SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We briefly go into more detail about formal differences, due to our specific assumptions, that delineate the differences to mediation analysis. Namely, our conditional exclusion restriction implies that  $Y_{1T_0} = Y_{T_0}$  and that  $Y_{0T_1} = Y_{1T_1}$  (in mediation notation with  $T_r = T(r)$  in our notation), so that so-called net direct effects are identically zero and the net indirect effect is the treatment effect (also called average encouragement effect here).
Human-in-the-loop in consequential domains. There is a great deal of interest in designing algorithms for the "human in the loop" and studying expertise and discretion in human oversight in consequential domains [14]. On the algorithmic side, recent work focuses on frameworks for learning to defer or human-algorithm collaboration. Our focus is prior to the design of these procedures for improved human-algorithm collaboration: we primarily hold fixed current human responsiveness to algorithmic recommendations. Therefore, our method can be helpful for optimizing local nudges. Incorporating these algorithmic design ideas would be interesting directions for future work.
Empirical literature on judicial discretion in the pretrial setting. Studying a slightly different substantive question, namely causal effects of pretrial decisions on later outcomes, a line of work uses individual judge decision-makers as a leniency instrumental variable for the treatment effect of (for example, EM) on pretrial outcomes [3, 2, 34]. And, judge IVs rely on quasi-random assignment of individual judges. We focus on the prescriptive question of optimal recommendation rules in view of patterns of judicial discretion, rather than the descriptive question of causal impacts of detention on downstream outcomes.
A number of works have emphasized the role of judicial discretion in pretrial risk assessments in particular [20, 15, 33]. In contrast to these works, we focus on studying decisions about electronic monitoring, which is an intermediate degree of decision lever to prevent FTA that nonetheless imposes costs. [23] study a randomized experiment of provision of the PSA and estimate (the sign of) principal causal effects, including potential group-conditional disparities. They are interested in a causal effect on the principal stratum of those marginal defendants who would not commit a new crime if recommended for detention. [9] study policy learning in the absence of positivity (since the PSA is a deterministic function of covariates) and consider a case study on determining optimal recommendation/detention decisions; however their observed outcomes are downstream of judicial decision-making. Relative to their approach, we handle lack of overlap via an exclusion restriction so that we only require ambiguity on treatment responsivity models rather than causal outcome models.
B Additional discussion on method
B.1 Additional discussion on constrained optimization
Feasibility program We can obtain upper/ lower bounds on  $\epsilon$  in order to obtain a feasible region for  $\epsilon$  by solving the below optimization over maximal/minimal values of the constraint:
$$
\bar {\epsilon}, \underline {{\epsilon}} \in \max  _ {\pi} / \min  _ {\pi} \mathbb {E} [ T (\pi) \mid A = a ] - \mathbb {E} [ T (\pi) \mid A = b ] \tag {3}
$$
$$
V _ {\epsilon} ^ {*} = \max  _ {\pi} \left\{\mathbb {E} [ c (\pi , T (\pi), Y (\pi)) ]: \mathbb {E} [ T (\pi) \mid A = a ] - \mathbb {E} [ T (\pi) \mid A = b ] \leq \epsilon \right\} \tag {4}
$$
B.2 Additional discussion on Algorithm 2 (general algorithm)
B.2.1 Additional fairness constraints and examples in this framework
In this section we discuss additional fairness constraints and how to formulate them in the generic framework. Much of this discussion is quite similar to [1] (including in notation) and is included in this appendix for completeness only. We only additionally provide novel identification results for another fairness measure on causal policies in Appendix B.2.2, concrete discussion of the reduction to weighted classification, and provide concrete descriptions of the causal fairness constraints in the more general framework.
We first discuss how to impose the treatment parity constraint. This is similar to the demographic parity example in [1], with different coefficients, but included for completeness. (Instead, recommendation parity in  $\mathbb{E}[\pi \mid A = a]$  is indeed nearly identical to demographic parity.)
Example 1 (Writing treatment parity in the general constrained classification framework.). We write the constraint
$$
\mathbb {E} [ T (\pi) \mid A = a ] - \mathbb {E} [ T (\pi) \mid A = b ] \tag {5}
$$
in this framework as follows:
$$
\mathbb {E} [ T (\pi) \mid A = a ] = \mathbb {E} [ \pi_ {1} (X) (p _ {1 | 1} (X, A) - p _ {1 | 0} (X, A)) + p _ {1 | 0} (X, A) \mid A = a ]
$$
For each  $u\in \mathcal{A}$  we enforce that
$$
\sum_ {r \in \{0, 1 \}} \mathbb {E} \left[ \pi_ {r} (X) p _ {1 | r} (X, A) \mid A = u \right] = \sum_ {r \in \{0, 1 \}} \mathbb {E} \left[ \pi_ {r} (X, A) p _ {1 | r} (X, A) \right]
$$
We can write this in the generic notation given previously by letting  $\mathcal{J} = \mathcal{A}\cup \{\circ \}$
$$
g _ {j} (O, \pi (X); \eta) = \pi_ {1} (X) \left(p _ {1 | 1} (X, A) - p _ {1 | 0} (X, A)\right) + p _ {1 | 0} (X, A), \forall j.
$$
We let the conditioning events  $\mathcal{E}_a = \{A = a\}$ ,  $\mathcal{E}_{\circ} = \{\mathrm{True}\}$ , i.e. conditioning on the latter is equivalent to evaluating the marginal expectation. Then we express Equation (5) as a set of equality constraints  $h_a(\pi) = h_{\circ}(\pi)$ , leading to pairs of inequality constraints,
$$
\left\{ \begin{array}{l} h _ {u} (\pi) - h _ {\circ} (\pi) \leq 0 \\ h _ {\circ} (\pi) - h _ {u} (\pi) \leq 0 \end{array} \right\} _ {u \in \mathcal {A}}
$$
The corresponding coefficients of  $M$  over this enumeration over groups  $(\mathcal{A})$  and epigraphical enforcement of equality  $\left(\{+, - \}\right)$  equation (1), gives  $\mathcal{K} = \mathcal{A}\times \{+, - \}$  so that  $M_{(a, + ),a^{\prime}} = \mathbf{1}\left\{a^{\prime} = a\right\}, M_{(a, + ),\star} = -1, M_{(a, - ),a^{\prime}} = -\mathbf{1}\left\{a^{\prime} = a\right\}, M_{(a, - ),\star} = 1,$  and  $\mathbf{d} = \mathbf{0}$ . Further we can relax equality to small amounts of constraint relaxation by instead setting  $d_k > 0$  for some (or all)  $k$ .
Next, we discuss a more complicated fairness measure. We first discuss identification and estimation before we also describe how to incorporate it in the generic framework.
B.2.2 Responder-dependent fairness measures
We consider a responder framework on outcomes (under our conditional exclusion restriction). Because the contribution to the AEE is indeed from the responder strata, this corresponds to additional estimation of the responder stratum.
We enumerate the four possible realizations of potential outcomes (given any fixed recommendation) as  $(Y(0(r)), Y(1(r)) \in \{0, 1\}^2$ . We call units with  $(Y(0(r)), Y(1(r))) = (0, 1)$  responders,  $(Y(0(r)), Y(1(r))) = (1, 0)$  anti-responders, and  $Y(0(r)) = Y(1(r))$  non-responders. Such a decomposition is general for the binary setting.
Assumption 8 (Binary outcomes, treatment).
$$
T, Y \in \{0, 1 \}
$$
Assumption 9 (Monotonicity).
$$
Y (T (1)) \geq Y (T (0))
$$
Importantly, the conditional exclusion restriction of Assumption 2 implies that responder status is independent of recommendation. Conditional on observables, whether a particular individual is a responder is independent of whether someone decides to treat them when recommended. In this way, we study responder status analogous to its use elsewhere in disparity assessment in algorithmic fairness [23, 28]. Importantly, this assumption implies that the conditioning event (of being a responder) is therefore independent of the policy  $\pi$ , so it can be handled in the same framework.
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).
$$
\mathbb {E} [ T (\pi) \mid Y (1 (R)) > Y (0 (R)), A = a ] - \mathbb {E} [ T (\pi) \mid Y (1 (R)) > Y (0 (R)), A = b ] \leq \epsilon
$$
We can obtain identification via regression adjustment:
Proposition 7 (Identification of treatment expenditure given responder status). Assume Assumptions 8 and 9.
$$
P (T (\pi) = 1 \mid A = a, Y (1 (\pi)) > Y (0 (\pi))) = \frac {\sum_ {r} \mathbb {E} [ \pi_ {r} (X) p _ {1 | r} (X , A) (\mu_ {1} (X , A) - \mu_ {0} (X , A)) \mid A = a ])}{\mathbb {E} [ (\mu_ {1} (X , A) - \mu_ {0} (X , A)) \mid A = a ]}
$$
Therefore this can be expressed in the general framework.
Example 3 (Writing treatment responder-conditional parity in the general constrained classification framework.). For each  $u \in \mathcal{A}$  we enforce that
$$
\frac {\sum_ {r} \mathbb {E} [ \pi_ {r} (X) p _ {1 | r} (X , A) (\mu_ {1} (X , A) - \mu_ {0} (X , A)) | A = u ])}{\mathbb {E} [ (\mu_ {1} (X , A) - \mu_ {0} (X , A)) | A = u ]} = \frac {\sum_ {r} \mathbb {E} [ \pi_ {r} (X) p _ {1 | r} (X , A) (\mu_ {1} (X , A) - \mu_ {0} (X , A)) ])}{\mathbb {E} [ (\mu_ {1} (X , A) - \mu_ {0} (X , A)) ]}
$$
We can write this in the generic notation given previously by letting  $\mathcal{I} = \mathcal{A}\cup \{\circ \}$
$$
g _ {j} (O, \pi (X); \eta) = \frac {\left\{\pi_ {1} (X) \left(p _ {1 | 1} (X , A) - p _ {1 | 0} (X , A)\right) + p _ {1 | 0} (X , A) \right\} \left(\mu_ {1} (X , A) - \mu_ {0} (X , A)\right)}{\mathbb {E} \left[ \left(\mu_ {1} (X , A) - \mu_ {0} (X , A)\right) \mid A = a \right]}, \forall j.
$$
Let  $\mathcal{E}_a^j = \{A = a_j\}$ ,  $\mathcal{E}_{\circ} = \{\text{True}\}$ , and we express Equation (5) as a set of equality constraints of the above moment  $h_a(\pi) = h_{\circ}(\pi)$ , leading to pairs of inequality constraints,
The corresponding coefficients of  $M$  proceed analogously as for treatment parity.
B.2.3 Best-response oracles
Best-responding classifier  $\pi$ , given  $\lambda$ :  $\mathrm{BEST}_{\pi}(\lambda)$  The best-response oracle, given a particular  $\lambda$  value, optimizes the Lagrangian given  $\pi$ :
$$
\begin{array}{l} L (\pi , \lambda) = \hat {V} (\pi) + \lambda^ {\top} (M \hat {h} (\pi) - \hat {d}) \\ = \hat {V} (\pi) - \lambda^ {\top} \hat {d} + \sum_ {k, j} \frac {M _ {k , j} \lambda_ {k}}{p _ {j}} \mathbb {E} _ {n} \left[ g _ {j} (O, \pi) 1 \left\{O \in \mathcal {E} _ {j} \right\} \right]. \\ \end{array}
$$
Best-responding Lagrange multiplier  $\lambda$ , given  $\pi$ :  $\mathrm{BEST}_{\lambda}(Q)$  is the best response of the  $\Lambda$  player. It can be chosen to be either 0 or put all the mass on the most violated constraint. Let  $\gamma(Q) := Mh(Q)$  denote the constraint values, then  $\mathrm{BEST}_{\lambda}(Q)$  returns
$$
\left\{ \begin{array}{l l} \mathbf {0} & \text {i f} \widehat {\gamma} (Q) \leq \widehat {\mathbf {c}} \\ B \mathbf {e} _ {k ^ {*}} & \text {o t h e r w i s e , w h e r e} k ^ {*} = \arg \max  _ {k} \left[ \widehat {\gamma} _ {k} (Q) - \widehat {c} _ {k} \right] \end{array} \right.
$$
B.2.4 Weighted classification reduction
There is a well-known reduction of optimizing the zero-one loss for policy learning to weighted classification. A cost-sensitive classification problem is
$$
\arg \min  _ {\pi_ {1}} \sum_ {i = 1} ^ {n} \pi_ {1} (X _ {i}) C _ {i} ^ {1} + (1 - \pi_ {1} (X _ {i})) C _ {i} ^ {0}
$$
The weighted classification error is  $\sum_{i=1}^{n} W_i 1\{h(X_i) \neq Y_i\}$  which is an equivalent formulation if  $W_i = |C_i^0 - C_i^1|$  and  $Y_i = 1\{C_i^0 \geq C_i^1\}$ .
The reduction to weighted classification is particularly helpful since taking the Lagrangian will introduce datapoint-dependent penalties that can be interpreted as additional weights. We can consider the centered regret  $J(\pi) = \mathbb{E}[Y(\pi)] - \frac{1}{2}\mathbb{E}[\mathbb{E}[Y\mid R = 1,X] + \mathbb{E}[Y\mid R = 0,X]]$ . Then
$$
J (\theta) = J (\mathrm {s g n} (g _ {\theta} (\cdot))) = \mathbb {E} [ \mathrm {s g n} (g _ {\theta} (X)) \{\psi \} ]
$$
where  $\psi$  can be one of, where  $\mu_r^R (X) = \mathbb{E}[Y\mid R = r,X]$
We can apply the standard reduction to cost-sensitive classification since  $\psi_i\operatorname{sgn}(g_\theta(X_i)) = |\psi_i|(1 - 2\mathbb{I}[\operatorname{sgn}(g_\theta(X_i)) \neq \operatorname{sgn}(\psi_i)])$ . Then we can use surrogate losses for the zero-one loss,
$$
L (\theta) = \mathbb {E} [ | \psi | \ell (g _ {\theta} (X), \operatorname {s g n} (\psi)) ]
$$
Although many functional forms for  $\ell(\cdot)$  are Fisher-consistent, the logistic (cross-entropy) loss will be particularly relevant:  $l(g,s) = 2\log(1 + \exp(g)) - (s + 1)g$ .
Example 4 (Treatment parity, continued (weighted classification reduction)). The cost-sensitive reduction for a vector of Lagrange multipliers can be deduced by applying the weighted classification reduction to the Lagrangian:
$$
L (\beta) = \mathbb {E} \left[ | \tilde {\psi} ^ {\lambda} | \ell \left(g _ {\beta} (X), \operatorname {s g n} (\tilde {\psi} ^ {\lambda})\right) \right], \qquad \text {w h e r e} \tilde {\psi} ^ {\lambda} = \psi + \frac {\lambda_ {A}}{p _ {A}} (p _ {1 | 1} - p _ {1 | 0}) - \sum_ {a \in \mathcal {A}} \lambda_ {a}.
$$
where  $p_a \coloneqq \hat{P}(A = a)$  and  $\lambda_a \coloneqq \lambda_{(a, +)} - \lambda_{(a, -)}$ , effectively replacing two non-negative Lagrange multipliers by a single multiplier, which can be either positive or negative.
Example 5 (Responder-conditional treatment parity, continued). The Lagrangian is  $L(\beta) = \mathbb{E}\left[|\tilde{\psi}^{\lambda}|\ell\left(g_{\beta}(X),\operatorname{sgn}(\tilde{\psi}^{\lambda})\right)\right]$  with weights:
$$
\tilde {\psi} ^ {\lambda} = \psi + \frac {\lambda_ {A}}{p _ {A}} \frac {(p _ {1 | 1} - p _ {1 | 0}) (\mu_ {1} - \mu_ {0})}{\mathbb {E} _ {n} [ (\mu_ {1} (X , A) - \mu_ {0} (X , A)) \mid A = a ]} - \sum_ {a \in \mathcal {A}} \lambda_ {a}.
$$
where  $p_a \coloneqq \hat{P}(A = a)$  and  $\lambda_a \coloneqq \lambda_{(a, +)} - \lambda_{(a, -)}$ .
Proof of Proposition 7.
$$
\begin{array}{l} P (T (\pi) = 1 \mid A = a, Y (1 (\pi)) > Y (0 (\pi))) \\ = \frac {P (T (\pi) = 1 , Y (1 (r)) > Y (0 (r)) \mid A = a)}{P (Y (1 (\pi)) > Y (0 (\pi)) \mid A = a)} \\ = \frac {P (T (\pi) = 1 , Y (1) > Y (0) \mid A = a)}{P (Y (1) > Y (0) \mid A = a)} \\ = \frac {\sum_ {r} \mathbb {E} [ \mathbb {E} [ \pi_ {r} (X) \mathbb {I} [ T (r) = 1 ] \mathbb {I} [ Y (1) > Y (0) ] \mid A = a , X ])}{P (Y (1) > Y (0) \mid A = a)} \\ \end{array}
$$
$$
\frac {\sum_ {r} \mathbb {E} [ \pi_ {r} (X) p _ {1 | r} (X , A) (\mu_ {1} (X , A) - \mu_ {0} (X , A)) \mid A = a ])}{\mathbb {E} [ (\mu_ {1} (X , A) - \mu_ {0} (X , A)) \mid A = a ]}
$$
by Bayes' rule
by Assumption 2
by iter.exp
by Proposition 1
C.1 Proofs for generalization under unconstrained policies
Proposition 8 (Policy value generalization). Assume the nuisance models  $\eta = [p_{1|0}, p_{1|1}, \mu_1, \mu_0, e_r(X)]^\top$ ,  $\eta \in H$  are consistent and well-specified with finite VC-dimension  $V_\eta$  over the product function class  $H$ . We provide a proof for the general case, including doubly-robust estimators, which applies to the statement of Proposition 8 by taking  $\eta = [p_{1|0}, p_{1|1}, \mu_1, \mu_0]$ .
Let  $\Pi = \{\mathbb{I}\{\mathbb{E}[L(\lambda, X, A; \eta) \mid X] > 0 \colon \lambda \in \mathbb{R}; \eta \in \mathcal{F}\}$ .
The generalization bound allows deducing risk bounds on the out-of-sample value:
$$
\mathbb {E} \left[ L (\hat {\lambda}, X, A) _ {+} \right] \leq \mathbb {E} \left[ L \left(\lambda^ {*}, X, A\right) _ {+} \right] + O _ {p} \left(n ^ {- \frac {1}{2}}\right)
$$
Proof of Proposition 8. We study a general Lagrangian, which takes as input pseudo-outcomes  $\psi^{t|r}(O;\eta),\psi^{y|t}(O;\eta),\psi^{1|0,\Delta A}$  where each satisfies that
$$
\mathbb {E} \left[ \psi^ {t | r} (O; \eta) \mid X, A \right] = p _ {1 | 1} (X, A) - p _ {1 | 0} (X, A)
$$
$$
\mathbb {E} [ \psi^ {y | t} (O; \eta) \mid X, A ] = \tau (X, A)
$$
$$
\mathbb {E} [ \psi^ {1 | 0, \Delta A} \mid X ] = p _ {1 | 0} (X, a) - p _ {1 | 0} (X, b)
$$
We make high-level stability assumptions on pseudoutcomes  $\psi$  relative to the nuisance functions  $\eta$  (these are satisfied by standard estimators that we will consider):
Assumption 10.  $\psi^{t|r},\psi^{y|t},\psi^{1|0,\Delta A}$  respectively are Lipschitz contractions with respect to  $\eta$  and bounded
We study a generalized Lagrangian of an optimization problem that took these pseudoutcome estimates as inputs:
$$
L (\lambda , X, A; \eta) = \psi_ {t | r} (O; \eta) \left\{\psi_ {y | t} (O; \eta) + \frac {\lambda}{p (A)} (\mathbb {I} [ A = a ] - \mathbb {I} [ A = b ]) \right\} + \lambda (\psi^ {1 | 0, \Delta A} (O; \eta))
$$
$$
\sup  _ {\pi \in \Pi , \lambda \in \mathbb {R}} | (\mathbb {E} _ {n} [ \pi L (\lambda , X, A) ] - \mathbb {E} [ \pi L (\lambda , X, A) ]) | = O _ {p} \left(n ^ {- \frac {1}{2}}\right)
$$
which, by applying the generalization bound twice gives that
$$
\mathbb {E} _ {n} [ \pi L (\lambda , X, A) ] = \mathbb {E} [ \pi L (\lambda , X, A) ]) + O _ {p} \left(n ^ {- \frac {1}{2}}\right)
$$
Write Lagrangian as
$$
\max  _ {\pi} \min  _ {\lambda} = \min  _ {\lambda} \max  _ {\pi} = \min  _ {\lambda} \mathbb {E} [ L (O, \lambda ; \eta) _ {+} ]
$$
Suppose the Rademacher complexity of  $\eta_{k}$  is given by  $\mathcal{R}(H_k)$ , so that [7, Thm. 12] gives that the Rademacher complexity of the product nuisance class  $H$  is therefore  $\sum_{k} \mathcal{R}(H_k)$ . The main result follows by applying vector-valued extensions of Lipschitz contraction of Rademacher complexity given in [36]. Suppose that  $\psi^{t|r}, \psi^{y|t}, \psi^{1|0,\Delta A}$  are Lipschitz with constants  $C_{t|r}^{L}, C_{y|t}^{L}, C_{1|0,\Delta A}^{L}$ .
We establish VC-properties of
$$
\mathcal {F} _ {L _ {1}} \left(O _ {1: n}\right) = \left\{\left(g _ {\eta} \left(O _ {1}\right), g _ {\eta} \left(O _ {i}\right), \dots g _ {\eta} \left(O _ {n}\right)\right): \eta \in H \right\}, \text {w h e r e} g _ {\eta} (O) = \psi_ {t | r} (O; \eta) \psi_ {y | t} (O; \eta)
$$
$$
\mathcal {F} _ {L _ {2}} \left(O _ {1: n}\right) = \left\{\left(h _ {\eta} \left(O _ {1}\right), h _ {\eta} \left(O _ {i}\right), \dots h _ {\eta} \left(O _ {n}\right)\right): \eta \in H \right\}, \text {w h e r e} h _ {\eta} (O) = \psi_ {t | r} (O; \eta) \frac {\lambda}{p (A)} \left(\mathbb {I} [ A = a ] - \mathbb {I} [ A = b ]\right)
$$
$$
\mathcal {F} _ {L _ {3}} \left(O _ {1: n}\right) = \left\{\left(m _ {\eta} \left(O _ {1}\right), m _ {\eta} \left(O _ {i}\right), \dots m _ {\eta} \left(O _ {n}\right)\right): \eta \in H \right\}, \text {w h e r e} m _ {\eta} (O) = \lambda \left(\psi^ {1 | 0, \Delta A} \left(O; \eta\right)\right)
$$
and the function class for the truncated Lagrangian,
$$
\mathcal {F} _ {L _ {+}} = \{\{(g _ {\eta} (O _ {i}) + h _ {\eta} (O _ {i}) + m _ {\eta} (O _ {i})) _ {+} \} _ {1: n} \colon g \in \mathcal {F} _ {L _ {1}} (O _ {1: n}), h \in \mathcal {F} _ {L _ {2}} (O _ {1: n}), m \in \mathcal {F} _ {L _ {3}} (O _ {1: n}), \eta \in H \}
$$
[36, Corollary 4] (and discussion of product function classes) gives the following: Let  $\mathcal{X}$  be any set,  $(x_{1},\ldots ,x_{n})\in \mathcal{X}^{n}$ , let  $F$  be a class of functions  $f:\mathcal{X}\to \ell_2$  and let  $h_i:\ell_2\to \mathbb{R}$  have Lipschitz norm  $L$ . Then
$$
\mathbb {E} \sup  _ {\eta \in H} \sum_ {i} \epsilon_ {i} \psi_ {i} (\eta (O _ {i})) \leq \sqrt {2} L \mathbb {E} \sup  _ {\eta \in H} \sum_ {i, k} \epsilon_ {i k} \eta (O _ {i}) \leq \sqrt {2} L \sum_ {k} \mathbb {E} \sup  _ {\eta_ {k} \in H _ {k}} \sum_ {i} \epsilon_ {i} \eta_ {k} (O _ {i}) \tag {6}
$$
where  $\epsilon_{ik}$  is an independent doubly indexed Rademacher sequence and  $f_{k}(x_{i})$  is the  $k$ -th component of  $f(x_{i})$ .
Applying Equation (6) to each of the component classes  $\mathcal{F}_{L_1}(O_{1:n}),\mathcal{F}_{L_2}(O_{1:n}),\mathcal{F}_{L_3}(O_{1:n})$  , and Lipschitz contraction [7, Thm. 12.4] of the positive part function  $\mathcal{F}_{L_+}$  , we obtain the bound
$$
\sup _ {\lambda , \eta} | \mathbb {E} _ {n} [ L (O, \lambda ; \eta) _ {+} ] - \mathbb {E} [ L (O, \lambda ; \eta) _ {+} ] | \leq \sqrt {2} (C _ {t | r} ^ {L} C _ {y | t} ^ {L} + C _ {t | r} ^ {L} B _ {p _ {a}} B + B C _ {1 | 0, \Delta A} ^ {L}) \sum_ {k} \mathcal {R} (H _ {k})
$$
Proposition 9 (Threshold solutions). Define
$$
\begin{array}{l} L (\lambda , X, A) = (p _ {1 | 1} (X, A) - p _ {1 | 0} (X, A)) \left\{\tau (X, A) + \frac {\lambda}{p (A)} (\mathbb {I} [ A = a ] - \mathbb {I} [ A = b ]) \right\} + \lambda (p _ {1 | 0} (X, a) - p _ {1 | 0} (X, b)) \\ \lambda^ {*} \in \arg \min _ {\lambda} \mathbb {E} [ L (\lambda , X, A) _ {+} ], \pi^ {*} (x, u) = \mathbb {I} \{L (\lambda^ {*}, X, u) > 0 \} \\ \end{array}
$$
$$
\lambda^ {*} \in \arg \min _ {\lambda} \mathbb {E} [ \mathbb {E} [ L (\lambda , X, A) \mid X ] _ {+} ], \pi^ {*} (x) = \mathbb {I} \{\mathbb {E} [ L (\lambda^ {*}, X, A) \mid X ] > 0 \}
$$
Proof of Proposition 9. The characterization follows by strong duality in infinite-dimensional linear programming [42]. Strict feasibility can be satisfied by, e.g. solving eq. (3) to set ranges for  $\epsilon$ .
C.2 Proofs for robust characterization
Proof of Proposition 5.
$$
\begin{array}{l} V (\pi) = \sum_ {t \in \mathcal {T}, r \in \{0, 1 \}} \mathbb {E} [ \pi_ {r} (X) \mathbb {E} [ c _ {r t} (Y (t)) ] [ T (r) = t ] \mid R = r, X ] ] \\ = \sum_ {t \in \mathcal {T}, r \in \{0, 1 \}} \mathbb {E} [ \pi_ {r} (X) \mathbb {E} [ c _ {r t} (Y (t)) \mid R = r, X ] P (T (r) = t \mid R = r, X) ] \qquad \text {u n c o n f .} \\ = \sum_ {t \in \mathcal {T}, r \in \{0, 1 \}} \mathbb {E} \left[ \pi_ {r} (X) \mathbb {E} \left[ c _ {r t} (Y (t)) \mid X \right] P (T (r) = t \mid R = r, X) \right] (7) \\ = \sum_ {t \in \mathcal {T}, r \in \{0, 1 \}} \mathbb {E} \left[ \pi_ {r} (X) \mathbb {E} \left[ c _ {r t} (Y (t)) \frac {\mathbb {I} [ T (r) = t ]}{p _ {t} (X)} \mid X \right] P (T (r) = t \mid R = r, X) \right] \quad \text {u n c o n f .} \\ = \sum_ {t \in \mathcal {T}, r \in \{0, 1 \}} \mathbb {E} \left[ \pi_ {r} (X) \left\{\mathbb {E} \left[ c _ {r t} (Y (t)) \frac {\mathbb {I} [ T (r) = t ]}{p _ {t} (X)} + \left(1 - \frac {T}{p _ {t} (X)}\right) \mu_ {t} (X) \mid X \right] p _ {t | r} (X) \right\} \right] \quad \text {c o n t r o l v a r i a t e} \\ = \sum_ {t \in \mathcal {T}, r \in \{0, 1 \}} \mathbb {E} \left[ \pi_ {r} (X) \left\{\left\{c _ {r t} (Y (t)) \frac {\mathbb {I} [ T (r) = t ]}{p _ {t} (X)} + \left(1 - \frac {T}{p _ {t} (X)}\right) \mu_ {t} (X) \right\} p _ {t | r} (X) \right\} \right] (LOTE) \\ \end{array}
$$
where  $p_t(X) = P(T = t \mid X)$  (marginally over  $R$  in the observational data) and (LOTE) is an abbreviation for the law of total expectation.
$$
\begin{array}{l} \overline {{V}} _ {n o} (\pi) := \max  _ {q _ {t r} (X) \in \mathcal {U}} \left\{\sum_ {t \in \mathcal {T}, r \in \{0, 1 \}} \mathbb {E} [ \pi_ {r} (X) \mu_ {t} (X) q _ {t r} (X) \mathbb {I} [ X \in \mathcal {X} ^ {\mathrm {n o}} ] ] ] \right\} \\ = \max  _ {q _ {t r} (X) \in \mathcal {U}} \left\{\sum_ {t \in \mathcal {T}, r \in \{0, 1 \}} \mathbb {E} [ \pi_ {r} (X) \mathbb {E} [ Y \mid T = t, X ] q _ {t r} (X) \mathbb {I} [ X \in \mathcal {X} ^ {\mathrm {n o}} ] ] ] \right\} \\ \end{array}
$$
Note the objective function can be reparametrized under a surjection of  $q_{t|r}(X)$  to its marginalization, i.e. marginal expectation over a  $\{T = t\}$  partition (equivalently  $\{T = t, A = a\}$  partition for a fairness-constrained setting).
$$
\beta_ {t | r} (a) := \mathbb {E} \left[ q _ {t | r} (X, A) \mid T = t, A = a \right], \beta_ {t | r} := \mathbb {E} \left[ q _ {t | r} (X, A) \mid T = t \right]
$$
Therefore we may reparametrize  $\overline{V}_{no}(\pi)$  as an optimization over constant coefficients (bounded by B):