Optimal and Fair Encouragement Policy Evaluation and Learning
Authors: Angela Zhou
Venue: NeurIPS 2023 poster (2023)
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
Angela Zhou
Department of Data Sciences and Operations University of Southern California choua@usc.ets
Abstract
In consequential domains, it is often impossible compel individuals take treat- ment, so that optimal treatment assignments are merely suggestions when humans make the final treatment decisions. On the other hand, there can be different hetero- geneity in both the actual response to treatment and final treatment decisions given recommendations. For example, in social services, persistent puzzle is the gap in take-up of beneficial services among those who may benefit from them the most. When makerss have equity- for fairness-minded preferences over both ac- cess 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 treatment take-up, and other constraints, via constrained opti- mization. We develop a two-stage, online learning- based algorithm for solving over parametrized policy classes under general constraints to obtain seriance-sensingg regret bounds. Our framework can be extended to handle algorithmic recommenda- tions under an reasnalee covariate-conditional exclusion restriction, using our robustness checks for lack of positivity in the recommendation.
Introduction
The intersection of causal inference and machine learning for heterogeneous treatment effect estima- tion can improve public health, increase revenue, and improve outcomes by personalizing treatment decisions, such as medications, commerce platform interactions, and social interventions, those But, in many important settings, we do not have direct control over treatment, and only optimize ment. For example, in e-commerce, companies can rarely compel users to sign up for certain services, rather nudge encourage users to sign up via promotions and offers. When in optimizing the effects of signing on important final outcomes such as revenue, we therefore need consider optimal encouragement designs. Often human expert oversight is required in the loop in important settings where ensuring fairnes machine learning 6 is also of interest: doctors preseribe treat- ment from recommendations [3I] managers and workers combine their expertise to act based on decision support [8], and in the social sector, caseworkers assign beneficial programs based on recommendations from risk scores that support triage [14] 19 48).
who benefit from it the most 4 30 35 49
encouragements, recommendations for treat-
interested or other voluntary actions beyond a platform's ''s control
The human in the loop requires new methodology for optimal encouragement designs because
when the human in the loop makes the final prescription, algorithmic recommenda- tions do not have direct causal effects on outcomes; they change the probability of treatment assignment.
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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On the other hand, this is analogous to the well-understood notion of non-coomliancchon--dleee ancheennce in randomized controlled trials the real world [22 21]. For example, patients who are preseribed treatment may not actually take medication. A common strategy is to conduct an intention-to--re -teteatt 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-infomme social settings, if we are concerned about access to the intervention in addition to utility of the policy over the populatioon finer-grained analysis is warranted. If an outcome-optimal policy results in wide disparities in access, for example 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 -eaiisssmmmakss..... 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.
decisions in
Our contributions are as follows: characterize optimal and resource fairness-constraines 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 faimesss 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-DME pretrial ais-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 33 studies an optimal individualized encouragement design, though their focus 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 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 regression adjustment are conditional on the recommended/not recommended partitions which would not allow our fairness constraints that introduce treatment- and group-dependent costs. 44 has studied uniform feasibility in constrained resource allocation, but without encouragement fairness. algorithmic recommendation, but not fairness. Our later case study is on supervised release, where there is lot of randomness in final treatment decisions, rather than pretrial detention.
49 35| 30
99 studies robust extrapolation in policy learning from
Fair off-policy learning We highlight some most closely related works in off- off-policy learning (omitting works in the sequential setting). (37] studies gigghhrobability fairness constraint satisfaction. 29 studies doubly-robust causal fair classification, while others have imposed deterministic resource constraints on the optimal policy formulation [3] 26 studies (robust) bounds for treatment responders in binary outcome settings; this desiderata 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-to- treat analyses in view of fairness constraints (which result in group-speeifie welfare weights). Our interest is in imposing separate desiderata on treatment realizations under non-compliance; but we don't 't conduct instrumental variable inference and we assume unconfoundedness holds. We argue ITT policy-relews
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. '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

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characterization. [27] studies multi-objective desiderata for pricing and notes intention-to to-trrt structure in pricing. but not fairness considerations in more general problems. related literature studies principal stratification [24], which has similar policy-relevance disadvantages regarding interpretability as complier analysis does.
Problem Setup
We briefty describe the problem setup. We work in the Neyman-Rubin potential outcomes framework We define the following:
for causal inference 40)
recommendation flag R E {0, use the terms encurngeenn/reommendiii
1}, where == means encouragee/recommenddd (We will interchangeably).
treatment T(R) E T, where T(r) 1 indicates the treatment decision was when the recommendation reported r.
outcome ((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 E (a. (For notational brevity, we may generically discuss identification/ restimatio/tiiiiioonn without additionally conditioning on the protected attribute). For example, recommendations arise from binary high-risk/low-rist 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 (((r)) = as the positive outcome, and when treatments are also binary, we may further develop analogues of fair classification criteria. We let c(r,t,)) t, y): {0. 133 + denote the cost function for r E {0, 1). T. y {0, 1}. which may sometimes be abbreviated cye (y). We discuss identification and estimation based on the following recommendation, treatment propensity, and outcome models:
(X, A) = P(R = ( X, Pap A) = P(T = R R=
My My, (X, 4) (Y) == t, A| = (Y) | T= T A] = Me A) (asn
We are generally instead interested in personalized recommendation rules T(r X) = (X) which describes the probability of assigning the recommendation to covariates X. The average encouragement effect the difference average outcomes if we refer everyone vs. no one, while the encouragement policy value (Tr) is the population expectation induced by the potential outcomes and treatment assignments realized under a recommendation policy π.
AEE E[((T()) -
V(*) B[c(m, (()),
Because algorithmic decision makers may be differentially responsive to recommendation, and treatment effects may be heterogeneous, the optimal recommendation rule may differ from the (infcasible) optimal treatment rule when taking constraints into account for simpler policy classes.
Assumption (Consistency and SUTVA ). Y; Y((T:(R)). Assumption 2 (Conditional exclusion restriction). (T))R) 1 T,X,A.
Assumption (Unconfoundedness) Y(T(r)) L T(r) X.A.
Assumption (Stable responsivities under new recommendations), P(T = R X) remains fixed from the observational to the future dataset.
Assumption 5 (Decomposable costs). c(r, t, t.y) = c,)rr + (t) + cv(v)
Assumption (Overlap). e, 5 e, (X, A) < 1 1,; S P.p (X) A) < Uz M; 0
Our key assumption beyond standard causal inference assumptions the conditional exclusion restriction assumption 2 i.e. that conditional on observable information X, the recommendation has no causal effect on 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 also a starting point for sensitivity analysis under violations of it (25]. Assuming assumption o is like assuming

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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 is structural assumption that limits our method to most appropriately re-optimize small changes existing algorithmic recommendations. This is also required for the validity of intention-to-tren analyses. For example, Poja (x) (disagreement with algorithmic recommendation) could be baseline algorithmic aversion. Not all settings are appropriate for this assumption. We don't assume focoo-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 via conducting sensitivity analysis, i.c. optimizing over unknown responsivity probabilities near known ones.
Later on, we will be particularly interested in constrained formulations on the intention-to-tren effect that impose separate desiderata outcomes under treatment, well 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 7 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/consttiine 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 optimal encouragement designs.
Proposition 1 (Regression adjustment identification).
E[c(T, (()))]
Proof of Proposition !
E[c(n, ((r)) = ( [[
R=
= -ieTT [| (X)
= (x)
=1 (Y (r, R = x,],
EEeT,re(0.1) E|T, (X) = | [B=r, (Y) T=
where the last line follows by the conditional exclusion restriction (Assumption 2 and consistency (Assumption T.
Resource-parity constrained optimal decision rules constraint:
We consider resource//urden parity faimess
V max [B[e(n, (( ( (T)(T) A (T) A b) <
(1)
Enforcing absolute values, etc. follows in the standard way. Not all values may be feasible; in the appendix we give an auxiliary program to compute feasible ranges of e. We first characterize a threshold solution when the policy class is unconstrained.
Proposition 2 (Threshold solutions). Detine
À L(, 4) = (x) T(X, A) + P(A) [1 a] -[A 6]) -X(eno((X,a-pp1o0/
X mrg X, x (I, "[L(X",X,))) > 0}
If instead d(x) is function of covariates z only,
X E arg min E[[[LA,, X, X]+), T")x) I[E[((A ,ABILAA",X,AAA X > 0}
X
Establishing this threshold structure (follows by duality of infnite-diimnnsinnal linear programming) allows us provide generalization bound argument.

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4.1 Generalization
(Policy value generalization). Assume the nuisance models ") [Pij::Pij] Pij1, M4, /o]""x" .1 H are consistent and well-specified with finite VC-dimension Vn over the product function class H. Let = (L, A: X] > 0: A R
Proposition
SUPREILAER (E, [T,, A)] - LL A)) = (p(
This bound 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 either model. We first consider the ideal setting when algorithmic recommendations are randomized so that (xX) P(R = r X).
Proposition 4 (Variance-reducee estimation).
I = (I[T = ()eri(Y) 1n (x))) 1n (X)
V(T)
Σ
E
Ir (X)
(X)
(E.1)
[[[(T)) = (re(
E
Tr Tx()
[R=r (T(r) - (x)) + P1)- (x)
Although similar characterization appears in [39] for the doubly-robust policy value alone, note that doubly-robust versions of the constraints study would result in differences in the Lagrangian so we retain the full expression rather than simplifying. For example, for regression adjustment, Proposition provides interpretability on how constraints affect the optimal decision rule. In the appendix we provide additional results describing extensions of Proposition g 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 requires only SUTVA and consistency, and the exclusion restriction assumption. Additional assumptions may be required to extrapolate (X) beyond regions of common support. On the other hand, supposing that positivity held with respect to I' given covariates X, given unconfoundedness, our finer-grained approach can be beneficial because we only require robust extrapolation of (X), response recommendations, rather than the outcome models (x))
first describe what can be done we allow ourselves parametric extrapolation on Pıt1 Pıj1(X) (X), treatment responsivity. In the case study later on, the support of X R = 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 Pt = P(T distribution of At in the observational data):
X) marginalizes over the
Proposition 5 (Control variate for alternative identification ). Assume that Y(T(r)) 1 T(r) R r, X.
-T (x( V(T) = Σ E ((OaL110)) IET, (Ef,))
+ (1 T[T (T=11) ((x]] (x)
p((X)
Robust extrapolation of (X) Let 2T P = = 0} denote the no overlap region; on this region there are no joint observations of v,), We consider uncertainty sets for ambiguous treatment recommendation probabilities. For example, one plausible structural assumption monotonicity, that is, making an algorithmic recommendation can only increase the probability of being treated.
5

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We define the following uncertainty set:
Upe (41)-(2'): 91 (z) 2 (z), Vx E x EiET 9t/r (z) 1, vx,r)
We could assume uniform bounds on unknown probabilities, or more refined bounds, such as Lipschitz-smootthess with respect to some distance metric d, or boundedness.
Uip {u')r(=') :e (((u))(a)) (1)(()) S Ld(a', x)), (x',ar)
E (2 x , == {a))(a"): b(x) S 2u)r(2)) S b(x)}
Let U denote the uncertainty set including any custom constraints, e.g. U = Ug AiApp For brevity we use Ino to denote I [X E x Loo to denote [X E x=I. Then we may obtain robust bounds by optimizing over regions of no overlap:
Define (T)) = EeeT.rE(o.)) (r X
((x) = (T) V (T)), (T)
max (x-(X)E
(InET B(((r) (X))
In the specialized, but practically relevant case of binary outcommssrrattmettt/ccmmeeetttte 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:
we
Lemma 1 (Binary outcomes, constant bound). Let Uebnd == {a)) B S (1)r((") s B} and 4 = Ugg- O Mebms Define (a)r(a) = E[ggr X, A) T = f, a). If E (0.1),
V (T) EieT.rE(o.)
B Blc
BY X) T =]]Tno]],
BT 11 = 1| BT i 0
y <r X) | T 1 20 iE T <0
where G+
BI [ - B
BT
We state the next result for simple uncertainty sets, like intervals, to deduce insights about the robust policy. In the appendix we include more general reformulation for polytopic uncertainty sets. Proposition 6 (Robust linear program
). Suppose R, T E {0, 1), and 9-1 u) E Lond, Vr, u. Define
T(E, = (1 a) Ho(æ, a), (z, = (z, u) (2r u)), Brdidd (I, u) (, u) + (z,
BIA.. [())] (T) A= = a] - ([T A = b], (T) = B(TTT,
Then the robust linear program is:
max Vov(T) + E[uo] + c1 (T) - E, B|T/ (X, A)
s.t. (B]T(r) (X, A == al
E B, A)l AA=6) + AT) Se
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 policy class of restricted functional form and 2) more general fairness constraints. We first introduce the fair-classification algorithm of [1] describe extensions to obtain variance-sensitive regret bounds and less conserva- tive policy optimization (inspired by regularized ERM argument given in [2],, and then provide sample complexity analysis.
Algorithm and setup In the following, to be consistent with standard form for lincar programs, phrase the saddle-point as minimization-
note that we consider outcomes Y to be costs so that maximization. Consider (CC linear constraints and J groups (values of protected attribute A), with coefficient matrix M E KK the constraint moment function hy (T). j E [J] (with J the number of groups), O = (X, A, R, T, Y) denoting our data observations, and the constraint constant vector:
(T) E (0, π (X)) Eg] for E J, Mh(T) < d
Importantly, 9j depends on π while the conditioning event &, cannot depend on n. Many important fairness constraints can nonetheless be written in this framework, such as burden/resource parity,

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Algorithm 1 MW2REDFAIR y, &, M.
Input: D {(X;, R, T, Y, A, 9,E, d. B, v. a, 01 0 R(K]
2: for f = 1,2,... do
exp( 3: Set λ1, B 1+ (0p) for all l K, B, BEST (2e) Qu t Ei== Br
+ H--LQQ. BEST (Q.)), d, t dr, (BEST, (
5: - m 21) L. L I-L(Q,, ), If 1 Σ v then return (Q Ac)
6:
1,+1.1 O. log(1 (M (hh) 2)
7: end for
parity in true positive rates, but not measures such as calibration whose conditioning event does depend on π. (See Appendix [.2 for examples omitted for brevity). We further consider a convexification of II via randomized policies Q E AIII, where (III) is the set of distributions over II, i.e. randomized classifier that samples a policy Q. Therefore we solve
min (V(T): Mh(T) s d} QEA(T))
On the other hand, the optimization is solved using sampled moments. so that we ought to interpret the constraint values du = dx + <e 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,
L(Q,A) = ( (MT)() -d),
min (V(T): Mh(T) S d} = min max L(Q, A). QEA(I))
QE AER
We simultancously solve for an approximate saddle point and bound the domain of A by B:
mingee max XERR MIRI. S A),,
max AER X MERR SJ minges L(Q,
We play no-regret (second-order multiplicative weights Hedge/exponentiated gradient (18) algorithm for the X-player, while using best-response ora- cles for the Q Q-player. Full details are in Algorithm [ Given A,, BEST, (Ae) 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-senssttiee/viiihhee classification [TO 49], which we
a slight variant of
describe in further detail below. Computing the best response over BEST (Q.)) 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 weighted classification. Taking the Lagrangian will introduce datapoint- dependent additional weights. This reduction requires π E 1-11, +1,. e { +1} (for notational convenience alone). We consider parameterized policy classes that ((x) sign (x)) for some index function g depending
parameter E Rdd Consider the centered regret ((r) = Y
EE R = 1, + E = 0, X]]. Then J(3) = J(sgn(gx())) = Elsgn( (g (X)) H
where va can be one of, where (X) = L R = X]
VDDM = Pio((()) -o VIPW RY en(X) - + +
Ra (x) ex(
We apply the standard reduction to cost-sensitive classification since 1hs (gg(gg(X,)) | (1
2Il [sgn(g) (X))) # (gn(w,)) )). Then we can use surrogate losses for the zero-one loss. Although many functional forms for E(.) are Fisher-consistent, one such choice of l is the logistic (cross-entropp) loss given below:
(()) B[ia ((gs(X), (1))),
1(g,s) 2 log(1 + exp(g)) - (s +
(2)
Two-stage
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 fcasibility slacks via generalization bounds that previously depended these worst-case constants.
7

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Algorithm 2 Two-stage localized fair classification via reductions
1: Randomly split the data into D1, D2
2: Obtain #1 and the index set of binding constraints Σn by learning nuisances 77 and running Algorithm 1 on Dj with (D, E,,,,,,,, M,d;m) m)
3: d3 < Vgr (O,
Vj; d d n
4: Augment additional constraints with G, t.--ooiccy-vluee and constraint slices relative to u1: define
an augmented system (where subscripting by I, indexes the corresponding matrix or vector):
hy E,,, H6g #i: of (0; Egl, Vj' EI, he = En, (VDR(O n.,
M [M; M2, 11, [h(::), h.T d en, enl"" = 0]
5: Obtain 2 by running Algorithm on D2 with MW2REDFAIR g. &, M, d, m2)
These challenges motivate the two-stage procedure, described formally in Algorithm 2 and ver- bally here. We adapt an out-of-sample regularization scheme developed in [[2]. which recovers puriance-sensitive regret bounds via a small modification to ERM. We split the data, learn nui- sance estimators 7), for use in our policy value and constraint estimates, and run Algorithm [] (MW2REDFATR(D) h, &, M, d: )) to obtain an estimate of the optimal policy #1, and the con- straint variances #1. Next, we augment the constraint matrix with additional constraints that require the second-stage π to achieve in close policy value and constraint moment values relative to क 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 #1, which will be less conservative than using worst-case bounds. Define
0), for (.) E (0, DR), 9;; = (9, O,E,]. ( = En (
so that (Q) = (Q)) and h (Q) = Elg, (0: Eg]. Define the function classes
Fn (;) т E 11, 1) E HJ = =(6(s m):): x 11, H}
and the empirical entropy integral n(r, .F) = inf (40 - + 10 V Hale.F.n) de} where H2(c, m) the Le empirical entropy, i.e. log of the |1-112 e-covering number. We make a mild assumption of learnable function class (bounded entropy integral) [46].
Assumption 7. The function classes Fr1, (Ji satisfy that for any constant r, ((r, - 0 as + 00. The function classes {J,}, comprise of L.p-iischitz contractions of æ.
We will assume that we are using doubly- robust/orthooonnaizzed estimation as proposition state results depending on estimation error of nuisance vector
hence
Theorem 1 (Variance-Baal Based Oracle Policy Regret), Suppose that the mean- mean-squen squared-error of the nuisance estimates is upper bounded w.p. 1-8/2 2 by x over the randommess of nuisance sample:
max [E T))ee == Xxa Let r = suprER [E[UDR(2; xt and c, = O (r. Fu) +
log(1/51), Moreover, let
Qx(c) = = A|Ii: V(Q.) - V(Q) sc, h(Q)) -(Q)) s d
denote an e-regret slice of the policy space. Let En = O(en + x2,5) and
V = sup [Var (vbn uhn (0:Q)) Q Q' E (x.))
denote the variance of the difference between any two policies in an regret slice, evaluated at the true nuisance quantities. (Define V2 analogously for the variance of constraint moments). Then, letting (Q)) == Mh(Q) denote the constraint values, the policy distribution Q2 returned by the out-of-sample regularized ERM, satisfies w.p. 1 - over the randomness of S
v(θ) - V(Q") = ( yohj conv (Fn)) + n
yobs log(3/5) + x2.6)
-
= 0(1( V/S conv(( + + V33 log(3/5) + +2,
8

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The benefits are that 1) the constants are improved from an absolute, structure-agnoste 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 only present a case study based the PSA-DMF for supervised release [38]. In the appendix we include additional experiments and robustness checks, including 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 iisk-scoreinfformed 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 not informed by empirical evidence. There are current policy concerns about disparities in increasing use of supervised release given mixed evidence outcomes [38;; e.g. Safety and Justice Challenge
[1]] concludes "targeted efforts to reduce racial disparities are necessary". First, we acknowledge data issues. We work with publicly available data that discretized for privacy (38) The final supervision decision does not include intensity, but different intensities are recommended in the data, here
which we collapse into single level. The PSA-DMF is algorithmic recommendation
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 fiiberryi empirical study.
Next in Figure I 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 failure to appear, so negative scores are beneficial. The second figure illustrates the difference in responsiveness: how much more likely decision-maker makers are to assign treatment when there is vs. isn' an algorithmic recommendation to do so. The last figure plots logistic regression of the lift in responsiveness on the causal effect Ma (z, a) the same treatment effect efficacy. This is importantly not claim of animus because didn't 't have access to causal effect estimates. Nonetheless, disparities persist.
(z. a). We observe disparities in how responsive decision makers are conditional on
In Figure 2
we highlight results from constrained policy optimization. The first plots in each set illustrate the objective function value and A = a average treatment cost, respectively; for A being race (nonwhite/white (failure to appear, 0 for Y = O, and 20 when T 1 (set arbitrarily). On the x-axis we plot the penalty À that we use to assess the solutions of Proposition g The vertical dashed line indicates the solution achieving e = 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- burdeeddd marginalized populations. We see that indeed, for race, surveillance-parit 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.
gender (female/male), respectively. We use costs of 100 for Y = 1
References
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---

r by gender
gender
P1. Pio VS 5, a= gender
0.30
17.5
Male Temale
0.25
0.20
12.5
0.10
0.03
5.0
-0.2
Figure Distribution of treatment effect by gender, lift treatment probabilities Pila Pola P(T = R = 1,A = a, X) P(T = R 0, A = X), and plot of Pila
Pola VS.
17.30
3.5
A-vhitt
Amnonshite
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A-male
Amfenale
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3.0
3.0
CITIe
ETTIe"M
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12

---

A Additional discussion
A.1 Additional related work
32 studies an optimal test-and-treat regime under a no-direct-effect assumption, that assigning diagnostic 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 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-makee and we consider generally nonparametric models.
Principal stratification and mediation analysis in causal inference
We briefly go into more detail about formal differences, due to our specific assumptions, that delineate the differences mediation analysis. Namely, our conditional exclusion restriction implies that YIT, = Yr, and that Yor; Yın, (in mediation notation with T,- = T(r) in our notation), so that so-called net direct effects are identically zero and the net indirect effect the treatment effect (also called average encouragement effect here).
Human-in-the-loop in consequential domains. There is great deal of interest in designing algorithms for the "human in the loop" and studying expertise and discretion human oversight in consequential domains [14]. On the algorithmic side, recent work focuses frameworks for learning to defer or human-algorithm collaboration. Our focus is prior to the design of these procedures for improved numan-algorithh collaboration: primarily hold fixed current human responsiveness 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 later outcomes, line of work uses individual judge decision-makers as 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 13,
33 In contrast to these works, we focus on studying decisions about electronic monitoring, which an intermediate degree of decision lever 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 causal effect on the principal stratum of those marginal defendants who would not commit new crime if recommended for detention. [9] study policy learning in the absence of positivity (since the PSA is deterministic function of covariates) and consider case study on determining optimal recommendaioo//etention decisions; however their observed outcomes are downstream judicial decision- cecision-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.
13

---

B Additional discussion on method
B.1 Additional discussion on constrained optimization
Feasibility program We can obtain upper/lower bounds on in order to obtain a feasible region for by solving the below optimization maximal/minimal values of the constraint:
E, max / min (TT AA= = - E[[(()) A -
(3)
V max ((Td,, (TT) | - A 5
(4)
B.2 Additional discussion on Algorithm (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 [ (including in notation) and is included in this appendix for completeness only. We only additionally provide novel identification results for another fairness measure causal policies in Appendix B.2.2 concrete discussion of the reduction to weighted classification, and provide concrete descriptions of the causal faireess constraints in the more general framework.
We first discuss how impose the treatment parity constraint. This similar to the demographic parity example in [1. with different coefficients, but included for completeness. (Instead, recommendation parity in TE(T / A is indeed nearly identical to demographic parity.)
Example 1 (Writing treatment parity in the general constrained classification framework.). We write the constraint
([((T)) | == a) -E[((>)) A A=
(5)
this framework as follows:
(TT(T) A = al = Elm PPijo (1)) + (i)((X
For each E we enforce that
Ere(o,1 (X))), A= Er(,) E [T, (X, (X,
We can write this in the generic notation given previously by letting J AU {o),
g, n (X): = - (X) A) Pijo A)) Pijo X, A), Vj.
We let the conditioning events E, {A a), (True), 3 i.c. conditioning on the latter is equivalent to evaluating the marginal expectation. Then we express Equation (5) as a set of equality constraints (T) ho (TT), leading to pairs of inequality constraints,
hna ho < (H((T)) (mT <
WEA
The corresponding coefficients of M over this enumeration enforcement equality ({+,-)) equation (1), gives = A x (+,-} so that M(a.)).'
groups (AA) and epigraphical
1 (a' = a} M(,,,),+ -1, M(a.))aa' = -1 (a' = a) M(,,)), = 1, and d can relax equality to small amounts of constraint relaxation by instead setting dę > 0 for some (or all) k.
Further we
Next, we discuss more complicated fairness measure. We first discuss identification and estimation before also describe how to incorporate in the generie framework.
14

---

B.2.2 Responder-dependent fairness measures
We consider a responder framework on outcomes (under our conditional exclusion restriction). Because the contribution 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 recommenda- tion) as (0(r)) (((r)) & (0,1)². We call units with (Y(rr)), (((r))) = (0, 1) responders, (O(r)), Y (1())) (1,0) anti-responders, and (((r)) (((rr) non-responders. Such
decomposition is general for the binary setting.
Assumption (Binary outcomes, treatment).
T,Y {0,1)
Assumption 9 (Monotonicity).
(((())) 2 (
Importantly, the conditional exclusion restriction of Assumption 2 implies that responder status is independent of recommendation. Conditional on observables, whether a particular individual is responder is independent of whether someone decides to treat them when recommended. In this way, we study responder status analogous to its use elsewhere disparity assessment algorithmic fairness [23 28 Importantly, this assumption implies that the conditioning event (of being responder) is therefore independent of the policy म, 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).
E[T(=) ((((R)) > Y(O())), A = a] - -T(((T) (()(R)) > (( Se
We can obtain identification via regression adjustment:
Proposition 7 (Identification of treatment expenditure given responder status). Assume Assump- andd
((((() Y(O(=))) Σ, T (X) P1 (X, (X, - Mo A)) A= a)) (X, A) (o A)) A=
E (1
Therefore this can be expressed in the general framework. Example 3 (Writing treatment responder-conditiona parity in the general constrained classification framework.). For each u E A we enforce that
Σ. (X)p( (X (X, ([(m)(X))) (X) A-x
E. E|x (x), (X.
We can write this in the generic notation given previously by letting J AU {a},
(X):n) (x) (X, A) (i A)) - P1j (X, A)) (X, A) - Ho A)) Vjj [[( A) No A)) A
Let E {A= o3 (True), and we express Equation 5 as set of equality constraints of the above moment (T) = (T)) leading to pairs of inequality constraints,
(ha(T) ho (T) (h(TT) (T)
<
WEA
The corresponding coefficients of M proceed analogously as for treatment parity.
B.2.3 Best-response oracles
Best-responding classifier given X: BEST, value, optimizes the Lagrangian given n::
The best-response oracle, given a particular À
L(m, A) (x)) T (Mh(*)=
dt)
= ( x Σ MR En (0, E,11
Pj
15

---

Best-responding Lagrange multiplier X, given हı the A player. can be chosen be either 0 or put all the mass on the most violated constraint. Let (Q) = Mh(Q) denote the constraint values, then BESTx(Q) returns So
B (Q) is the best response of
i غ
Beg otherwise, where k maxk (Tx(Q) [i.]
B.2.4 Weighted classification reduction
There is well-known reduction of optimizing the zero-one loss for policy learning to weighted classification. A cost-sensitive classification problem
arg min
(X,) C. + (1- C?
The weighted classification error Σ W.1 (h (X.) * Yi} which an equivalent formulation W; C C. and Y; ( 2 C.).
The reduction to weighted classification is particularly helpful since taking the Lagrangian will introduce datapoint-dependeer penalties that be interpreted as additional weights. We can
consider the centered regret ((T) = [x)(=)] - EE 1, + B|Y R 0, Then
(() = (sgn(gg = Eg (X)) [
where v can be one of, where xx(()= EE[Y R R=
RY
Ru ex((X)
dnn (pip (X) (x) k1Pw
e DR PDM-
We can apply the standard reduction to cost-sensitive classification since v sgn(go (X.)) = lts 211 (go (gx().)) * sgn(w.))). Then we can use surrogate losses for the zero-one loss.
L(O) = Bl|bb ((g)())
Although many functional forms for {(.) are Fisher-consistent, the logistic (cross-entropy) loss will be particularly relevant: ((g s) log 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:
L(B) = E 1ibA1e (gs(x),sgn((})))
where Jtd
+ +
AA (Pip P1j0) - Σ Aa.
PA
aEA
where Pa := P(A a) and A := A(a+t) A effectively replacing two non-negative Lagrange multipliers by single multiplier, which can be either positive negative.
Example 5 cossooddr-conditiion treatment parity, continued). The Lagrangian is I(3)
1e (x)(X), sgn(((>))) with weights:
Aa
(pn (11 11)
JR 2b
Σ >a.
DA En (M A) (o(X,AA A)) A
aEAA
where Pa P(A = a) and Xo = \(a.t) A(a-))-
16

---

B.3 Proofs
Proof of Proposition
P(T(T) [
by Bayes' rule
= ( =
by Assumption
P(Y(1) > A=0)
Σ, EE F (X] a)
A= x|I)
by iter. exp
E, (X JPI (X, () (X, A)) AA a])
by Proposition |
(1) (X, A) no (X, A)) A=
17

---

C Proofs
C.1 Proofs for generalization under unconstrained policies
(Policy value generalization). Assume the nuisance models [Pijo: Pijo-Piji= /11, Po: e, ex())] , H consistent and well-specified with finite VC-dimension V over the product function class H. We provide proof for the general case, including doubly-robust estimators, which applies to the statement Proposition 8 by taking 7) [Pijo P1|1s M1, Ho]
Proposition
Ler II = [T[B]L(X, X, A;r) X] > 0: X E R; Fj.
AEER (E,, [ L X, A) H[mL(A, X,)) = Op(n
The generalization bound allows deducing risk bounds on the out-of-sample value: Corollary 2.
E[L(,, x, A)++] E[L(",, x, A)-
Proof of Proposition 8 We study a general Lagrangian, which takes as input pseudo-outcomes z/t x (O:n), sbait (O: n) 2b 10, where each satisfies that
El ((; X,
Pip (X, A) (1
Elebal (O; n) X, A] TXX, A)
Blv X] = Pijo(X, a) (X.)
We make high-level stability assumptions on pseudooutcomes v relative to the nuisance functions (these are satisfied by standard estimators that we will consider):
Assumption 10. th ebale, 10..0 respectively Lipschitz contractions with respect to r) and bounded
We study generalized Lagrangian of an optimization problem that took these pseudooutcome estimates as inputs:
L,A,X, = (a: ( + P(A) (T[A-
I[A- [[AA= b1) ((0)110AAAOOn))
We will show that
[B X, A)] E(m X, =
sup EERRAER
which, by applying the generalization bound twice gives that
= E/T = +
Write Lagrangian
max min min max min E|L(O,A:
Suppose the Rademacher complexity of 77K is given by R(Hx). so that 7 Thm. 12] gives that the Rademacher complexity of the product nuisance class // therefore Σx R(Hx). The main result follows by applying vector-valued extensions of Lipschitz contraction of Rademacher complexity
given in 36) Suppose that yi gallt 3110,A44 are Lipschitz with constants Clp- Ct Cl..AA
We establish VC-properties of (01..) =
H}, where (0) = (0)
F1.(0 =
ho,)) H}, where (0) = thir (O;
[I[A = a] [[A b)
p(4)
FL, (01..) ((m) H), where m, (0) = (0.AA..A(0
18

---

and the function class for the truncated Lagrangian.
FL4 = =(;))((O)) + M,, + m. (m:))) 9 E FL, (O1m), FL, (O1m), m E FL, (11m)) ; n E H}
36 Corollary 4] (and discussion of product function classes) gives the following: Let X be any set,
(T,,..., ,xn) ) E X", let F be class of functions f + E2 and let h4 : l2 - R have Lipschitz
norm L. Then
E sup ne
Σ Eiils (O.)) S V2LE sup
Eewn (O.) S V2L Σ
ΣEAR E sup Σ ETT (O.)
(6)
7EH
where 6s is an independent doubly indexed Rademacher sequence and fi (x;) is the k-th component of f (2:).
Fia Applying Equation (6) to each of the component classes Fr, (0 (0 1,)), FLa (Otm), and
Lipschitz contraction 7 Thm. 12.4] of the positive part function Fr.,, we obtain the bound
sup (L( BIL(O, s v2(C C. Ctf. Bp. B + BC E R(Hs) Xm
Proposition 9 (Threshold solutions). Define
L(A, X, A)
A)) A) p(A)
A
[I a] II[A -[[A=b] b]] +))p1j((,,a) -P1o))X)
X E mig EIL(A,X, A)+], (I = I(L(X ,,,) >
If instead d(x) is function of covariates x only,
X arg min (. x]+). (' = I(E X, A) X > 0}
Proof of Proposition The characterization follows by strong duality in infinite-dimensinee dimensional linear
programming 42 Striet feasibility can be satisfied by, e.g. solving eq. (3) to set ranges for e. 0
19

---

C.2 Proofs for robust characterization
Proof of Proposition
((x) Σ
(YX -=,;;
(ET,)E(O.))
Σ
(X)
| = - r,
unconf.
(E..)
Σ
(X)
(XP(()) =
x)P(T(()
R r(
Assumption 2 (ER)
1ET.rE
(v()) (IT(r) =
E
Try (X)E
X P(T(r) = R= R= x)
unconf.
Pc(X)
(ET,1
(r)(Y(t)) [T((r) =
(1- Pt(X) ) mx(x) xx] Pin(X0)
T
Σ
(X)
control variate
(x(
rEf01,
{{ {f (Y((Y())) [T6) =0]
T mm(x] (1-1((x))
Σ
my Tx((
p(())
(LOTE)
(x)
(E..)
where Px( P(T x) (marginally over R in the observational data) and (LOTE) is an abbreviation for the law of total expectation.
Proof of Lemma 7
V (T)
max (X)
Σ (0,
(xxx E xT"]]]]
max (X)(
Σ
(X)[[
T -t E x=III]
Note the objective function be reparametrized under surjection of gejr (X) to its marginalization, iec. marginal expectation over {T' t} partition (equivalently {T t, A a} partition for faimmes--cosstrained setting).
Define
(a) (X, A Bej E(g, T
Therefore may reparametrize V (T) optimization constant coefficients (bounded by B):
max
EX T == ==t,x|i[X x=II]] B < B,C 11 (0T.)
Σ
max
1E T 4] 1]] B < < B, 1-
LOTE
(0.1)
Σ
S/ct E T
T= xX E 1
(E7,))
BT [t = 1] BT | 0)
ir (X) T= = 2
where oh
BI [t = 0) + BI it
iF (X) - <
20

---

Proof of proposition 6
((T) -
(8)
B[T 1 A a| - [ & | A
(9)
+ [ E X
A = a] -T) X E 2
A b) S c Aar E e4 (10)
Define
(r u) (Mr1 (2, u) (,o u))
then we can rewrite this further and apply the standard epigraph transformation:
max t
Σ Σ
(T) + Eluo), 4g,1 Eu
TEX
WE (a,b) (E(0.1)
(f(z I a)
(
Tr (r f(x 16) Σ (2, b)um (x, b1)) BIAm. S e, Van E u
Project the uncertainty set onto the direct product of uncertainty sets:
max t
t J DEX wE(a,b) rE(U.1)
Σ Σ
(f, (0 dr & Voo(T) + Bluol, 4ar1 E U60
I J
(f(x a)
E T= a)) f(x 18) b T. (1, b])) E e, v4r U
Clearly robust feasibility of the resource parity constraint over the interval is obtained by the highest/lowest bounds for groups a, h, respectively:
max t
t
Σ
Σ
< (o) + Bluo), Vga E Uoo
TEX uE(a,)) rE(0,1)
[ J
(f(e a)
Tr (,, f(x n. (x, (BA
define
(2, 2(qr1(0, u) (,(x, (Br(r, (, (, (.,
then
(B, u) S 4rl (2, u) < BB, u)} = (1 (e,,,,,,olll loe 5
and
1
(x, u) = (x, u) + (B, u) - (z, +1).
2
For brevity denote AB (B, u) (2, u)), so
max t
t+
max (6.12. rE(o.))uElad! I Nxx
Σ
Σ (o (x,, (x, w)
Ab) u)dr cI( - Vov(T) +1
wa(a,b) rE(0.1)
(f(x a)
TT f(x b) b) n.
w)) + +()) Se,
21

---

where
(T(T)
=
Σ
Σ
(gr(a, (I) + 2 (B, u) - (x, u)
2
(a,b) rE(0,1)
This is equivalent
max t
++ Σ
Σ
ux w)l AB A (()) s (T) + Bluo]
JEXP (E(a, rE(0.1)
JEEN (f(2
f(x क. 71
Undoing the epigraph tramsformation, obtain:
max (T) + Bluo] + cI(T) - 1 Σ
Σ (t ~
AB(x, u)dr
wE(a,b) rE(0.1)
J (f(x I a) ②) (=, f(x
n. (1
and simplifying the absolute value:
max (T) + Eluo] + (TT -
/
Σ
Σ
lg, uZT, wx, u) AB(E u)dr
FEXX (e( (o.1)
T. () T(..T Se f J0EX (f(x a) [f(x|a)(2 () T (T f(x b){
22

---

C.3 Proofs for general fairness-constrainee policy optimization algorithm and analysis
We begin with some notation that will simplify some statemetns. Define, for observation tuples O ~ (X, A, R. Y, the value estimate n) given some pseudo-outcome (5(O;) n) dependent on observation information and nuisance functions 7). (We often suppress notation of r7 for brevity). We let estimators sub/super-scripted by denote estimators from the first dataset.
(Q) E T O0]. for (.) E DR)
(Q) E
(Q) E,, (e)
(): = E (00, 10,8,
hy( Elg, (0: Eg
[)(()= E,, (0:()) &g]
C.3.1 Preliminaries: results from other works used without proof
3. [ (saddle point generalization bound (non-localized)) ) Let p max ([M čllco- Let Assumption hold for C² 2 2C + 2 + /m((4/8/22 where > 0. Let Q' minimize V(Q) subject to MuQ)) S Then Algorithm with v x n n B x n and 1) p n2n 20 terminates in O (p2n In KC]) iterations and returns Q. np3 2 8 8log(2/)) ( for all
Theorem 3 (Thm.
j. then with probability at least 1 - (IJ] + 1)8 then for all k Q satisfies:
v(Q) < V (Q*) Ō (n-a)
7A (2) S c +
1 2v
Σ JEJ
| Ō ((mp)) -a)
B
The proof of [I] Thm. 3] is modular in invoking Rademacher complexity bounds on the objective function and constraint moments, so that invoking standard Rademacher complexity bounds for off-
policy evaluation/learning [5 45 yields the above statement for V(Tr) (and analogously, randomized policies by 7 Thm. 12.2] giving stability for convex hulls of policy classes).
Lemma (Lemma 4, [77. Consider a fufunction class F : X + Rd, with sup ll & 1 and
pick any f E F. Assume that (TTT) is L-Lipschitz in its first argument with respect to the & norm and let:
(r) sup ((E, ((Q)) (()(Q) vI(")|| -l
<r)
QEQ
Then for universal constants C1, C
{
Cynuu
Pr Zn(r) 2 16L Σ (r. conv(( Qi) + u S c1 exp
1,2,2 + 2L
Moreover, if 8n is any solution the inequalities:
VI (1....,d) d} R (6; star (conv (II,) - -Qi) < 82
then for each r 2 6n
C2rvu² 2
(Z.(r) 2 16Ldrō, + c1 exp
T22 + 2Tu
Lemma (Concentration of conditional moments (1 47 1 5, for all Q
For any j E T, with probability least
2
(n(28))
( S 2Rn, (H) +
nmj
2n.
If npp 2 8log(2/5), then with probability at least 1 - 5, for all Q.
In In(4/8) np}
(,,(Q) - h,,( < 2 (2( +
np'
23

---

Lemma 4 (Orthogonality (analogous to 12 (Lemma 8), others)). Suppose the muisance estimates satisfy mean
mean-squarees squared-error bound
max (() [het = =xn
Then w.p. 1 - å over the randomness of the policy sample,
V(Qn) - V(Q) V(()) - O(Rn,s + Xn)
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 (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. ô by h and suppose hi,5 S Ep. Then:
V22 =
sup
Var (v)R(I; ()) -DDR (x; x'))
Then V/2 < V2 + O (hn.8).
C.5 Proof of Theorem
Proof of Theorem 7
We first study the meta-algorithm with "oracle" nuisance functions = 770-
Define
Il (Gx) a II: E., [o(Q: u(Q1: Cn En, (0, no) 9j (O; (1; mo) [E] s cn, E 23}
Q2 (fn) = (Q E A (T2 (c.)))
Q" (en) - (Q E A(I): E[(v(Q;m)) - uQQ': c. Elg [ Elg, ; [9 s {en
In the following, we suppress notational dependence on Tjo-
Note that Q E Q2 (en).
Step 1: First we argue that w.p. 1 - 5/6, Q" 22.
3 on the output of the first stage of the algorithm, yields that with probability 1 5 over the randomness in D2, by choice of en ((n
Invoking Theorem
6
v(Q1) V(Q") + en/2
m2(2)) & de + Σ
Mk.i Ô (mpp))aa) s ds cn/2 for all k
jET
Further, by Lemma
E m [(((Q) - [((2)) -((()) x(Q"))] S 6n/2 El(((( - ((::00))) -((o(00))) s cn/2
sup
Q
sup
E,, -(000))))00000)))))))
QEQ
Therefore, with high probability the good event, Q* E Q2.
Step 2: Again invoking Theorem 3 this time on the output of the second stage of the algorithm with function space II, (hence implicitly Q2), and conditioning the "good event" that Q" Q2, obtain the bound that with probability 2 1 - 8/3 over the randomness of the second sample D2.
v(42) (VQ")=+++ - 2n/2
7h (Q2) s (Q()) + c./2
24

---

Step 3: empirical small-regret slices relate to population small-regret slices, and variance bounds
We show that if Q 22, then with high probability Q 22 (defined on small population value- and constraint-regret slices relative to Q1 rather than small empirical regret slices)
2 = (Q E conv(II: V(Q) - ((Q)))((()) e., E,/ -;:00)0)))0(00))) 1&) S cns Vj}
that w.h.p. 22 C 22-
Note that for Q E Q, w.h.p. 1 5/6 over the first sample, we have that
En [n(u(Q) - v(Q1) B(v(Q) v(Q+)) < 2 sup [E, B[((Q))] s,,
sup
QEQ
QEo
- 9, Em, -0000))))((000100 Q1) -
9, (000) &
sup QEQ
2 sup EE,, [g (0; Q) Eg] E(g, & <
QEe
The second bound follows from 77 Theorem 12.2] (equivalence of Rademacher complexity 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 Qı 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 Q2, with high probability 21 1-83:
[B((((Q) v(12))] < (((((Q) -(c)) ((00 - e
-9,(0:0))))) -000(0))))) | <
E + (0:,)(0:11) g9,0))))) en
Define the space of distributions over policies that achieve value and constraint regret in the population of at most 6n
Q. (c.) (Q E Q: v(Q) - VVQ") S Cn. ( c, Vj}. so that on that high-probability event,
2)(en) C Q.(en).
(11)
Then on that event with probability 2 1/3.
2 sup
QE [[(x(Q) - u(Q))) (EQ)(())
sup B[(v(Q) x(Q'))²
= sup Var(u(Q) v((")) + [[(((Q - 1(2")))
(c.)
< sup Var(u(Q) - v(())) +
(c.)((..)
Therefore:
Var (e(Q) - v(Q*)) + 2e, = vV2 + 2
T2 S sup (c.)
V
Combining this with the local Rademacher complexity bound, we obtain that:
-(u((a)~v(Q') = =0 (VVs+ 2cn, Q. (Gr)) +
V/ log(3/8)
-
n
These same arguments apply for the variance of the constraints
V sup (Var (g, () 9j (0:Q)) Q. (.)).)
25

---

C.6 Proofs of auxiliary/adapted lemmas
Proof of Lemma The proof is analogous that of 12 Lemma except for the step establishing that T E O +0,+ 5) 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 muisances):
X (Q0,0- Ao.u) E arg min max Elxxn A (TDa(Q:no) d) L(Q, A: no) = L(Q.)),
(Q7 N7,0) E arg min max Baxx (ypnl
(y d),
Q
(Q", x²) E arg min max Elx (TDe d).
(7DR(Q:n)
We have that:
Blunn(Q')) [(Q", (-;;,,
L(Q", mo) +V X2,
$ L[Q"" 10) + Xn.s
by Lemma 4
3 L(Q", N1, 10) + + X
by saddlepoint prop. triangle ineq.
L(Q No 100 + L [( no) L(Q", No no) x
L(Q X 77000 + + 1/ Xn.6
assuming en 2 Xn.6
BlUDR(Q
)) 7)
+ c + 2y + x
apx. complementary slackness suboptimality
BluDn(Qo,0) 1) 6n 2v Xn.5
Hence
El EL (Q"; :n] Ellnn-Elpnnoo 1)) S en + 2v + X2.
We generally assume that the saddlepoint suboptimality is of lower order than en (since is under our computational control).
Applying Lemma gives;
V(Q") - V(Qū,) S 6, + 2v + 2x
Define policy classes with respect small-population regret slices (with a muisance-estimation enlarged radius):
(c) e A(L); v(Q6) (QQ) 56,((6)) -(Q) se}
Then have that
obj
sup (E.)(e.)
Var(vDR(O;*)) "DR(O;=*))
where we have shown that n" (e + 2n + 2x2
Following the result the argumentation in 12 Lemma 9] from here on out gives the result.
26

---

D Case Studies
D.1
Oregon Health Insurance Study
The Oregon Health Insurance Study [66] is an important study on the causal effect of expanding public health insurance on healthcare utilization, outcomes, and other outcomes. It based on randomized controlled trial made possible by resource limitations, which enabled the use of randomized lottery 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 individuals 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 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 as follows:
covariates (baseline household information, socioeconomic demographics, health infor- mation)
A race (non-white/white), gender (female/male)
These protected attributes were binarized.
R encouragement: lottery status of expanded eligibility (i.e. invitation to enroll when individual was previously incligible to enroll)
T: whether the individual is enrolled insurance ever
Note that for = this can be either Medicaid or private insurance while for R = this is still well-defined as this can be private insurance.
Y: number of doctor visits
This outcome used measure of healthcare utilization. Overall, the study found statistically significant effects on healthcare utilization. An implicit assumption is that increased healthcare utilization leads better health outcomes.
We subsetted the data include complete cases only (i.e. without missing covariates). We learned propensity and treatment propensity models via logistic regression for each group, and used gradient- boosted regression for the outcome model. We first include results for regression adjustment identifi- cation.
In Figure 3 we plot descriptive statistics. We include histograms of the treatment responsivity lifts Pijaa(r (I, a) Piina (x,a), a). We see some differences distributions of responsivity by gender and race. We then regress treatment responsivity more heterogeneity
the outcome- modell estimate of r. We find substantially treatment responsivity by race than by gender: whites are substantially more likely to take up insurance when made eligible, conditional heterogeneity in increase in healthcare utilization. (This is broadly consistent with health policy discussions regarding mistrust of the healthcare system).
the same expected treatment effect
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 B[T(TT) A a). for gender and race, respectively, enumerated over values of the constraint. We use costs of for the number of doctors visits and for enrollment in Medicaid (so [[T()) A = a] is on the scale of probability of
27

---

am gender
Pio Pru VS gender
V5.5.
25-
25
male
2.0
0.5
0.4
1,5
0.3
1.0
0.2
6.5
6.1
6.1
ronwhite
0.0
0.0
6.0
white
0.2
0.6
0.2
0.4
0.6
10
10
10
10
Figure 3: Distribution of lift in treatment probabilities Pitt,e
Pt 0,0 = P(T = 1 R = 1,4
x - PP = 1 R= 0, = a, X), and plot of P1j1,a
P1 0,a VS.
4.27
0.35
426
0.30
4.2
09
4.24
6.32
0.34
4.25
4.23
0.30
Figure Policy value ((T*), treatment value ETT(T>)) A a). for race, gender.
enrollment). These costs were chosen arbitrarily, Finding optimal policies that improve disparities group-conditional access can be done with relatively little impact the overall objective value. These cooup-oonditionll access disparities can be reduced from 4 percentage points (0.04) for gender and about 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/comprrmiis 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 À however, we can also see that small values can have relatively large improvements in access for blacks before these improvements saturate, and larger values lead smaller increases in access for blacks vs. larger decreases access for whites.
in access value
28


Reviews:
[{"review_id": "3IZbJyqyqp", "reviewer": "Reviewer_2wiP", "summary": "The authors characterize optimal and resource fairness-constrained optimal decision rules, and develop a doubly-robust estimator for the optimal decision rules.", "questions": "See **Weaknesses** section.", "limitations": "See **Weaknesses** section.", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 3, "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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for the review! See global rebuttal where we propose to reiterate the limitations we discuss throughout the paper (e.g. our discussion right after assumptions) in a new concluding paragraph at the end. \n\n\n> “In summary, we provide theoretical characterization of fair encouragement designs with a human-in-the-loop, algorithms, and empirical demonstration. On the sociotechnical side, the limitations of our work include that interpreting any additional constraints implemented in our framework as improving fairness will depend on the context. On the technical side, our methodology is especially tailored to Assumption 6, about extrapolating responsivity to algorithms from the training data to the deployment environment. Although we can develop robustness checks to the violation of Assumption 6, this means we are really operating in one regime of “human-AI” collaboration and this method is not necessarily appropriate in all settings. Empirical work in different applications is required to verify the appropriateness of this assumption. Interesting directions for future work include algorithms that can handle intermediate settings, or use limited online learning to assess validity of assumptions.”"}, {"review_id": "sHiLpiC3pP", "reviewer": "Reviewer_QSPP", "summary": "This 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", "questions": "In proposition 2, L looks complicated. Is it explainable?", "limitations": "I may raise my score if the questions/weaknesses are properly explained.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. 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.", "weaknesses": "1. 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\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your review! Below we make some clarifications, which we hope will help clear up that these are not weaknesses of the paper. We will add these explanations to clarify. \n\nWeaknesses\n\n- 1: See lines 40-46 which discuss scenarios where the fairness constraint should be on the expectation of treatment but not recommendation. Line 288 also quotes a report citing this motivation in the case study. \n\nAlgorithmic auditing on the recommendation does not actually guarantee that there will not be disparities in realized treatment decisions, e.g. who actually finally redeems an offer or signs up for health insurance or receives preventive/punitive services. In a perfect world where treatment decisions by human decisions were optimal, there would be no difference in auditing recommendations vs. treatment decisions. But they are not, and this can be a central source of disparities; see e.g. discussion of administrative burden/cognitive burden affecting marginalized populations the most, making it more difficult to sign up for beneficial services, or due to discriminatory behavior of human decision makers. Imposing fairness constraints on treatment is a stronger guarantee of actual reductions in disparities. \n\n- 2. We will add an extra citation to causal inference textbooks here [21] (Hernan and Robins’ Causal Inference) and Imbens and Rubin [1] for the standard assumptions 1,3,6. \n\n[1] Imbens, Guido W., and Donald B. Rubin. Causal inference in statistics, social, and biomedical sciences. Cambridge University Press, 2015.\n\nOverall we propose to add the following sentences to the main text to further explain standard causal inference assumptions (we also included this in the rebuttal to all authors): \n> In the standard Neyman-Rubin potential outcomes framework, individuals have a random vector of potential outcomes $Y(t)$ (indexed by treatment levels), but in observational data, under Assumption 1 (consistency and stable unit treatment values assumption) we only observe outcomes masked by the actual treatment assignment, aka $Y_i=Y(T_i)$, and under Assumption 3 (unconfoundedness) treatment is as-if randomized conditional on covariates. Assumption 3 is satisfied by design in randomized trials and otherwise is an assumption about the data-generating process, i.e. human decision-makers weren’t basing treatment decisions on unmeasured confounders unavailable in our dataset.  \n\nAlso please note that in lines 128-142 we spend a lot of time explaining the non-standard assumptions specific to our paper. \n\nFinally, Re: Assumption 5: note that this doesn’t impose any functional form restrictions on the conditional cost function given covariates, i.e. $\\mathbb{E}[c(Y)\\mid T=1,X]$ is unrestricted. This just means that we can’t directly handle counterfactual cost functions, e.g. we rule out cases like $c(r,t,y) = T * Y(1-T)$. This is because of the fundamental problem of causal inference that the joint distribution of $(Y(1),Y(0))$ is unidentified and we don’t know what the counterfactual distribution of $P(Y(1) | T=0,X)$ is. \n\nFor example, standard settings where firms get utility on the causal outcomes (i.e. revenue gained from a customer redeeming a 10 dollar off coupon and spending it) and some cost on the use of treatment (i.e. paying the 10 dollar discount) satisfy Assumption 5. \n\nSo, Assumption 5 is relatively mild; it says that we have separate cost functions for treatment outcomes and causal $Y$ outcomes. Ultimately A5 is not very restrictive; we can handle nonstandard classification-type constraints/disparities by first applying the identification argument as in prop. 7 of the appendix and treating this as a covariate-conditional treatment cost. \n\nThanks for your question on this; we’ll add this to the paper. \n\n3. Thanks; that’s a minor typo, will be updated to $\\nu_r, \\nu_t \\geq 0$. We use cost/utility interchangeably (difference is in sign). We list the assumptions together but will add which assumptions each prop. depends on. Prop 1,2,3 depend on A1-5. Prop 4 depends on A1-6. Prop 5,6 depend on A1-5 (no overlap). We will also clarify that all the propositions depend on A1-5 and what changes in different regimes is whether we assume overlap (A6) or not. \n\nQuestions: \n\n- \"L looks complicated... is it explainable?\" \n\nYes, it's explainable. Regression adjustment identification for causal inference (in Prop. 1) says that the optimal policy is to treat if $(\\mu_1(X)-\\mu_0(X)) (p_{1 \\mid 1}(X)-p_{1 \\mid 0}(X)) < 0$ (when outcomes are costs) if we didn't have any constraints. Think of this as a compliance-weighted CATE (conditional average-treatment effect), i.e. we weight $\\tau$ (CATE =$\\mu_1(X)-\\mu_0(X)$) by the compliance effect of recommendations $ (p_{1 \\mid 1}(X)-p_{1 \\mid 0}(X))$. This is the unconstrained case. If we do have constraints, then Lagrange duality says that the optimal solution is given by the optimal unconstrained solution with an additional $\\lambda$-multiplier on the constraint violations. The term $\\left(p_{1 \\mid 1}(X, A)-p_{1 \\mid 0}(X, A)\\right)\\frac{\\lambda}{p(A)}(\\mathbb{I}[A=a]-\\mathbb{I}[A=b])$ is just the integrand of the fairness constraint $\\mathbb{E}[T(\\pi) \\mid A=a]-\\mathbb{E}[T(\\pi) \\mid A=b]$ (by iterated expectations). So the $\\lambda$ penalizes constraint violations. The term multiplying it is the contribution of a datapoint to estimating the constraint violation. This characterization of optimization over $\\lambda$ is via Lagrange duality. \n\nWe'll add this to the paper, thanks! "}, {"review_id": "DAK6xlPasD", "reviewer": "Reviewer_Cr9c", "summary": "The 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\n\n", "questions": "There 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\n\n", "limitations": "A discussion or outlook section is missing. The authors do address limitations of some of their assumptions in lines 126", "rating": 3, "confidence": 3, "soundness": 2, "presentation": 1, "contribution": 1, "strengths": "The 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.", "weaknesses": "While 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\"", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We respond point-by-point below. Note many of these points are *already* in the paper. \n\n1. Adding section numbers to our narrative description is straightforward and we will do so.\n2. No, our background is split between the related work and problem setup. \nWe introduce policy value functions in lines 113-118. Re: assumptions, see response to Reviewer QSPP, weakness 2. We will go into greater description in the appendix, and include the following sentences about the standard DR estimator: \n> “For complete background we describe estimation improvements in standard causal inference in estimating the average treatment effect. The celebrated doubly-robust estimator for the ATE is [Robins and Rotnitzky 1995]: $$V_{DR}(\\pi) = \\sum_{t \\in \\mathcal{T}} E[ \\pi(t\\mid X)( \\mu_t(X) +  \\mathbb{I}[T=t] (Y-\\mu_t(X)) / e_t(X))  ] $$ Doubly robust estimators use both the outcome model and propensity score model and enjoy 1) two chances at model misspecification: only one of the propensity score and outcome model need to be well-specified and 2) rate double-robustness: we only require the product of the MSE convergence rates to achieve $n^{-\\frac 12}$ consistency. Finally, one last perspective on doubly-robust estimators is that they are control variate estimators. This last perspective is closest to the developments in our paper. Looking at the form of the doubly-robust estimator, it is the sum of a zero-mean term to the regression adjustment estimator. “\n3. Sec. 3 is a succinct formal problem setup. We will add, \n> “The cost function puts recommendations, treatments, and causal outcomes in the same “currency”, for example, a common cost basis to compare utility of, for example, the final purchase value if someone uses a 10 dollar off coupon vs. the cost of someone using treatment, e.g. the 10 dollar discount.”  \n4. Adding equation numbers is straightforward. But we *already* have assumption/proposition labels, e.g. line 112, line 173, 177, 247, 255, 292. \n5. No, we *already* have an introduction there in lines 146-153: we summarize that we study two regimes, causal identification and estimation. We will add a summary sentence.\n6. We discuss limitations throughout. See response to 2WiP. \n\nClarity: \n1. Algorithmic auditing looks at disparities in $P(R=1|A=1)-P(R=1|A=0)$ instead of $P(T=1|A=1)-P(T=1|A=0)$. This can be an issue whether it is the subject not complying with the treatment, or another decision-maker. We describe this in lines 46-48 (fairness constraints should be on realized treatments, not algorithmic recommendations) and lines 118-120 (the optimal decision rule can be different). \n2.  No, after most works mentioned, we exactly describe how our work is similar and different. See line 65, 66-68; where we introduce a related work, we contrast with “but (without/not), however, rather than, our focus is instead on”. Counterfactual risk assessment still assesses fairness of the high-risk/low-risk labels (recommendations), not the realized decisions. \n3. We provided multiple examples of utility of outcomes vs. benefits from treatment access alone in the introduction in 25-31, 36-49. \n4. The other assumptions are standard in causal inference, as we mentioned. See response to Structure 2). \n5. We already did this. Every proposition has a sentence right before it describing the result in words: “we discuss causal identification (Prop. 1), we characterize a threshold solution.. (Prop. 2)”. E.g. lines 163-164 say that we use Prop. 2 to obtain the the next proposition. We can make these transitions longer. \n6. Lines 261-287 and lines 293-300 (interpretation of fairness/treatment costs) describe the story (summarized for space constraints). We will add a few sentences:\n> Judges can choose to detain, (unconditionally) release, or release with supervision (supervised release) defendants when they are arrested before their trial. … lines 262 - lines 267 … Supervised release is an example of the second regime: recommendations are made with a human in the loop who makes the final decision, and we are concerned about disparities in outcomes and treatments.\n7. See line 285. $\\tau = \\mu_1(x) - \\mu_0(x)$. Will add. \n8. We use the terms “algorithmic recommendation, recommendation for treatment, encouragement/recommendation” to avoid confusion. \n\nCitations: \n1. Many papers audit fair binary classification in a consequential domain such as social services (with caseworkers) [1], healthcare [2], criminal justice [3] where decisions are not automated. \n[1] Chouldechova, Alexandra, et al. \"A case study of algorithm-assisted decision making in child maltreatment hotline screening decisions.\" FAT*, 2018.\n[2] Pfohl, Stephen R., Agata Foryciarz, and Nigam H. Shah. \"An empirical characterization of fair machine learning for clinical risk prediction.\" Journal of biomedical informatics 2021. \n[3] Stevenson, Megan T., and Jennifer L. Doleac. \"Algorithmic risk assessment in the hands of humans.\" (2022).\n2. No, [21] is a book: Hernán MA, Robins JM (2020). Causal Inference: What If. Boca Raton: Chapman & Hall/CRC. See 22.1.\nPages 12-15 of [20] discuss encouragement designs related to Appendix D.1. \n\nQuestions: \n1. No, we said what they are on line 112: recommendation ($e_r$), treatment propensity ($p_{t\\mid r}$) and outcome ($u_{t}$) models for causal identification and estimation. \n2. line 175 says these are from deterministic binary classifiers (as common in the fairness literature). \n3. Doubly robust estimators are standard important estimators in causal inference that reduce variance in estimation. See our response to your structure 2. \n4. Equation (1) has the main fairness constraint we use as an example in the main text. After line 218 we refer to more detail in appendix B.2.1 and B.2.2. We described what fairness constraints are relevant in lines 36-29. \n5. We said this in lines 272-275 but can add numbers. The ones relevant to treatment realizations are Assumptions 3 (unconfoundedness), 4 (responsivity) and 6 (overlap). \n"}, {"review_id": "buyiNBk1Dc", "reviewer": "Reviewer_h3Xk", "summary": "The 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.", "questions": "See weaknesses", "limitations": "See weakensses", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "\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", "weaknesses": "- 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", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": " Thanks for the summary and assessment of strengths! \n\nWe respond to questions/limitations/weaknesses. \n\n1. Baselines: \n\nGood question. Note that we actually plot the full curve of objective value and constraint reduction in Fig. 2. To your question about what happens if we wrongly assume a 100% compliance rate: optimizing over an unconstrained policy class, the policy decision is the same, assuming monotonicity (that recommending treatment only makes treatment more, not less, likely). Therefore, assuming 100% compliance is equivalent to an unconstrained policy (i.e. \\lambda = 0). Although we increase the objective function, we incur great disparities in treatment across groups, because we wrongly assumed 100% compliance. \n\nHowever, the optimal policy value could indeed be different when we optimize over a *constrained* policy class, even without any _fairness_ constraints. It depends on the magnitude of the “compliance weights” in addition to $\\tau=\\mu_1(x)-\\mu_0(x)$ and the exact functional form. For example, consider a simple two-dimensional case with linear decision rule. Suppose for $x_1 > 0, \\tau = -1, x_1<0, \\tau=0$ (good reduction in costs when $x_1>0$) but compliance is different: for $ \\{x_1>0, x_2 > 0\\}$, we have that $p_{1\\mid 1}-p_{1\\mid 0} = -\\epsilon$ but for the other subregion, $ \\{x_1>0 , x_2 < 0\\}$, we have that $p_{1\\mid 1}-p_{1\\mid 0} = 1$. Then the optimal encouragement-aware linear decision rule treats  $ \\{x_1>0 , x_2 < 0\\}$, but wrongly assuming 100% compliance treats $x_1>0$ which does worse! We will add this simple example to illustrate what can go wrong. \n2. Running-time analysis or discussion: \nGood point; we will add this discussion in the appendix. \nThe approach we suggest only has twice the computational complexity of the method of Agarwal et al. because our two-stage variance reduced approach essentially runs their method twice. That’s because policy learning can be reduced to cost-sensitive learning, which is the key computational oracle used in Agarwal et al. That also means that improvements in cost-sensitive learning, such as using stochastic gradient techniques can equivalently be used here and reduce computational complexity. Computationally, one iteration of the approach can be as difficult as solving a mixed-integer program (aka solvable in practice but without poly-time guarantees), or can be solved computationally efficiently with stochastic gradient descent on a surrogate-loss relaxation; both of these are common approaches in practice. \n"}, {"review_id": "o6xHOlOUFJ", "reviewer": "Reviewer_xiA7", "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.", "questions": "N/A", "limitations": "Yes.", "rating": 7, "confidence": 1, "soundness": 3, "presentation": 4, "contribution": 4, "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.\n\nThe 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. \n", "weaknesses": "I have not identified any core weaknesses in this paper.\n\nThere’s an unattributed quote block on line 33.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ""}]

Author rebuttal:
Thanks to all the reviewers for feedback and suggestions. We are encouraged that the reviewers find that we provide comprehensive theoretical/empirical results in tackling an important problem! 

We respond point-by-point below and remark on some common points here. We believe these very minor  few sentences will improve clarity along these points. Thanks to the reviewers for identifying opportunities to clarify further.

- 2wiP and Cr9c note we don’t have a separate limitations paragraph. We propose to add the following concluding paragraph (which reiterates main limitations we discussed in the text, but explicitly labels them as such): 
 > “*Conclusion and Limitations*: In summary, we provide theoretical characterization of fair encouragement designs with a human-in-the-loop, algorithms, and empirical demonstration. On the sociotechnical side, the limitations of our work include that interpreting any additional constraints implemented in our framework as improving fairness will depend on the context. On the technical side, our methodology is especially tailored to Assumption 6, about extrapolating responsivity to algorithms from the training data to the deployment environment. Although we can develop robustness checks to the violation of Assumption 6, this means we are really operating in one regime of “human-AI” collaboration and this method is not necessarily appropriate in all settings. Empirical work in different applications is required to verify the appropriateness of this assumption. Interesting directions for future work include algorithms that can handle intermediate settings, or use limited online learning to assess validity of assumptions.”

- Reviewers QSPP and Cr9c note that although we take care to carefully discuss non-standard assumptions specific to our paper, in the main text we don’t explain standard causal inference assumptions (consistency/SUTVA/unconfoundedness). This is a good point: assumptions are absolutely central to causal inference and they should be explained. As mentioned in specific reviewer response, we will add the following explanation of assumptions to the main text: 

> "In the standard Neyman-Rubin potential outcomes framework, individuals have a random vector of potential outcomes $Y(t)$ (indexed by treatment levels), but in observational data. Under Assumption 1 (consistency and stable unit treatment values assumption) we only observe outcomes masked by the actual treatment assignment, aka $Y_i=Y(T_i)$, and under Assumption 3 (unconfoundedness) treatment is as-if randomized conditional on covariates. Assumption 3 is satisfied by design in randomized trials and otherwise is an assumption about the data-generating process, i.e. human decision-makers weren’t basing treatment decisions on unmeasured confounders unavailable in our dataset."

This is somewhat boilerplate for causal inference in general, which is why it wasn’t in the main text before; so adding this extra explanation doesn’t change anything about the paper. 

Finally, Cr9c notes that there are opportunities to further include background material for readers unfamiliar with causal inference. Upon reflection, we recognize the writing is dense to fit our comprehensive results and we omit some background for readers unfamiliar with causal inference. But also in our response to Cr9c we note many areas where we have in fact included the information in question. We can certainly add additional background material on standard settings in the appendix, because we want to be as clear as possible to all audiences. Thanks to Cr9c for noting opportunities to improve clarity for audiences new to causal inference. On the other hand, multiple reviewers (h3Xk and xiA7) explicitly acknowledge the paper is well-written, and xiA7 even acknowledges the writing was already useful for someone new to causality. While we think adding additional emphases can help, we don’t at all think that the line edits that we included in our comprehensive response to Cr9c point to major technical flaws, and/or poor evaluation, limited impact, poor reproducibility and mostly unaddressed ethical considerations”. 

We thank reviewers for the questions and noting opportunities for minor clarifications to further improve clarity of the camera-ready paper and we are absolutely confident that the minor adjustments we have laid out will do so in a camera-ready.

Meta-review: The paper presents a fairness problem in algorithmic decision making, where different social groups may respond differently to the same (positive) decision: e.g., when given the same treatment recommendation, some groups are more likely to adhere to it than others, thereby creating different treatment outcomes, even though the decision made by the system/algorithm was the same. This leads to disparity in the actual benefits received by these groups.  The paper suggests a method to learn a decision policy that assigns recommendations to individuals so as to fulfill certain fairness criteria and reduce the disparity in the treatment realized.  

The proposed problem was viewed by all reviewers as an interesting problem setting worth studying.  Although there is some disagreement on whether the paper's clarity and quality of writing is ready for publication, on balance the reviews are more positive than negative.
Decision: Accept (poster)