effects, with each $\yen$1 of government subsidy inducing $\yen$3.4 in consumer spending; (2) consumption responses vary substantially, driven by both demand-side factors (e.g., wealth) and supply-side factors (e.g., local consumption amenities); (3) The largest spending increases occur among consumers whose baseline spending already exceeds coupon thresholds and for whom coupon subsidies should be equivalent to cash, suggesting behavioral motivations; (4) high-response consumers disproportionately direct their spending toward large businesses, leading to a regressive allocation of stimulus benefits; and (5) targeting the most responsive consumers can double total stimulus effects. A hybrid design combining targeted distribution with direct support to small businesses improves both the efficiency and equity of the program.
A collector samples coupons with replacement from a pool containing $g$ \textit{uniform} groups of coupons, where "uniform group" means that all coupons in the group are equally likely to occur. For each $j = 1, \dots, g$ let $T_j$ be the number of trials needed to detect Group $j$, namely to collect all $M_j$ coupons belonging to it at least once. We derive an explicit formula for the probability that the $l$-th group is the first one to be detected (symbolically, $P\{T_l = \bigwedge_{j=1}^g T_j\}$). We also compute the asymptotics of this probability in the case $g=2$ as the number of coupons grows to infinity in a certain manner. Then, in the case of two groups we focus on $T := T_1 \vee T_2$, i.e. the number of trials needed to collect all coupons of the pool (at least once). We determine the asymptotics of $E[T]$ and $V[T]$, as well as the limiting distribution of $T$ (appropriately normalized) as the number of coupons becomes very large.
In the UK betting market, bookmakers often offer a free coupon to new customers. These free coupons allow the customer to place extra bets, at lower risk, in combination with the usual betting odds. We are interested in whether a customer can exploit these free coupons in order to make a sure gain, and if so, how the customer can achieve this. To answer this question, we evaluate the odds and free coupons as a set of desirable gambles for the bookmaker. We show that we can use the Choquet integral to check whether this set of desirable gambles incurs sure loss for the bookmaker, and hence, results in a sure gain for the customer. In the latter case, we also show how a customer can determine the combination of bets that make the best possible gain, based on complementary slackness. As an illustration, we look at some actual betting odds in the market and find that, without free coupons, the set of desirable gambles derived from those odds avoids sure loss. However, with free coupons, we identify some combinations of bets that customers could place in order to make a guaranteed gain.
In this note, we introduce a distributed twist on the classic coupon collector problem: a set of $m$ collectors wish to each obtain a set of $n$ coupons; for this, they can each sample coupons uniformly at random, but can also meet in pairwise interactions, during which they can exchange coupons. By doing so, they hope to reduce the number of coupons that must be sampled by each collector in order to obtain a full set. This extension is natural when considering real-world manifestations of the coupon collector phenomenon, and has been remarked upon and studied empirically [Hayes and Hannigan 2006, Ahmad et al. 2014, Delmarcelle 2019]. We provide the first theoretical analysis for such a scenario. We find that "coupon collecting with friends" can indeed significantly reduce the number of coupons each collector must sample, and raises interesting connections to the more traditional variants of the problem. While our analysis is in most cases asymptotically tight, there are several open questions raised, regarding finer-grained analysis of both "coupon collecting with friends", and of a long-studied variant of the original problem in which a collector requires multiple full sets of co
This paper is about the Coupon collector's problem. There are some coupons, or baseball cards, or other plastic knick-knacks that are put into bags of chips or under soda bottles, etc. A collector starts collecting these trinkets and wants to form a complete collection of all possible ones. Every time they buy the product however, they don't know which coupon they will "collect" until they open the product. How many coupons do they need to collect before they complete the collection? In this paper, we explore the mean and variance of this random variable, $N$ using various methods. Some of them work only for the special case with the coupons having equal probabilities of being collected, while others generalize to the case where the coupons are collected with unequal probabilities (which is closer to a real world scenario).
Motivated by a problem in the theory of randomized search heuristics, we give a very precise analysis for the coupon collector problem where the collector starts with a random set of coupons (chosen uniformly from all sets). We show that the expected number of rounds until we have a coupon of each type is $nH_{n/2} - 1/2 \pm o(1)$, where $H_{n/2}$ denotes the $(n/2)$th harmonic number when $n$ is even, and $H_{n/2}:= (1/2) H_{\lfloor n/2 \rfloor} + (1/2) H_{\lceil n/2 \rceil}$ when $n$ is odd. Consequently, the coupon collector with random initial stake is by half a round faster than the one starting with exactly $n/2$ coupons (apart from additive $o(1)$ terms). This result implies that classic simple heuristic called \emph{randomized local search} needs an expected number of $nH_{n/2} - 1/2 \pm o(1)$ iterations to find the optimum of any monotonic function defined on bit-strings of length $n$.
We consider the problem of identifying current coupons for Agency backed To-be-Announced (TBA) Mortgage Backed Securities. In a doubly stochastic factor based model which allows for prepayment intensities to depend upon current and origination mortgage rates, as well as underlying investment factors, we identify the current coupon with solutions to a degenerate elliptic, non-linear fixed point problem. Using Schaefer's theorem we prove existence of current coupons. We also provide an explicit approximation to the fixed point, valid for compact perturbations off a baseline factor-based intensity model. Numerical examples are provided which show the approximation performs remarkably well in estimating the current coupon.
In this study, we consider traveler coupon redemption behavior from the perspective of an urban mobility service. Assuming traveler behavior is in accordance with the principle of utility maximization, we first formulate a baseline dynamical model for traveler's expected future trip sequence under the framework of Markov decision processes and from which we derive approximations of the optimal coupon redemption policy. However, we find that this baseline model cannot explain perfectly observed coupon redemption behavior of traveler for a car-sharing service. To resolve this deviation from utility-maximizing behavior, we suggest a hypothesis that travelers may not be aware of all coupons available to them. Based on this hypothesis, we formulate an inattention model on unawareness, which is complementary to the existing models of inattention, and incorporate it into the baseline model. Estimation results show that the proposed model better explains the coupon redemption dataset than the baseline model. We also conduct a simulation experiment to quantify the negative impact of unawareness on coupons' promotional effects. These results can be used by mobility service operators to desig
The Coupon Collector Problem (CCP) is a well-known combinatorial problem that seeks to estimate the number of random draws required to complete a collection of $n$ distinct coupon types. Various generalizations of this problem have been applied in numerous engineering domains. However, practical applications are often hindered by the computational challenges associated with deriving numerical results for moments and distributions. In this work, we present three algorithms for solving the most general form of the CCP, where coupons are collected under any arbitrary drawing probability, with the objective of obtaining $t$ copies of a subset of $k$ coupons from a total of $n$. The First algorithm provides the base model to compute the expectation, variance, and the second moment of the collection process. The second algorithm utilizes the construction of the base model and computes the same values in polynomial time with respect to $n$ under the uniform drawing distribution, and the third algorithm extends to any general drawing distribution. All algorithms leverage Markov models specifically designed to address computational challenges, ensuring exact computation of the expectation a
We generalize the well-known Coupon Collector Problem (CCP) in combinatorics. Our problem is to find the minimum and expected number of draws, with replacement, required to recover $n$ distinctly labeled coupons, with each draw consisting of a random subset of $k$ different coupons and a random ordering of their associated labels. We specify two variations of the problem, Type-I in which the set of labels is known at the start, and Type-II in which the set of labels is unknown at the start. We show that our problem can be viewed as an extension of the separating system problem introduced by Rényi and Katona, provide a full characterization of the minimum, and provide a numerical approach to finding the expectation using a Markov chain model, with special attention given to the case where two coupons are drawn at a time.
Currently, many e-commerce websites issue online/electronic coupons as an effective tool for promoting sales of various products and services. We focus on the problem of optimally allocating coupons to customers subject to a budget constraint on an e-commerce website. We apply a robust portfolio optimization model based on customer segmentation to the coupon allocation problem. We also validate the efficacy of our method through numerical experiments using actual data from randomly distributed coupons. Main contributions of our research are twofold. First, we handle six types of coupons, thereby making it extremely difficult to accurately estimate the difference in the effects of various coupons. Second, we demonstrate from detailed numerical results that the robust optimization model achieved larger uplifts of sales than did the commonly-used multiple-choice knapsack model and the conventional mean-variance optimization model. Our results open up great potential for robust portfolio optimization as an effective tool for practical coupon allocation.
For status update systems operating over unreliable energy-constrained wireless channels, we address Weaver's long-standing Level-C question: do my packets actually improve the plant's behavior? Each fresh sample carries a stochastic expiration time -- governed by the plant's instability dynamics -- after which the information becomes useless for control. Casting the problem as a coupon-collector variant with expiring coupons, we (i) formulate a two-dimensional average-reward MDP, (ii) prove that the optimal schedule is doubly thresholded in the receiver's freshness timer and the sender's stored lifetime, (iii) derive a closed-form policy for deterministic lifetimes, and (iv) design a Structure-Aware Q-learning algorithm (SAQ) that learns the optimal policy without knowing the channel success probability or lifetime distribution. Simulations validate our theoretical predictions: SAQ matches optimal Value Iteration performance while converging significantly faster than baseline Q-learning, and expiration-aware scheduling achieves up to 50% higher reward than age-based baselines by adapting transmissions to state-dependent urgency -- thereby delivering Level-C effectiveness under tig
This article addresses a conjecture by Schilling concerning the optimality of the uniform distribution in the generalized Coupon Collector's Problem (CCP) where, in each round, a subset (package) of $s$ coupons is drawn from a total of $n$ distinct coupons. While the classical CCP (with single-coupon draws) is well understood, the group-draw variant - where packages of size $s$ are drawn - presents new challenges and has applications in areas such as biological network models. Schilling conjectured that, for $2 \leq s \leq n-1$, the uniform distribution over all possible packages minimizes the expected number of rounds needed to collect all coupons if and only if $s = n-1$. We prove Schilling's conjecture in full by presenting, for all other values of $s$, "natural" non-uniform distributions yielding strictly lower expected collection times. Explicit formulas and asymptotic analyses are provided for the expected number of rounds under these and related distributions. The article further explores the behavior of the expected collection time as $s$ varies under the uniform distribution, including the cases where $s$ is constant, proportional to $n$, or nearly $n$. Keywords: Coupon Co
We initiate the study of the Careless Coupon Collector's Problem (CCCP), a novel variation of the classical coupon collector, that we envision as a model for information systems such as web crawlers, dynamic caches, and fault-resilient networks. In CCCP, a collector attempts to gather $n$ distinct coupon types by obtaining one coupon type uniformly at random in each discrete round, however the collector is \textit{careless}: at the end of each round, each collected coupon type is independently lost with probability $p$. We analyze the number of rounds required to complete the collection as a function of $n$ and $p$. In particular, we show that it transitions from $Θ(n \ln n)$ when $p = o\big(\frac{\ln n}{n^2}\big)$ up to $Θ\big((\frac{np}{1-p})^n\big)$ when $p=ω\big(\frac{1}{n}\big)$ in multiple distinct phases. Interestingly, when $p=\frac{c}{n}$, the process remains in a metastable phase, where the fraction of collected coupon types is concentrated around $\frac{1}{1+c}$ with probability $1-o(1)$, for a time window of length $e^{Θ(n)}$. Finally, we give an algorithm that computes the expected completion time of CCCP in $O(n^2)$ time.
We study a labeled variant of the classical Coupon Collector Problem (CCP), recently introduced by Tan et al., where coupons arrive in groups and only the set of labels is revealed. The goal is to determine the expected number of group drawings required to uniquely identify the labeling of all coupons. We focus on the case where groups consist of pairs ($k=2$), and provide rigorous proofs for two conjectures posed by Tan et al.
We extend the Coupon Collector's Problem (CCP) and present a novel generalized model, referred as the k-LCCP problem, where one is interested in recovering a bipartite graph with a perfect matching, which represents the coupons and their matching labels. We show two extra-extensions to this variation: the heterogeneous sample size case (K-LCCP) and the partly recovering case.
Traditionally, firms have offered coupons to customer groups at predetermined discount rates. However, advancements in machine learning and the availability of abundant customer data now enable platforms to provide real-time customized coupons to individuals. In this study, we partner with Meituan, a leading shopping platform, to develop a real-time, end-to-end coupon allocation system that is fast and effective in stimulating demand while adhering to marketing budgets when faced with uncertain traffic from a diverse customer base. Leveraging comprehensive customer and product features, we estimate Conversion Rates (CVR) under various coupon values and employ isotonic regression to ensure the monotonicity of predicted CVRs with respect to coupon value. Using calibrated CVR predictions as input, we propose a Lagrangian Dual-based algorithm that efficiently determines optimal coupon values for each arriving customer within 50 milliseconds. We theoretically and numerically investigate the model performance under parameter misspecifications and apply a control loop to adapt to real-time updated information, thereby better adhering to the marketing budget. Finally, we demonstrate throug
We consider a generalisation of the classical coupon collector problem. We define a super-coupon to be any $s$-subset of a universe of $n$ coupons. In each round, a random $r$-subset from the universe is drawn and all its $s$-subsets are marked as collected. We show that the time to collect all super-coupons is $\binom{r}{s}^{-1}\binom{n}{s} \log \binom{n}{s}(1 + o(1))$ on average and has a Gumbel limit after a suitable normalisation. In a similar vein, we show that for any $α\in (0, 1)$, the expected time to collect $(1 - α)$ proportion of all super-coupons is $\binom{r}{s}^{-1}\binom{n}{s} \log \big(\frac{1}α\big)(1 + o(1))$. The $r = s$ case of this model is equivalent to the classical coupon collector model. We also consider a temporally dependent model where the $r$-subsets are drawn according to the following Markovian dynamics: the $r$-subset at round $k + 1$ is formed by replacing a random coupon from the $r$-subset drawn at round $k$ with another random coupon from outside this $r$-subset. We link the time it takes to collect all super-coupons in the $r = s$ case of this model to the cover time of random walk on a certain finite regular graph and conjecture that in general
Coupon distribution is a critical marketing strategy used by online platforms to boost revenue and enhance user engagement. Regrettably, existing coupon distribution strategies fall far short of effectively leveraging the complex sequential interactions between platforms and users. This critical oversight, despite the abundance of e-commerce log data, has precipitated a performance plateau. In this paper, we focus on the scene that the platforms make sequential coupon distribution decision multiple times for various users, with each user interacting with the platform repeatedly. Based on this scenario, we propose a novel marketing framework, named \textbf{S}equence-\textbf{A}ware \textbf{C}onstrained \textbf{O}ptimization (SACO) framework, to directly devise coupon distribution policy for long-term revenue boosting. SACO framework enables optimized online decision-making in a variety of real-world marketing scenarios. It achieves this by seamlessly integrating three key characteristics, general scenarios, sequential modeling with more comprehensive historical data, and efficient iterative updates within a unified framework. Furthermore, empirical results on real-world industrial da