Introduction

Many empirical questions concern a treatment’s probability of making a unit better off on an ordered scale. In a training-program setting, for example, a randomized offer can shift actual program receipt, employment determines whether wages are observed, and the outcome of interest may be a prespecified ordered wage category. A natural principal-stratum target is then the probability that program receipt raises the wage category among offer compliers who would be employed under either receipt state. This paper gives sharp bounds and guarded inference for that same-unit ordinal benefit probability in a finite ordered-outcome instrumental-variables model with treatment-induced selection.

The model combines a binary instrument $Z$, treatment receipt $D$, selection $S$, covariates $X$, and an ordered outcome $Y\in \{0,\dots ,K-1\}$ observed among selected units. The complier event $C$ is the group with $D_1>D_0$, where $D_0,D_1$ are potential treatment receipts under the two instrument states. The survivor-complier target conditions further on $S_0=S_1=1$, where $S_0,S_1$ are potential selection indicators under the two treatment states. The estimand is $$\theta =P(Y_1>Y_0\mid S_1=S_0=1,C),$$ the probability that the treatment-one potential ordered outcome exceeds the treatment-zero potential ordered outcome among survivor compliers. In the Job Corps-style interpretation, $Z$ is offer assignment, $D$ is program receipt, $S$ is employment, and $Y$ is an ordered wage category; the numerical example below is a synthetic compatible law.

The identifying restrictions follow the local-average-treatment-effect and principal-stratification traditions (Imbens et al., 1994; Angrist et al., 1996; Frangakis et al., 2002). Conditional on covariates, the instrument is independent of potential treatments, potential selections, and potential ordered outcomes. Observed treatment equals treatment under the realized instrument, observed selection equals selection under the realized treatment, and observed outcomes equal potential outcomes under the realized treatment on selected units. Treatment receipt is monotone in the instrument, and instrument propensities satisfy cellwise overlap. Selection is governed by a covariate-specific weak monotonicity direction: within each complier cell, either treatment weakly raises selection or treatment weakly lowers selection. These conditions match the selected-complier logic used in IV sample-selection work (Chen et al., 2015; Kennedy et al., 2019; Dong et al., 2026).

The first result identifies the observable capacities that enter the benefit problem. For each covariate cell $x$, the lower-arm capacities ${\ell }_i(x)$ and upper-arm capacities $h_j(x)$ are observed selected-complier outcome submeasures for $Y_0=i$ and $Y_1=j$, respectively. Their totals $q_0(x)$ and $q_1(x)$ can differ because treatment changes selection. Weak selection monotonicity makes $$m(x)=\min \{q_0(x),q_1(x)\}$$ the survivor-complier mass in cell $x$. Thus the observed law supplies two possibly unequal nonnegative capacities and an exact amount of mass that must be paired between them.

The construction follows a four-step route. Observed conditional instrument contrasts first produce the lower- and upper-arm selected-complier capacities. Their totals determine the cellwise exact survivor-complier mass. The threshold cuts then solve the exact-mass transport problem in each cell. Finally, the cellwise cut values are averaged with covariate-cell weights and normalized by aggregate survivor-complier mass to obtain the sharp interval.

The sharp-bound problem is a finite exact-mass partial-transport problem. In each cell, a feasible coupling ${\gamma }_x$ places mass on ordered pairs $(i,j)$, respects the row capacities ${\ell }_i(x)$, respects the column capacities $h_j(x)$, and has total mass $m(x)$. The cellwise benefit functional $$b_x({\gamma }_x)=\sum _{i<j}{\gamma }_{ij,x}$$ counts strict ordinal improvement. The paper proves that the feasible projections of full latent IV-selection laws are exactly these exact-mass transport polytopes, and that the sharp lower and upper values are the threshold-cut quantities $B_L(x)$ and $B_U(x)$. Aggregating by covariate-cell probabilities $p_x$ gives $${\Theta }_I(P_{\mathrm{obs}})=\left[\frac{{\sum }_x{p}_x{B}_L(x)}{{\sum }_x{p}_{x}m(x)},\frac{{\sum }_x{p}_x{B}_U(x)}{{\sum }_x{p}_{x}m(x)}\right].$$ Endpoint-attaining full latent laws show that the interval is sharp under the maintained structural law class.

The threshold formulas connect this construction to existing ordinal benefit bounds. Makarov-type and transport bounds characterize feasible joint distributions from marginal information (Makarov, 1982; Williamson et al., 1990; Fan et al., 2010; Villani, 2009). For complete ordered marginals, Lu et al. (2018) give closed-form sharp bounds for the strict-benefit probability. The present exact-mass formulation recovers that complete-marginal formula on the no-selection face where survivor-complier status coincides with complier status and the two capacity totals agree. Away from that face, the same threshold logic applies to selected-complier submeasures generated by noncompliance and treatment-induced selection. Related ordinal-benefit and probability-of-causation analyses study graphical, interventional, or unconfoundedness-based restrictions (Tian et al., 2000; Sachs et al., 2023; Duarte et al., 2023; Gabriel et al., 2024; de Aguas et al., 2025; Chen et al., 2026); the contribution here is the sharp survivor-complier IV-selection interval with endpoint-attaining latent laws.

The constructive proof is also an algorithm. The strict-benefit graph $i<j$ and its complement have nested threshold structure on a finite ordered support. A sparse threshold-flow routine computes all cellwise endpoint values and positive allocation traces in $O(|\mathcal{X}|K)$ operations. Writing full lower and upper $K\times K$ coupling matrices adds the expected $O(|\mathcal{X}|K^2)$ materialization cost. A three-level witness makes the calculation explicit: one covariate cell, three ordered outcomes, full compliance, and treatment-increased selection produce capacities $(0.075,0.10,0.075)$ and $(0.10,0.25,0.15)$, exact survivor-complier mass $1/4$, and the sharp interval $[0,0.7]$.

The estimation and inference results treat the sharp interval as a finite-dimensional functional of the observed-data law. The estimator replaces observed probabilities with empirical analogues, projects raw capacities onto the nonnegative cone, screens cells by empirical survivor-complier mass, and evaluates the same threshold formulas. At a fixed observed law, under independent sampling, overlap, the structural restrictions, positive aggregate survivor-complier mass, and a screening threshold ${\eta }_n$ with ${\eta }_n\to 0$ and $\sqrt{n}{\eta }_n\to \infty$, the retained support is recovered with probability tending to one, the plug-in endpoints are consistent, and the scaled endpoint error converges to the Hadamard directional derivative of the reduced-support endpoint map applied to the centered multinomial Gaussian limit. This accommodates simultaneous active threshold cuts and zero selected-complier capacity gaps through the directional derivative framework for nonsmooth maps (van der Vaart, 1998; Fang et al., 2019; Hong et al., 2018).

For finite-sample containment, the paper gives a deterministic guard. Over a finite event collection determined by $\mathcal{X}$ and $K$, a union-bound deviation threshold yields an explicit capacity-and-screening envelope. Under fixed instrument overlap and a uniform lower bound $M\ge m_{\star }>0$ on aggregate survivor-complier mass, padding the screened plug-in endpoints by this envelope produces a guarded interval that contains the entire sharp identified interval with probability at least $1-\alpha$, uniformly over the indexed finite-slate law family and every sample size. The guard radius is $O(n^{-1/2}+{\eta }_n)$, and the choice ${\eta }_n=n^{-1/2}{L}_n$ with $L_n\to \infty$ and $L_n=o(\sqrt{n})$ gives radius $O(n^{-1/2}{L}_n)$. The technical appendix records supporting derivations and cited inputs.

The paper proceeds as follows. Section 2 discusses the related econometric literature. Section 3 introduces the potential-outcome setup, the structural restrictions, and the observable capacity identities. Section 4 states the exact-mass partial-transport representation, the threshold-cut endpoint formulas, the endpoint-attaining law construction, and the no-selection reduction. Section 5 gives the sparse threshold-flow algorithm and the three-level witness. Section 6 develops the projected screened estimator, the fixed-law directional limit, and the deterministic guarded confidence interval. Section 7 records limitations and future work, and the technical appendix collects supporting derivations.

Related work

This paper works in a potential-outcome setting with a binary instrument, treatment-induced selection, and a finite ordered outcome. Its target combines three strands of econometric work: instrumental variables and principal strata, sample selection bounds, and sharp bounds for ordinal benefit probabilities.

The instrumental-variables component follows the local-treatment-effect tradition in which a binary instrument shifts treatment take-up and identifies causal information for compliers (Imbens et al., 1994; Angrist et al., 1996). Principal stratification gives the language for conditioning on joint potential statuses, including compliance and post-treatment selection strata (Frangakis et al., 2002). Partial-identification results for IV models, including sharp bounds under latent response restrictions, provide the surrounding identification perspective (Balke et al., 1997; Manski, 1990; Manski, 1997; Manski et al., 2000; Manski, 2003). The present analysis uses that tradition to study survivor compliers, a principal stratum defined jointly by compliance behavior and selection under treatment states, and asks for sharp bounds on the probability that treatment strictly improves an ordered outcome among that group.

Selection is central because the ordered outcome is observed on selected units. Classical sample-selection models motivate the econometric concern that outcome comparisons among selected units mix treatment effects with selection effects (Heckman, 1979). Lee-style trimming bounds give a transparent solution under monotone selection (Lee, 2009), and subsequent work develops this approach for instrumental variables and principal strata in settings with attrition or post-treatment selection (Chen et al., 2015; Blanco et al., 2013; Chen et al., 2018; Kennedy et al., 2019). The closest selection-bounds comparison is Dong et al. (2026), who study covariate-specific weak selection monotonicity and selected-complier outcome distributions. Closest on the benefit side under selection is Possebom et al. (2025), who bound the probability of causation for a binary outcome when selection determines whether the outcome is recorded, and who identify their target on the always-observed stratum. The present paper differs on both axes that generate its mathematics: the outcome is ordered with $K\ge 3$ levels, so the target is a same-unit comparison $Y_1>Y_0$ whose sharp value is a coupling problem rather than a pair of marginal probabilities; and the conditioning stratum is the survivor compliers, whose two arms carry unequal capacity totals when treatment shifts selection. That inequality is what forces the exact-mass partial-transport formulation below: the two arms must be coupled subject to a common exact-mass constraint, so the sharp value solves a transport problem rather than following from a pair of marginal probabilities. The present paper uses the same type of covariate-specific selection direction as an input to a sharp ordinal-benefit analysis for survivor compliers.

The ordinal outcome component connects to work on distribution-free bounds for functions of paired potential outcomes. Makarov-type bounds characterize feasible joint distributions given marginal distributions (Makarov, 1982; Williamson et al., 1990), and optimal-transport formulations give a useful finite-dimensional view of the same coupling problem (Villani, 2009; Fan et al., 2010). For ordered categorical outcomes, the benefit probability is a functional of the unobserved joint distribution of two potential ordered outcomes; complete marginal information yields closed-form formulas for this functional (Lu et al., 2018; Lu et al., 2020). This paper adapts that coupling logic to the selected-complier capacities generated by an IV model with treatment-induced selection: within each covariate cell, the relevant margins are selected-complier subprobability capacities with potentially unequal totals.

Recent work also develops sharp ordinal benefit bounds under other observable restrictions. Gabriel et al. (2024) study sharp ordinal-benefit bounds including response-function IV restrictions, and de Aguas et al. (2025) establish sharp bounds for tiered benefit under monotonicity with inference. Related causal-graphical analyses of probability-of-causation and benefit-type estimands include Tian et al. (2000); Sachs et al. (2023); Duarte et al. (2023). Chen et al. (2026) study principal generalized causal effects under treatment unconfoundedness and ignorability conditions. The contribution here is a sharp identified interval for a survivor-complier strict-benefit probability under the paper’s IV, consistency, overlap, no-defiers, weak-selection-monotonicity, and positive survivor-mass conditions, together with a finite-dimensional selected-complier capacity and partial-transport representation used for computation and inference.

The inference part of the paper belongs to the econometric literature on partially identified parameters and directionally nonregular maps. Confidence procedures for identified sets motivate coverage statements for an interval-valued target (Imbens et al., 2004; Chernozhukov et al., 2007; Romano et al., 2010; Tamer, 2010; Molinari, 2020; Chernozhukov et al., 2013). Directional differentiability and bootstrap validity for nonsmooth functionals provide the local asymptotic framework (van der Vaart, 1998; Fang et al., 2019; Hong et al., 2018). Recent contributions on inference for bounds and partially identified causal parameters, including Russell (2021), Levis et al. (2025), Ben-Michael (2025), Khandamiryan et al. (2026), Noack (2026), and Lee et al. (2024), frame the inferential goal: to report uncertainty for a set identified by observable distributions and maintained structural restrictions. Here that goal is implemented through a guarded finite-sample construction tailored to the threshold-cut representation of the survivor-complier benefit interval.

Setup and assumptions

Throughout the paper, covariate support is finite and sums over covariate cells range over $\mathcal{X}$. Outcome levels are ordered as $0,\dots ,K-1$, and indices $i,j,t$ range over this ordered support unless stated otherwise. Generic constants are positive and finite, Euclidean norms are used for finite-dimensional vectors, and asymptotic notation is always along the sampling sequence introduced in the inference section.

We begin with the potential-outcome structure. The observed system has covariates $X$, a binary instrument $Z$, treatment receipt $D$, selection $S$, and an ordered outcome $Y$ observed on selected units. The model also fixes the latent probability law $P$, the finite covariate carrier $\mathcal{X}$, the outcome-support size $K$, and the ordered outcome set $\mathcal{Y}$.

The section proceeds from variables to restrictions to capacities: the slate fixes the variables, the consistency and exclusion restrictions connect observed cells to potential outcomes, monotonicity identifies the complier and survivor-complier components, and the observable-capacity definition packages the two selected-complier submeasures used by the transport problem.

Potential treatments, selections, and outcomes are indexed by the intervention that defines them. The treatment potentials $D_0,D_1$ vary the instrument, while the selection potentials $S_0,S_1$ and outcome potentials $Y_0,Y_1$ vary treatment receipt.

The first group of identifying restrictions is the standard IV structure with treatment-mediated selection and outcomes. It separates instrument assignment from latent response types within covariate cells, and it connects the observed variables to their potential-outcome counterparts.

Assumption 1 is the conditional instrument independence condition: after conditioning on $X$, variation in $Z$ is as good as randomly assigned for the potential treatments, selections, and ordered outcomes (Angrist et al., 1996).

Assumption 2 is the treatment consistency condition, which links realized receipt to the potential treatment indexed by the realized instrument (Angrist et al., 1996). Together with independence, it gives observed first-stage contrasts their usual complier interpretation.

Assumption 3 is the selection exclusion through received treatment condition: instrument assignment affects selection through the treatment actually received (Kennedy et al., 2019).

Assumption 4 is the outcome exclusion through received treatment condition on selected units (Kennedy et al., 2019). Because outcomes are observed when $S=1$, this condition fixes the outcome content of selected observed cells.

The observed-data law records the variables available to the analyst, with the outcome marked missing outside the selected sample. We write the observed tuple as $O$ and its law as $P_{\mathrm{obs}}$.

Cell weights enter every aggregate target below. The covariate-cell probability $p_x$ is defined before the complier and survivor-complier quantities are formed.

The complier event $C$ is the latent stratum whose treatment receipt is shifted by the instrument. The benefit parameter studied later is restricted further to compliers who would be selected under both treatment states.

For compact reference, the following definition collects the finite slate, the observed law, the monotonicity structure, and the cellwise capacity objects used throughout the rest of the paper.

The remaining assumptions impose overlap and monotonicity. The instrument propensity $\pi (x)$ and the overlap constant ${\epsilon }_Z$ determine where conditional instrument contrasts are well defined.

Assumption 5 is the conditional instrument overlap condition (Kennedy et al., 2019). It keeps both instrument states present in every supported covariate cell, so the observed conditional contrasts used to form capacities are defined cell by cell.

Assumption 6 is the instrument monotonicity condition (Angrist et al., 1996). It orients the first stage so that the shifted latent group is the complier stratum in Definition 10.

Assumption 7 is the covariate-specific weak sample-selection monotonicity condition of Dong et al. (2026). The direction $d(x)$ is allowed to vary across covariate cells, and the condition makes the sign of the selected-complier selection contrast interpretable within cells that contain compliers.

Assumption 8 is the positive target-stratum probability condition (Kennedy et al., 2019). It ensures that the aggregate survivor-complier mass $M$ can normalize the benefit probability studied in the next section.

The maintained structural law class $\mathcal{M}$ is the collection of latent laws satisfying the restrictions just stated.

The observable selected-complier capacities are the bridge from the observed law to the partial-transport problem. For each cell, ${\ell }_i(x)$ records lower-arm selected-complier mass at outcome level $i$, while $h_j(x)$ records upper-arm selected-complier mass at level $j$. Their totals $q_0(x)$ and $q_1(x)$, their difference $\Delta q(x)$, and their exact survivor-complier mass $m(x)$ determine the feasible cellwise coupling sets introduced in the next section.

The section closes with the identification statement for these capacities. It converts the observed IV contrasts into latent selected-complier submeasures and identifies the exact survivor-complier mass cell by cell.

The intuition is that the IV independence and consistency restrictions express selected outcome contrasts by latent complier components, while treatment monotonicity fixes which treatment response type contributes to the contrast. Weak selection monotonicity then aligns the selection contrast so that the smaller selected-complier total is exactly the always-selected complier mass in that cell. Consequently, Theorem 1 supplies the two nonnegative marginal submeasures that feed the exact-mass partial-transport construction below.

Sharp benefit bounds

The capacity identification result in Theorem 1 reduces the target parameter to a finite collection of cellwise coupling problems. In each covariate cell, the lower-arm and upper-arm selected-complier outcome capacities provide two submarginals, while the exact survivor-complier mass fixes how much of these capacities must be paired. The resulting object is the exact-mass partial-transport polytope ${\Gamma }_x^{\ast }$, whose elements are cellwise survivor-complier coupling arrays ${\gamma }_x$. This is the finite ordinal analogue of a partial transport problem with capacity constraints (Villani, 2009).

The comparison polytopes in Definition 15 make explicit which side of the two selected-complier capacity totals binds. When the upper-arm total exceeds the lower-arm total, the row constraints can bind; when the lower-arm total exceeds the upper-arm total, the column constraints can bind. The branch-free notation ${\Gamma }_x$ records that the paper’s identified coupling set is always the exact-mass set, with the operative branch determined directly by the observed capacity totals.

The endpoint calculations use ordinal threshold cuts. The lower endpoint is obtained by forcing enough mass onto weakly adverse pairs to limit strict benefit, while the upper endpoint packs as much mass as possible onto pairs with treatment-one outcome above treatment-zero outcome. The finite support lets these constraints be summarized by prefixes and tails, in the spirit of sharp distributional bounds for ordinal and discrete outcomes (Makarov, 1982; Williamson et al., 1990; Fan et al., 2010).

The lower threshold-cut value $B_L(x)$ and upper threshold-cut value $B_U(x)$ are unnormalized benefit-mass bounds within a cell. Their formulas combine the ordinal cut constraints with the exact survivor-complier mass, including zero-gap cells where the two capacity totals coincide.

Definition 17 aggregates the cellwise cut values into the endpoint map $\Psi (P_{\mathrm{obs}})$. The lower endpoint ${\theta }_L$, upper endpoint ${\theta }_U$, and interval ${\Theta }_I(P_{\mathrm{obs}})$ are normalized by the aggregate survivor-complier mass from Assumption 8, so the estimand is the fraction benefiting among survivor compliers.

The aggregate notation in Definition 18 records the prefix and tail capacities used by the threshold formulas. The additional upper tail ${\ell }_{>t}(x)$, together with ${\ell }_{\le t}(x)$, ${\ell }_{<t}(x)$, $h_{\le t}(x)$, and $h_{>t}(x)$, gives the algebraic identities needed to compare the exact-mass problem with complete-marginal ordinal bounds.

Theorem 2 characterizes the zero-gap face of the partial-transport problem. At $\Delta q(x)=0$, the exact survivor-complier mass equals both selected-complier capacity totals, so the row-exact, column-exact, and doubly exact comparison polytopes coincide with the maintained coupling polytope. The threshold identity then expresses the same balance through prefixes and tails, which lets implementations evaluate zero-gap cells using the branch-free formulas in Definition 16.

The next construction turns the threshold formulas into full latent laws. Cellwise, it first builds extremal couplings for the strict-benefit and nonbenefit graphs; globally, it embeds those couplings into potential-outcome laws that retain the observed law and the structural restrictions.

The lower endpoint law $P^L$ and upper endpoint law $P^U$ are constructed cell by cell and then completed with the compliance and selection strata. The residual-mass step aligns the partial-transport coupling with the weak selection-monotonicity direction from Assumption 7, while the unchanged never-taker and always-taker components preserve the observed IV law.

Theorem 3 establishes that the cellwise extremal couplings are attainable by full potential-outcome laws. The construction first preserves the observed capacities identified in Theorem 1, then assigns the unmatched selected-complier capacity to the one-sided selected stratum allowed by weak selection monotonicity. Because the theorem attains the two endpoint probabilities under the maintained structural restrictions, the threshold formulas are realized by feasible full latent laws.

The exact-mass formulation also contains the familiar complete-marginal ordinal benefit problem on the no-selection face. In that submodel, survivor-complier status coincides with complier status in every relevant cell, so the two selected-complier capacity totals agree and the partial-transport polytope becomes a doubly exact coupling set.

The normalized formulas in Theorem 4 match the closed-form strict-benefit bounds for two complete ordinal marginals in Lu et al. (2018). The theorem identifies the precise face of the present model where that comparison applies: every relevant complier is always selected, the capacity gap is zero, and the exact-mass polytope has both margins fixed.

It remains to name the cellwise objective optimized over the coupling set.

The functional $b_x({\gamma }_x)$ is linear in the coupling and counts strict ordinal improvement. With this objective, the threshold cuts in Definition 16 become the exact lower and upper values of the cellwise optimization problem.

Theorem 5 is the section’s sharpness result. It characterizes the projection of feasible full laws onto the exact-mass partial-transport polytope, identifies the sign-specific comparison polytope selected by the capacity gap, and evaluates the lower and upper values of the strict-benefit objective by the threshold cuts. Together with endpoint law attainment in Theorem 3, these statements deliver the sharp identified interval for the survivor-complier benefit probability $\theta$ under the structural law class in Definition 13.

Computation and witness

The threshold-cut formulas in Definition 16 and the endpoint construction in Algorithm 1 have two practical consequences. First, the construction can be implemented cell by cell by sorting the ordered outcome levels once and moving through nested lower and upper threshold sets. Second, the same calculation produces explicit sparse couplings, so the numerical endpoint is accompanied by a witness allocation in the partial-transport polytope.

We begin with a three-level law that makes the objects in Definition 14 concrete. The example has one covariate cell, three ordered outcomes, and full compliance. Selection differs across treatment states, so the lower- and upper-arm selected-complier capacities have unequal totals before the exact survivor-complier mass is imposed.

The observed witness law $P_{\mathrm{obs}}^{\star }$ displays all observed cells needed to recover the capacity vector, while the latent witness law $P^{\star }$ supplies a compatible potential-outcome system. The selected always-survivor mass is diagonal in the ordered outcome, and the additional selected mass under treatment one raises the upper-arm capacities. The resulting calculation is small enough to audit directly but still contains the capacity imbalance that drives a nondegenerate sharp interval.

Theorem 6 shows how the observed margins translate into the interval $[0,0.7]$. In the witness, the exact survivor-complier mass is $1/4$, while the upper threshold cut is $7/40$. Dividing the upper cut by the aggregate survivor-complier mass gives $7/10$, and the lower cut gives the lower endpoint $0$. Endpoint-attaining latent laws therefore realize both ends of the interval within the same IV-selection model.

The computational routine implements the same threshold logic used in the sharpness theorem. It computes the two cut values, records the positive allocations that attain them, and separates the sparse work needed for endpoint evaluation from the additional work required to materialize full coupling matrices.

These definitions count the operations at the level used by the constructive proof of Theorem 5. The sparse trace is the economically relevant certificate: it lists the row–column masses that attain the endpoint. Dense matrices are useful when an implementation or diagnostic display requires a complete $K\times K$ array, and their cost includes the explicit zero writes.

Theorem 7 establishes that exact endpoint computation scales linearly in the number of outcome levels when the output is kept sparse, uniformly across covariate cells. The sparse bound follows the nested-threshold structure: each pass updates a prefix or tail quantity, and each pointer movement is charged once. A quadratic term enters precisely when the implementation writes full $K\times K$ lower and upper matrices. The magnitude-invariance clause is useful for repeated evaluation because the charged schedule depends on the support size and outcome grid, while the returned cut values and allocation masses depend on the realized capacities.

Estimation and guarded inference

The sharp interval in Definition 17 is a finite-dimensional functional of the observed-data law. Estimation therefore proceeds by replacing the observable probabilities in Definition 14 with empirical analogues, projecting the resulting capacities onto their nonnegative range, and evaluating the same threshold-cut formula cell by cell. The inferential problem is the usual one for nonsmooth partially identified endpoints: max, min, screening, and quotient operations can all determine the local behavior of the endpoint map (Imbens et al., 2004; Chernozhukov et al., 2007; Tamer, 2010; Molinari, 2020; Fang et al., 2019). The construction below keeps these operations explicit.

The finite-sample guard uses two sampling conditions. The first keeps the target stratum uniformly away from zero in aggregate, so that the endpoint quotients remain stable. The second fixes the sampling scheme for the empirical observed law.

The requirement in Assumption 9 is a uniform positive target-stratum probability condition: the aggregate survivor-complier mass entering the denominator of the identified endpoints is bounded below by $m_{\star }$. This is the standard positivity requirement for inference on survivor-type causal targets (Kennedy et al., 2019).

Assumption 10 is the independent sampling condition. It supplies the empirical-process input for the finite observed support and is the conventional baseline for the delta-method and multiplier arguments used below (van der Vaart, 1998).

The class Definition 23 collects the laws for which the structural restrictions in Definition 13, the lower aggregate-mass condition, and independent sampling hold at the sample-size index. The notation ${\mathcal{P}}_n$ will be used for uniform probability statements indexed by $n$.

The empirical covariate-cell probability ${\hat{p}}_x$ is the sample analogue of the cell weight $p_x$ in Definition 17. It is the outer weight in the plug-in sums and also enters the screening score.

Algorithm 2 mirrors the population construction in Definition 17. The raw empirical capacity vector ${\tilde{c}}_n$ is projected to the nonnegative capacity cone, giving the projected empirical capacity vector ${\hat{c}}_n$. The retained-cell score ${\hat{r}}_x$ and screening threshold ${\eta }_n$ restrict the quotient to cells with empirical survivor-complier mass large enough to enter stably. The resulting estimator ${\hat{\Psi }}_n$ is the sample analogue of the endpoint map.

For diagnostics and finite-sample containment, the guard works with a finite collection of observable events and a multiplier process over observed cells. The operational event collection consists of cell events $\{X=x\}$, arm events $\{X=x,Z=z\}$, and selected-outcome events $\{X=x,Z=z,D=d,S=1,\tilde{Y}=k\}$, indexed by $x\in \mathcal{X}$, $z,d\in \{0,1\}$, and $k\in \{0,\dots ,K-1\}$. Hence $$|\mathcal{E}|=|\mathcal{X}|+2|\mathcal{X}|+4K|\mathcal{X}|=|\mathcal{X}|(4K+3).$$

The event collection $\mathcal{E}$ is finite because both the covariate and outcome supports are finite. Empirical frequencies ${\hat{P}}_n$ control every sample input to the capacity construction, while the multiplier process ${\mathbb{G}}_n^{\xi }$ records the local face diagnostics associated with simultaneous max, min, and projection operations.

The construction below names three objects that play different roles, and it is worth separating them before reading it. The first is the reported interval: it is implementable from ${\hat{\Psi }}_n$, ${\eta }_n$, $\alpha$, ${\epsilon }_Z$, $m_{\star }$, and the support dimensions $(|\mathcal{X}|,K)$, which between them determine $|\mathcal{E}|$ and $b_{n,\alpha }$. Everything an analyst computes is in that list. The second is ${\delta }_n$, the maximal deviation of the empirical frequencies from $P_{\mathrm{obs}}$ over $\mathcal{E}$. It depends on the unknown $P_{\mathrm{obs}}$, so it belongs to the analysis rather than to the computation: it is the quantity the concentration argument bounds, while the padding radius actually applied is computed from $b_{n,\alpha }$ alone. The third is the multiplier face envelope, a diagnostic for the local nonsmooth behaviour of the endpoint map: it reports which threshold cuts and projection faces are near-active, and it is read on its own, alongside the reported interval. The coverage guarantee rests on the deterministic padding; uniform multiplier calibration under changing survivor support remains open, and is posed in Remark 1.

The confidence set in Algorithm 3 uses the deterministic deviation threshold $b_{n,\alpha }$ to form a capacity-and-screening envelope $A_{n,\alpha }$. Dividing by the aggregate-mass lower bound yields the endpoint guard $g_{n,\alpha }$, and padding the plug-in endpoints gives the interval ${\mathcal{C}}_{1-\alpha ,n}$. The multiplier component supplies a face-aware diagnostic for nonsmooth local behavior; the stated confidence set is the deterministic guarded interval.

The first large-sample result fixes one observed-data law and characterizes the local endpoint limit on the recovered positive-survivor support.

Theorem 8 establishes consistency of the screened support and of the endpoint estimator, then identifies the fixed-law directional limit. The Gaussian input ${\mathbb{Z}}_{P_{\mathrm{obs}}}$ is the centered multinomial limit for the observed cells, and the derivative is taken after the positive-survivor support ${\mathcal{X}}_{+}(P)$ has been recovered. The branch-free threshold representation handles simultaneous max and min faces through a Hadamard directional derivative; this derivative is the econometric object governing nonsmooth endpoint asymptotics (Fang et al., 2019; Hong et al., 2018).

The next three algebraic facts record how the cellwise masses and threshold cuts scale with nonnegative cell weights. They justify treating covariate-cell probabilities as outer weights in the aggregate endpoint formula and in the empirical analogue.

The proof is deferred to Section B.

The proof is deferred to Section B.

The proof is deferred to Section B.

Lemmas 1, 2, and 3 show that multiplying all capacities in a cell by the same nonnegative weight multiplies the exact mass and both benefit cut values by that weight. Consequently, aggregation over cells preserves the threshold-cut form of the endpoint numerators and denominator.

For the uniform statement, laws are indexed by $\lambda$. The empirical law and auxiliary multipliers are defined law by law, while the deterministic guard keeps the same finite-event form.

The empirical law ${\hat{P}}_{n,\lambda }$ and auxiliary process in Definitions 28 and 29 provide the indexed-sample notation for applying the same construction uniformly over $\Lambda$. The multiplier process is diagnostic for the finite face envelope, while containment is delivered by the deterministic padding rule in Algorithm 3.

Theorem 9 gives the advertised guarded inference guarantee. For every sample size, the interval from Algorithm 3 contains the full sharp identified interval with probability at least $1-\alpha$, uniformly over the indexed law family. The same result establishes uniform endpoint consistency under the deterministic guard, with radius $O(n^{-1/2}+{\eta }_n)$; choosing ${\eta }_n=n^{-1/2}{L}_n$ for any $L_n\to \infty$ and $L_n=o(\sqrt{n})$ yields a guard of order $O(n^{-1/2}{L}_n)$. The construction is conservative in the sense of finite-sample containment, and its rate statement shows that the extra screening term can vanish arbitrarily close to the root-$n$ scale.

Limitations and future work

The most important practical limitation is the width of the deterministic guard. The guarantee in Theorem 9 is a union bound over the finite event collection $\mathcal{E}$, and it buys uniformity at the price of large constants. Taking the smallest configuration the model allows — $K=3$, $|\mathcal{X}|=1$, ${\epsilon }_Z=1/4$, $\alpha =0.05$, and $m_{\star }=1/4$ — gives $|\mathcal{E}|=15$, so $$b_{n,\alpha }\approx 8.7n^{-1/2},A_{n,\alpha }\approx 2.7\times 10^4{n}^{-1/2},g_{n,\alpha }\approx 4.3\times 10^5{n}^{-1/2}$$ before the screening term is added. Since $g_{n,\alpha }$ is capped at $1$, the reported interval equals the whole of $[0,1]$ for every $n$ below roughly $10^{11}$, and its half-width $g_{n,\alpha }$ first falls below $0.1$ — a reported width of about $0.2$ plus the width of the identified interval itself — at $n$ of order $10^{13}$. At any sample size an applied researcher will encounter, the guarded interval is therefore uninformative: it is a proof that a finite-sample, uniformly valid procedure exists in this model, not a procedure we recommend reporting on its own. The rate $O(n^{-1/2}+{\eta }_n)$ is the honest content of the result; the constants are not sharp, and we have not attempted to optimize them. Reducing them — by replacing the union bound over $\mathcal{E}$ with a self-normalized or empirical-process concentration argument, and by exploiting that the capacity map is $1$-Lipschitz in far fewer directions than $|\mathcal{E}|$ suggests — is the most valuable next step for making the guarantee usable, and a simulation study establishing the achievable widths is a prerequisite for any empirical recommendation. We make no claim about practical performance here.

The sharp bounds and guarded confidence set above also leave one calibrated-inference problem for future work. The issue arises when threshold-cut faces intersect, selection-direction signs are close to changing, and the retained covariate support varies with the screening threshold. A useful strengthening of the inference theory would combine the finite deterministic guard in Theorem 9 with a first-order calibration based on the near-active faces recorded by Algorithm 3. The additional sign separation needed for such a direction-stable analysis is the following standard strong variant of the covariate-specific selection-direction condition.

Assumption 11 is imposed for a strictly positive constant ${\kappa }_{\sigma }>0$; read that way it is a sign-separation condition on the covariate-specific selected-complier capacity gap, requiring the selection direction to stay at least ${\kappa }_{\sigma }$ away from zero on cells with positive survivor-complier mass. The displayed inequality alone carries no content when ${\kappa }_{\sigma }\le 0$, since $|\Delta q(x)|\ge {\kappa }_{\sigma }$ then holds automatically; positivity of the margin is what the direction-stable analysis below uses, and it is not implied by the inequality as displayed. This is the analogue, for the capacity representation used here, of the strong direction conditions used to stabilize sign-sensitive inference under sample selection (Semenova, 2025). It gives the multiplier program a fixed local direction in cells that contribute to the survivor-complier target, while retaining the threshold-cut ties that make the endpoint map nonsmooth.

Remark 1 formulates the corresponding open calibration target. The desired confidence set would cover ${\Theta }_I(P_{\mathrm{obs}})$, the sharp identified interval for the survivor-complier benefit probability, uniformly over direction-separated triangular arrays, using ${\mathcal{C}}_{1-\alpha ,n}$, the guarded interval constructed from the finite face diagnostic. The challenge is to preserve first-order calibration when several threshold cuts bind at once and when cells enter or leave the retained support as their empirical survivor mass crosses ${\eta }_n$. This problem is closely connected to modern inference for partially identified treatment effects and selected samples, where uniformity at boundary and contact points is central (Lee et al., 2024; Bartalotti et al., 2023).

Several empirical and modeling extensions are also natural. The Job Corps application that motivates the ordinal wage target can be revisited with the sharp survivor-complier bounds and the guarded interval reported jointly. Continuous outcomes can be handled through discretizations or through an analogue of the threshold-transport construction on richer ordered supports, with the estimand and support conditions stated directly for that setting. Sensitivity analysis for weak selection monotonicity would index departures from Assumption 7 and trace their effect on the endpoint formulas in Definition 17. Additional concordance restrictions, such as restrictions tying potential wage ranks across treatment states, would sharpen the cellwise transport polytopes in Definition 15 and produce a different identified interval whose sharpness would need to be established under the enriched model.

Appendices

Technical derivations and verification note

The appendix collects the supporting derivations for the sharp survivor-complier benefit bounds, the computational construction, and the guarded inference result. The first part works through the capacity algebra behind Definition 16 and the residual completion used by Algorithm 1. The second part records the comparison argument behind Theorem 4. The final technical part gives the concentration calculations that enter the deterministic guard in Algorithm 3 and Theorem 9.

Threshold-cut algebra and residual completion

The threshold-cut calculations organize the cellwise problem in Definition 15 by ordered lower and upper outcome levels. For the lower endpoint, the relevant comparisons aggregate lower-arm mass weakly below a threshold and upper-arm mass weakly below the same threshold; for the upper endpoint, they compare lower-arm mass below a threshold with upper-arm mass above it. These finite ordered-support identities are the algebraic content behind Definition 16 and explain why the endpoint formulas in Definition 17 depend only on observed capacities and cell probabilities.

The residual completion step fills the remaining rows and columns after the threshold mass has been assigned. Its role is to turn the cut values into feasible cellwise couplings in the exact-mass transport set, preserving the row and column inequalities while attaining the desired strict-benefit mass. This is the constructive bridge from the formulas in Definition 16 to the endpoint-attaining laws in Theorem 3. The same finite-dimensional structure also underlies the sparse implementation summarized by Theorem 7.

No-selection reduction

The no-selection reduction specializes the general selected-survivor construction to the setting in which the selection constraint is inactive. Under the hypotheses of Theorem 4, the cellwise capacity arrays collapse to the ordinary complier outcome margins, and the same threshold formulas recover the established ordinal benefit bounds. This comparison fixes the relationship between the survivor-complier estimand studied here, sharp complete-marginal ordinal-outcome benefit bounds (Lu et al., 2018), and selection-bounds analyses of intensive-margin effects under endogenous treatment (Dong et al., 2026).

Guard calculations

The inference appendix derives the deterministic inequalities used by the guarded confidence set in Algorithm 3. The calculations combine finite-support empirical deviations over the event collection in that construction with the screening rule in Algorithm 2. Standard empirical-process and delta-method arguments for finite-dimensional laws supply the pointwise Gaussian approximation underlying Theorem 8 (van der Vaart, 1998). The guard in Theorem 9 then converts uniform control of the observed event probabilities into endpoint padding for the full identified interval.

The face-aware diagnostic records the threshold cuts that are near active at the empirical law. This finite diagnostic is useful because the endpoint map is piecewise linear in the capacity vector, with different linear pieces selected by the active threshold comparisons. The deterministic guard separates two tasks: it bounds capacity and screening errors by an explicit envelope, and it expands the resulting endpoint interval by a finite-sample padding term. This organization follows the econometric logic of inference for nonsmooth and partially identified parameters, where contact sets and binding inequalities determine the local approximation (Fang et al., 2019).

Verification note

The formal assumptions, definitions, propositions, algorithms, theorem statements, and proof dependencies in the paper are machine-checked in Lean 4 for the finite ordered-outcome model stated in the main text. The checking covers the capacity-identification formulas, the threshold-cut endpoint characterization, endpoint attainment by the threshold-flow construction, the no-selection reduction, the three-level witness, the sparse threshold-flow calculation, the pointwise directional limit statement, and the deterministic guarded-inference guarantee.

Scope, toolchain, and commit.

The development is checked with Lean 4, toolchain leanprover/lean4:v4.33.0, at commit 8f179a26a191, in the module tree CausalSmith/PartialID/PID_SlateBenefitPartialtransport_Research. This paper displays $56$ numbered formal objects. Of these, $48$ are matched to the Lean development in that tree, $7$ are presentation-level restatements carrying no Lean declaration of their own (listed below), and $1$ — the direction margin of Assumption 11 — is stated only to pose the open calibration problem and is used by no verified theorem. Every declaration certifying a matched object is free of sorry and of added axioms beyond the ambient Lean and Mathlib development.

Objects with no matching declaration.

The seven displayed definitions above are presentation-level restatements introduced for readability rather than separate Lean declarations: the slate-benefit partial-transport system (Definition 11), the threshold-cut capacity aggregates (Definition 18), the cellwise benefit-mass functional (Definition 19), the costed threshold-flow routines (Definition 21), the sparse threshold-flow operation counts (Definition 22), the empirical observed-data law for the indexed sample (Definition 28), and the auxiliary multiplier process (Definition 29). Each names or aggregates quantities that the matched declarations define; none carries a verified claim of its own, and no theorem in this paper is certified through one of them. Readers should treat the verification claim as attaching to the matched statements and their proofs, not to these expository restatements.

Cited inputs.

Published econometric and probability results invoked through citations are bibliographic dependencies; their source proofs are used in the ordinary way through the cited literature. Where a displayed result depends on such an input, the result carries a footnote naming the cited conclusion and stating that the portions depending on it are certified conditional on that conclusion.

Proofs of the main results

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