FunnelCausalNet: Funnel-aware Joint Conversion-Revenue Uplift for Multi-tier Coupon Allocation

FunnelCausalNet: Funnel-aware Joint Conversion-Revenue Uplift for Multi-tier Coupon Allocation

Abstract

Coupon campaigns aim to lift both conversion and revenue, but gross merchandise value (GMV) inherits a deterministic funnel structure from conversion and conditional order value and is typically zero-inflated and heavy-tailed. We propose FunnelCausalNet, an uplift estimator that couples a binary conversion head with a nonnegative conditional-value head under the funnel composition mu gmv = muconv mu val. Under explicit RCT, support, rate-gap, and cross-head covariance-control assumptions, we derive an idealized leading-order MSE-ratio comparison that identifies a variance regime in which funnel composition can reduce pointwise estimation variance; it is a regime heuristic rather than a guarantee for the shared-representation neural implementation. The estimator is paired with marginal split-conformal summaries on each outcome's CATE (Bonferroni union for two-number joint coverage, treated as audit/monitoring bands) and a Lagrangian budgeted allocator that consumes RCT-anchored estimates for subsidy-aware ROI accounting. On semi-synthetic multi-tier Criteo-MT7 with oracle individualized treatment effects, FunnelCausalNet's mean AUUC GMV is within one seed standard deviation of the leading recent feature-interaction baseline among eleven baselines, and a controlled funnel-coupling ablation reduces GMV effect error against direct GMV regression by 18–48% across the tested zero-inflation regimes. On de-identified industrial Hotel-Coupon RCT logs with ≈ 4.9 × 106 hold-out exposure records per seed, RCT-consistent expected-outcome (EOM) evaluation sweeps full LP frontiers, and FunnelCausalNet attains the best seed-averaged mean ΔROI at 7/7 correlated EOM anchors in 10%–60%; we treat this as descriptive frontier consistency rather than independent-anchor significance. On sparse binary-spend public benchmarks, revenue-focused rankers can dominate uplift-curve proxies; we foreground this regime boundary explicitly.

Introduction

Digital coupon programs aim to lift both the probability that a user converts and the revenue generated conditional on conversion. In practice, marketing teams often estimate conversion uplift and revenue-related quantities through loosely coupled pipelines—separate models for conversion probability and order value—and combine predictions downstream for targeting or budget allocation. This decoupled workflow ignores three structural features that routinely appear in coupon randomized controlled trials (RCTs). (i) Funnel identity: GMV satisfies Y g = 0 whenever Y c = 0, so GMV decomposes algebraically into conversion mass and conditional spend. Treating GMV as an unconstrained continuous response under extreme zero inflation produces variance-dominated estimates of heterogeneous treatment effects (HTE) on revenue. (ii) Ranking divergence: Ordering users by estimated conversion uplift can disagree substantially with ordering by estimated GMV uplift. Under tight budgets, this inconsistency directly translates into lost incremental GMV relative to revenue-aligned objectives. (iii) Multi-tier action space with tier-specific conversion- and revenue-elasticities: Retailers choose among multiple discount tiers; in our industrial RCT logs different coupon strengths exhibit quantitatively different elasticities on conversion probability and on conditional spend—a heavier coupon may convert more users while also reshaping the spend distribution among converters in ways that a single binary lift cannot pin down. Decision-making therefore requires assigning limited subsidy budgets across users and arms, and a binary-encouragement formulation cannot answer who should receive which tier under explicit cost-of-promotion constraints.

These observations motivate funnel-aware multi-outcome uplift modeling: jointly estimate heterogeneous causal effects on conversion and GMV while respecting the funnel structure, quantify uncertainty to support conservative deployment, and feed estimates into scalable budgeted multi-tier allocation. We treat the contribution as an end-to-end stack—point estimates, audit-oriented uncertainty, and budgeted assignment with subsidy accounting—rather than a single estimator.

Contributions

(1) Funnel-structured uplift estimation under zero inflation, with a regime-guided variance analysis. We present FunnelCausalNet, an estimator that couples a binary conversion head with a nonnegative spending head consistent with the deterministic zero mass on GMV. Under explicit assumptions on RCT identification, support, convergence rates, and covariance control (Sec. 4.2, Proposition 2), we derive an idealized leading-order MSE-ratio comparison for the high-zero-inflation regime. The comparison provides directional guidance; it does not guarantee dominance for the shared-representation neural implementation or across datasets. (2) Budgeted multi-tier allocation with RCT anchoring. We combine funnel estimates with Lagrangian relaxation for large-scale multi-tier assignment under subsidy budgets, and absorb additive shifts estimated from RCT arm averages on a held-in slice to mitigate systematic GMV-level bias before forming allocator rewards. (3) Auditable joint uncertainty layer. We supply marginal split-conformal intervals on each outcome's CATE summary together with a Bonferroni union (a finite-sample valid two-number joint coverage statement) and a Top-K boundary screen flagging users with unstable cross-objective rankings; these are positioned as audit/monitoring bands for compliance review rather than allocator inputs (Sec. 4.3).

Empirical scope. We evaluate against eleven baselines—meta-learners (S-, T-, X-Learner), causal forests, a dual-head network, CFRNet, DragonNet, EFIN, DESCN, ECUP, and RERUM—on (i) semi-synthetic Criteo-MT7 with oracle individualized treatment effects (ITEs), (ii) the public Hillstrom RCT (3-arm encouragement collapsed to a single send-vs-control contrast as in standard uplift evaluation), (iii) controlled funnel ablations, (iv) joint conformal coverage and computational scaling to 106 users, and (v) a large industrial Hotel-Coupon multi-arm RCT with ≈ 4.9 × 106 hold-out exposure records per seed under RCT-consistent expected-outcome (EOM) evaluation that sweeps full LP frontiers, with ΔGMV% anchors chosen so realized ΔROI values straddle the break-even band predicted by a commission-rate sensitivity scan (Sec. 5.8).

Honest benchmarking. Funnel coupling targets multi-tier RCT regimes in which heterogeneous coupon strengths induce distinct conversion- and revenue-elasticities. Public benchmarks built around a single send-vs-control encouragement (e.g., Hillstrom) instantiate a different decision problem—there is no tier-strength axis along which conversion- and spend-elasticities can differ—so revenue-focused rankers tuned for binary AUUC proxies can score higher. We surface this scope boundary in Section 6 rather than suppressing it. The anchored allocator and the audit conformal layer nonetheless provide a uniform pipeline aligned with subsidy accounting in all regimes.

Problem Formulation

Observables and funnel support. Consider a coupon RCT with observables (X,T,Y c,Y g), where X ∈ X are user features, T ∈ 0, 1,..., K is a discrete treatment indicator (T = 0 is control), Y c ∈ 0, 1 is conversion, and Y g ∈ R ≥0 is GMV. Here T indexes the coupon offers randomized in the logs; the method does not interpolate a continuous dose–response curve between those arms. Throughout we impose the funnel support restriction Y g = 0 whenever Y c = 0, so GMV is undefined as a positive outcome until conversion occurs and conditional spend is only meaningful on the converting subpopulation.

Causal targets. Let Y c(t), Y g(t) denote potential outcomes under assignment t. For each non-control arm t ∈ 1,..., K, define arm-specific CATEs tautc(x) = E[Y c(t)−Y c(0) X = x], tautg(x) = E[Y g(t)−Y g(0) X = x]. We identify tautc and tautg under randomized T X (RCT) as in standard analyses; we do not claim identification from purely observational logs.

Tower identity. For any fixed arm t, the law of iterated expectations gives E[Y g(t)] = E[Y c(t) · E[Y g(t) Y c(t)=1, X]], separating level calibration of GMV (anchoring metrics in Sec. 4.4) from heterogeneous ordering (PEHE/AUUC in Sec. 5).

Decision problem: budgeted multi-tier allocation. A deterministic policy pi: X → 0,..., K assigns each user to control or one tier. Let c(x, k) denote the predicted incremental subsidy cost of assigning x to tier k, derived from the tier-specific coupon terms and the campaign-specific accounting base. Feasible policies satisfy a total budget B > 0: Σi c(xi, pi(xi)) ≤ B, pi(xi) ∈ 0,..., K. Objectives include maximizing incremental GMV Σi E[taupi(xi)g(xi)] or its ROI-style surrogate ΔROI:= Σi taûpi(xi)g(xi) / Σi ĉi,pi(xi).

Dual-objective tension. When rankings induced by tauc and taug disagree, no single scalar objective is universally aligned with business preferences. The conflict diagnostic in Sec. 4.3 does not solve a general multi-objective program; it flags individuals whose objective-wise rankings and intervals jointly indicate instability near budgeted Top-K cuts.

Methodology

Funnel-structured uplift estimation. We estimate multi-arm conversion probabilities muconv(t)(x) and nonnegative conditional order-value expectations muval(t)(x) with shared representations. GMV expectations obey the funnel composition mugmv(t)(x) = muconv(t)(x) muval(t)(x) after numerical stabilization (clipping, nonnegative projections). Training combines Bernoulli conversion losses with squared error on log(1+GMV) among converters; inference maps normalized logits back to currency units using a LogNormal-style mean correction. The total objective allows optional consistency and monotonicity terms: Ltotal = Lconv + alpha Lval + beta Lconsist + gamma Lmono. Soft funnel variants replace the hard product with large penalties. They can be preferable when stage labels are asynchronously logged, missing, or otherwise make the support relation approximate. In our RCT logs the support identity is verified by construction, and Sec. 5.3 shows that retaining violations through a soft penalty does not match hard composition under extreme zero inflation.

Variance decomposition and leading-order MSE ratio. Fix (X,T) = (x,t). For the following propositions, all moments are conditional on this event; let p:= E[Yc], muv:= E[Yg Yc=1], and sigmav2:= Var(Yg Yc=1).

Proposition 1 (Variance decomposition): Var(Yg X=x,T=t) = psigmav2 + p(1−p)muv2. The first term is the within-converter variance; the second is the Bernoulli switching variance contributed by the zero mass.

Proposition 2 (Idealized leading-order MSE ratio under a rate gap): Let mûgdirect be a direct nonparametric squared-error estimator of E[Yg X=x,T=t] and mûgfunnel:= mûconv mûval the funnel composition estimator. Assume:

(A1) (RCT identification.) T ⊥ (Yc(·), Yg(·)) X and the propensity Pr(T=t X) is bounded away from 0 at x.

(A2) (Funnel support.) Yg = 0 whenever Yc = 0, with sigmav2 ∈ (0, ∞) and muv ∈ (0, ∞).

(A3) (Conv-head parametric rate.) mûconv is fit by a (correctly-specified) parametric Bernoulli model on the full n-sample, so p̂−p = Op(n−1/2) at (x,t).

(A4) (Value-head and direct nonparametric variance.) mûval is fit nonparametrically on the converter subsample and mûgdirect is fit nonparametrically on the full sample, both with negligible bias under standard undersmoothing and asymptotic pointwise variances scaling as 1/rn for an effective-sample-size sequence rn → ∞ with rn = o(n): Var(mûval) ∼ sigmav2/(prn) and Var(mûgdirect) ∼ Var(Yg X=x,T=t)/rn.

(A5) (Cross-head covariance control.) Either independent sample splitting makes Cov(p̂, mûv) = 0 in the idealized analysis, or the covariance is o(rn−1). This condition is not guaranteed by a shared-representation neural implementation.

Then, applying the delta method to (p, muv) ↦→ pmuv, the leading-order pointwise MSEs satisfy limn→∞ MSE(mûgfunnel)/MSE(mûgdirect) = psigmav2/(psigmav2 + p(1−p)muv2) = 1/(1 + (1−p)muv2/sigmav2).

Sufficient-regime reading. Eq. (8) is an idealized pointwise variance comparison, not a universal optimality theorem or a guarantee for CATE ranking. It relies on the parametric rate gap (A3) and covariance control (A5), which make the Bernoulli switching and cross-head terms vanish faster than the within-converter contribution psigmav2/rn. If both heads are estimated nonparametrically at the same rate, the ratio collapses to one. Shared neural representations can also induce correlated finite-sample errors, and systematic biases in the two heads can be multiplied by the product composition. We therefore use (8) only as a regime indicator; the controlled E2 stress test (Sec. 5.3) probes whether its predicted direction appears empirically across p̂ ∈ [5, 45]% without validating the asymptotic assumptions.

Operational regime. Within these idealized assumptions, the ratio in (8) is below one whenever (1−p)muv2/sigmav2 > 0, and shrinks as the zero mass (1−p) grows or muv dominates sigmav. This is the same hurdle/two-part structural intuition long studied in econometrics; our contribution is to connect the leading-order ratio under the rate-gap regime to the coupon-uplift setting, not to claim a new funnel identity or universal dominance.

Joint conformal intervals and conflict screening. Scope: We compose marginal split-conformal intervals on each outcome's CATE summary and apply a Bonferroni union across the two margins. Under standard split-conformal assumptions on a disjoint calibration fold, both intervals jointly cover their respective targets with probability at least 1−alpha at nominal level alpha/2 per margin. This is a finite-sample valid two-number coverage statement, not a simultaneous band over the bivariate CATE surface. Deployment stance: In practice, Bonferroni splits, finite-sample CQR offsets, and heavy-tailed GMV residuals drive empirical joint coverage above the nominal 1−alpha (Sec. 5.6). We therefore treat intervals primarily as auditable monitoring bands for compliance and risk review, recommend wider nominal alpha ∈ [0.10, 0.20] when widths must remain actionable, and pair intervals with anchored point estimates when feeding optimizers, because marginally valid lower-conformal bounds for taug at narrow alpha can be so pessimistic under zero inflation that budgeted LCB policies collapse to all-control assignments (Sec. 5.5). Conflict diagnostic: A Top-K boundary screen flags users with (i) disagreement between tauc and taug rankings, (ii) wide dual intervals or predictions near decision cutoffs, and (iii) instability near budgeted thresholds. The rule is a risk-disclosure layer for manual review or conservative assignment, not a precision-calibrated detector of latent business conflicts.

Budgeted multi-tier allocation with anchoring. Given predicted incremental rewards taûtg(x) and costs c(x,t), we maximize the budgeted assignment using Lagrangian relaxation: dual updates over a scalar multiplier lambda approximately enforce (4) while inner problems decouple across users for scalability. Moderate-scale LP relaxations serve as references where memory permits and provide the headline industrial allocator in Sec. 5.8. Anchoring: Predicted GMV levels ca…

Based on the paper, here are specific improvements to AI systems and what the improved systems can do:

Improvement: Enforce algebraic composition constraints between related outcomes (e.g., GMV = conversion probability × conditional spend) rather than modeling them independently.

Capability: The AI system can produce internally consistent predictions across hierarchical outcomes, reducing variance in high-zero-inflation regimes by 18–48% compared to direct regression, and avoiding impossible predictions (e.g., positive revenue with zero conversion).

Improvement: When one outcome head (e.g., conversion) can be estimated at parametric rates while another (e.g., conditional spend) requires nonparametric rates, compose them multiplicatively to exploit the faster-converging head.

Improvement: Combine Lagrangian relaxation for scalable budget-constrained assignment with additive calibration shifts estimated from randomized controlled trial arm averages.

Improvement: Provide marginal split-conformal intervals per outcome with Bonferroni union for joint coverage, explicitly positioned as monitoring bands rather than allocator inputs.

Improvement: Detect individuals where rankings induced by different objectives (e.g., conversion uplift vs. GMV uplift) disagree, especially near decision boundaries.

Improvement: Explicitly identify when the model's assumptions (multi-tier treatment, funnel structure) are not realized in the evaluation data, and report performance separately across regimes.

Improvement: Integrate training, conformal calibration (<1s at 1M users), inference, and Lagrangian allocation into a unified pipeline with sublinear scaling.

Improvement: Evaluate allocation policies using Hájek IPW on policy-matched RCT subsets, sweeping dual multipliers to trace full (ΔGMV%, ΔROI) frontiers.