SlowGuess/NeurIPS_2023_OCR huggingface dataset:
all  $u \in U$  s.t.  $(X, Z, W)(u) = (x, z, w)$ . The proof of the claim in Eq. 21 is given in Appendix B. Using this connection of canonical types and the notion of benefit, we can formulate a solution to the problem in Def. 1, given in Alg. 1. In particular, Alg. 1 takes as input the observational distribution  $P(V)$ , but its adaptation to inference from finite samples follows easily. In Step 2, we check whether we are operating under resource scarcity, and if not, the optimal policy simply treats everyone who stands to benefit from the treatment. Otherwise, we find the  $\delta_b > 0$  which uses the entire budget (Step 3) and separate the interior  $\mathcal{I}$  of those with the highest benefit (all of whom are treated), and the boundary  $\mathcal{B}$  (those who are to be randomized) in Step 4. The budget remaining to be spent on the boundary is  $b - P(\mathcal{I})$ , and thus individuals on the boundary are treated with probability  $\frac{b - P(\mathcal{I})}{P(\mathcal{B})}$ . Importantly, male and female groups are treated separately in this random selection process (Eq. 20), reflecting BF. The BF criterion can be seen as a minimal fairness requirement in decision-making, and is often aligned with maximizing utility, as seen from the following theorem:
Theorem 2 (Alg. 1 Optimality). Among all feasible policies  $D$  for the optimization problem in Eqs. 1-2, the result of Alg. 1 is optimal and satisfies benefit fairness.
A key extension we discuss next relates to the cases in which the benefit itself may be deemed as discriminatory towards a protected group.
3 Fairness of the Benefit
As discussed above, benefit fairness guarantees that at each fixed level of the benefit  $\Delta = \delta$ , the protected attribute plays no role in the treatment assignment. However, benefit fairness does not guarantee that treatment probability is equal between groups, i.e., that  $P(d \mid x_1) = P(d \mid x_0)$ . In fact, benefit fairness implies equal treatment allocation in cases where the distribution of the benefit  $\Delta$  is equal across groups (shown formally in Appendix F), which may not always be the case:
Example (Cancer Surgery - continued). After applying benefit fairness and implementing the optimal policy  $D^{*} = \mathbb{1}\big(W > \frac{1}{2}\big)$ , the clinicians compute that  $P(d\mid x_1) - P(d\mid x_0) = -50\%$ , that is, females are  $50\%$  less likely to be treated than males.
In our example benefit fairness results in a disparity in resource allocation. Whenever this is the case, it implies that the benefit  $\Delta$  differs between groups. In Alg. 2 we describe a formal procedure that helps the decision-maker to obtain a causal understanding of why that is, i.e., which underlying causal mechanisms (direct, indirect, spurious) lead to the difference in the benefit. In Appendix G we discuss in detail how the terms appearing in the decompositions in Eqs. 22 and 23 are defined. The key subtlety here is how to define counterfactuals with respect to the random variable  $\Delta$ . In the appendix we define this formally and show that the variable  $\Delta$  can be considered as an auxiliary variable in the structural causal model. Furthermore, we show that the notions of direct, indirect, and spurious effects are identifiable under the SFM in Fig. 1 and provide the identification expressions for them, allowing the data analyst to compute the decompositions in Alg. 2 in practice (see Appendix G). We next ground the idea behind Alg. 2 in our example:
(a) Density of benefit  $\Delta$  by group.
(b) Density of counterfactual benefit  $\Delta_C$ .
Figure 4: Elements of analytical tools from Alg. 2 in Ex. 3.
Example (Decomposing the disparity). Following Alg. 2, the clinicians first decompose the observed disparities into their direct, indirect, and spurious components:
$$
P \left(d \mid x _ {1}\right) - P \left(d \mid x _ {0}\right) = \underbrace {0 \%} _ {D E} + \underbrace {- 50 \%} _ {I E} + \underbrace {0 \%} _ {S E}, \tag{24}
$$
$$
\mathbb {E} (\Delta \mid x _ {1}) - \mathbb {E} (\Delta \mid x _ {0}) = \underbrace {0} _ {D E} + \underbrace {- \frac {1}{9}} _ {I E} + \underbrace {0} _ {S E}, \tag {25}
$$
showing that the difference between groups is entirely explained by the levels of illness severity, that is, male patients are on average more severely ill than female patients (see Fig. 4a). Direct and spurious effects, in this example, do not explain the difference in benefit between the groups.
Based on these findings, the clinicians realize that the main driver of the disparity in the benefit  $\Delta$  is the indirect effect. Thus, they decide to compute the distribution of the benefit  $P(\Delta_{x_1,W_{x_0}}\mid x_1)$  which corresponds to the distribution of the benefit had  $X$  been equal to  $x_0$  along the indirect effect. The comparison of this distribution, with the distribution  $P(\Delta \mid x_0)$  is shown in Fig. 4b, indicating that the two distributions are in fact equal.
In the above example, the difference between groups is driven by the indirect effect, although generally, the situation may be more complex, with a combination of effects driving the disparity. Still, the tools of Alg. 2 equip the reader for analyzing such more complex cases. The key takeaway here is that the first step in analyzing a disparity in treatment allocation is to obtain a causal understanding of why the benefit differs between groups. Based on this understanding, the decision-maker may decide that the benefit  $\Delta$  is unfair, which is what we discuss next.
3.1 Controlling the Gap
A causal approach. The first approach for controlling the gap in resource allocation takes a counterfactual perspective. We first define what it means for the benefit  $\Delta$  to be causally fair:
Definition 3 (Causal Benefit Fairness). Suppose  $\mathcal{C} = (C_0, C_1)$  describes a causal pathway from  $X$  to  $Y$  which is deemed unfair, with  $C_0, C_1$  representing possibly counterfactual interventions. The pair  $(Y, D)$  satisfies counterfactual benefit fairness (CBF) if
$$
\mathbb {E} \left(y _ {C _ {1}, d _ {1}} - y _ {C _ {1}, d _ {0}} \mid x, z, w\right) = \mathbb {E} \left(y _ {C _ {0}, d _ {1}} - y _ {C _ {0}, d _ {0}} \mid x, z, w\right) \forall x, z, w \tag {26}
$$
$$
P (d \mid \Delta , x _ {0}) = P (d \mid \Delta , x _ {1}). \tag {27}
$$
To account for discrimination along a specific causal pathway (after using Alg. 2), the decision-maker needs to compute an adjusted version of the benefit  $\Delta$ , such that the protected attribute has no effect along the intended causal pathway  $\mathcal{C}$ . For instance,  $\mathcal{C} = (\{x_0\}, \{x_1\})$  describes the total causal effect, whereas  $\mathcal{C} = (\{x_0\}, \{x_1, W_{x_0}\})$  describes the direct effect. In words, CBF requires that treatment benefit  $\Delta$  should not depend on the effect of  $X$  on  $Y$  along the causal pathway  $\mathcal{C}$ , and this condition is covariate-specific, i.e., holds for any choice of covariates  $x, z, w$ . Additionally, the decision policy  $D$  should satisfy BF, meaning that at each degree of benefit  $\Delta = \delta$ , the protected attribute plays no
role in deciding whether the individual is treated or not. This statement is for fixed value of  $\Delta = \delta$ , and possibly considers individuals with different values of  $x, z, w$ . CBF can be satisfied using Alg. 3. In Step 1, the factual benefit values  $\Delta$ , together with the adjusted, counterfactual benefit values  $\Delta_C$  (that satisfy Def. 3) are computed. Then,  $\delta_{CF}$  is chosen to match the budget  $b$ , and all patients with a counterfactual benefit above  $\delta_{CF}$  are treated<sup>5</sup>, as demonstrated in the following example:
Example (Cancer Surgery - Counterfactual Approach). The clinicians realize that the difference in illness severity comes from the fact that female patients are subject to regular screening tests, and are therefore diagnosed earlier. The clinicians want to compute the adjusted benefit, by computing the counterfactual values of the benefit  $\Delta_{x_1,W_{x_0}}(u)$  for all  $u$  such that  $X(u) = x_{1}$ . For the computation, they assume that the relative order of the illness severity for females in the counterfactual world would have stayed the same (which holds true in the underlying SCM). Therefore, they compute that
$$
\Delta_ {x _ {1}, W _ {x _ {0}}} (u) = \frac {1}{3} \sqrt {1 - (1 - W (u)) ^ {2}}, \tag {28}
$$
for each unit  $u$  with  $X(u) = x_{1}$ . After applying Alg. 1 with the counterfactual benefit values  $\Delta_C$ , the resulting policy  $D^{CF} = \mathbb{1}(\Delta_C > \frac{1}{4})$  has a resource allocation disparity of 0.
The above example illustrates the core of the causal counterfactual approach to discrimination removal. BF was not appropriate in itself, since the clinicians are aware that the benefit of the treatment depends on sex in a way they deemed unfair. Therefore, to solve the problem, they first remove the undesired effect from the benefit  $\Delta$ , by computing the counterfactual benefit  $\Delta_C$ . After this, they apply Alg. 3 with the counterfactual method (CF) to construct a fair decision policy. Notably, the causal approach to reducing the disparity relies on the counterfactual values of the benefit  $\Delta_C(u)$ , as opposed to the factual benefit values  $\Delta(u)$ . As it turns out, measuring (or removing) the covariate-specific direct effect of  $X$  on the benefit  $\Delta$  is relatively simple in general, while the indirect effect may be more challenging to handle:
Remark 3 (Covariate-Specific Direct Effect of  $X$  on Benefit  $\Delta$  is Computable). Under the assumptions of the SFM, the potential outcome  $\Delta_{x_1,W_{x_0}}(u)$  is identifiable for any unit  $u$  with  $X(u) = x_0$  for which the attributes  $Z(u) = z, W(u) = w$  are observed. However, the same is not true for the indirect effect. While the counterfactual distribution of the benefit when the indirect effect is manipulated, written  $P(\Delta_{x_0,W_{x_1}}|x_0)$ , is identifiable under the SFM, the covariate-level values  $\Delta_{x_0,W_{x_1}}(x_0,z,w)$  are not identifiable without further assumptions.
A utilitarian/factual approach. An alternative, utilitarian (or factual) approach to reduce the disparity in resource allocation uses the factual benefit  $\Delta(u)$ , instead of the counterfactual benefit  $\Delta_C(u)$  used in the causal approach. This approach is also described in Alg. 3, with the utilitarian (UT) method. Firstly, in Step 5, the counterfactual values  $\Delta_C$  are used to compute the disparity that would arise from the optimal policy in the hypothetical, counterfactual world:
$$
M := \left| P \left(\Delta_ {C} \geq \delta_ {C F} \mid x _ {1}\right) - P \left(\Delta_ {C} \geq \delta_ {C F} \mid x _ {0}\right) \right|. \tag {29}
$$
The idea then is to introduce different thresholds  $\delta^{(x_0)},\delta^{(x_1)}$  for  $x_0$  and  $x_{1}$  groups, such that they introduce a disparity of at most  $M$ . In Step 6 we check whether the optimal policy introduces a disparity bounded by  $M$ . If the disparity is larger than  $M$  by an  $\epsilon$ , in Step 7 we determine how much slack the disadvantaged group requires, by finding thresholds  $\delta^{(x_0)},\delta^{(x_1)}$  that either treat everyone in the disadvantaged group, or achieve a disparity bounded by  $M$ . The counterfactual (CF) approach focused on the counterfactual benefit values  $\Delta_C$  and used a single threshold. The utilitarian (UT) approach focuses on the factual benefit values  $\Delta$ , but uses different thresholds within groups. However, the utilitarian approach uses the counterfactual values  $\Delta_C$  to determine the maximum allowed disparity. Alternatively, this disparity can be pre-specified, as shown in the following example:
Example (Cancer Surgery - Utilitarian Approach). Due to regulatory purposes, clinicians decide that  $M = 20\%$  is the maximum allowed disparity that can be introduced by the new policy  $D$ . Using Alg. 3, they construct  $D^{UT}$  and find that for  $\delta^{(x_0)} = 0.21$ ,  $\delta^{(x_1)} = 0.12$ ,
$$
P \left(\Delta > \delta^ {\left(x _ {0}\right)} \mid x _ {0}\right) \approx 60 \%, P \left(\Delta > \delta^ {\left(x _ {1}\right)} \mid x _ {1}\right) \approx 40 \%, \tag{33}
$$
which yields  $P(d^{UT}) \approx 50\%$ , and  $P(d^{UT} \mid x_1) - P(d^{UT} \mid x_0) \approx 20\%$ , which is in line with the hospital resources and the maximum disparity allowed by the regulators.
In this section, for clarity of exposition we assume that distribution of the benefit admits a density, although the methods are easily adapted to the case when this does not hold.
Algorithm 3 Causal Discrimination Removal for Outcome Control
- Inputs: Distribution  $P\left( V\right)$  ,Budget  $b$  ,Intervention  $C$  ,Max. Disparity  $M$  ,Method  $\in  \{ {CF},{UT}\}$
1: Compute  $\Delta (x,z,w),\Delta_C(x,z,w)$  for all  $(x,z,w)$
2: If  $P(\Delta > 0) \leq b$ , set  $D = \mathbb{1}(\Delta(x, z, w) > 0)$  and  $\mathbf{RETURN}(D)$ .
3: Find  $\delta_{CF} > 0$  such that
$$
P \left(\Delta_ {C} \geq \delta_ {C F}\right) = b. \tag {30}
$$
4: If Method is CF, set  $D^{CF} = \mathbb{1}(\Delta_C(x,z,w) \geq \delta_{CF})$  and  $\mathbf{RETURN}(D^{CF})$ .
5: If  $M$  not pre-specified, compute the disparity
$$
M := P \left(\Delta_ {C} \geq \delta_ {C F} \mid x _ {1}\right) - P \left(\Delta_ {C} \geq \delta_ {C F} \mid x _ {0}\right). \tag {31}
$$
6: Find  $\delta_{UT}$  such that  $P(\Delta \geq \delta_{UT}) = b$ . If
$$
\left| P \left(\Delta \geq \delta_ {U T} \mid x _ {1}\right) - P \left(\Delta \geq \delta_ {U T} \mid x _ {0}\right) \right| \leq M,
$$
set  $D^{UT} = \mathbb{1}(\Delta (x,z,w)\geq \delta_{UT})$  and  $\mathbf{RETURN}(D^{UT})$
7: Otherwise, suppose w.l.o.g. that  $P(\Delta \geq \delta_b \mid x_1) - P(\Delta \geq \delta_b \mid x_0) = M + \epsilon$  for  $\epsilon > 0$ . Define  $l := \frac{P(x_1)}{P(x_0)}$ , and let  $\delta_{lb}^{(x_0)}$  be such that
$$
P (\Delta \geq \delta_ {l b} ^ {(x _ {0})} \mid x _ {0}) = P (\Delta \geq \delta_ {b} \mid x _ {0}) + \epsilon \frac {l}{1 + l}.
$$
Set  $\delta^{(x_0)} = \max (\delta_{lb}^{(x_0)},0)$ , and  $\delta^{(x_1)}$  s.t.  $P(\Delta \geq \delta^{(x_1)}\mid x_1) = \frac{b}{P(x_1)} -\frac{1}{l} P(\Delta \geq \delta^{(x_0)}\mid x_0)$ .
8: Construct and RETURN the policy  $D^{UT}$ :
$$
D ^ {U T} := \left\{ \begin{array}{l} 1 \text {f o r} (x _ {1}, z, w) \text {s . t .} \Delta (x _ {1}, z, w) \geq \delta^ {(x _ {1})}, \\ 1 \text {f o r} (x _ {0}, z, w) \text {s . t .} \Delta (x _ {0}, z, w) \geq \delta^ {(x _ {0})}, \\ 0 \text {o t h e r w i s e .} \end{array} \right. \tag {32}
$$
Finally, we describe the theoretical guarantees for the methods in Alg. 3 (proof given in Appendix C):
Theorem 4 (Alg. 3 Guarantees). The policy  $D^{CF}$  is optimal among all policies with a budget  $\leq b$  that in the counterfactual world described by intervention  $C$ . The policy  $D^{UT}$  is optimal among all policies with a budget  $\leq b$  that either introduce a bounded disparity in resource allocation  $|P(d \mid x_1) - P(d \mid x_0)| \leq M$  or treat everyone with a positive benefit in the disadvantaged group.
We remark that policies  $D^{CF}$  and  $D^{UT}$  do not necessarily treat the same individuals in general. In Appendix D, we discuss a formal condition called counterfactual crossing that ensures that  $D^{CF}$  and  $D^{UT}$  treat the same individuals, therefore explaining when the causal and utilitarian approaches are equivalent [29]. In Appendix E we provide an additional application of our outcome control framework to the problem of allocating respirators [5] in intensive care units (ICUs), using the MIMIC-IV dataset [17].
In this paper we developed causal tools for understanding fairness in the task of outcome control. We introduced the notion of benefit fairness (Def. 2), and developed a procedure for achieving it (Alg. 1). Further, we develop a procedure for determining which causal mechanisms (direct, indirect, spurious) explain the difference in benefit between groups (Alg. 2). Finally, we developed two approaches that allow the removal of discrimination from the decision process along undesired causal pathways (Alg. 3). The proposed framework was demonstrated through a hypothetical cancer surgery example (see vignette) and a real-world respirator allocation example (see Appendix E). We leave for future work the extensions of the methods to the setting of continuous decisions  $D$ , and the setting of performing decision-making under uncertainty or imperfect causal knowledge.
[39] J. L. Vincent, R. Moreno, J. Takala, S. Willatts, A. De Mendonça, H. Bruining, C. Reinhart, P. Suter, and L. G. Thijs. The sofa (sepsis-related organ failure assessment) score to describe organ dysfunction/failure: On behalf of the working group on sepsis-related problems of the european society of intensive care medicine (see contributors to the project in the appendix), 1996.
[40] D. B. White and B. Lo. A framework for rationing ventilators and critical care beds during the COVID-19 pandemic. Jama, 323(18):1773-1774, 2020.
[41] Y. Wu, L. Zhang, X. Wu, and H. Tong. Pc-fairness: A unified framework for measuring causality-based fairness. Advances in neural information processing systems, 32, 2019.
[42] H. Wunsch, A. D. Hill, N. Bosch, N. K. Adhikari, G. Rubenfeld, A. Walkey, B. L. Ferreyro, B. W. Tillmann, A. C. Amaral, D. C. Scales, et al. Comparison of 2 triage scoring guidelines for allocation of mechanical ventilators. JAMA network open, 3(12):e2029250-e2029250, 2020.
[43] J. Zhang and E. Bareinboim. Equality of opportunity in classification: A causal approach. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems 31, pages 3671-3681, Montreal, Canada, 2018. Curran Associates, Inc.
[44] J. Zhang and E. Bareinboim. Fairness in decision-making—the causal explanation formula. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.
Acknowledgements This research was supported in part by the NSF, ONR, AFOSR, DoE, Amazon, JP Morgan, and The Alfred P. Sloan Foundation.
Supplementary Material for Causal Fairness for Outcome Control
The source code for reproducing all the experiments can be found in our code repository. Furthermore, the vignette accompanying the main text can be found here.
A Principal Fairness
We start with the definition of principal fairness:
Definition 4 (Principal Fairness [14]). Let  $D$  be a decision that possibly affects the outcome  $Y$ . The pair  $(Y, D)$  is said to satisfy principal fairness if
$$
P \left(d \mid y _ {d _ {0}}, y _ {d _ {1}}, x _ {1}\right) = P \left(d \mid y _ {d _ {0}}, y _ {d _ {1}}, x _ {0}\right), \tag {34}
$$
for each principal stratum  $(y_{d_0}, y_{d_1})$ , which can also be written as  $D \perp X \mid Y_{d_0}, Y_{d_1}$ . Furthermore, define the principal fairness measure (PFM) as:
$$
P F M _ {x _ {0}, x _ {1}} \left(d \mid y _ {d _ {0}}, y _ {d _ {1}}\right) = P \left(d \mid y _ {d _ {0}}, y _ {d _ {1}}, x _ {1}\right) - P \left(d \mid y _ {d _ {0}}, y _ {d _ {1}}, x _ {0}\right). \tag {35}
$$
The above notion of principal fairness aims to capture the intuition described in oracle example in Sec. 2.1. However, unlike in the example, the definition needs to be evaluated under imperfect knowledge, when only the collected data is available<sup>6</sup>. An immediate cause for concern, in this context, is the joint appearance of the potential outcomes  $Y_{d_0}, Y_{d_1}$  in the definition of principal fairness. As is well-known in the literature, the joint distribution of the potential outcomes  $Y_{d_0}, Y_{d_1}$  is in general impossible to obtain, which leads to the lack of identifiability of the principal fairness criterion:
Proposition 5 (Principal Fairness is Not Identifiable). The Principal Fairness (PF) criterion from Eq. 34 is not identifiable from observational or experimental data.
The implication of the proposition is that principal fairness, in general, cannot be evaluated, even if an unlimited amount of data was available. One way to see why PF is not identifiable is the following construction. Consider an SCM consisting of two binary variables  $D, Y \in \{0,1\}$  and the simple graph  $D \to Y$ . Suppose that we observe  $P(d) = p_d$ , and  $P(y \mid d_1) = m_1$ ,  $P(y \mid d_0) = m_0$  for some constants  $p_d, m_1, m_0$  (additionally assume  $m_0 \leq m_1$  w.l.o.g.). It is easy to show that these three values determine all of the observational and interventional distributions of the SCM. However, notice that for any  $\lambda \in [0,1 - m_1]$  the SCM given by
$$
D \leftarrow U _ {D} \tag {36}
$$
$$
Y \leftarrow \mathbb {1} \left(U _ {Y} \in [ 0, m _ {0} - \lambda ]\right) + D \mathbb {1} \left(U _ {Y} \in [ m _ {0} - \lambda , m _ {1} ]\right) + \tag {37}
$$
$$
(1 - D) \mathbb {1} \left(U _ {Y} \in \left[ m _ {1}, m _ {1} + \lambda \right]\right),
$$
$$
U _ {Y} \sim \operatorname {U n i f} [ 0, 1 ], U _ {D} \sim \operatorname {B e r n o u l l i} \left(p _ {d}\right), \tag {38}
$$
satisfies  $P(d) = p_d, P(y \mid d_1) = m_1$ , and  $P(y \mid d_0) = m_0$ , but the joint distribution  $P(y_{d_0} = 0, y_{d_1} = 1) = m_1 - m_0 + \lambda$  depends on the  $\lambda$  parameter and is therefore non-identifiable.
A.1 Monotonicity Assumption
To remedy the problem of non-identifiability of principal fairness, [14] proposes the monotonicity assumption:
Definition 5 (Monotonicity). We say that an outcome  $Y$  satisfies monotonicity with respect to a decision  $D$  if
$$
Y _ {d _ {1}} (u) \geq Y _ {d _ {0}} (u). \tag {39}
$$
In words, monotonicity says that for every unit, the outcome with the positive decision  $(D = 1)$  would not be worse than with the negative decision  $(D = 0)$ . We now demonstrate how monotonicity aids the identifiability of principal fairness.
As implied by the definition of the SCM, we almost never have access to the unobserved sources of variation  $(u)$  that determine the identity of each unit.
Proposition 6. Under the monotonicity assumption (Eq. 39), the principal fairness criterion is identifiable under the Standard Fairness Model (SFM).
Proof. The main challenge in PF is to obtain the joint distribution  $P(y_{d_0}, y_{d_1})$ , which is non-identifiable in general. Under monotonicity, however, we have that
$$
Y _ {d _ {0}} (u) = 0 \wedge Y _ {d _ {1}} (u) = 0 \Longleftrightarrow Y _ {d _ {1}} (u) = 0, \tag {40}
$$
$$
Y _ {d _ {0}} (u) = 1 \wedge Y _ {d _ {1}} (u) = 1 \Longleftrightarrow Y _ {d _ {0}} (u) = 1. \tag {41}
$$
Therefore, it follows from monotonicity that
$$
P \left(y _ {d _ {0}} = 1, y _ {d _ {1}} = 0\right) = 0, \tag {42}
$$
$$
P \left(y _ {d _ {0}} = 0, y _ {d _ {1}} = 0\right) = P \left(y _ {d _ {1}} = 0\right), \tag {43}
$$
$$
P \left(y _ {d _ {0}} = 1, y _ {d _ {1}} = 1\right) = P \left(y _ {d _ {0}} = 1\right), \tag {44}
$$
$$
P \left(y _ {d _ {0}} = 0, y _ {d _ {1}} = 1\right) = 1 - P \left(y _ {d _ {1}} = 0\right) - P \left(y _ {d _ {0}} = 1\right), \tag {45}
$$
thereby identifying the joint distribution whenever the interventional distributions  $P(y_{d_0}), P(y_{d_1})$  are identifiable.
In the cancer surgery example, the monotonicity assumption would require that the patients have strictly better survival outcomes when the surgery is performed, compared to when it is not. Given the known risks of surgical procedures, the assumption may be rightfully challenged in such a setting. In the sequel, we argue that the assumption of monotonicity is not really necessary, and often does not help the decision-maker, even if it holds true. To fix this issue, in the main text we discuss a relaxation of the PF criterion that suffers from neither of the above two problems but still captures the essential intuition that motivated PF.
B Canonical Types & Bounds
Definition 6 (Canonical Types for Decision-Making). Let  $Y$  be the outcome of interest, and  $D$  a binary decision. We then consider four canonical types of units: