Title: Optimal and Fair Encouragement Policy Evaluation and Learning
Venue: NeurIPS 2023
Accepted: yes
Keywords: causal inference

fairness in machine learning

algorithmic fairness

criminal justice

policy learning

off-policy evaluation
Abstract: In consequential domains, it is often impossible to compel individuals to take treatment, so that optimal policy rules are merely suggestions in the presence of human non-adherence to treatment recommendations. In these same domains, there may be heterogeneity both in who responds in taking-up treatment, and heterogeneity in treatment efficacy. For example, in social services, a persistent puzzle is the gap in take-up of beneficial services among those who may benefit from them the most. When in addition the decision-maker has distributional preferences over both access and average outcomes, the optimal decision rule changes. We study identification, doubly-robust estimation, and robust estimation under potential violations of positivity. We consider fairness constraints such as demographic parity in treatment take-up, and other constraints, via constrained optimization. Our framework can be extended to handle algorithmic recommendations under an often-reasonable covariate-conditional exclusion restriction, using our robustness checks for lack of positivity in the recommendation. We develop a two-stage, online learning-based algorithm for solving over parametrized policy classes under general constraints to obtain variance-sensitive regret bounds. We assess improved recommendation rules in a stylized case study of optimizing recommendation of supervised release in the PSA-DMF pretrial risk-assessment tool while reducing surveillance disparities.

Optimal and Fair Encouragement Policy Evaluation and Learning

Abstract

In consequential domains, it is often impossible to compel individuals to take treatment, so that optimal treatment assignments are merely suggestions when humans make the final treatment decisions. On the other hand, there can be different heterogeneity in both the actual response to treatment and final treatment decisions given recommendations. For example, in social services, a persistent puzzle is the gap in take-up of beneficial services among those who may benefit from them the most. When decision-makers have equity- for fairness-minded preferences over both access and average outcomes, the optimal decision rule changes due to these differing heterogeneity patterns. We study identification and improved/robust estimation under potential violations of positivity. We consider fairness constraints such as demographic parity in treatment take-up, and other constraints, via constrained optimization. We develop a two-stage, online learning-based algorithm for solving over parametrized policy classes under general constraints to obtain variance-sensitive regret bounds. Our framework can be extended to handle algorithmic recommendations under an often-reasonable covariate-conditional exclusion restriction, using our robustness checks for lack of positivity in the recommendation.

1 Introduction

The intersection of causal inference and machine learning for heterogeneous treatment effect estimation can improve public health, increase revenue, and improve outcomes by personalizing treatment decisions, such as medications, e-commerce platform interactions, and social interventions, to those who benefit from it the most [4, 30, 35, 49]. But, in many important settings, we do not have direct control over treatment, and can only optimize over encouragements , or recommendations for treatment. For example, in e-commerce, companies can rarely compel users to sign up for certain services, rather nudge or encourage users to sign up via promotions and offers. When we are interested in optimizing the effects of signing up - or other voluntary actions beyond a platform's control on important final outcomes such as revenue, we therefore need to consider fairness-constrained optimal encouragement designs . Often human expert oversight is required in the loop in important settings where ensuring fairness in machine learning [6] is also of interest: doctors prescribe treatment from recommendations [31], managers and workers combine their expertise to act based on decision support [8], and in the social sector, caseworkers assign to beneficial programs based on recommendations from risk scores that support triage [14, 19, 48].

The human in the loop requires new methodology for optimal encouragement designs because when the human in the loop makes the final prescription, algorithmic recommendations do not have direct causal effects on outcomes; they change the probability of treatment assignment.

On the other hand, this is analogous to the well-understood notion of non-compliance/non-adherence in randomized controlled trials in the real world [22, 21]. For example, patients who are prescribed treatment may not actually take medication. A common strategy is to conduct an intention-to-treat analysis: under assumptions of no unobserved confounders affecting treatment take-up and outcome, we may simply view encouragement as treatment. But, in the case of prediction-informed decisions in social settings, if we are concerned about access to the intervention in addition to utility of the policy over the population , finer-grained analysis is warranted. If an outcome-optimal policy results in wide disparities in access, for example in marginalized populations not taking up incentives for healthy food due to lack of access in food deserts, or administrative burden that screens out individuals applying for social services that could benefit the most, this could be a serious concern for decision-makers. We ultimately may seek optimal decision rules that improve disparities in treatment access. In contrast, previous work in algorithmic accountability primarily focuses on auditing recommendations , but not both the access and efficacy achieved under the final decision rule. Therefore, previous methods can fall short in mitigating potential disparities.

Our contributions are as follows: we characterize optimal and resource fairness-constrained optimal decision rules, develop statistically improved estimators and robustness checks for the setting of algorithmic recommendations with sufficiently randomized decisions. We also develop methodology for optimizing over a constrained policy class with less conservative out-of-sample fairness constraint satisfaction by a two-stage procedure, and we provide sample complexity bounds. We assess improved recommendation rules in a stylized case study of optimizing recommendation of supervised release in the PSA-DMF pretrial risk-assessment tool while reducing surveillance disparities.

2 Related Work

In the main text, we briefly highlight the most relevant methodological and substantive work and defer additional discussion to the appendix.

Optimal encouragement designs/policy learning with constraints. There is extensive literature on off-policy evaluation and learning, empirical welfare maximization, and optimal treatment regimes [5, 49, 35, 30]. [39] studies an optimal individualized encouragement design, though their focus is on optimal individualized treatment regimes with instrumental variables. [27] study fairness in pricing, and some of the desiderata in that setting on revenue (here, marginal welfare) and demand (take-up) are again relevant here, but in a more general setting beyond pricing. The most closely related work in terms of problem setup is the formulation of 'optimal encouragement designs" in [39]. However, they focus on knapsack resource constraints, which have a different solution structure than fairness constraints. Their outcome models in regression adjustment are conditional on the recommended/not recommended partitions which would not allow our fairness constraints that introduce treatmentand group-dependent costs. [44] has studied uniform feasibility in constrained resource allocation, but without encouragement or fairness. [9] studies robust extrapolation in policy learning from algorithmic recommendation, but not fairness. Our later case study is on supervised release, where there is a lot of randomness in final treatment decisions, rather than pretrial detention.

Fair off-policy learning We highlight some most closely related works in off-policy learning (omitting works in the sequential setting). [37] studies high-probability fairness constraint satisfaction. [29] studies doubly-robust causal fair classification, while others have imposed deterministic resource constraints on the optimal policy formulation [13]. [26] studies (robust) bounds for treatment responders in binary outcome settings; this desiderata is coupled to classification notions of direct treatment. Again, our focus is on modeling the fairness implications of non-adherence. Indeed, in order to provide general algorithms and methods, we do build on prior fair classification literature. A different line of work studies 'counterfactual" risk assessments which models a different concern.

Other causal methodology for intention-to-treat We focus on deriving estimators for intention-totreat analyses in view of fairness constraints (which result in group-specific welfare weights). Our interest is in imposing separate desiderata on treatment realizations under non-compliance; but we don't conduct instrumental variable inference and we assume unconfoundedness holds. We argue ITT is policy-relevant whereas complier-strata specific analysis is less policy-relevant since the compliers are unknown. Since our primary interest is in characterizing fair optimal decision rules, we don't model this as a mediation analysis problem (isolating the impact of recommendation even under the same ultimate treatment in a nested counterfactual), which may be more relevant for descriptive

characterization. [27] studies multi-objective desiderata for pricing and notes intention-to-treat structure in pricing, but not fairness considerations in more general problems. A related literature studies principal stratification [24], which has similar policy-relevance disadvantages regarding interpretability as complier analysis does.

3 Problem Setup

We briefly describe the problem setup. We work in the Neyman-Rubin potential outcomes framework for causal inference [40]. We define the following:

recommendation flag R ∈ { 0 , 1 } , where R = 1 means encouraged/recommended. (We will use the terms encouragement/recommendation interchangeably).

treatment T ( R ) ∈ T , where T ( r ) = 1 indicates the treatment decision was 1 when the recommendation reported r .

outcome Y ( t ( r )) is the potential outcome under encouragement r and treatment t .

Regarding fairness, we will be concerned about disparities in utility and treatment benefits (resources or burdens) across different groups, denoted A ∈ { a, b } . (For notational brevity, we may generically discuss identification/estimation without additionally conditioning on the protected attribute). For example, recommendations arise from binary high-risk/low-risk labels of classifiers. In practice, in consequential domains, classifier decisions are rarely automated, rather used to inform humans in the loop. The human expert in the loop decides whether or not to assign treatment. For binary outcomes, we will interpret Y ( t ( r )) = 1 as the positive outcome, and when treatments are also binary, we may further develop analogues of fair classification criteria. We let c ( r, t, y ): { 0 , 1 } 3 ↦→ R denote the cost function for r ∈ { 0 , 1 } , t ∈ T , y ∈ { 0 , 1 } , which may sometimes be abbreviated c rt ( y ) . We discuss identification and estimation based on the following recommendation, treatment propensity, and outcome models:

We are generally instead interested in personalized recommendation rules π ( r | X ) = π r ( X ) which describes the probability of assigning the recommendation r to covariates X . The average encouragement effect is the difference in average outcomes if we refer everyone vs. no one, while the encouragement policy value V ( π ) is the population expectation induced by the potential outcomes and treatment assignments realized under a recommendation policy π.

Because algorithmic decision makers may be differentially responsive to recommendation, and treatment effects may be heterogeneous, the optimal recommendation rule may differ from the (infeasible) optimal treatment rule when taking constraints into account or for simpler policy classes.

Assumption 1 (Consistency and SUTVA ) . Y i = Y i ( T i ( R i )) .

Assumption 2 (Conditional exclusion restriction) . Y ( T ( R )) ⊥ ⊥ R | T, X, A.

Assumption 3 (Unconfoundedness) . Y ( T ( r )) ⊥ ⊥ T ( r ) | X,A.

Assumption 4 (Stable responsivities under new recommendations) . P ( T = t | R = r, X ) remains fixed from the observational to the future dataset.

Assumption 5 (Decomposable costs) . c ( r, t, y ) = c r ( r ) + c t ( t ) + c y ( y )

Assumption 6 (Overlap) . ν r ≤ e r ( X,A ) ≤ 1 -ν r ; ν t ≤ p t | r ( X,A ) ≤ 1 -ν t ; ν r , ν t ≤ 0

Our key assumption beyond standard causal inference assumptions is the conditional exclusion restriction assumption 2, i.e. that conditional on observable information X , the recommendation has no causal effect on the outcome beyond its effect on increasing treatment probability. This assumes that all of the covariate information that is informative of downstream outcomes is measured. Although this may not exactly hold in all applications, stating this assumption is also a starting point for sensitivity analysis under violations of it [25]. Assuming assumption 6 is like assuming

we consider a randomized controlled trial with nonadherence. But later we give arguments using robustness to go beyond this, leveraging our finer-grained characterization.

Assumption 4 is a structural assumption that limits our method to most appropriately re-optimize over small changes to existing algorithmic recommendations. This is also required for the validity of intention-to-treat analyses. For example, p 0 | 1 ( x ) (disagreement with algorithmic recommendation) could be a baseline algorithmic aversion. Not all settings are appropriate for this assumption. We don't assume micro-foundations on how or why human decision-makers were deviating from algorithmic recommendations, but take these patterns as given. One possibility for relaxing this assumption is via conducting sensitivity analysis, i.e. optimizing over unknown responsivity probabilities near known ones.

Later on, we will be particularly interested in constrained formulations on the intention-to-treat effect that impose separate desiderata on outcomes under treatment, as well as treatment.

4 Method

We consider two settings: in the first, R is (as-if) randomized and satisfies overlap. Then R can be interpreted as intention to treat or prescription, whereas T is the actual realization thereof. We study identification of optimal encouragement designs with potential constraints on treatment or outcome utility patterns by group membership. We characterize optimal unconstrained/constrained decisions under resource parity. In the second, R is an algorithmic recommendation that does not satisfy overlap in recommendation (but there is sufficient randomness in human decisions to satisfy overlap in treatment): we derive robustness checks in this setting by being robust. First we discuss causal identification in optimal encouragement designs.

Proposition 1 (Regression adjustment identification) .

Proof of Proposition 1.

where the last line follows by the conditional exclusion restriction (Assumption 2) and consistency (Assumption 1).

Resource-parity constrained optimal decision rules Weconsider a resource/burden parity fairness constraint:

Enforcing absolute values, etc. follows in the standard way. Not all values of ϵ may be feasible; in the appendix we give an auxiliary program to compute feasible ranges of ϵ . We first characterize a threshold solution when the policy class is unconstrained.

Proposition 2 (Threshold solutions) . Define

If instead d ( x ) is a function of covariates x only,

Establishing this threshold structure (follows by duality of infinite-dimensional linear programming) allows us to provide a generalization bound argument.

4.1 Generalization

Proposition 3 (Policy value generalization) . Assume the nuisance models η = [ p 1 | 0 , p 1 | 1 , µ 1 , µ 0 ] ⊤ , η ∈ H are consistent and well-specified with finite VC-dimension V η over the product function class H . Let Π = { I { E [ L ( λ, X, A ; η ) | X ] > 0: λ ∈ R ; η ∈ F} .

This bound is stated for known nuisance functions: verifying stability under estimated nuisance functions further requires rate conditions.

Doubly-robust estimation We may improve statistical properties of estimation by developing doubly robust estimators which can achieve faster statistical convergence when both the probability of recommendation assignment (when it is random), and the probability of outcome are consistently estimated; or otherwise protect against misspecification of either model. We first consider the ideal setting when algorithmic recommendations are randomized so that e r ( X ) = P ( R = r | X ) .

Proposition 4 (Variance-reduced estimation) .

Although similar characterization appears in [39] for the doubly-robust policy value alone, note that doubly-robust versions of the constraints we study would result in differences in the Lagrangian so we retain the full expression rather than simplifying. For example, for regression adjustment, Proposition 9 provides interpretability on how constraints affect the optimal decision rule. In the appendix we provide additional results describing extensions of Proposition 8 with improved estimation.

4.2 Robust estimation with treatment overlap but not recommendation overlap

When recommendations are e.g. the high-risk/low-risk labels from binary classifiers, we may not satisfy the overlap assumption, since algorithmic recommendations are deterministic functions of covariates. However, note that identification in Proposition 1 requires only SUTV A and consistency, and the exclusion restriction assumption. Additional assumptions may be required to extrapolate p t | r ( X ) beyond regions of common support. On the other hand, supposing that positivity held with respect to T given covariates X , given unconfoundedness, our finer-grained approach can be beneficial because we only require robust extrapolation of p t | r ( X ) , response to recommendations, rather than the outcome models µ t ( X ) .

We first describe what can be done if we allow ourselves parametric extrapolation on p 1 | 1 ( X ) , treatment responsivity. In the case study later on, the support of X | R = 1 is a superset of the support of X | R = 0 in the observational data. Given this, we derive the following alternative identification based on marginal control variates (where p t = P ( T = t | X ) marginalizes over the distribution of R in the observational data):

Proposition 5 (Control variate for alternative identification ) . Assume that Y ( T ( r )) ⊥ T ( r ) | R = r, X .

Robust extrapolation of p t | r ( X ) Let X no = { x : P ( R = 1 | X ) = 0 } denote the no overlap region; on this region there are no joint observations of ( t, r, x ) . We consider uncertainty sets for ambiguous treatment recommendation probabilities. For example, one plausible structural assumption is monotonicity , that is, making an algorithmic recommendation can only increase the probability of being treated.

We define the following uncertainty set:

We could assume uniform bounds on unknown probabilities, or more refined bounds, such as Lipschitz-smoothness with respect to some distance metric d , or boundedness.

Define V ov ( π ) := ∑ t ∈T ,r ∈{ 0 , 1 } E [ π ( r | X ) p t | r ( X ) µ t ( X ) I ov ] . Let U denote the uncertainty set including any custom constraints, e.g. U = U q t | r ∩U lip. For brevity we use I no to denote I [ X ∈ X no ] , I ov to denote I [ X ∈ X ov ] . Then we may obtain robust bounds by optimizing over regions of no overlap:

In the specialized, but practically relevant case of binary outcomes/treatments/recommendations, we obtain the following simplifications for bounds on the policy value, and the minimax robust policy that optimizes the worst-case overlap extrapolation function. In the special case of constant uniform bounds, it is equivalent (in the case of binary outcomes) to consider marginalizations:

Lemma 1 (Binary outcomes, constant bound) . Let U cbnd := { q t | r ( x ′ ): B ≤ q 1 | r ( x ′ ) ≤ B } and U = U q t | r ∩ U cbnd . Define β t | r ( a ) := E [ q t | r ( X,A ) | T = t, A = a ] . If T ∈ { 0 , 1 } ,

We state the next result for simple uncertainty sets, like intervals, to deduce insights about the robust policy. In the appendix we include a more general reformulation for polytopic uncertainty sets.

Proposition 6 (Robust linear program ) . Suppose R,T ∈ { 0 , 1 } , and q r 1 ( · , u ) ∈ U bnd , ∀ r, u. Define

Then the robust linear program is:

5 Additional fairness constraints and policy optimization

We previously discussed policy optimization, over unrestricted decision rules, given estimates. We now introduce general methodology to handle 1) optimization over a policy class of restricted functional form and 2) more general fairness constraints. We first introduce the fair-classification algorithm of [1], describe our extensions to obtain variance-sensitive regret bounds and less conservative policy optimization (inspired by a regularized ERM argument given in [12]), and then provide sample complexity analysis.

Algorithm and setup In the following, to be consistent with standard form for linear programs, note that we consider outcomes Y to be costs so that we can phrase the saddle-point as minimizationmaximization. Consider |K| linear constraints and J groups (values of protected attribute A ), with coefficient matrix M ∈ R K × J , the constraint moment function h j ( π ) , j ∈ [ J ] (with J the number of groups), O = ( X,A,R,T,Y ) denoting our data observations, and d the constraint constant vector:

Importantly, g j depends on π while the conditioning event E j cannot depend on π . Many important fairness constraints can nonetheless be written in this framework, such as burden/resource parity,

Algorithm 1 MW2REDFAIR ( D , g, E , M, d )

1: Input: D = { ( X i , R i , T i , Y i , A i ) } n i =1 , g, E , M, ̂ d , B, ν , α , θ 1 = 0 ∈ R |K|

2: for t = 1 , 2 , . . . do

3: Set λ t,k = B exp { θ k } 1+ ∑ k ′ ∈X exp { θ k ′ } for all k ∈ K , β t ← BEST β ( λ t ) , ̂ Q t ← 1 t ∑ t t ′ =1 β t ′

5: ν t ← max { L ( ˆ Q t , ˆ λ t ) -L, ¯ L -L ( ˆ Q t , ˆ λ t ) } , If ν t ≤ ν then return ( ˆ Q t , ˆ λ t )

6: θ t +1 ,i = θ t +log(1 -η ( M ˆ µ ( h t ) -ˆ c ) j ) , ∀ i

7: end for

parity in true positive rates, but not measures such as calibration whose conditioning event does depend on π . (See Appendix B.2 for examples omitted for brevity). We further consider a convexification of Π via randomized policies Q ∈ ∆(Π) , where ∆(Π) is the set of distributions over Π , i.e. a randomized classifier that samples a policy π ∼ Q . Therefore we solve

On the other hand, the optimization is solved using sampled moments, so that we ought to interpret the constraint values ˆ d k = d k + ϵ k , for all k . The overall algorithmic scheme is similar: we seek an approximate saddle point so that the constrained solution is equivalent to the Lagrangian,

We simultaneously solve for an approximate saddle point and bound the domain of Λ by B :

We play a no-regret (second-order multiplicative weights [11, 43], a slight variant of Hedge/exponentiated gradient [18]) algorithm for the λ -player, while using best-response oracles for the Q -player. Full details are in Algorithm 1. Given λ t , BEST β ( λ t ) computes a best response over Q ; since the worst-case distribution will place all its weight on one classifier, this step can be implemented by a reduction to cost-sensitive/weighted classification [10, 49], which we describe in further detail below. Computing the best response over BEST λ ( ˆ Q t )) selects the most violated constraint. We include further details in Appendix B.2.

Weighted classification reduction. There is a well-known reduction of optimizing the zero-one loss for policy learning to weighted classification. Taking the Lagrangian will introduce datapointdependent additional weights. This reduction requires π ∈ {-1 , +1 } , T ∈ {-1 , +1 } (for notational convenience alone). We consider parameterized policy classes so that π ( x ) = sign( g θ ( x )) for some index function g depending on a parameter β ∈ R d . Consider the centered regret J ( π ) = E [ Y ( π )] -1 2 E [ E [ Y | R = 1 , X ] + E [ Y | R = 0 , X ]] . Then J ( β ) = J (sgn( g β ( · ))) = E [sgn( g β ( X )) { ψ } ] where ψ can be one of, where µ R r ( X ) = E [ Y | R = r, X ] ,

̸

We can apply the standard reduction to cost-sensitive classification since ψ i sgn( g β ( X i )) = | ψ i | (1 -2 I [sgn( g β ( X i )) = sgn( ψ i )] ). Then we can use surrogate losses for the zero-one loss. Although many functional forms for ℓ ( · ) are Fisher-consistent, one such choice of ℓ is the logistic (cross-entropy) loss given below:

Two-stage variance-constrained algorithm. We seek to improve upon this procedure so that we may obtain regret bounds on policy value and fairness constraint violation that exhibit more favorable dependence on the maximal variance over small-variance slices near the optimal policy, rather than worst-case constants over all policies, [12, 5]. Further, note that the algorithm sets the constraint feasibility slacks via generalization bounds that previously depended on these worst-case constants.

Algorithm 2 Two-stage localized fair classification via reductions

1: Randomly split the data into D 1 , D 2

2: Obtain ˆ π ∗ 1 and the index set of binding constraints ˆ I 1 by learning nuisances η 1 and running Algorithm 1 on D 1 with MW2REDFAIR ( D 1 , h, E , M, d ; η 1 )

3: ˆ σ 2 j ← ˆ Var( g j ( O, ˆ π ∗ 1 ) I [ E j ] /p j ) , ∀ j ; ˆ d ← d +2ˆ σ 2 n -α

4: Augment additional constraints with ϵ n -policy-value and constraint slices relative to ˆ π 1 : define an augmented system (where subscripting by ˆ I 1 indexes the corresponding matrix or vector):

5: Obtain ˆ π ∗ 2 by running Algorithm 1 on D 2 with MW2REDFAIR ( D , ˜ g, ˜ E , ˜ M, ˜ d, η 2 )

These challenges motivate the two-stage procedure, described formally in Algorithm 2 and verbally here. We adapt an out-of-sample regularization scheme developed in [12], which recovers variance-sensitive regret bounds via a small modification to ERM. We split the data, learn nuisance estimators η 1 for use in our policy value and constraint estimates, and run Algorithm 1 ( MW2REDFAIR( D 1 , h, E , M, d ; η 1 ) ) to obtain an estimate of the optimal policy ˆ π 1 , and the constraint variances at ˆ π 1 . Next, we augment the constraint matrix with additional constraints that require the second-stage π to achieve ϵ n close policy value and constraint moment values relative to ˆ π 1 . Since errors concentrate fast, this can be viewed as variance regularization. And, we set the constraint slacks ˜ d in the second stage using estimated variance constants from ˆ π 1 , which will be less conservative than using worst-case bounds. Define

and the empirical entropy integral κ ( r, F ) = inf α ≥ 0 { 4 α +10 ∫ r α √ H 2 ( ϵ, F ,n ) n dϵ } where H 2 ( ϵ, F , n ) is the L 2 empirical entropy, i.e. log of the ∥·∥ 2 ϵ -covering number. We make a mild assumption of a learnable function class (bounded entropy integral) [46].

Assumption 7. The function classes F Π , {F j } j ∈J satisfy that for any constant r, κ ( r, F ) → 0 as n →∞ . The function classes {F j } j comprise of L j -Lipschitz contractions of π.

We will assume that we are using doubly-robust/orthogonalized estimation as in proposition 4, hence state results depending on estimation error of nuisance vector η .

Theorem 1 (Variance-Based Oracle Policy Regret) . Suppose that the mean-squared-error of the 1 -δ/ 2 χ 2

nuisance estimates is upper bounded w.p. by n,δ , over the randomness of the nuisance sample:

denote an ϵ -regret slice of the policy space. Let ˜ ϵ n = O ( ϵ n + χ 2 n,δ ) and

denote the variance of the difference between any two policies in an ϵ n -regret slice, evaluated at the true nuisance quantities. (Define V j 2 analogously for the variance of constraint moments). Then, letting γ ( Q ) := Mh ( Q ) denote the constraint values, the policy distribution Q 2 returned by the out-of-sample regularized ERM, satisfies w.p. 1 -δ over the randomness of S :

The benefits are that 1) the constants are improved from an absolute, structure-agnostic bound to depending on the variance of low-regret policies, which also reflects improved variance from using doubly-robust estimation as in proposition 4, and 2) less-conservative out-of-sample fairness constraint satisfaction.

6 Experiments

Due to space constraints, in the main text we only present a case study based on the PSA-DMF for supervised release [38]. In the appendix we include additional experiments and robustness checks, including a case study of fully-randomized recommendations and non-adherence.

PSA-DMF case study. Our case study is on a dataset of judicial decisions on supervised release based on risk-score-informed recommendations [38]. The PSA-DMF (Public Safety Assessment Decision Making Framework) uses a prediction of failure to appear for a future court data to inform pretrial decisions, including our focus on supervised release (i.e. electronic monitoring in addition to release) where judges make the final decision. Despite a large literature on algorithmic fairness of pretrial risk assessment, to the best of our knowledge, recommendations about supervised release are not informed by empirical evidence. There are current policy concerns about disparities in increasing use of supervised release given mixed evidence on outcomes [38]; e.g. Safety and Justice Challenge [41] concludes "targeted efforts to reduce racial disparities are necessary". First, we acknowledge data issues. We work with publicly available data that was discretized for privacy [38]. The final supervision decision does not include intensity, but different intensities are recommended in the data, which we collapse into a single level. The PSA-DMF is an algorithmic recommendation so here we are appealing to overlap in treatment recommendations, but using parametric extrapolation in responsivity. Finally, unconfoundedness is likely untrue, but sensitivity analysis could address this. So, this analysis should be considered exploratory, to illustrate the relevance of the methods. Future work will seek proprietary data for a higher-fidelity empirical study.

Next in Figure 1 we provide descriptive information illustrating heterogeneity (including by protected attribute) in adherence and effectiveness. We observe wide variation in judges assigning supervised release beyond the recommendation. We use logistic regression to estimate outcome models and treatment response models. The first figure shows estimates of the causal effect for different groups, by gender (similar heterogeneity for race). The outcome is failure to appear, so negative scores are beneficial. The second figure illustrates the difference in responsiveness: how much more likely decision-makers are to assign treatment when there is vs. isn't an algorithmic recommendation to do so. The last figure plots a logistic regression of the lift in responsiveness on the causal effect µ 1 ( x, a ) -µ 0 ( x, a ) . We observe disparities in how responsive decision-makers are conditional on the same treatment effect efficacy. This is importantly not a claim of animus because decision-makers didn't have access to causal effect estimates. Nonetheless, disparities persist.

In Figure 2 we highlight results from constrained policy optimization. The first two plots in each set illustrate the objective function value and A = a average treatment cost, respectively; for A being race (nonwhite/white) or gender (female/male), respectively. We use costs of 100 for Y = 1 (failure to appear, 0 for Y = 0 , and 20 when T = 1 (set arbitrarily). On the x-axis we plot the penalty λ that we use to assess the solutions of Proposition 9. The vertical dashed line indicates the solution achieving ϵ = 0 , i.e. parity in treatment take-up. Near-optimal policies that reduce treatment disparity can be of interest due to advocacy concerns about how the expansion of supervised release could increase the surveillance of already surveillance-burdened marginalized populations. We see that indeed, for race, surveillance-parity constrained policies can substantially reduce disparities for nonwhites while not increasing surveillance on whites that much: the red line decreases significantly at low increase to the blue line (and low increases to the objective value). On the other hand, for gender, the opportunity for improvement in surveillance disparity is much smaller. See the appendix for further experiments and computational details.

References

A. Agarwal, A. Beygelzimer, M. Dudík, J. Langford, and H. Wallach. A reductions approach to fair classification. In International Conference on Machine Learning , pages 60-69. PMLR, 2018.

Tbygender

P1a-Poavss,a=gender

10

Male

Female

17.5

15.0

Male

Female

0.25

0.30

Male

Female

8

0.20

12.5

0.15

6

10.0

0.10

7.5

4

0.05

5.0

2

0.00

2.5

0

0.0

0.2

0.0

0.2

0.0

0.0

0.1

0.2

0.3

S

0.2

0.0

0.2

17.10

17.15

17.20

17.25

17.30

Obj. reduction

0.2

0.0

0.2

1.0

1.5

2.0

2.5

3.0

3.5

E

[

T

(

)|

A

=

a

]

A=nonwhite

A=white

0.2

0.0

0.2

17.075

17.100

17.125

17.150

17.175

17.200

17.225

Obj. reduction (A=gender)

0.2

0.0

0.2

1.5

2.0

2.5

3.0

3.5

E

[

T

(

)|

A

=

a

]

A=female

A=male

D. Arnold, W. Dobbie, and C. S. Yang. Racial bias in bail decisions. The Quarterly Journal of Economics , 133(4):1885-1932, 2018.

D. Arnold, W. Dobbie, and P. Hull. Measuring racial discrimination in bail decisions. American Economic Review , 112(9):2992-3038, 2022.

S. Athey. Beyond prediction: Using big data for policy problems. Science , 2017.

S. Athey and S. Wager. Policy learning with observational data. Econometrica , 89(1):133-161, 2021.

S. Barocas, M. Hardt, and A. Narayanan. Fairness and Machine Learning . fairmlbook.org, 2018. http://www.fairmlbook.org .

P. L. Bartlett and S. Mendelson. Rademacher and gaussian complexities: Risk bounds and structural results. Journal of Machine Learning Research , 3(Nov):463-482, 2002.

H. Bastani, O. Bastani, and W. P. Sinchaisri. Improving human decision-making with machine learning. arXiv preprint arXiv:2108.08454 , 2021.

E. Ben-Michael, D. J. Greiner, K. Imai, and Z. Jiang. Safe policy learning through extrapolation: Application to pre-trial risk assessment. arXiv preprint arXiv:2109.11679 , 2021.

A. Beygelzimer and J. Langford. The offset tree for learning with partial labels. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining , pages 129-138, 2009.

N. Cesa-Bianchi, Y. Mansour, and G. Stoltz. Improved second-order bounds for prediction with expert advice. Machine Learning , 66:321-352, 2007.

V. Chernozhukov, M. Demirer, G. Lewis, and V. Syrgkanis. Semi-parametric efficient policy learning with continuous actions. Advances in Neural Information Processing Systems , 32, 2019.

A. Chohlas-Wood, M. Coots, H. Zhu, E. Brunskill, and S. Goel. Learning to be fair: A consequentialist approach to equitable decision-making. arXiv preprint arXiv:2109.08792 , 2021.

M. De-Arteaga, R. Fogliato, and A. Chouldechova. A case for humans-in-the-loop: Decisions in the presence of erroneous algorithmic scores. In Proceedings of the 2020 CHI Conference on Human Factors in Computing Systems , pages 1-12, 2020.

J. L. Doleac and M. T. Stevenson. Algorithmic risk assessments in the hands of humans. Salem Center , 2020.

A. Finkelstein, S. Taubman, B. Wright, M. Bernstein, J. Gruber, J. P. Newhouse, H. Allen, K. Baicker, and O. H. S. Group. The oregon health insurance experiment: evidence from the first year. The Quarterly journal of economics , 127(3):1057-1106, 2012.

D. J. Foster and V. Syrgkanis. Orthogonal statistical learning. arXiv preprint arXiv:1901.09036 , 2019.

Y. Freund and R. E. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. Journal of computer and system sciences , 55(1):119-139, 1997.

B. Green and Y. Chen. Disparate interactions: An algorithm-in-the-loop analysis of fairness in risk assessments. In Proceedings of the conference on fairness, accountability, and transparency , pages 90-99, 2019.

B. Green and Y. Chen. Algorithmic risk assessments can alter human decision-making processes in high-stakes government contexts. Proceedings of the ACM on Human-Computer Interaction , 5(CSCW2):1-33, 2021.

K. Heard, E. O'Toole, R. Naimpally, and L. Bressler. Real world challenges to randomization and their solutions. Boston, MA: Abdul Latif Jameel Poverty Action Lab , 2017.

M. A. Hernán and J. M. Robins. Causal inference.

K. Imai, Z. Jiang, J. Greiner, R. Halen, and S. Shin. Experimental evaluation of algorithmassisted human decision-making: Application to pretrial public safety assessment. arXiv preprint arXiv:2012.02845 , 2020.

Z. Jiang, S. Yang, and P. Ding. Multiply robust estimation of causal effects under principal ignorability. arXiv preprint arXiv:2012.01615 , 2020.

N. Kallus and A. Zhou. Confounding-robust policy improvement. In Advances in Neural Information Processing Systems , pages 9269-9279, 2018.

N. Kallus and A. Zhou. Assessing disparate impact of personalized interventions: identifiability and bounds. Advances in neural information processing systems , 32, 2019.

N. Kallus and A. Zhou. Fairness, welfare, and equity in personalized pricing. In Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency , pages 296-314, 2021.

N. Kallus, X. Mao, and A. Zhou. Assessing algorithmic fairness with unobserved protected class using data combination. arXiv preprint arXiv:1906.00285 , 2019.

K. Kim, E. Kennedy, and J. Zubizarreta. Doubly robust counterfactual classification. Advances in Neural Information Processing Systems , 35:34831-34845, 2022.

T. Kitagawa and A. Tetenov. Empirical welfare maximization. 2015.

W. Lin, S.-H. Kim, and J. Tong. Does algorithm aversion exist in the field? an empirical analysis of algorithm use determinants in diabetes self-management. An Empirical Analysis of Algorithm Use Determinants in Diabetes Self-Management (July 23, 2021). USC Marshall School of Business Research Paper Sponsored by iORB, No. Forthcoming , 2021.

L. Liu, Z. Shahn, J. M. Robins, and A. Rotnitzky. Efficient estimation of optimal regimes under a no direct effect assumption. Journal of the American Statistical Association , 116(533): 224-239, 2021.

J. Ludwig and S. Mullainathan. Fragile algorithms and fallible decision-makers: lessons from the justice system. Journal of Economic Perspectives , 35(4):71-96, 2021.

K. Lum, E. Ma, and M. Baiocchi. The causal impact of bail on case outcomes for indigent defendants in new york city. Observational Studies , 3(1):38-64, 2017.

C. Manski. Social Choice with Partial Knoweldge of Treatment Response . The Econometric Institute Lectures, 2005.

A. Maurer. A vector-contraction inequality for rademacher complexities. In Algorithmic Learning Theory: 27th International Conference, ALT 2016, Bari, Italy, October 19-21, 2016, Proceedings 27 , pages 3-17. Springer, 2016.

B. Metevier, S. Giguere, S. Brockman, A. Kobren, Y. Brun, E. Brunskill, and P. S. Thomas. Offline contextual bandits with high probability fairness guarantees. Advances in neural information processing systems , 32, 2019.

Office of the Chief Judge. Bail reform in cook county: An examination of general order 18.8a and bail in felony cases. 2019.

H. Qiu, M. Carone, E. Sadikova, M. Petukhova, R. C. Kessler, and A. Luedtke. Optimal individualized decision rules using instrumental variable methods. Journal of the American Statistical Association , 116(533):174-191, 2021.

D. B. Rubin. Comments on 'randomization analysis of experimental data: The fisher randomization test comment'. Journal of the American Statistical Association , 75(371):591-593, 1980.

Safety and C. f. C. I. Justice Challenge. Expanding supervised release in new york city. 2022. URL https://safetyandjusticechallenge.org/resources/ expanding-supervised-release-in-new-york-city/ .

A. Shapiro. On duality theory of conic linear problems. Semi-Infinite Programming: Recent Advances , pages 135-165, 2001.

J. Steinhardt and P. Liang. Adaptivity and optimism: An improved exponentiated gradient algorithm. In International conference on machine learning , pages 1593-1601. PMLR, 2014.

H. Sun, E. Munro, G. Kalashnov, S. Du, and S. Wager. Treatment allocation under uncertain costs. arXiv preprint arXiv:2103.11066 , 2021.

A. Swaminathan and T. Joachims. Counterfactual risk minimization. Journal of Machine Learning Research , 2015.

A. W. Van Der Vaart, J. A. Wellner, A. W. van der Vaart, and J. A. Wellner. Weak convergence . Springer, 1996.

B. Woodworth, S. Gunasekar, M. I. Ohannessian, and N. Srebro. Learning non-discriminatory predictors. In Conference on Learning Theory , pages 1920-1953. PMLR, 2017.

Y. Yacoby, B. Green, C. L. Griffin Jr, and F. Doshi-Velez. 'if it didn't happen, why would i change my decision?': How judges respond to counterfactual explanations for the public safety assessment. In Proceedings of the AAAI Conference on Human Computation and Crowdsourcing , volume 10, pages 219-230, 2022.

Y. Zhao, D. Zeng, A. J. Rush, and M. R. Kosorok. Estimating individualized treatment rules using outcome weighted learning. Journal of the American Statistical Association , 107(499): 1106-1118, 2012.

A Additional discussion

A.1 Additional related work

Principal stratification and mediation analysis in causal inference [32] studies an optimal test-and-treat regime under a no-direct-effect assumption, that assigning a diagnostic test has no effect on outcomes except via propensity to treat, and studies semiparametric efficiency using Structural Nested-Mean Models. Though our exclusion restriction is also a no-direct-effect assumption, our optimal treatment regime is in the space of recommendations only as we do not have control over the final decision-maker, and we consider generally nonparametric models.

We briefly go into more detail about formal differences, due to our specific assumptions, that delineate the differences to mediation analysis. Namely, our conditional exclusion restriction implies that Y 1 T 0 = Y T 0 and that Y 0 T 1 = Y 1 T 1 (in mediation notation with T r = T ( r ) in our notation), so that so-called net direct effects are identically zero and the net indirect effect is the treatment effect (also called average encouragement effect here).

Human-in-the-loop in consequential domains. There is a great deal of interest in designing algorithms for the 'human in the loop" and studying expertise and discretion in human oversight in consequential domains [14]. On the algorithmic side, recent work focuses on frameworks for learning to defer or human-algorithm collaboration. Our focus is prior to the design of these procedures for improved human-algorithm collaboration: we primarily hold fixed current human responsiveness to algorithmic recommendations. Therefore, our method can be helpful for optimizing local nudges. Incorporating these algorithmic design ideas would be interesting directions for future work.

Empirical literature on judicial discretion in the pretrial setting. Studying a slightly different substantive question, namely causal effects of pretrial decisions on later outcomes, a line of work uses individual judge decision-makers as a leniency instrumental variable for the treatment effect of (for example, EM) on pretrial outcomes [3, 2, 34]. And, judge IVs rely on quasi-random assignment of individual judges. We focus on the prescriptive question of optimal recommendation rules in view of patterns of judicial discretion, rather than the descriptive question of causal impacts of detention on downstream outcomes.

A number of works have emphasized the role of judicial discretion in pretrial risk assessments in particular [20, 15, 33]. In contrast to these works, we focus on studying decisions about electronic monitoring, which is an intermediate degree of decision lever to prevent FTA that nonetheless imposes costs. [23] study a randomized experiment of provision of the PSA and estimate (the sign of) principal causal effects, including potential group-conditional disparities. They are interested in a causal effect on the principal stratum of those marginal defendants who would not commit a new crime if recommended for detention. [9] study policy learning in the absence of positivity (since the PSA is a deterministic function of covariates) and consider a case study on determining optimal recommendation/detention decisions; however their observed outcomes are downstream of judicial decision-making. Relative to their approach, we handle lack of overlap via an exclusion restriction so that we only require ambiguity on treatment responsivity models rather than causal outcome models.

B Additional discussion on method

B.1 Additional discussion on constrained optimization

Feasibility program We can obtain upper/lower bounds on ϵ in order to obtain a feasible region for ϵ by solving the below optimization over maximal/minimal values of the constraint:

B.2 Additional discussion on Algorithm 2 (general algorithm)

B.2.1 Additional fairness constraints and examples in this framework

In this section we discuss additional fairness constraints and how to formulate them in the generic framework. Much of this discussion is quite similar to [1] (including in notation) and is included in this appendix for completeness only. We only additionally provide novel identification results for another fairness measure on causal policies in Appendix B.2.2, concrete discussion of the reduction to weighted classification, and provide concrete descriptions of the causal fairness constraints in the more general framework.

Wefirst discuss how to impose the treatment parity constraint. This is similar to the demographic parity example in [1], with different coefficients, but included for completeness. (Instead, recommendation parity in E [ π | A = a ] is indeed nearly identical to demographic parity.)

Example 1 (Writing treatment parity in the general constrained classification framework.) . We write the constraint

in this framework as follows:

For each u ∈ A we enforce that

We can write this in the generic notation given previously by letting J = A∪{◦} ,

We let the conditioning events E a = { A = a } , E ◦ = { True } , i.e. conditioning on the latter is equivalent to evaluating the marginal expectation. Then we express Equation (5) as a set of equality constraints h a ( π ) = h ◦ ( π ) , leading to pairs of inequality constraints,

The corresponding coefficients of M over this enumeration over groups ( A ) and epigraphical enforcement of equality ( { + , -} ) equation (1), gives K = A × { + , -} so that M ( a, +) ,a ′ = 1 { a ′ = a } , M ( a, +) ,⋆ = -1 , M ( a, -) ,a ′ = -1 { a ′ = a } , M ( a, -) ,⋆ = 1 , and d = 0 . Further we can relax equality to small amounts of constraint relaxation by instead setting d k > 0 for some (or all) k .

Next, we discuss a more complicated fairness measure. We first discuss identification and estimation before we also describe how to incorporate it in the generic framework.

B.2.2 Responder-dependent fairness measures

We consider a responder framework on outcomes (under our conditional exclusion restriction). Because the contribution to the AEE is indeed from the responder strata, this corresponds to additional estimation of the responder stratum.

We enumerate the four possible realizations of potential outcomes (given any fixed recommendation) as ( Y (0( r )) , Y (1( r )) ∈ { 0 , 1 } 2 . We call units with ( Y (0( r )) , Y (1( r ))) = (0 , 1) responders, ( Y (0( r )) , Y (1( r ))) = (1 , 0) anti-responders, and Y (0( r )) = Y (1( r )) non-responders. Such a decomposition is general for the binary setting.

Assumption 8 (Binary outcomes, treatment) .

Importantly, the conditional exclusion restriction of Assumption 2 implies that responder status is independent of recommendation. Conditional on observables, whether a particular individual is a responder is independent of whether someone decides to treat them when recommended. In this way, we study responder status analogous to its use elsewhere in disparity assessment in algorithmic fairness [23, 28]. Importantly, this assumption implies that the conditioning event (of being a responder) is therefore independent of the policy π , so it can be handled in the same framework. s

We may consider reducing disparities in resource expenditure given responder status.

We may be interested in the probability of receiving treatment assignment given responder status.

Example 2 (Fair treatment expenditure given responder status) .

We can obtain identification via regression adjustment:

Proposition 7 (Identification of treatment expenditure given responder status) . Assume Assumptions 8 and 9.

Therefore this can be expressed in the general framework.

Example 3 (Writing treatment responder-conditional parity in the general constrained classification framework.) . For each u ∈ A we enforce that

We can write this in the generic notation given previously by letting J = A∪{◦} ,

Let E j a = { A = a j } , E ◦ = { True } , and we express Equation (5) as a set of equality constraints of the above moment h a ( π ) = h ◦ ( π ) , leading to pairs of inequality constraints,

The corresponding coefficients of M proceed analogously as for treatment parity.

B.2.3 Best-response oracles

Best-responding classifier π , given λ : BEST π ( λ ) The best-response oracle, given a particular λ value, optimizes the Lagrangian given π :

Assumption 9 (Monotonicity) .

Best-responding Lagrange multiplier λ , given π : BEST λ ( Q ) is the best response of the Λ player. It can be chosen to be either 0 or put all the mass on the most violated constraint. Let γ ( Q ) := Mh ( Q ) denote the constraint values, then BEST λ ( Q ) returns {

B.2.4 Weighted classification reduction

There is a well-known reduction of optimizing the zero-one loss for policy learning to weighted classification. A cost-sensitive classification problem is

̸

The weighted classification error is ∑ n i =1 W i 1 { h ( X i ) = Y i } which is an equivalent formulation if W i = ∣ ∣ C 0 i -C 1 i ∣ ∣ and Y i = 1 { C 0 i ≥ C 1 i } .

The reduction to weighted classification is particularly helpful since taking the Lagrangian will introduce datapoint-dependent penalties that can be interpreted as additional weights. We can consider the centered regret J ( π ) = E [ Y ( π )] -1 2 E [ E [ Y | R = 1 , X ] + E [ Y | R = 0 , X ]] . Then

where ψ can be one of, where µ R r ( X ) = E [ Y | R = r, X ] ,

̸

We can apply the standard reduction to cost-sensitive classification since ψ i sgn( g θ ( X i )) = | ψ i | (1 -2 I [sgn( g θ ( X i )) = sgn( ψ i )] ). Then we can use surrogate losses for the zero-one loss,

Although many functional forms for ℓ ( · ) are Fisher-consistent, the logistic (cross-entropy) loss will be particularly relevant: l ( g, s ) = 2 log(1 + exp( g )) -( s +1) g .

Example 4 (Treatment parity, continued (weighted classification reduction)) . The cost-sensitive reduction for a vector of Lagrange multipliers can be deduced by applying the weighted classification reduction to the Lagrangian:

where p a := ˆ P ( A = a ) and λ a := λ ( a, +) -λ ( a, -) , effectively replacing two non-negative Lagrange multipliers by a single multiplier, which can be either positive or negative.

Example 5 (Responder-conditional treatment parity, continued) . The Lagrangian is L ( β ) = E [ | ˜ ψ λ | ℓ ( g β ( X ) , sgn( ˜ ψ λ ) )] with weights:

where p a := ˆ P ( A = a ) and λ a := λ ( a, +) -λ ( a, -) .

B.3 Proofs

Proof of Proposition 7.

C Proofs

C.1 Proofs for generalization under unconstrained policies

Proposition 8 (Policy value generalization) . Assume the nuisance models η = [ p 1 | 0 , p 1 | 1 , µ 1 , µ 0 , e r ( X )] ⊤ , η ∈ H are consistent and well-specified with finite VC-dimension V η over the product function class H . We provide a proof for the general case, including doubly-robust estimators, which applies to the statement of Proposition 8 by taking η = [ p 1 | 0 , p 1 | 1 , µ 1 , µ 0 ] .

The generalization bound allows deducing risk bounds on the out-of-sample value:

Corollary 2.

Write Lagrangian as

Proof of Proposition 8. We study a general Lagrangian, which takes as input pseudo-outcomes ψ t | r ( O ; η ) , ψ y | t ( O ; η ) , ψ 1 | 0 , ∆ A where each satisfies that

We make high-level stability assumptions on pseudooutcomes ψ relative to the nuisance functions η (these are satisfied by standard estimators that we will consider):

Assumption 10. ψ t | r , ψ y | t , ψ 1 | 0 , ∆ A respectively are Lipschitz contractions with respect to η and bounded

We study a generalized Lagrangian of an optimization problem that took these pseudooutcome estimates as inputs:

We will show that

which, by applying the generalization bound twice gives that

Suppose the Rademacher complexity of η k is given by R ( H k ) , so that [7, Thm. 12] gives that the Rademacher complexity of the product nuisance class H is therefore ∑ k R ( H k ) . The main result follows by applying vector-valued extensions of Lipschitz contraction of Rademacher complexity given in [36]. Suppose that ψ t | r , ψ y | t , ψ 1 | 0 , ∆ A are Lipschitz with constants C L t | r , C L y | t , C L 1 | 0 , ∆ A .

We establish VC-properties of

and the function class for the truncated Lagrangian,

[36, Corollary 4] (and discussion of product function classes) gives the following: Let X be any set, ( x 1 , . . . , x n ) ∈ X n , let F be a class of functions f : X → ℓ 2 and let h i : ℓ 2 → R have Lipschitz norm L . Then

where ϵ ik is an independent doubly indexed Rademacher sequence and f k ( x i ) is the k -th component of f ( x i ) .

Applying Equation (6) to each of the component classes F L 1 ( O 1: n ) , F L 2 ( O 1: n ) , F L 3 ( O 1: n ) , and Lipschitz contraction [7, Thm. 12.4] of the positive part function F L + , we obtain the bound

Proposition 9 (Threshold solutions) . Define

If instead d ( x ) is a function of covariates x only,

Proof of Proposition 9. The characterization follows by strong duality in infinite-dimensional linear programming [42]. Strict feasibility can be satisfied by, e.g. solving eq. (3) to set ranges for ϵ .

C.2 Proofs for robust characterization

Proof of Proposition 5.

where p t ( X ) = P ( T = t | X ) (marginally over R in the observational data) and (LOTE) is an abbreviation for the law of total expectation.

Proof of Lemma 1.

Note the objective function can be reparametrized under a surjection of q t | r ( X ) to its marginalization, i.e. marginal expectation over a { T = t } partition (equivalently { T = t, A = a } partition for a fairness-constrained setting).

Define

Therefore we may reparametrize V no ( π ) as an optimization over constant coefficients (bounded by B):

Assumption 2 (ER)

Proof of proposition 6.

Define

then we can rewrite this further and apply the standard epigraph transformation: max t

Project the uncertainty set onto the direct product of uncertainty sets: max t

Clearly robust feasibility of the resource parity constraint over the interval is obtained by the highest/lowest bounds for groups a, b , respectively:

We define

then and

For brevity we denote ∆ B = ( B r ( x, u ) -B r ( x, u )) , so max t

where

This is equivalent to:

Undoing the epigraph transformation, we obtain:

and simplifying the absolute value:

C.3 Proofs for general fairness-constrained policy optimization algorithm and analysis

We begin with some notation that will simplify some statemetns. Define, for observation tuples O ∼ ( X,A,R,T,Y ) , the value estimate v ( Q ; η ) given some pseudo-outcome ψ ( O ; η ) dependent on observation information and nuisance functions η . (We often suppress notation of η for brevity). We let estimators sub/super-scripted by 1 denote estimators from the first dataset.

C.3.1 Preliminaries: results from other works used without proof

Theorem 3 (Thm. 3, [1] (saddle point generalization bound (non-localized)) ) . Let ρ := max h ∥ M ˆ µ ( h ) -ˆ c ∥ ∞ . Let Assumption 1 hold for C ′ ≥ 2 C + 2 + √ ln(4 /δ ) / 2 , where δ > 0 . Let Q ⋆ minimize V ( Q ) subject to Mµ ( Q ) ≤ c . Then Algorithm 1 with ν ∝ n -α , B ∝ n α and η ∝ ρ -2 n -2 α terminates in O ( ρ 2 n 4 α ln |K| ) iterations and returns ˆ Q . If np ⋆ j ≥ 8 log(2 /δ ) for all j , then with probability at least 1 -( |J | +1) δ then for all k ˆ Q satisfies:

The proof of [1, Thm. 3] is modular in invoking Rademacher complexity bounds on the objective function and constraint moments, so that invoking standard Rademacher complexity bounds for offpolicy evaluation/learning [5, 45] yields the above statement for V ( π ) (and analogously, randomized policies by [7, Thm. 12.2] giving stability for convex hulls of policy classes).

Lemma 2 (Lemma 4, [17]) . Consider a function class F : X → R d , with sup f ∈F ∥ f ∥ ∞ , 2 ≤ 1 and pick any f ∗ ∈ F . Assume that v ( π ) is L-Lipschitz in its first argument with respect to the ℓ 2 norm and let:

Then for some universal constants c 1 , c 2 :

Moreover, if δ n is any solution to the inequalities:

then for each r ≥ δ n :

Lemma 3 (Concentration of conditional moments ([1, 47]) . For any j ∈ J , with probability at least 1 -δ , for all Q ,

If np ⋆ j ≥ 8 log(2 /δ ) , then with probability at least 1 -δ , for all Q ,

Lemma 4 (Orthogonality (analogous to [12] (Lemma 8), others)) . Suppose the nuisance estimates satisfy a mean-squared-error bound

Then w.p. 1 -δ over the randomness of the policy sample,

C.4 Adapted lemmas

In this subsection we collect results similar to those that have appeared previously, but that require substantial additional argumentation in our specific saddle point setting.

Lemma 5 (Feasible vs. oracle nuisances in low-variance regret slices ([12], Lemma 9) ) . Consider the setting of Corollary 7. Suppose that the mean squared error of the nuisance estimates is upper bounded w.p. 1 -δ by h 2 n,δ and suppose h 2 n,δ ≤ ϵ n . Then:

Then V 2 ≤ V 0 2 + O ( h n,δ ) .

C.5 Proof of Theorem 1

Proof of Theorem 1. We first study the meta-algorithm with 'oracle" nuisance functions η = η 0 .

Define

In the following, we suppress notational dependence on η 0 .

Step 1: First we argue that w.p. 1 -δ/ 6 , Q ∗ ∈ Q 2 .

Invoking Theorem 3 on the output of the first stage of the algorithm, yields that with probability 1 -δ 6 over the randomness in D 1 , by choice of ϵ n = ¯ O ( n -α )) ,

Therefore, with high probability on the good event, Q ∗ ∈ Q 2 .

Step 2: Again invoking Theorem 3, this time on the output of the second stage of the algorithm with function space Π 2 (hence implicitly Q 2 ), and conditioning on the 'good event" that Q ∗ ∈ Q 2 , we obtain the bound that with probability ≥ 1 -δ/ 3 over the randomness of the second sample D 2 ,

Step 3: empirical small-regret slices relate to population small-regret slices, and variance bounds

We show that if Q ∈ Q 2 , then with high probability Q ∈ Q 0 2 (defined on small population value- and constraint-regret slices relative to ˆ Q 1 rather than small empirical regret slices)

so that w.h.p. Q 2 ⊆ Q 0 2

Note that for Q ∈ Q , w.h.p. 1 -δ/ 6 over the first sample, we have that

The second bound follows from [7, Theorem 12.2] (equivalence of Rademacher complexity over convex hull of the policy class) and linearity of the policy value and constraint estimators in π, and hence Q .

On the other hand since Q 1 achieves low policy regret, the triangle inequality implies that we can contain the true policy by increasing the error radius. That is, for all Q ∈ Q 2 , with high probability ≥ 1 -δ/ 3 :

Define the space of distributions over policies that achieve value and constraint regret in the population of at most ϵ n :

so that on that high-probability event,

Then on that event with probability ≥ 1 -δ/ 3 ,

Therefore:

Combining this with the local Rademacher complexity bound, we obtain that:

These same arguments apply for the variance of the constraints

C.6 Proofs of auxiliary/adapted lemmas

Proof of Lemma 5. The proof is analogous to that of [12, Lemma 9] except for the step of establishing that π ∗ ∈ Q 0 ϵ n + O ( χ 2 n,δ ) : in our case we must establish relationships between saddlepoints under estimated vs. true nuisances. We show an analogous version below.

Define the saddle points to the following problems (with estimated vs. true nuisances):

We have that:

Hence

We generally assume that the saddlepoint suboptimality ν is of lower order than ϵ n (since it is under our computational control).

Applying Lemma 4 gives;

Define policy classes with respect to small-population regret slices (with a nuisance-estimation enlarged radius):

Then we have that

where we have shown that π ∗ ∈ Q 0 ( ϵ +2 ν +2 χ 2 n,δ ) .

Following the result of the argumentation in [12, Lemma 9] from here on out gives the result.

D Case Studies

D.1 Oregon Health Insurance Study

The Oregon Health Insurance Study [16] is an important study on the causal effect of expanding public health insurance on healthcare utilization, outcomes, and other outcomes. It is based on a randomized controlled trial made possible by resource limitations, which enabled the use of a randomized lottery to expand Medicaid eligibility for low-income uninsured adults. Outcomes of interest included health care utilization, financial hardship, health, and labor market outcomes and political participation.

Because the Oregon Health Insurance Study expanded access to enroll in Medicaid, a social safety net program, the effective treatment policy is in the space of encouragement to enroll in insurance (via access to Medicaid) rather than direct enrollment. This encouragement structure is shared by many other interventions in social services that may invest in nudges to individuals to enroll, tailored assistance, outreach, etc., but typically do not automatically enroll or automatically initiate transfers. Indeed this so-called administrative burden of requiring eligible individuals to undergo a costly enrollment process, rather than automatically enrolling all eligible individuals, is a common policy design lever in social safety net programs. Therefore we expect many beneficial interventions in this consequential domain to have this encouragement structure.

We preprocess the data by partially running the Stata replication file, obtaining a processed data file as input, and then selecting a subset of covariates. These covariates include household information that affected stratified lottery probabilities, socioeconomic demographics, medical status and other health information.

In the notation of our framework, the setup of the optimal/fair encouragement policy design question is as follows:

X covariates (baseline household information, socioeconomic demographics, health information)

A race (non-white/white), or gender (female/male)

These protected attributes were binarized.

R encouragement: lottery status of expanded eligibility (i.e. invitation to enroll when individual was previously ineligible to enroll)

T : whether the individual is enrolled in insurance ever

R = 1 this can be either Medicaid or private insurance while for R = 0 this is

Note that for still well-defined as this can be private insurance.

Y : number of doctor visits

This outcome was used as a measure of healthcare utilization. Overall, the study found statistically significant effects on healthcare utilization. An implicit assumption is that increased healthcare utilization leads to better health outcomes.

We subsetted the data to include complete cases only (i.e. without missing covariates). We learned propensity and treatment propensity models via logistic regression for each group, and used gradientboosted regression for the outcome model. We first include results for regression adjustment identification.

In Figure 3 we plot descriptive statistics. We include histograms of the treatment responsivity lifts p 1 | 1 a ( x, a ) -p 1 | 0 a ( x, a ) . We see some differences in distributions of responsivity by gender and race. We then regress treatment responsivity on the outcome-model estimate of τ . We find substantially more heterogeneity in treatment responsivity by race than by gender: whites are substantially more likely to take up insurance when made eligible, conditional on the same expected treatment effect heterogeneity in increase in healthcare utilization. (This is broadly consistent with health policy discussions regarding mistrust of the healthcare system).

Next we consider imposing treatment parity constraints on an unconstrained optimal policy (defined on these estimates). In Figure 4 we display the objective value, and E [ T ( π ) | A = a ] , for gender and race, respectively, enumerated over values of the constraint. We use costs of 2 for the number of doctors visits and 1 for enrollment in Medicaid (so E [ T ( π ) | A = a ] is on the scale of probability of

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0.30

0.32

0.34

0.36

0.38

0.40

0.42

E

[

T

(

)|

A

=

a

]

A=male

A=female

1.0

0.5

0.0

0.5

4.24

4.25

4.26

4.27

4.28

Obj.  (A=race)

1.0

0.5

0.0

0.5

0.15

0.20

0.25

0.30

0.35

0.40

E

[

T

(

)|

A

=

a

]

A=nonwhite

A=white

enrollment). These costs were chosen arbitrarily. Finding optimal policies that improve disparities in group-conditional access can be done with relatively little impact to the overall objective value. These group-conditional access disparities can be reduced from 4 percentage points ( 0 . 04 ) for gender and about 6 percentage points (0 . 06) for race at a cost of 0 . 01 or 0 . 02 in objective value (twice the number of doctors' visits). On the other hand, relative improvements/compromises in access value for the 'advantaged group" show different tradeoffs. Plotting the tradeoff curve for race shows that, consistent with the large differences in treatment responsivity we see for whites, improving access for blacks. Looking at this disparity curve given λ however, we can also see that small values of λ can have relatively large improvements in access for blacks before these improvements saturate, and larger λ values lead to smaller increases in access for blacks vs. larger decreases in access for whites.

Peer reviews:
comments: SUMMARY:
There exist (many) healthcare programs where it is impossible to compel individuals to take treatment; rather, the problem of solving for an optimal policy takes the form of providing beneficial services and recommendations to patients. This paper formalizes a causal framework for these cases and develops statistically improved estimators and robustness checks for the setting of algorithmic recommendations with sufficiently randomized decisions. The contributions of this paper are both theoretical and empirical in nature.

STRENGTHS:
I believe that this paper is clearly written. While I struggled to understand the work in this paper, I believe that my challenges were due to not being an expert in this field. In fact, I have learned something about causality from reading this paper.

The problem that this paper is attempting to address is very interesting to me, and the empirical results on a variety of applications illustrated the use and significance of this paper’s contributions.

WEAKNESSES:
I have not identified any core weaknesses in this paper.

There’s an unattributed quote block on line 33.

QUESTIONS:
N/A

LIMITATIONS:
Yes.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-05

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors study the problem of providing treatment recommendations in systems where those receiving recommendations posses the ability to dissent or deviant from the suggested course of action. Within these types of problems the authors develop a method for providing treatment recommendations which can both account for this ability in recomendees and adhere to notions of fairness in terms of successful treatment adoption. The authors provide several theoretical results characterizing the problem space and their approach as well as experimentally demonstrating its efficacy.  On top of this the authors adapt the classic two-player cost-sensitive learning optimization technique found in prior works to their problem and introduce several interesting additions to account for the additional considerations of their problem.\n\nSTRENGTHS:\n- The problem studied in the paper is important and helps bridge the gap between works on treatment recommendation which ignore autonomy of the recomendee and the real-world in which humans will inevitably express dissatisfaction or apathy towards recommendations. \n\n- The paper is well written and the authors clearly outline their approach, relevant background, and possible limitations.\n\n- The authors provide a mix of theoretical results (characterizing both the problem and priorities of their approach) and experimental results.\n\n- The assumptions made in the authors model are reasonable. The least standard assumption appearing to be Assumption 2, which may not always hold in practice (as the authors point out), but this assumptions is a good jumping-off point for the type of problem the authors wish to investigate. Moreover, in most “reasonable” domains I would expect that the majority of the effect of the recommendation is its ability to alter treatment adoption rates.\n\n- The modifications to the traditional reductions approach are interesting and provide some useful new ideas from a technical standpoint.\n\nWEAKNESSES:\n- The experimental results are missing baseline comparisons. For example, what would happen if we assumed a 100% compliance rate in terms of recommendations; applying such an approach would provide information on how much we gain by considering agent autonomy. \n\n- It would be informative to see some type of running time analysis or discussion on how easy it is to solve the objectives presented on line 219. For example, computing a best response amounts to cost sensitive learning at each round, which I would expect is extremely costly unless only a few epochs of cost sensitive learning are performed at each round. Although this approach is heavily inspired by (and similar to) the method in Agarwal et al., which is very fast in practice, it is hard to see how much extra compute is required as a result of the difference between the two approaches. The authors approach feels more similar to adversarial training which is known to be a quite expensive and scale poorly with larger models and feature spaces (but perhaps I am wrong about it). Any clarification would be appreciated!\n\nQUESTIONS:\nSee weaknesses\n\nLIMITATIONS:\nSee weakensses", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper addresses consequential decision making situation where we have a decision-making policy that outputs decisions (referred to as recommendatinos) R, individuals who may adhere to the decisions and realize a treatment T, which then leads to an outcome Y.  The problem arises, when individuals do not adhere to the decisions (i.e., follow the recommendations) and thus do not realize a treatment T, thus not realize any outcome Y. The decision making policy does not have direct control over treatment, but can only provide decisions (recommendations for treatment). Fainress comes into play, when amongst those poeple that received a positive decision (recommendation) certain social groups are less likely to adhere to the decision and thus realize an outcome. The paper suggests a method to learn a decision making policy that assigns recommendations to individuals as to fulfill certain fairness criteria in the treatment realized, e.g., reducing the disparity in treatment. The authors present a case study on \"failure to appear before court\" data (PSA-DMF).\n\nSTRENGTHS:\nThe paper studies an interesting problem, that can be relevant to the ML community working on consequential decision making). In the classical loan example, we generally assume that an individual decided to be granted a loan will utilize it, overlooking the possibility that some individuals might choose not to realize the loan. When individuals from certain demographics systematically underutilize positive decisions, this has fairness implications that are important to be considered, when designing decision-making policies.\n\nWEAKNESSES:\nWhile I believe the problem tackled is interesting, I strongly believe the paper is not yet ready for publication. The paper lacks structure and clarity. I am enlisting some important points below. \n\nOn structure: \n1. Section 1: The last paragraph of the introduction summarizes the contribution. It would help, if the authors could point to the specific sections, where these contributions are made and provide an overview of the structure of the paper.\n2. Section 3: There is no background section. The paper would benefit from a background section to understand the introduced method better. For example, I do not find the Neyman-Rubin potnetial outcomes framework introduced that the work claims to use (ln. 95-96). There is also no introduction to policy value functions, doubly robust optimization/policies.\n3. Section 3: The introduction of notation and concepts may benefit from introducing concepts as definitions, e.g., the cost function. There is also a lack of introduction of notation, e.g., what $p_{t|r}$ refers to. \n4. Section 3: The mathematical definitions / equations do not carry any equation numbers. The work could benefit from adding equation numbers, such that certain formulas or definitions are easier accessible. \n5. Section 4: This section could benefit from an introduction, what this section is about. I am not understanding, why this is called \"Method\" and then section 5 also introduces a method (\"We now introduce general methodology to handle ...\", ln 214)\n6. Finally, there is no discussion/limitations/outlook section. The paper would benefit from a section that summarizes the work, discusses its contribution and limiations and provides an outlook. \n\nClairty: I find it, in general, very difficult to follow the authors in their main idea. I am sharing a selection of difficulties I had with respect to clarity:\n1. Section 1: I am missing clarity in the pipeline of decisions, why is human in the loop that makes the final prescription a problem? I understood the problem is that people, even if they receive a prescription, may not comply with it. So I understood the problem to be rather on the subject side that receives the treatment. Could you clarify?\n2. Section 2: The related work section cites work relevant to the field. However, I am missing clarity in how the previous work relates to the current work. There is a list of work and description of it, but often I am missing how this relates. For example, work [37] is cited twice in line 62 and then again in line 65, I am not understanding that split. In line 72 there is also a mentioning of \"supervised release\", without explaining it. You aslo write ln. 81 \"a different line of work studies counterfactual risk assesment, which models a different concern\", how is that different?\n3. Section 3: There is an introduction of social groups \"regarding fairness, we will be concerned about disparities in utiltiy and treatment benefits across different groups\", but then I find \"utility\" and \"treatment benefits\" not explained.\n3. Section 3: While assumption 2, 4, 6 are addressed in lines 126 ff. I am missing an explanation of the rest of the assumptions.\n4. Section 4 & 5: I am failing to follow the story and propositions. The paper would benefit from a clear explanation in words of i) what the proposition says / what the formulas express and ii) why it is important for the remainer/goal of the paper. \n5. Section 6: I am not understanding the motivation or story of the case study in the experimental evaluation. I am missing a description of the experimental setup.\n7. Section 6: The figures are missing an explanation what the different parameters are, e.g., $\\tau$ etc.\n6. A general point, I find the usage of the word \"recommendation\" difficult in the context of fair decision making (and here the authors use binary decisions as far as I understand). This is because there is a different field of fairness literature that concerns recommendations in teh context of ranking and that is different from decision making.\n\nIn addition I have the following comments/concerns about citations:\n1. I am missing citations in the introduction. For example, in line 38ff. \"a common strategy is to conduct an intention-to-treat analysis\"; in line 47 \"previous work in algorithmic accountability primarily focuses on auditing recommendations\".\n2. I looked up citations [20, 21] and did not found \"the well-understood notion of non-compliance/non-adherence\". Can you point me to this one? Also I found [21] to be a presentation on causal inference, not a (published) paper or book, is this correct? I am not sure about the quality of that citation. \n\nAbout formatting: \n1. line 33 ff. is differently formated than the text below, I am not understanding why.\n2. Margin violations, at proposition 2, 5, and 6\n\nTypos: \n* ln. 85 \"don't\" -> do not\n* ln. 108 do you mean $\\mathcal{T} = \\{0,1\\}$ instead of $\\mathcal{T} \\in = \\{0,1\\}$\n* ln. 129 a blank to much after X\n* ln. 291 \"arbitrarily\"\n\nQUESTIONS:\nThere might be parts of their methodology or approach I simply did not understand due to the lack of explanation. \n\n1. Section 3: I find the equations in line 112 not introduced, what is $e_r$, what is $\\mu_{r, t}$ and what do they mean, what will we need them for? \n2. Section 4: line 176 \"since algorithmic recommendations are deterministic functions of covariates\" - what do you mean by this? Decision policies can be probabilistic [1, 2], as far as I am concerned. Or am I missing the point?\n3. Why are you using doubly robust estimation? From the text, I am not understanding why this is necessary. Also it may be worth stating, why you do not simply use inverse propensity scoring.\n4. In Section 4 and 5 I am failing to undestand the fairness understanding present in the work. Could you point me to where this is defined and detailled? \n5. What are you assumptions in the experimental section as to how individuals realize treatments? \n\n\n[1] Bechavod, Yahav, et al. \"Equal opportunity in online classification with partial feedback.\" Advances in Neural Information Processing Systems 32 (2019).\n[2] Kilbertus, Niki, et al. \"Fair decisions despite imperfect predictions.\" International Conference on Artificial Intelligence and Statistics. PMLR, 2020.\n\nLIMITATIONS:\nA discussion or outlook section is missing. The authors do address limitations of some of their assumptions in lines 126", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-16"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper focuses on fair optimal decision rules, enhancing statistical estimators, and robustness checks for algorithmic recommendations with randomized decisions. It introduces a two-stage procedure with a complexity bound for optimizing within a constrained policy class, ensuring less conservative out-of-sample fairness constraint satisfaction.\n\nSTRENGTHS:\n1. The paper addresses an interesting topic, considering the randomness caused by humans in the loop and providing fairness guarantees. The technical aspects of the paper are robust and well-founded.\n\n2. The paper explores two settings: one where R is randomized and satisfies overlap, and the other where R is deterministic but does not satisfy overlap. Comprehensive results are presented for both settings.\n\nWEAKNESSES:\n1. The fairness constraint considers the expectation of treatment but not recommendations which are the outcome of the algorithm. The reason for this is not clearly explained.\n\n2. The assumptions made in the paper are not adequately cited or explained. While the author states that most assumptions are standard, some of them appear to be quite strong, such as assumption 5, which requires a strong decomposable linear function, and it's unclear if it can be satisfied in general.\n\n3. There are some minor points that need clarification:\na. In Assumption 6, the symbol '\\leq' may need to be changed to '\\geq.'\nb. Line 110 defines a cost function, but it seems like it should be referred to as a utility function, as indicated in line 157.\nc. The conditions of each proposition are not clearly stated within the propositions, which causes some confusion, considering the numerous assumptions and settings.\n\nQUESTIONS:\nIn proposition 2, L looks complicated. Is it explainable?\n\nLIMITATIONS:\nI may raise my score if the questions/weaknesses are properly explained.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-18"}

comments: SUMMARY:
The authors characterize optimal and resource fairness-constrained optimal decision rules, and develop a doubly-robust estimator for the optimal decision rules.

STRENGTHS:
I'm not very familiar with the areas of doubly-robust policy learning and am unable to assess the paper adequately.

WEAKNESSES:
The authors have not discussed the limitations of their work.

QUESTIONS:
See **Weaknesses** section.

LIMITATIONS:
See **Weaknesses** section.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-24