Identifying the desert decision rule to assess and achieve fairness

Identifying the desert decision rule to assess and achiev e fairness Ping Zhang, Naiwen Y ing and W ang Miao Depar tment of Probability and Statistics, P eking Univ ersity , 5 Summer P alace Road, Haidian District, Beijing 100871, China Summary . We study fairness in decision-making when the data may encode systematic bias. Existing approaches typically impose f airness constraints while predicting the observed decision, which may itself be unf air . W e propose a nov el frame work for char acterising and addressing fairness issues by intro- ducing the notion of desert decision, a latent variable representing the decision an individual rightfully deser v es based on their actions, eff or ts, or abilities. This formulation shifts the prediction target from the potentially biased obser v ed decision to the deser t decision. W e advocate achie ving fair decision- making by predicting the deser t decision and assessing unfairness by the discrepancy between deser t and obser v ed decisions. W e establish nonparametric identiﬁcation results under causally inter pretab le assumptions on the f air ness of the deser t decision and the unfairness mechanism of the obser v ed de- cision. For estimation, we dev elop a sie ve maximum likelihood estimator for the deser t decision rule and an inﬂuence-function-based estimator for the degree of unf air ness . Sensitivity analysis procedures are fur ther proposed to assess robustness to violations of identifying assumptions. Our frame work con- nects fairness with measurement error models, aligning predictive accuracy with fairness relative to an appropriate target, and providing a structural approach to modelling the unfairness mechanism. Ke ywor ds : Deser t decision; F airness; Identiﬁcation; Inﬂuence function; Measurement error 1. Introduction Statistical and machine learning methods have achiev ed substantial success in recent years and are no w widely deployed in domains including hiring (Pessach et al., 2020), lending (Thomas et al., 2017), medicine (Shehab et al., 2022), education (Luan and Tsai, 2021), and criminal justice (Brennan et al., 2009). Decisions that once relied primarily on human judgement are increasingly supported by predicti ve models. Alongside these advances, concerns regarding fairness hav e become more promi- nent (e.g., Nature Editorials, 2016; Kleinberg et al., 2018; V anderford, 2022). Unfairness often arises as systematic dif ferences in treatment or outcomes across demographic groups. A widely cited exam- ple is a risk assessment tool used in the U.S. criminal justice system, which has been reported to assign higher predicted recidivism risk to Black defendants than to White defendants (Angwin et al., 2022). Such disparities can shape individuals’ life chances and entrench existing social inequities. Ensuring fairness should therefore be treated not as a peripheral adjustment, but as a central requirement for statistical and machine learning methods that are deployed in socially consequential settings. Despite widespread concerns regarding algorithmic f airness, most statistical and machine learning methods are, in themselves, neutral and are not designed to discriminate intentionally against partic- ular individuals or groups. Rather, unfairness commonly originates in the training data: recorded de- cisions may reﬂect historical biases, including discrimination against minority groups. F or example, in the labour market, Black applicants may face greater barriers to employment than White applicants e ven when they hav e comparable skills or qualiﬁcations (Bertrand and Mullainathan, 2004). When a model is trained to predict the observed decision, it may learn and propagate the bias, thereby pro- ducing unfair or unwarranted predictions. Consequently , in settings where unfairness is likely to be present, a fair decision rule should not merely replicate the observed decision as a predictiv e target. Among philosophical accounts of fairness, one inﬂuential vie w holds that fairness is achiev ed when each individual receiv es the decision they rightfully deserve, giv en their actions, ef forts, or abilities; see the literature on desert (Feinberg, 1970; Sadurski, 1985; Miller, 2001). Under this perspectiv e, the appropriate target is the desert decision rather than the observ ed decision, with the latter acting as a potentially biased proxy for the former . The e xtensi ve literature on algorithmic fairness has produced a wide range of criteria and methods for improving fairness in decision-making, suited to different application settings; see Mehrabi et al. (2021); Mitchell et al. (2021); Barocas et al. (2023); He and Li (2025) for comprehensiv e revie ws. De- spite their differences, most approaches share a common objectiv e: to predict the observed decision while enforcing a pre-speciﬁed fairness criterion. Enforcing fairness constraints can reduce dispari- ties in predicted decisions, but the target remains the observed decision, which may itself be unfair . Consequently , ef forts to improve fairness often entail a reduction in predictiv e accuracy , yielding the well-recognised utility–fairness trade-off (Kleinberg et al., 2016; Zhao and Gordon, 2022). More- ov er , without an explicit characterisation of the mechanism generating unfairness, imposing fairness constraints while predicting the observ ed decision may produce a form of “balance” that lacks a clear causal interpretation and substantiv e justiﬁcation in practice. A more principled alternativ e is to shift the target from the observed decision to the desert decision, which is intended to be fair . Under this shift, fairness is preserved by construction while predictive accuracy is pursued with respect to an appropriate target, thereby reducing the need to trade utility ag ainst fairness. In this paper , we propose a novel framework for assessing and achieving fairness centred on the desert decision. The paper has three main contributions: 2 • Conceptual contribution. W e introduce the desert decision as the prediction target and deﬁne fairness at the le vel of the desert decision. This perspectiv e shifts the central question from “how to predict the observed decision while enforcing a fairness constraint” to “how to recover and predict the desert decision”, so that fairness and predicti ve accuracy are aligned by construction. In this frame work, unfairness is quantiﬁed by the discrepancy between the desert and observed decisions rather than by constraints imposed on the observed decision or model output. • Identiﬁcation and estimation methods. Because the desert decision is unobserved, identiﬁcation is challenging. W e establish identiﬁcation of the desert decision rule and the degree of unfair - ness, under transparent and causally interpretable assumptions on (i) the fairness property of the desert decision, (ii) the mechanism through which unfairness distorts the observ ed decision, and (iii) an auxiliary variable that is informativ e about the desert decision but does not af fect the unfairness mechanism. W e then dev elop a siev e maximum likelihood estimator for the desert decision rule and an inﬂuence-function-based estimator for the degree of unfairness, together with sensiti vity analysis procedures for assumption violations. • Unfairness mechanism. W e explicitly characterise and estimate the mechanism of unfairness, providing a structural interpretation of discrimination and preferential treatment. In contrast, although the mechanism of unfairness is sometimes discussed in speciﬁc examples (Barocas et al., 2023), much of the algorithmic fairness literature does not formally deﬁne, identify , or estimate an explicit unf airness-generating mechanism. Besides, this work also contributes new identiﬁcation and estimation methods to the measurement error literature. The remainder of the paper is org anised as follo ws. In Section 2, we re view existing f airness crite- ria and methods, introduce the notion of the desert decision, and deﬁne parameters for assessing the degree of unfairness and for achie ving fair decision-making. In Section 3, we describe and illustrate assumptions on the mechanism of unfairness and establish identiﬁcation results. Estimation methods for the parameters of interest are dev eloped in Section 4. In Section 5, we discuss extensions that accommodate violations of the identifying assumptions and propose sensitivity analysis procedures. For illustration, we conduct numerical simulations in Section 6 and apply the proposed methods to a 3 real-data example (Bertrand and Mullainathan, 2004) in Section 7. Section 8 concludes and outlines directions for future work. 2. F airness and the desert decision framework 2.1. Existing fairness criteria and methods Throughout the paper , let Y denote the observed decision, with Y = 1 indicating the fav ourable decision (e.g., hiring, loan appro val) and Y = 0 the unfa vourable decision. Let S denote the sensitiv e attribute, with S = 1 for the advantaged group and S = 0 for the disadvantaged group. Here, the adv antaged group refers to individuals who are historically privile ged (e.g., men, ethnic majority groups), whereas the disadvantaged group refers to those who typically face discrimination (e.g., women, ethnic minority groups). Let V denote a vector of nonsensitiv e cov ariates that are rele vant for decision-making. W e use f to denote a generic probability mass or density function. The observed decision Y may be subject to unf airness with respect to the sensiti ve attrib ute S . For example, Y may depend on S (i.e., Y ⊥  ⊥ S ), or it may remain associated with S e ven after conditioning on e xplanatory characteristics (i.e., Y ⊥  ⊥ S | V ). W e brieﬂy re view representative fairness criteria and methods in the algorithmic fairness liter- ature. Let d ( S, V ) denote a decision rule or score taking values in [0 , 1] . The predicted decision induced by d ( S, V ) is ˆ Y = I { d ( S, V ) > t } , where I ( · ) denotes the indicator function and t is a gi ven threshold. T able 1 summarises widely used fairness criteria expressed as statistical indepen- dence conditions. Statistical parity and conditional statistical parity require independence between the predicted decision and the sensiti ve attribute (Dwork et al., 2012; Kamiran and ˇ Zliobait ˙ e, 2013). Equalised odds and equal opportunity require the predicted decision to be independent of the sen- siti ve attribute conditional on the observed decision (Hardt et al., 2016). Calibration and predictiv e parity require independence between the observ ed decision and the sensiti ve attrib ute gi ven the score or predicted decision (Chouldechova, 2017). In addition to criteria based on independence, causal notions of fairness ha ve been proposed, including path-speciﬁc fairness (Zhang et al., 2017; Nabi and Shpitser, 2018), counterfactual fairness (Kusner et al., 2017), and principal fairness (Imai and Jiang, 2023), each corresponding to the absence of certain causal ef fects. These criteria provide useful perspecti ves on dif ferent aspects of fairness, tailored to different application settings. 4 T able 1: Deﬁnitions of fairness criteria Criterion Deﬁnition Statistical Parity (Dw ork et al., 2012) ˆ Y ⊥ ⊥ S Conditional Statistical Parity (Kamiran and ˇ Zliobait ˙ e, 2013) ˆ Y ⊥ ⊥ S | V Equalised Odds (Hardt et al., 2016) ˆ Y ⊥ ⊥ S | Y Equal Opportunity (Hardt et al., 2016) ˆ Y ⊥ ⊥ S | Y = 1 Calibration (Chouldechov a, 2017) Y ⊥ ⊥ S | d ( S, V ) Predicti ve P arity (Chouldechov a, 2017) Y ⊥ ⊥ S | ˆ Y = 1 Representati ve fairness-enhancing methods include: adjusting the training data to mitigate bias (Kamiran and Calders, 2012; Calmon et al., 2017; Chen et al., 2024); incorporating fairness require- ments into model training via regularisations or constraints (Donini et al., 2018; Zafar et al., 2019); and modifying the outputs of trained models by group-speciﬁc decision thresholds for sensitiv e at- tributes (Hardt et al., 2016; Fan et al., 2023). Despite differences in implementation, these approaches typically share the objecti ve of predicting the observ ed decision Y while enforcing fairness. This ob- jecti ve can be formulat ed as the follo wing constrained statistical learning problem (Nabi et al., 2024): arg min d ∈D : C ( d )=0 R ( d ) , where R ( d ) is a risk function for predicting Y , C ( d ) = 0 is the constraint induced by a chosen fairness criterion, and D denotes the set of measurable functions of ( S , V ) taking v alues in [0 , 1] . For example, choosing the constraint C ( d ) = E { d (1 , V ) − d (0 , V ) } , which enforces a zero av erage causal ef fect of S on Y , Nabi et al. (2024) deriv ed the best predictors of Y under various risk functions. See the supplementary material for further details. Decision rules obtained from such constrained learning problems can reduce disparities in decisions. Ho wev er , when unfairness is present, predicting the observed decision Y may be an inappropriate target; the goal should instead be to recover the desert decision, rather than the potentially unfair decision Y itself. 2.2. Desert decision and parameters of interest W e introduce the desert decision Y ∗ , a latent v ariable representing the deserved outcome that accords with an indi vidual’ s actions, efforts, or abilities, drawing on the notion of “desert” in philosophy (e.g., Feinberg, 1970; Sadurski, 1985; Miller, 2001). In the algorithmic fairness literature, the concept of desert has been discussed as a general fairness principle (Barocas et al., 2023); in this paper , we 5 deﬁne the desert decision as an indi vidual-le vel v ariable. Unlike the observ ed decision Y , which may be distorted by unfairness, the desert decision Y ∗ corresponds to the decision that would have been made in the absence of unfairness, that is, the appropriate decision in an idealised setting in which indi viduals recei ve what they rightfully deserve. Ho wev er , the desert decision is not realised in the observed world, or at least cannot be veriﬁed as such. T o further illustrate the concept of the desert decision, we pro vide an example based on bank loan approv al using credit scoring (Hand and Henley, 1997; Thomas et al., 2017). Example 1. F igur e 1 illustrates a loan appr oval pr ocedur e wher eby a bank collects applicant charac- teristics, computes a scor e based on a pr edeﬁned rule, and appr oves the loan when the scor e exceeds a thr eshold. F ormally , the desert decision is char acterised by the following gener ative model: U ∗ = g ( V , ε ∗ ) , Y ∗ = I ( U ∗ > 0) , wher e the function g stands for a pr e-speciﬁed rule, ε ∗ is an independent err or term, and U ∗ r epr esents the scor e minus the thr eshold of loan appr oval. This pr ocedur e can also be interpr eted thr ough discr ete choice theory (McF adden, 2001; T rain, 2009), wher e U ∗ r epr esents the utility of appr oving the loan for an applicant with nonsensitive c haracteristics V , suc h as annual income, debt-to-income ratio, and the number of past over due payments. Importantly , the utility for the desert decision only depends on nonsensitive char acteristics that ar e associated with default risk; and legislation explicitly pr events the use of sensitive attribute S such as gender or race (Hand and Henle y, 1997). Applicant characteristics ( S, V ) Compute score or utility U ∗ Desert decision Y ∗ Figure 1: Desert decision in a standard loan approv al procedure. If the decision rule deﬁning the desert decision were implemented faithfully in practice, the deci- sion recorded in the data would be independent of S gi ven V . Ho wev er , in practice the loan approv al process may be contaminated by unfairness: the realised decision may be inﬂuenced by the sensitiv e attribute S , and hence may deviate from the desert decision. The desert decision is intended to be both fair and substanti vely justiﬁed; we encode its fairness property through the follo wing assumption. Assumption 1. Y ∗ ⊥ ⊥ S | V . Assumption 1 requires the desert decision Y ∗ to be conditionally independent of S giv en V . Its plausibility is supported by a le gal characterisation of employment discrimination (7th Circuit Court, 6 1996): “The central question in any employment-discrimination case is whether the employer would hav e taken the same action had the employee been of a different race (age, sex, religion, national origin, etc.) and ev erything else had been the same”. In the bank loan Example 1, the desert decision does not depend on the sensiti ve attribute, so Assumption 1 holds. W e emphasise that this assumption is a basic requirement b ut not suf ﬁcient to characterise Y ∗ ; for example, a coin-ﬂip decision also satisﬁes the conditional independence yet cannot represent what indi viduals rightfully deserve. T o achieve fair decision-making, we advocate basing decisions on predictions of the desert deci- sion Y ∗ . Speciﬁcally , we consider the decision rule τ ( V ) = f ( Y ∗ = 1 | V ) , the conditional probability of the fav ourable desert decision gi ven the predictiv e cov ariates. The desert decision rule τ ( V ) has the follo wing property . Proposition 1. Under Assumption 1, (i) τ ( V ) is the best pr edictor of the desert decision under the cr oss-entr opy risk: τ ( V ) = arg min d ∈D E [ − Y ∗ log d ( S, V ) − (1 − Y ∗ ) log { 1 − d ( S , V ) } ]; (ii) τ ( V ) satisﬁes calibration for pr edicting the desert decision: Y ∗ ⊥ ⊥ S | τ ( V ) . Although Proposition 1 is immediate, its implications are substanti ve. First, τ ( V ) attains the min- imal cross-entropy risk for predicting Y ∗ among all decision rules based on ( S, V ) . More generally , τ ( V ) is optimal for predicting Y ∗ under a broad class of common loss functions (e.g., squared error). Second, τ ( V ) satisﬁes calibration for predicting the desert decision: conditional on τ ( V ) , the distribu- tion of Y ∗ does not depend on the sensiti ve attribute, so indi viduals with the same score τ ( V ) hav e the same probability of the fa vourable desert decision regardless of group membership (Chouldechov a, 2017). Third, the predicted decision based on τ ( V ) satisﬁes the f airness requirement by construction, because it does not depend on S . T aken together , using τ ( V ) allows us to pursue predictiv e accuracy for Y ∗ while preserving fairness with respect to S , thereby av oiding the con ventional utility–fairness trade-of f that arises when predicting Y under fairness constraints. T o quantify the degree of unfairness in the observed decision, we consider the probability that the 7 observed decision dif fers from the desert decision, that is, θ = f ( Y  = Y ∗ ) . When θ > 0 , the observed decision is subject to unfairness; larger values of θ indicate more sev ere unfairness. If Y ∗ = 1 but Y = 0 , individuals are denied the fav ourable decision they deserve, which can be interpreted as discrimination; con versely , if Y ∗ = 0 but Y = 1 , indi viduals recei ve more fa vourable outcomes than they deserve, suggesting preferential treatment. Thus, θ summarises the ov erall degree of unfairness in the observed decision. One may also consider more speciﬁc param- eters, such as f ( Y = 0 | Y ∗ = 1) to quantify discrimination and f ( Y = 1 | Y ∗ = 0) to quantify preferential treatment. The key distinction between our proposal and existing frameworks is the explicit introduction of the desert decision. Our approach tar gets the prediction of Y ∗ and uses Y ∗ to characterise the mechanism of unfairness. When unfairness is present, Y ∗ and Y are distinct variables. Condi- tional statistical parity is deﬁned for the observed decision Y or the predicted decision ˆ Y and can be assessed empirically , whereas Assumption 1 concerns the unobserved Y ∗ . Like wise, existing measures of unfairness typically rely on observed, predicted, or counterfactual decisions without reference to the desert decision, whereas our measures are grounded in Y ∗ and directly ev aluate discrepancies between the observed and desert decisions. For example, the demographic disparity f ( Y = 1 | S = 1) − f ( Y = 1 | S = 0) captures dif ferences in the observed decision across sensiti ve-attrib ute groups, but it does not re veal the extent to which the observed decision departs from what individuals rightfully deserve. Moreo ver , predicting Y subject to imposed fairness con- straints may yield an enforced balance rather than the decision that indi viduals rightfully deserv e; the supplementary material provides an e xample comparing τ ( V ) with representati ve methods that target Y , illustrating that a single decision rule generally cannot deliv er strong predictiv e performance for both Y ∗ and Y simultaneously . Relativ e to existing frameworks, howe ver , determining the decision rule τ ( V ) and the parameter θ is more challenging because both in volv e the unobserved v ariable Y ∗ . In the next section, we establish identiﬁcation of τ ( V ) and θ by exploiting Assumption 1 together with additional assumptions on the unfairness mechanism of the observ ed decision Y . 8 3. Identiﬁcation 3.1. Assumptions on the mechanism of unfairness W e make the following assumptions about the mechanism of unf airness in the observed decision. Assumption 2. (i) f ( Y = 1 | Y ∗ = 0 , S = 0 , V ) = 0 ; (ii) f ( Y = 0 | Y ∗ = 1 , S = 1 , V ) = 0 ; and (iii) f ( Y = 0 | Y ∗ = 1 , S = 0 , V ) < 1 and f ( Y = 1 | Y ∗ = 0 , S = 1 , V ) < 1 . Assumption 3. The covariates vector V includes a binary auxiliary variable Z such that (i) Z ⊥ ⊥ Y | ( S, X , Y ∗ ) ; and (ii) Z ⊥  ⊥ Y ∗ | ( S, X ) , wher e X is the r emainder of V , i.e., V = ( Z , X ) . Here f ( Y = 0 | Y ∗ = 1 , S, V ) and f ( Y = 1 | Y ∗ = 0 , S, V ) encode the mechanism of unfair- ness: the former corresponds to discrimination and the latter corresponds to preferential treatment. Assumption 2 (i)–(ii) rule out preferential treatment for the disadvantaged group and discrimina- tion against the advantaged group; equiv alently , the realised decision cannot be upgraded (from un- fa vourable ( Y ∗ = 0) to fa vourable ( Y = 1) ) for S = 0 or downgraded (from unfa vourable ( Y ∗ = 1) to fav ourable ( Y = 0) ) for S = 1 . Although these restrictions may not hold exactly , they are often a reasonable approximation in many conte xts: the primary sources of unfairness are typically discrimi- nation against the disadv antaged group and preferential treatment for the adv antaged group, captured by f ( Y = 0 | Y ∗ = 1 , S = 0 , V ) and f ( Y = 1 | Y ∗ = 0 , S = 1 , V ) , respecti vely . These two quantities are unrestricted in our framew ork. From a measurement error perspectiv e, the unfairness mechanism can be viewed as misclassiﬁcation probabilities linking the latent variable Y ∗ to its mis- measured counterpart Y . Under this interpretation, Assumption 2(i)–(ii) corresponds to the one-sided misclassiﬁcation condition, a common feature in the literature (Nguimkeu et al., 2019; Millimet and Parmeter, 2022; Mondal and W ang, 2024). Condition (iii) is a standard positi vity assumption that rules out extreme unf airness under which Y = 0 for S = 0 or Y = 1 for S = 1 almost surely . Assumption 3 posits a binary auxiliary variable Z satisfying the non-dif ferentiality Condition (i), or exclusion restriction, namely that Z does not affect the mechanism of unfairness. Condition (ii) is a relev ance requirement: Z must be predictiv e of the desert decision Y ∗ gi ven ( S , X ) . When the auxiliary v ariable Z is multi-v alued or continuous, it can provide additional restrictions that strengthen identiﬁcation. Howe ver , a binary Z sufﬁces for our identiﬁcation analysis, and we focus on the binary case in this paper for simplicity . In the measurement error literature, variables satisfying Assumption 3 are sometimes referred to as instruments or secondary measurements for the latent variable; see, for 9 example, Mahajan (2006); Le wbel (2007); Hu (2008); Mondal and W ang (2024). Under Assumptions 2 and 3, the mechanism of unfairness is characterised by α ( X ) = f ( Y = 0 | Y ∗ = 1 , S = 0 , X ) , β ( X ) = f ( Y = 1 | Y ∗ = 0 , S = 1 , X ) , (1) where α ( X ) quantiﬁes the extent of discrimination against the disadvantaged group ( S = 0 ) and β ( X ) quantiﬁes the degree of preferential treatment towards the advantaged group ( S = 1 ), conditional on cov ariates X . The desert decision rule can be equiv alently written as τ ( Z , X ) = (1 − Z ) τ 0 ( X ) + Z τ 1 ( X ) , with τ z ( X ) = f ( Y ∗ = 1 | Z = z , X ) for z = 0 , 1 . Figure 2 presents se veral causal diagrams corresponding to Assumptions 1 and 3. In all diagrams, the sensiti ve attribute S does not directly af fect the desert decision Y ∗ , and the auxiliary variable Z does not directly affect the observed decision Y . In panel (a), Z plays the role of an instrument that precedes Y ∗ and is conditionally independent of Y given ( X , Y ∗ ) . Panel (b) depicts a setting in which Z is a proxy of Y ∗ subject to non-dif ferential measurement error . Panels (c) and (d) consider more general settings than panel (a), allowing S and Z to be dependent. The measurement error literature has also studied identiﬁcation in models consistent with these causal diagrams; howe ver , existing re- sults either rely on parametric speciﬁcations (Nguimkeu et al., 2019) or yield partial identiﬁcation (Mondal and W ang, 2024). In contrast, we will establish nonparametric identiﬁcation. In the supple- mentary material, we further illustrate the proposed framew ork and assumptions using a generativ e model for the observed decision subject to unf airness. (a) S Y ∗ Z Y (b) S Y ∗ Z Y (c) S Y ∗ Z Y (d) S Y ∗ Z Y Figure 2: Causal diagrams satisfying Assumptions 1 and 3, conditional on X . W e end this subsection with a practical example. W e discuss the plausibility of Assumptions 1–3 in this setting and further analyse the data in Section 7. Example 2. In a study of racial discrimination in the labour market, Bertrand and Mullainathan (2004) sent ﬁctitious r esumes to help-wanted advertisements in newspapers. The observed decision 10 is whether an applicant r eceives a callback for an interview . The sensitive attribute is the randomly assigned Black- or White-sounding name on the r esume, which serves as a pr oxy for the applicant’ s race . The r esume content other than the name was sampled fr om templates constructed by the r e- sear chers and classiﬁed into two gr oups of high and low quality . F or high quality r esumes, additional featur es wer e selectively added to match job r equirements, including extra experience, honours, or skills, while avoiding over qualiﬁcation. The covariates include city , occupation type, and job-seeker char acteristics listed on the r esume, such as labour market e xperience and educational attainment. In this example, the object of interest is the unobserved desert decision that each applicant right- fully deserves based on the generated job-rele v ant information. Assumption 1 requires the desert decision to be independent of race giv en the other characteristics. Assumption 2 is particularly com- pelling in this setting: Bertrand and Mullainathan (2004) report no evidence of re verse discrimination and ﬁnd that callbacks are uniformly more responsiv e to resumes with White names than for Black- sounding names, indicating discrimination against Black names and/or preferential treatment tow ards White names. T o justify Assumption 3, we use resume quality as the auxiliary variable. Resume quality is assigned and recorded by the researchers but is not explicitly rev ealed on the resume and therefore is not directly observed by employers. Consequently , conditional on the other resume co- v ariates, resume quality should not affect the unfairness mechanism. At the same time, it is reasonable that resume quality is predicti ve of the desert decision. 3.2. Nonparametric identiﬁcation For notational con venience, we let µ sz ( X ) = f ( Y = 1 | S = s, Z = z , X ) for ( s, z ) ∈ { 0 , 1 } 2 . Assumptions 1–3 suf ﬁce to identify the parameters of interest. Theorem 1. Under Assumptions 1–3, the joint distribution f ( S , Z , X , Y ∗ , Y ) is identiﬁed and τ z ( X ) = T z ( X ) , T z ( X ) = µ 0 z ( X ) { µ 11 ( X ) − µ 10 ( X ) } µ 01 ( X ) { 1 − µ 10 ( X ) } − µ 00 ( X ) { 1 − µ 11 ( X ) } , z = 0 , 1 . (2) Theorem 1 establishes identiﬁcation of the joint distribution and provides a closed-form identiﬁ- cation formula for the desert decision rule τ z ( X ) under Assumptions 1–3, by noting that T z ( X ) is av ailable from the observed data. The identiﬁcation formula T z ( X ) admits a bias-correction inter- pretation: it equals the observed probability µ 0 z ( X ) multiplied by a correction factor that accounts for bias arising from unfairness. In particular , µ 0 z ( X ) is do wnward biased relativ e to τ z ( X ) due to 11 discrimination against the disadv antaged group. The correction factor can be viewed as a comparison of dif ferences in µ sz ( X ) across le vels of Z , with the numerator gi ven by the dif ference in µ 1 z ( X ) and the denominator by the difference in µ 0 z ( X ) { 1 − µ 1(1 − z ) ( X ) } across z ∈ { 0 , 1 } . Because the aux- iliary variable Z shifts the desert decision Y ∗ without affecting the unfairness mechanism, this term isolates and corrects for the bias in µ 0 z ( X ) , thereby reco vering the distribution of the desert decision. The denominator of T z ( X ) is guaranteed to be nonzero under the relev ance condition in Assump- tion 3(ii). An immediate consequence of Theorem 1 is that the degree of unfairness θ is identiﬁed. The follo wing result enables identiﬁcation and estimation of θ using τ ( Z , X ) , α ( X ) , and β ( X ) . Theorem 2. Under Assumptions 1–3, we can identify θ with θ = E [(1 − S ) τ ( Z , X ) α ( X ) + S { 1 − τ ( Z , X ) } β ( X )] . (3) The intuition can be con ve yed with a parameter-counting argument. W e start from the factorisation of the joint distribution of Y ∗ and Y giv en ( S, Z , X ) : f ( Y ∗ , Y | S, Z , X ) = f ( Y ∗ | S, Z , X ) f ( Y | Y ∗ , S, Z , X ) , where f ( Y ∗ | S, Z , X ) denotes the conditional distribution of the desert decision and f ( Y | Y ∗ , S, Z , X ) characterises the mechanism of unf airness. W ithout additional restrictions, for an y ﬁxed value x , f ( Y ∗ | S, Z , X = x ) in volves four unkno wn parameters and f ( Y | Y ∗ , S, Z , X = x ) eight unknown parameters. By contrast, the observed data distribution yields only four constraints, encoded in f ( Y = 1 | S, Z, X ) = f ( Y = 1 | Y ∗ = 0 , S, Z , X ) f ( Y ∗ = 0 | S, Z , X ) + f ( Y = 1 | Y ∗ = 1 , S, Z , X ) f ( Y ∗ = 1 | S, Z , X ) , ( S, Z ) ∈ { 0 , 1 } 2 . (4) These four restrictions are insufﬁcient for identiﬁcation. Howe ver , under Assumption 1, f ( Y ∗ | S, Z , X ) depends only on two parameters, τ 0 ( X ) and τ 1 ( X ) ; under Assumption 2, f ( Y = 1 | Y ∗ = 0 , S = 0 , Z = z , X ) = 0 and f ( Y = 0 | Y ∗ = 1 , S = 1 , Z = z , X ) = 0 for z = 0 , 1 , and under Assumption 3, f ( Y | Y ∗ , S, Z , X ) further depends only on two parameters, α ( X ) and β ( X ) deﬁned in Equation (1). Consequently , (4) can be rewritten as µ 0 z ( X ) = τ z ( X ) { 1 − α ( X ) } , µ 1 z ( X ) = β ( X ) + τ z ( X ) { 1 − β ( X ) } , z = 0 , 1 . (5) For any ﬁx ed v alue x , this system comprises four equations in four unkno wn quantities, which clariﬁes 12 the role of the assumptions in identiﬁcation. Although the equations are nonlinear , the solution is unique in this setting; see the proof of Theorem 1 and the identiﬁcation formula for α ( X ) and β ( X ) in the supplementary material. Theorem 1 does not rely on speciﬁc parametric models or impose functional form restrictions on τ z ( X ) , and the impact of cov ariates X on the desert decision and the unfairness mechanism can be arbitrary . It opens the way to the estimation of the decision rule τ z ( X ) and other parameters us- ing v arious parametric or nonparametric models, ev en though the desert decision Y ∗ is unobserved. Compared with existing methods that seek to predict the observed decision Y under fairness con- straints, our decision rule T z ( X ) takes a markedly dif ferent form. The difference arises from shifting the target of prediction from Y to Y ∗ and from in voking structural assumptions about the unfairness mechanism. A related yet distinct line of work is the label debiasing method proposed by Jiang and Nachum (2020), which similarly views the observed decision Y as a biased proxy for an unobserved decision Y ∗ (though without interpreting Y ∗ as the desert decision) and aims to recover the distri- bution of Y ∗ . Ho wev er , their framework relies on the assumption that the conditional distrib ution of Y is the closest approximation to that of Y ∗ among all candidates exhibiting the same le vel of bias, whereas our assumptions on the unfairness mechanism can be more transparent and causally interpretable. Moreover , it is not entirely clear whether the distribution of Y ∗ is identiﬁed under the assumption of Jiang and Nachum (2020), whereas we provide rigorous identiﬁcation results. W e note that Assumptions 1–3 impose restrictions on the observed data distribution. The following result states testable implications in terms of the observed functions µ sz ( X ) . Proposition 2. Under Assumptions 1–3, we have that (i) µ 1 z ( X ) ≥ µ 0 z ( X ) for z ∈ { 0 , 1 } ; (ii) µ s 1 ( X )  = µ s 0 ( X ) for s ∈ { 0 , 1 } ; and (iii) µ 11 ( X ) − µ 10 ( X ) and µ 01 ( X ) − µ 00 ( X ) have the same sign. If any of these restrictions are violated, then at least one of Assumptions 1–3 must fail. Howe ver , e ven if the observed data are consistent with these restrictions, this does not establish that Assump- tions 1–3 hold. Their plausibility depends on domain-speciﬁc kno wledge and must be assessed on a case-by-case basis. The logic is analogous to the testable implications of the assumptions underpin- ning instrumental variable methods for causal effect estimation (Angrist et al., 1996; Balke and Pearl, 1997; Kitagawa, 2015). Sensitivity analysis for Assumptions 1–3 is therefore important for applying the proposed frame work in practice, which we will elaborate in Section 5. 13 4. Estimation 4.1. Estimation of the desert decision rule In this section, we estimate the desert decision rule τ ( Z , X ) , equiv alently the functions τ 0 ( X ) and τ 1 ( X ) . A natural approach is to ﬁrst estimate the conditional probabilities µ sz ( X ) for ( s, z ) ∈ { 0 , 1 } 2 and then plug in these estimates into (2). A key limitation, ho we ver , is that the resulting estimators of τ 0 ( X ) and τ 1 ( X ) are not guaranteed to lie in [0 , 1] in ﬁnite samples, and hence may fail to produce v alid probabilities. W e therefore adopt a nonparametric siev e maximum likelihood approach. For notational simplicity , we suppress the cov ariates X when no confusion arises; for example, we write τ 0 , τ 1 , α , β for τ 0 ( X ) , τ 1 ( X ) , α ( X ) , β ( X ) , respectiv ely . Let ξ = ( τ 0 , τ 1 , α, β ) and ˜ ξ = ( ˜ τ 0 , ˜ τ 1 , ˜ α, ˜ β ) , where ˜ τ 0 , ˜ τ 1 , ˜ α , and ˜ β denote generic approximating functions. Let O = ( S , Z , X , Y ) denote the observed data. The conditional log-likelihood of Y giv en ( S , Z , X ) is l ( O ; ˜ ξ ) = Y log f ( Y = 1 | S, Z , X ; ˜ ξ ) + (1 − Y ) log { 1 − f ( Y = 1 | S, Z , X ; ˜ ξ ) } , where f ( Y = 1 | S, Z, X ; ˜ ξ ) = (1 − S )(1 − Z ) µ 00 ( X ; ˜ τ 0 , ˜ α ) + (1 − S ) Z µ 01 ( X ; ˜ τ 1 , ˜ α ) + S (1 − Z ) µ 10 ( X ; ˜ τ 0 , ˜ β ) + S Z µ 11 ( X ; ˜ τ 1 , ˜ β ) , (6) with the component functions speciﬁed according to (5), i.e., µ 0 z ( X ; ˜ τ z , ˜ α ) = ˜ τ z ( X ) { 1 − ˜ α ( X ) } , µ 1 z ( X ; ˜ τ z , ˜ β ) = ˜ β ( X ) + ˜ τ z ( X ) { 1 − ˜ β ( X ) } , z = 0 , 1 . The siev e maximum likelihood method estimates the parameters by maximising a criterion function; here, the population criterion function is E { l ( O ; ˜ ξ ) } , which is uniquely maximised at the true param- eter ξ . Giv en n independent and identically distributed copies of O . The siev e maximum likelihood estimator ˆ ξ = ( ˆ τ 0 , ˆ τ 1 , ˆ α, ˆ β ) is obtained by maximising the empirical criterion function ov er a sequence of approximating spaces F n , i.e., ˆ ξ = arg max ˜ ξ ∈F n ˆ E { l ( O ; ˜ ξ ) } , (7) where ˆ E ( · ) denotes the empirical mean operator . Then, the estimator of τ ( Z , X ) is ˆ τ ( Z, X ) = (1 − Z ) ˆ τ 0 ( X ) + Z ˆ τ 1 ( X ) . 14 For the siev e spaces F n , we employ a series logit speciﬁcation, which is simple to implement. Let { ψ j ( X ) } ∞ j =1 be a sequence of basis functions (e.g., po wer series, splines, or wa velets), and deﬁne ϕ ( X ) = ( ψ 1 ( X ) , . . . , ψ J n ( X )) T as the vector of the ﬁrst J n basis functions, where J n increases with the sample size n . Denote expit ( · ) = exp( · ) / { 1 + exp( · ) } . W e approximate τ 0 ( X ) , τ 1 ( X ) , α ( X ) , and β ( X ) with logistic series expansions of the form e xpit { γ T ϕ ( X ) } . T o establish the asymptotic properties of the estimators ˆ ξ and ˆ τ , we introduce some notation and regularity conditions. For a generic vector of functions g ( O ) = ( g 1 ( O ) , . . . , g q ( O )) , let ∥ g ∥ 2 = [ P q i =1 E { g 2 i ( O ) } ] 1 / 2 denote the L 2 norm with respect to the observed data distrib ution. For a generic function h ( x ) , let ∥ h ∥ ∞ ,p denote the Sobole v norm, with p > 0 characterises the smoothness. Let π sz ( X ) = f ( S = s, Z = z | X ) for ( s, z ) ∈ { 0 , 1 } 2 denote the joint probabilities of the sensitiv e attribute and auxiliary variable giv en cov ariates. W e make the follo wing conditions for the distrib ution and the sie ve spaces, which are standard in sie ve estimation (Chen, 2007). Assumption 4. (i) The vector of covariates X ∈ R d has a compact support X = [0 , 1] d ; (ii) P os- itivity: π sz ≥ c for ( s, z ) ∈ { 0 , 1 } 2 , c ≤ τ 0 , τ 1 , α, β ≤ 1 − c , and | τ 1 − τ 0 | ≥ c for some c > 0 ; (iii) Smoothness: ∥ τ 0 ∥ ∞ ,p , ∥ τ 1 ∥ ∞ ,p , ∥ α ∥ ∞ ,p , ∥ β ∥ ∞ ,p < ∞ for some p > 0 . Assumption 5. (i) The smallest eigen value of E { ϕ ( X ) ϕ ( X ) T } is bounded away fr om zer o for all J n ; (ii) letting H n = { expit { γ T ϕ ( X ) } , γ ∈ R J n } , the sieve spaces F n ar e F n = { ˜ ξ = ( ˜ τ 0 , ˜ τ 1 , ˜ α, ˜ β ) : ˜ τ 0 , ˜ τ 1 , ˜ α, ˜ β ∈ H n , c ≤ ˜ τ 0 , ˜ τ 1 , ˜ α, ˜ β ≤ 1 − c, and | ˜ τ 1 − ˜ τ 0 | ≥ c } ; (iii) ther e exist an operator P n such that P n ξ ∈ F n and ∥P n ξ − ξ ∥ 2 = O ( J − p/d n ) . Theorem 3. Under Assumptions 1–5, letting J n = O ( n d/ (2 p + d ) ) , we have ∥ ˆ ξ − ξ ∥ 2 = O p ( n − p/ (2 p + d ) ) , ∥ ˆ τ − τ ∥ 2 = O p ( n − p/ (2 p + d ) ) . The proof of Theorem 3 adopts the local Rademacher complexity techniques in statistical learning theory (see e.g., W ainwright, 2019), which also pro vides ﬁnite sample error bounds. In line with the classical bias–v ariance trade-off, the estimation error can be decomposed into two parts: an ap- proximation error of order J − p/d n and a standard deviation term of order ( J n /n ) 1 / 2 . By choosing J n = O ( n d/ (2 p + d ) ) to balance these two components, the estimators achie ve the best con ver gence rate O p ( n − p/ (2 p + d ) ) , which matches the minimax optimal rate for standard nonparametric estimation (Tsybako v, 2009). In addition to the sie ve maximum likelihood estimator , it is of great interest to 15 employ other ﬂexible or nonparametric estimation methods such as random forests, neural networks, and reproducing kernel Hilbert space methods. W e remark that, as in many fairness-enhancing methods, the proposed estimator ˆ τ is obtained by solving an optimisation problem. A ke y dif ference is that the optimisation problem in (7) does not impose additional fairness constraints, whereas existing approaches typically encode fairness re- quirements via regularisations or explicit constraints and therefore trade utility against fairness. In our framework, fairness is satisﬁed by construction when targeting the desert decision Y ∗ , thereby aligning predicti ve accurac y with fairness. 4.2. Estimation of the degree of unfair ness The parameter θ is a complicated functional of the observed data distrib ution, depending on nuisance functions τ , α , and β as sho wn in (3). After obtaining estimators ˆ τ , ˆ α , and ˆ β , θ can be estimated by plugging in them into the sample analogue of (3). Howe ver , the plug-in estimator need not be asymptotically normal, which complicates uncertainty quantiﬁcation and inference for θ . W e there- fore de velop an inﬂuence-function-based estimator using semiparametric efﬁcienc y theory (Bickel et al., 1993; Tsiatis, 2006) to obtain valid inference under standard regularity conditions. W e begin by deri ving an inﬂuence function for θ . Theorem 4. Under Assumptions 1–3, an inﬂuence function for θ is IF( O ; η ) = φ ( O ; η ) − θ , wher e η = ( τ 0 , τ 1 , α, β , π 00 , π 01 , π 10 , π 11 ) denotes the vector of nuisance functions, and φ ( O ; η ) = (1 − S )(1 − Z ) { C 00 ( Y − τ 0 + τ 0 α ) + τ 0 α } + (1 − S ) Z { C 01 ( Y − τ 1 + τ 1 α ) + τ 1 α } + S (1 − Z ) { C 10 ( Y − β − τ 0 + τ 0 β ) + β − τ 0 β } + S Z { C 11 ( Y − β − τ 1 + τ 1 β ) + β − τ 1 β } , with C sz for ( s, z ) ∈ { 0 , 1 } 2 being functions of X given in the supplementary material. Theorem 4 provides a closed-form inﬂuence function for θ under the identifying assumptions. The inﬂuence function comprises four components corresponding to four v alues of ( S, Z ) . Relati ve to (3), each component contains an additional mean-zero augmentation term of the form, I ( S = s, Z = z ) C sz ( Y − µ sz ) for ( s, z ) ∈ { 0 , 1 } 2 . This construction is in the same spirt to the augmented in verse probability weighted estimator in causal inference and missing data literature (Bang and Robins, 2005). Although the coefﬁcients C sz appear to be algebraically complicated, they are functions ob- tained by arithmetic operations on the nuisance functions in η . As a result, the ﬁrst-order impact of 16 nuisance estimation error is removed, and the remaining bias of the inﬂuence-function-based estima- tor is of second order under standard conditions. Let ˆ η = ( ˆ τ 0 , ˆ τ 1 , ˆ α, ˆ β , ˆ π 00 , ˆ π 01 , ˆ π 10 , ˆ π 11 ) denote the vector of nuisance estimators. The proposed inﬂuence-function-based estimator is ˆ θ = ˆ E { φ ( O ; ˆ η ) } . The nuisance functions in η can be partitioned into two components: ξ = ( τ 0 , τ 1 , α, β ) and the con- ditional probabilities π sz for ( s, z ) ∈ { 0 , 1 } 2 . In Section 4.1, we hav e described a siev e maximum likelihood method for estimating ξ . Estimation of π sz can be formulated as a multi-class classiﬁca- tion problem with four classes indexed by ( s, z ) ∈ { 0 , 1 } 2 . W e may apply standard classiﬁers such as multinomial logistic regression or more ﬂexible methods. The follo wing theorem establishes the asymptotic normality of the proposed estimator . Theorem 5. Under the conditions of Theor em 3, suppose that the nuisance estimators further satisfy that (i) ˆ π sz ≥ c for some c > 0 , ( s, z ) ∈ { 0 , 1 } 2 ; (ii) ∥ ˆ η − η ∥ 2 = o p ( n − 1 / 4 ) ; and (iii) ˆ η and η ar e in a Donsker class, then we have ˆ θ − θ = ˆ E { IF( O ; η ) } + o p ( n − 1 / 2 ) . Theorem 5 implies that ˆ θ is n 1 / 2 -consistent and asymptotically normal, provided the nuisance estimators con verge faster than n − 1 / 4 . By the con vergence rate in Theorem 3, this requirement for ( ˆ τ 0 , ˆ τ 1 , ˆ α, ˆ β ) holds when the smoothness parameter p exceeds half the cov ariate dimension d . Rates faster than n − 1 / 4 are standard in inﬂuence-function-based estimation of complex functionals and can be achie ved by a range of ﬂexible methods, including random forests, neural netw orks, and ensemble procedures (Chernozhuko v et al., 2018). Theorem 5 also imposes a Donsker condition to control the comple xity of the nuisance models (v an der V aart and W ellner, 1996). This condition can be relaxed, allo wing more complex nuisance estimators, by using cross-ﬁtting (e.g., Robins et al., 2008; Chernozhuko v et al., 2018). Under the conditions of Theorem 5, the asymptotic v ariance of ˆ θ is E { IF( O ; η ) 2 } , which can be consistently estimated with ˆ σ 2 = ˆ E [ { φ ( O ; ˆ η ) − ˆ θ } 2 ] . W e can construct the 1 − α conﬁdence interval for θ , [ ˆ θ − q α/ 2 ˆ σ n − 1 / 2 , ˆ θ + q α/ 2 ˆ σ n − 1 / 2 ] , where q α/ 2 denotes the 1 − α/ 2 quantile of standard normal distribution function. 17 5. Extensions: legitimate differential treatment and sensiti vity analysis 5.1. Incorporating legitimate differ ential treatment In this section, we conduct sensiti vity analysis with respect to violations of the key identifying As- sumptions 1–3. W e begin with violations of the fairness Assumption 1. In some settings, fairness considerations may justify providing stronger support to the disadvantaged group rather than apply- ing a uniformly fair treatment rule, in order to improv e circumstances and promote ﬂourishing more rapidly . For example, Rawls’ difference principle argues that social and institutional arrangements should work to the greatest beneﬁt of the least advantaged (Ra wls, 1971). Practical instances include the WTO’ s “special and differential treatment” provisions designed to support dev eloping countries (W orld T rade Org anization, 2001) and “div ersity , equity , and inclusion” (DEI) policies intended to expand opportunities for historically disadv antaged groups (Kalev et al., 2006). W e therefore extend the proposed framew ork to more general settings in which the sensiti ve at- tribute S may inﬂuence the desert decision Y ∗ and may enter the decision rule. Let τ sz ( X ) = f ( Y ∗ = 1 | S = s, Z = z , X ) , ( s, z ) ∈ { 0 , 1 } 2 , denote the desert decision rule that depends on the sensitiv e attribute. The inﬂuence of S on Y ∗ conditional on ( Z, X ) is quantiﬁed by κ z ( X ) = τ 1 z ( X ) − τ 0 z ( X ) for z = 0 , 1 . W e assume that κ 0 and κ 1 are speciﬁed a priori by the decision-maker based on e xternal kno wledge, representing the le vel of legitimate support for the disadv antaged group. W e hav e the following result. Theorem 6. Under Assumptions 2 and 3 and given ( κ 0 , κ 1 ) , if τ 01 (1 − κ 0 )  = τ 00 (1 − κ 1 ) , then f ( S, Z , X , Y ∗ , Y ) is identiﬁed and τ sz = µ 0 z ( µ 11 − µ 10 ) + κ 0 µ 0 z (1 − µ 11 ) − κ 1 µ 0 z (1 − µ 10 ) µ 01 (1 − µ 10 ) − µ 00 (1 − µ 11 ) + sκ z , ( s, z ) ∈ { 0 , 1 } 2 . Theorem 6 establishes identiﬁcation of the joint distrib ution f ( S, Z , X , Y ∗ , Y ) and provides an identiﬁcation formula for τ sz ( X ) in general settings that allow the sensitiv e attribute to hav e a speci- ﬁed inﬂuence on the desert decision. It also facilitates sensiti vity analysis for assessing the rob ustness of the identiﬁcation and estimation results in Sections 3–4 by specifying and varying the sensitivity parameters κ 0 and κ 1 . Theorem 1 is a special case of Theorem 6 with κ 0 = κ 1 = 0 . The condition τ 01 (1 − κ 0 )  = τ 00 (1 − κ 1 ) is imposed to ensure that the denominator is nonzero. Equiv alently , it can be written as τ 01 / (1 − τ 11 )  = τ 00 / (1 − τ 10 ) , representing a generalised relev ance condition under 18 which τ 0 z / (1 − τ 1 z ) varies with z ∈ { 0 , 1 } . Under Assumptions 2 and 3, a sufﬁcient condition for this inequality is κ 0 = κ 1 , that is, when the inﬂuence of S on Y ∗ is homogeneous across le vels of Z . The method in Section 4 extends to estimation of τ sz ( X ) by incorporating the prescribed le vel of legitimate differential treatment and modifying the criterion function accordingly . Given κ 0 and κ 1 , the siev e maximum likelihood estimator ( ˆ τ 00 , ˆ τ 01 , ˆ α, ˆ β ) is obtained by solving the optimisation problem in (7), with the component functions in (6) replaced by µ 0 z ( X ; ˜ τ 0 z , ˜ α ) = ˜ τ 0 z ( X ) { 1 − ˜ α ( X ) } , µ 1 z ( X ; ˜ τ 0 z , ˜ β ) = ˜ β ( X ) + { ˜ τ 0 z ( X ) + κ z ( X ) }{ 1 − ˜ β ( X ) } . The resulting estimator of the decision rule is ˆ τ sz ( X ) = ˆ τ 0 z ( X ) + sκ z ( X ) . T o estimate the degree of unfairness θ , we deriv e an inﬂuence function under the conditions of Theorem 6 and construct the corresponding estimator . Details are provided in the supplementary material. 5.2. Sensitivity analysis against violations of the unfair ness mechanism W e ne xt consider violations of Assumption 2. Let δ 0 ( X ) = f ( Y = 1 | Y ∗ = 0 , S = 0 , X ) and δ 1 ( X ) = f ( Y = 0 | Y ∗ = 1 , S = 1 , X ) denote, respectiv ely , the probability of preferential treatment for the disadv antaged group and the probability of discrimination against the adv antaged group. These quantities serve as sensiti vity parameters that quantify departures from Assumption 2. Proposition 3. Under Assumptions 1 and 3 and given ( δ 0 , δ 1 ) , if α + δ 0 < 1 and β + δ 1 < 1 , then f ( S, Z , X , Y ∗ , Y ) is identiﬁed and τ z = ( µ 0 z − δ 0 )( µ 11 − µ 10 ) µ 01 (1 − µ 10 ) − µ 00 (1 − µ 11 ) − δ 0 ( µ 11 − µ 10 ) − δ 1 ( µ 01 − µ 00 ) , z = 0 , 1 . The conditions α + δ 0 < 1 and β + δ 1 < 1 ensure that the denominator is nonzero. Equiv alently , they can be written as δ 0 < 1 − α and β < 1 − δ 1 , implying that Y and Y ∗ are positiv ely associated conditional on the other v ariables; in the present context, the y mean that preferential treatment occurs with smaller probability than a fair fav ourable decision. Analogous conditions are standard in the measurement error literature to ensure that an observed proxy is informati ve for the latent v ariable (Mahajan, 2006; Le wbel, 2007). Proposition 3 has two substantiv e implications. First, it establishes identiﬁcation of the joint dis- tribution and the desert decision rule τ z ( X ) ev en when Assumption 2 is violated, by incorporating the sensiti vity parameters δ 0 and δ 1 . It thus generalises Theorem 1: under Assumption 2, δ 0 = δ 1 = 0 , 19 and hence τ z ( X ) = T z ( X ) . Although δ 0 and δ 1 are typically unkno wn in practice, Proposition 3 facil- itates sensiti vity analysis for violations of Assumption 2 by specifying and varying these parameters. Gi ven δ 0 and δ 1 , the sie ve maximum lik elihood estimator ˆ ξ = ( ˆ τ 0 , ˆ τ 1 , ˆ α, ˆ β ) is obtained by solving the optimisation problem in (7), with the component functions in (6) replaced by µ 0 z ( X ; ˜ τ z , ˜ α ) = δ 0 ( X ) + ˜ τ z ( X ) { 1 − δ 0 ( X ) − ˜ α ( X ) } , µ 1 z ( X ; ˜ τ z , ˜ β ) = ˜ β ( X ) + ˜ τ z ( X ) { 1 − δ 1 ( X ) − ˜ β ( X ) } . In the supplementary material, we deri ve an inﬂuence function for the degree of unfairness θ under the conditions of Proposition 3 and construct the corresponding estimator . Second, Proposition 3 characterises the rob ustness of T z ( X ) in Theorem 1 that is constructed under Assumption 2 to violations of that assumption. In the supplementary material, we quantify the bias of T z ( X ) in terms of the sensiti vity parameters when Assumption 2 is violated. In particular , we show that when δ 0 and δ 1 are small, T z − τ z ≈ δ 0 (1 − τ z ) 1 − α − δ 1 τ z 1 − β . Consequently , the bias of T z ( X ) is approximately linear in these sensitivity parameters, suggesting that T z ( X ) is robust to mild or moderate departures from Assumption 2. Regarding violations of Assumption 3, let α z ( X ) = f ( Y = 0 | Y ∗ = 1 , S = 0 , Z = z , X ) and β z ( X ) = f ( Y = 1 | Y ∗ = 0 , S = 1 , Z = z , X ) for z = 0 , 1 denote the unfairness mechanism. These functions generalise α ( X ) and β ( X ) in Section 3 by allo wing the mechanism to vary with Z , thereby capturing violations of the non-differentiality requirement. Let ζ 0 ( X ) = { 1 − α 1 ( X ) } / { 1 − α 0 ( X ) } − 1 and ζ 1 ( X ) = { 1 − β 1 ( X ) } / { 1 − β 0 ( X ) } − 1 denote the the dif ferential impact of Z on the unfairness mechanism and serv e as sensitivity parameters encoding departures from Assumption 3. Proposition 4. Under Assumptions 1 and 2 and given ( ζ 0 , ζ 1 ) , if τ 1  = τ 0 and 0 ≤ α z , β z < 1 for z = 0 , 1 , then f ( S, Z , X , Y ∗ , Y ) is identiﬁed and τ z = (1 + ζ 0 ) 1 − z µ 0 z { µ 11 − µ 10 + ζ 1 (1 − µ 10 ) } (1 + ζ 1 ) µ 01 (1 − µ 10 ) − (1 + ζ 0 ) µ 00 (1 − µ 11 ) , z = 0 , 1 . Proposition 4 establishes identiﬁcation of the joint distribution f ( S, Z, X , Y ∗ , Y ) and the desert decision rule τ z ( X ) under violations of Assumption 3 by incorporating the sensitivity parameters ζ 0 and ζ 1 . It generalises Theorem 1: under Assumption 3, ζ 0 = ζ 1 = 0 , and hence τ z ( X ) = T z ( X ) . 20 Sensiti vity analysis can therefore be conducted by specifying and v arying ζ 0 and ζ 1 . Gi ven ζ 0 and ζ 1 , we obtain the sie ve maximum likelihood estimator ˆ ξ = ( ˆ τ 0 , ˆ τ 1 , ˆ α 0 , ˆ β 0 ) by solving the optimisation problem in (7), with the component functions in (6) replaced by µ 0 z ( X ; ˜ τ z , ˜ α 0 ) = { 1 + ζ 0 ( X ) } z ˜ τ z ( X ) { 1 − ˜ α 0 ( X ) } , µ 1 z ( X ; ˜ τ z , ˜ β 0 ) = 1 − { 1 + ζ 1 ( X ) } z { 1 − ˜ τ z ( X ) }{ 1 − ˜ β 0 ( X ) } . In the supplementary material, we deriv e an inﬂuence function for θ under the conditions of Propo- sition 4 and construct the corresponding estimator . W e also show that, when ζ 0 and ζ 1 are small, the bias of T z ( X ) is approximately linear in these parameters, suggesting robustness to mild violations of Assumption 3. Moreov er , when both Assumptions 2 and 3 are violated, the bias of T z ( X ) is also quantiﬁed and sho wn to be approximately linear in the magnitude of the violations. The robustness is further supported by the simulation results in Section 6. 6. Simulation W e ev aluate the performance of the proposed methods via numerical simulations. W e generate two cov ariates X = ( X 1 , X 2 ) , a binary sensiti ve attrib ute S , a binary auxiliary v ariable Z , a binary desert decision Y ∗ , and a binary observed decision Y from the follo wing model: X 1 ⊥ ⊥ X 2 and X 1 , X 2 ∼ U(0 , 1) , S ⊥ ⊥ Z | X , f ( S = 1 | X ) = expit (2 − 2 sin X 1 − 2 X 2 ) , f ( Z = 1 | X ) = expit (1 − X 1 − sin X 2 ) , f ( Y ∗ = 1 | S, Z , X ) = (1 − Z ) expit ( − 3 + 5 X 1 + sin X 2 ) + Z expit ( − 3 + X 1 + 6 sin X 2 ) , f ( Y = 0 | Y ∗ = 1 , S, Z , X ) = (1 − S ) expit {− 1 − sin X 1 + 2 exp( − X 2 ) } + S δ , f ( Y = 1 | Y ∗ = 0 , S, Z , X ) = (1 − S ) δ + S expit {− 1 + 2 exp( − X 1 ) − X 2 } , The abov e setting satisﬁes Assumptions 1 and 3, with τ 0 ( X ) = expit ( − 3 + 5 X 1 + sin X 2 ) , τ 1 ( X ) = expit ( − 3 + X 1 + 6 sin X 2 ) , α ( X ) = e xpit {− 1 − sin X 1 + 2 exp( − X 2 ) } , β ( X ) = expit {− 1 + 2 exp( − X 1 ) − X 2 } . It further implies f ( Y = 0 | Y ∗ = 1 , S = 1 , Z , X ) = f ( Y = 1 | Y ∗ = 0 , S = 0 , Z, X ) = δ . The sensitivity parameter δ quantiﬁes departures from Assumption 2: it holds for δ = 0 but is violated for δ > 0 , with larger values indicating more se vere violations. W e consider three v alues, δ ∈ { 0 , 0 . 05 , 0 . 1 } . W e apply the sie ve approach proposed in Section 4.1 to obtain the estimator ˆ τ . T o estimate ξ = 21 ( τ 0 , τ 1 , α, β ) , we solv e the optimisation problem in (7) using polynomial basis functions of order 3 . The conditional probabilities π sz for ( s, z ) ∈ { 0 , 1 } 2 are estimated using a standard series logit speciﬁcation with the same polynomial basis. W e refer to the proposed method as DSD, as decisions are made based on ˆ τ , which targets the desert decision Y ∗ . W e estimate the degree of unfairness θ and construct conﬁdence interv als using the inﬂuence-function-based method proposed in Section 4.2. For comparison, we implement the follo wing four methods: (i) UML—an unconstrained machine learning predictor of the observ ed decision Y based on ( S , Z , X ) , without fairness adjustment; (ii) FTU—the fairness-through-unaw areness method that excludes S and predicts Y from the non- sensiti ve v ariables ( Z , X ) ; (iii) MLC—a machine learning predictor of Y based on ( S, Z, X ) subject to a fairness constraint enforcing a zero causal ef fect of S on Y (Nabi et al., 2024); (i v) LD—the label debiasing method based on re weighting (Jiang and Nachum, 2020). Implementation details for competing methods are provided in the supplementary material. For each setting, we replicate 1000 simulations at sample sizes n = 2000 , 4000 . Figure 3(a) reports the estimation error of the proposed estimator ˆ τ , measured by the L 2 norm ∥ ˆ τ − τ ∥ 2 and e valuated on an independent test set of size 10 6 generated from the same model. When δ = 0 , so that Assumptions 1–3 hold, the estimator has small error that decreases as the sample size increases. When δ > 0 , the error increases with δ due to violations of Assumption 2. The estimator remains relati vely stable for moderate v alues of δ , b ut the error can become substantial under more se vere violations of Assumption 2. The estimation errors for ˆ τ 0 , ˆ τ 1 , ˆ α , and ˆ β exhibit similar patterns, and we report these results in the supplementary material. T able 2 reports the area under the recei ver operating characteristic curve (A UC) for each method when predicting Y ∗ and Y . For predicting the desert decision Y ∗ , the proposed DSD method attains the highest A UC across all combinations of n and δ , whereas the unconstrained machine learning predictor UML performs worst. In contrast, for predicting the observed decision Y , UML achie ves the highest A UC, while the DSD method performs worst. The fairness-oriented methods FTU, MLC, and LD reduce A UC for Y relati ve to UML, consistent with an utility–fairness trade-off, and they 22 also reduce A UC for Y ∗ relati ve to the DSD method. T aken together , these results indicate a trade- of f between predicting Y and predicting Y ∗ when the observed decision is affected by unfairness, so that a single method generally cannot achiev e optimal performance for both tar gets. Notably , the supplementary material presents an example in which the A UC values of UML, FTU, and MLC for predicting Y ∗ fall belo w 0 . 5 , indicating performance worse than random guessing. When the identifying assumptions are plausible, these ﬁndings support using ˆ τ to guide decision-making rather than methods that target the observ ed decision. Figure 3(b) reports the bias of the proposed estimator ˆ θ , and T able 3 reports the cov erage prob- ability of the corresponding conﬁdence interval across settings with varying δ . When δ = 0 , so that Assumptions 1–3 hold, the bias is small and decreases with sample size, and the nominal 95% conﬁdence interval attains coverage close to 0 . 95 . When δ > 0 , Assumption 2 is violated; the bias increases with δ , and cov erage falls substantially below 0 . 95 for larger values (e.g., δ = 0 . 1 ). These results indicate that the proposed estimation and inference procedure for θ is v alid under the identify- ing assumptions and is reasonably robust to mild violations, but may be unreliable under more se vere departures. W e also conduct additional simulations to assess sensitivity to violations of Assumptions 1 and 3, with results reported in the supplementary material due to space constraints. The proposed estima- tors exhibit small error or bias under mild or moderate violations, b ut performance can deteriorate under more severe departures. In practice, we therefore recommend sensiti vity analysis to ev aluate robustness to potential violations of the identifying assumptions. δ = 0 δ = 0.05 δ = 0.1 0 0.1 0.2 0.3 (a) δ = 0 δ = 0.05 δ = 0.1 −0.4 0 0.4 (b) Figure 3: (a) Estimation error of ˆ τ and (b) bias of ˆ θ . Note: White boxes are for sample size n = 2000 and grey for 4000 . 23 T able 2: A UC values for dif ferent methods in predicting Y ∗ and Y Y ∗ Y Prediction methods: DSD UML FTU MLC LD DSD MLC FTU MLC LD δ = 0 n = 2000 0.803 0.604 0.757 0.724 0.789 0.603 0.768 0.624 0.622 0.671 n = 4000 0.820 0.603 0.766 0.735 0.795 0.606 0.772 0.628 0.628 0.676 δ = 0 . 05 n = 2000 0.787 0.608 0.751 0.716 0.779 0.593 0.738 0.614 0.614 0.656 n = 4000 0.812 0.607 0.761 0.729 0.789 0.597 0.743 0.618 0.621 0.662 δ = 0 . 1 n = 2000 0.776 0.610 0.744 0.706 0.767 0.582 0.709 0.603 0.606 0.641 n = 4000 0.803 0.612 0.756 0.721 0.779 0.587 0.715 0.608 0.613 0.647 T able 3: Coverage rate of the 95% conﬁdence interval for θ δ 0 0 . 05 0 . 1 n = 2000 0.954 0.938 0.851 n = 4000 0.961 0.914 0.836 7. A pplication W e apply the proposed methods to data from the ﬁeld experiment of Bertrand and Mullainathan (2004), introduced in Example 2. The data are publicly av ailable from https://doi.org/10. 3886/E116023V1 . The sample size is n = 4 , 870 . The study e valuates the impact of an applicant’ s race on labour market outcomes. Speciﬁcally , ﬁctitious resumes with randomly assigned Black- or White-sounding names were sent to job advertisements in newspapers, and callback for an interview was recorded. A direct examination of the data sho ws that the callback rate is 0 . 0965 for resumes with white-sounding names and 0 . 0645 for those with Black-sounding names. The estimated dif ference is 0 . 0320 , with a 95% conﬁdence interv al of (0 . 0168 , 0 . 0473) . Bertrand and Mullainathan (2004) highlighted that White names recei ve nearly 50% more callbacks. These results indicate signiﬁcantly fe wer callbacks for resumes with Black-sounding names, consistent with the presence of unfairness in this setting. By re-analysing the data using our methods, we aim to recov er a fair decision rule that targets the desert decision, quantify the degree of unfairness in the observed decision, and examine the under- lying unfairness mechanism. Let S and Z denote the randomly assigned White- or Black-sounding name and resume quality , with S = 1 for a White-sounding name and S = 0 for a Black-sounding name, and Z = 1 for high quality and Z = 0 for low quality . Let Y = 1 indicate a callback and Y = 0 indicate no callback. W e use resume quality as the auxiliary v ariable. Cov ariates X include the number of prior jobs, years of work experience, and ten discrete variables: educational attain- 24 ment, honours receiv ed, volunteer experience, military experience, presence of gaps in the resume, work during schooling, computer skills, special skills, city of the job, and occupation type. For this application, the plausibility of Assumptions 1–3 was discussed and justiﬁed in Example 2. The parameters of interest include the target desert decision rule τ ( Z, X ) = f ( Y ∗ = 1 | Z , X ) , the degree of unfairness θ = f ( Y  = Y ∗ ) , and the unfairness mechanism α ( X ) = f ( Y = 0 | Y ∗ = 1 , S = 0 , X ) , β ( X ) = f ( Y = 1 | Y ∗ = 0 , S = 1 , X ) , W e apply the methods in Section 4 to estimate these quantities. Speciﬁcally , τ 0 , τ 1 , α , and β are estimated via the sie ve maximum likelihood approach using polynomial basis functions of order 3 , and π sz for ( s, z ) ∈ { 0 , 1 } 2 are estimated using a standard series logit speciﬁcation with the same polynomial basis. T o obtain a classiﬁer based on ˆ τ that preserves the o verall rate of f av ourable decisions (i.e., the total number of callbacks), we deﬁne ˆ Y ∗ = I { ˆ τ ( Z , X ) > t ∗ } with t ∗ = sup { t ∈ [0 , 1] : ˆ E [ I { ˆ τ ( Z , X ) ≥ t } ] ≥ ˆ E ( Y ) } , where ˆ E ( Y ) = 0 . 0805 . For comparison, we implement the four methods UML, FTU, MLC, and LD introduced in Section 6. For each competing method, we construct an analogous classiﬁer that preserves the ov erall rate of fav ourable decisions, yielding corresponding predictions. Implementation details are provided in the supplementary material. T ables 4 and 5 report the conditional proportions of decisions predicted by the UML, FTU, MLC, and LD methods gi ven ˆ Y ∗ among all resumes and resumes with Black-sounding names, respectiv ely , thereby comparing the proposed predictions with those from the four competing methods. When the proposed DSD method predicts no callback ( ˆ Y ∗ = 0) , the competing methods agree with this predic- tion with probabilities close to one. In contrast, when the DSD method predicts a callback ( ˆ Y ∗ = 1) , the competing methods often disagree. Among them, UML exhibits the largest discrepancy from the DSD method, whereas the remaining three methods show more moderate differences. For resumes with Black-sounding names, a large share of applicants predicted a callback by the DSD method ( ˆ Y ∗ = 1) are predicted by the UML, MLC and LD methods to receiv e no callback, with proportions e ven exceeding 0.5. This discrepancy indicates that the competing methods may fail to recognise a considerable proportion of deserving indi viduals from the disadvantaged group compared with the DSD method. In the supplementary material, we also compare the DSD method with competing methods when the resume quality Z is not used for prediction; the results are similar . The estimated degree of unfairness is ˆ θ = 0 . 0161 , which is signiﬁcantly nonzero with a 95% 25 conﬁdence interval of (0 . 0085 , 0 . 0236) . Figure 4 displays histograms of the estimated values ˆ α ( X i ) and ˆ β ( X i ) across applicants. F or ˆ β ( X i ) , almost all values concentrate around zero, but for ˆ α ( X i ) , in addition to the concentration around zero, there exist a number of units with nonzero or large v alues. Speciﬁcally , a distinct cluster appears around 0.6 for approximately 100–200 individuals. Further e x- amination rev eals that this cluster consists almost entirely of applicants targeting sales representati ve positions and constitutes nearly one-third of nonzero v alues in this subgroup, as shown in Figure 5. This pattern is consistent with existing discrimination theories (Darity Jr and Mason, 1998; Becker, 2010) concerning the nature of such occupations, which typically in v olve extensi ve customer interac- tion and may therefore leav e greater scope for discriminatory behaviour . This ﬁnding suggests that the degree of unfairness is heterogeneous, and certain occupations may be associated with higher lev el of discrimination, highlighting the importance of moving beyond aggregate measures to understand ho w the nature of unfairness varies across subgroups. T aken together, these results indicate statis- tically signiﬁcant unfairness in the observed decisions, with the dominant contribution arising from discrimination against Black applicants, particularly for certain occupations. Preferential treatment to wards White applicants appears comparati vely mild. W e further conduct sensitivity analyses to assess robustness to potential violations of Assump- tions 1–3; details are provided in the supplementary material. The results indicate that the predicted decision ˆ Y ∗ and the estimated degree of unfairness ˆ θ are relativ ely insensiti ve to mild or moderate departures from these assumptions, whereas more se vere violations can lead to substantial changes. T able 4: Conditional proportions of predicted decisions among all resumes UML FTU MLC LD DSD 0 1 0 1 0 1 0 1 0 0 . 9665 0 . 0335 0 . 9736 0 . 0264 0 . 9692 0 . 0308 0 . 9692 0 . 0308 1 0.3852 0.6148 0 . 3010 0 . 6990 0 . 3520 0 . 6480 0 . 3520 0 . 6480 T able 5: Conditional proportions of predicted decisions among resumes with Black-sounding names UML FTU MLC LD DSD 0 1 0 1 0 1 0 1 0 0 . 9933 0 . 0067 0 . 9767 0 . 0233 0 . 9781 0 . 0219 0 . 9790 0 . 0210 1 0.6535 0.3465 0 . 3366 0 . 6634 0 . 5149 0 . 4851 0 . 5149 0 . 4851 26 Frequency 0.0 0.2 0.4 0.6 0.8 1.0 0 500 1000 1500 2000 2500 α ^ Frequency 0.0 0.2 0.4 0.6 0.8 1.0 0 1000 2000 3000 4000 β ^ Figure 4: Histograms of the estimated unfairness mechanism ˆ α and ˆ β . Frequency 0.0 0.2 0.4 0.6 0.8 1.0 0 50 100 150 200 250 α ^ Figure 5: Histogram of ˆ α for applicants targeting sales representati ve positions. 8. Discussion W e establish a framework for characterising and addressing fairness that builds on the notion of the desert decision. Although some readers may vie w this construct as metaphysical or un veriﬁable, we argue that introducing the desert decision can deepen our understanding of fairness and motiv ate the de velopment of statistical methods that are useful in practice, including the approach proposed here. Building on the desert decision, we deﬁne parameters for assessing and achieving fairness, estab- lish identiﬁcation under transparent and causally interpretable assumptions, and dev elop correspond- ing estimation procedures. In contrast to most existing approaches that target the observed decision 27 while imposing fairness constraints, our framework targets the desert decision directly . T o achiev e identiﬁcation of the desert decision rule, we rely on assumptions on the fairness property of the desert decision and the unfairness mechanism generating the observed decision, which differs fundamen- tally from approaches that deﬁne target decision rules through constrained statistical learning. For estimation, we propose a sieve maximum likelihood estimator for the desert decision rule τ and the unfairness mechanism ( α, β ) , and an inﬂuence-function-based estimator of the de gree of unfairness θ . W e further dev elop sensitivity analysis procedures that allo w for departures from Assumptions 1–3; it remains of interest to relax these assumptions further . Several e xtensions merit in vestigation in the future, including ﬂexible estimation of τ in high-dimensional settings and more precise inference for θ via the efﬁcient inﬂuence function and the corresponding semiparametrically ef ﬁcient estimator . Our proposal also contributes to the measurement error literature. Under this interpretation, the desert decision is an unobserved “true” v ariable, while the observ ed decision is a potentially misclas- siﬁed proxy . T o the best of our knowledge, our identiﬁcation results under Assumptions 1–3 provide ne w insights for binary choice models with misclassiﬁcation. It would also be of interest to dev elop partial identiﬁcation results when these assumptions are relaxed (see, e.g., Molinari, 2008; Mondal and W ang, 2024). As ongoing work, we are extending our identiﬁcation arguments beyond binary v ariables to accommodate more general outcomes and decision spaces. Supplementary material Supplementary material online includes a generativ e model for the observed decision subject to un- fairness, characterisations of the bias of T z ( X ) under violations of the unfairness mechanism, sen- siti vity analysis procedures for the degree of unfairness θ , proof of theorems and important results, implementation details of comparing methods, and additional results for the simulation and applica- tion studies. References 7th Circuit Court (1996). Carson vs Bethlehem Steel Corp. 70 FEP Cases 921. Angrist, J. D., G. W . Imbens, and D. B. Rubin (1996). Identiﬁcation of causal ef fects using instru- mental v ariables. J ournal of the American Statistical Association 91 , 444–455. 28 Angwin, J., J. Larson, S. Mattu, and L. Kirchner (2022). Machine bias. In Ethics of data and analytics , pp. 254–264. Auerbach Publications. Balke, A. and J. Pearl (1997). Bounds on treatment effects from studies with imperfect compliance. J ournal of the American Statistical Association 92 , 1171–1176. Bang, H. and J. M. Robins (2005). Doubly robust estimation in missing data and causal inference models. Biometrics 61 , 962–973. Barocas, S., M. Hardt, and A. Narayanan (2023). F airness and Machine Learning: Limitations and Opportunities . MIT Press. Becker , G. S. (2010). The Economics of Discrimination . Univ ersity of Chicago Press. Bertrand, M. and S. Mullainathan (2004). Are Emily and Greg more employable than Lakisha and Jamal? A ﬁeld experiment on labor market discrimination. American Economic Review 94 , 991– 1013. Bickel, P . J., C. A. J. Klaassen, Y . Ritov , and J. A. W ellner (1993). Efﬁcient and Adaptive Estimation for Semiparametric Models . Johns Hopkins Univ ersity Press Baltimore. Brennan, T ., W . Dieterich, and B. Ehret (2009). Ev aluating the predictiv e validity of the COMP AS risk and needs assessment system. Criminal J ustice and Behavior 36 , 21–40. Calmon, F ., D. W ei, B. V inzamuri, K. Natesan Ramamurthy , and K. R. V arshney (2017). Opti- mized pre-processing for discrimination prev ention. Advances in Neural Information Pr ocessing Systems 30 . Chen, H., W . Lu, R. Song, and P . Ghosh (2024). On learning and testing of counterfactual fairness through data preprocessing. J ournal of the American Statistical Association 119 , 1286–1296. Chen, X. (2007). Large sample sie ve estimation of semi-nonparametric models. Handbook of Econo- metrics 6 , 5549–5632. Chernozhuko v , V ., D. Chetveriko v , M. Demirer , E. Duﬂo, C. Hansen, W . Ne wey , and J. Robins (2018). Double/debiased machine learning for treatment and structural parameters. The Econometrics Jour - nal 21 , C1–C68. Chouldechov a, A. (2017). Fair prediction with disparate impact: A study of bias in recidivism pre- diction instruments. Big Data 5 , 153–163. 29 Darity Jr , W . A. and P . L. Mason (1998). Evidence on discrimination in employment: Codes of color , codes of gender . Journal of Economic P erspectives 12 , 63–90. Donini, M., L. Oneto, S. Ben-David, J. S. Shawe-T aylor , and M. Pontil (2018). Empirical risk mini- mization under fairness constraints. Advances in Neural Information Pr ocessing Systems 31 . Dwork, C., M. Hardt, T . Pitassi, O. Reingold, and R. Zemel (2012). Fairness through aw areness. In Pr oceedings of the 3r d Innovations in Theor etical Computer Science Confer ence , pp. 214–226. Fan, J., X. T ong, Y . W u, and S. Y ao (2023). Neyman-pearson and equal opportunity: when ef ﬁciency meets fairness in classiﬁcation. arXiv pr eprint arXiv:2310.01009 . Feinberg, J. (1970). Justice and personal desert. In Rights and r eason: Essays in honor of Carl W ellman , pp. 221–250. Springer . Hand, D. J. and W . E. Henley (1997). Statistical classiﬁcation methods in consumer credit scoring: a re view . Journal of the Royal Statistical Society: Series A 160 , 523–541. Hardt, M., E. Price, and N. Srebro (2016). Equality of opportunity in supervised learning. Advances in Neural Information Pr ocessing Systems 29 . He, X. and Y . Li (2025). Fairness in machine learning: A re view for statisticians. Journal of the American Statistical Association 120 , 2834–2851. Hu, Y . (2008). Identiﬁcation and estimation of nonlinear models with misclassiﬁcation error using instrumental v ariables: A general solution. J ournal of Econometrics 144 , 27–61. Imai, K. and Z. Jiang (2023). Principal fairness for human and algorithmic decision-making. Statis- tical Science 38 , 317–328. Jiang, H. and O. Nachum (2020). Identifying and correcting label bias in machine learning. In International Confer ence on Artiﬁcial Intelligence and Statistics , pp. 702–712. Kale v , A., F . Dobbin, and E. K elly (2006). Best practices or best guesses? Assessing the efﬁcac y of corporate af ﬁrmativ e action and div ersity policies. American Sociological Re view 71 , 589–617. Kamiran, F . and T . Calders (2012). Data preprocessing techniques for classiﬁcation without discrim- ination. Knowledge and Information Systems 33 , 1–33. 30 Kamiran, F . and I. ˇ Zliobait ˙ e (2013). Explainable and non-explainable discrimination in classiﬁcation. In Discrimination and Privacy in the Information Society: Data Mining and Pr oﬁling in Lar ge Databases , pp. 155–170. Springer . Kitagaw a, T . (2015). A test for instrument v alidity . Econometrica 83 , 2043–2063. Kleinberg, J., J. Ludwig, S. Mullainathan, and C. R. Sunstein (2018). Discrimination in the age of algorithms. J ournal of Le gal Analysis 10 , 113–174. Kleinberg, J., S. Mullainathan, and M. Ragha van (2016). Inherent trade-of fs in the fair determination of risk scores. arXiv pr eprint arXiv:1609.05807 . Kusner , M. J., J. Loftus, C. Russell, and R. Silva (2017). Counterfactual f airness. Advances in Neural Information Pr ocessing Systems 30 . Le wbel, A. (2007). Estimation of av erage treatment effects with misclassiﬁcation. Econometrica 75 , 537–551. Luan, H. and C.-C. Tsai (2021). A revie w of using machine learning approaches for precision educa- tion. Educational T echnolo gy & Society 24 , 250–266. Mahajan, A. (2006). Identiﬁcation and estimation of regression models with misclassiﬁcation. Econo- metrica 74 , 631–665. McFadden, D. (2001). Economic choices. American Economic Revie w 91 , 351–378. Mehrabi, N., F . Morstatter , N. Saxena, K. Lerman, and A. Galstyan (2021). A survey on bias and fairness in machine learning. ACM Computing Surve ys 54 , 1–35. Miller , D. (2001). Principles of Social J ustice . Harvard Uni versity Press. Millimet, D. L. and C. F . P armeter (2022). Accounting for ske wed or one-sided measurement error in the dependent v ariable. P olitical Analysis 30 , 66–88. Mitchell, S., E. Potash, S. Barocas, A. D’Amour , and K. Lum (2021). Algorithmic fairness: Choices, assumptions, and deﬁnitions. Annual Re view of Statistics and Its Application 8 , 141–163. Molinari, F . (2008). Partial identiﬁcation of probability distrib utions with misclassiﬁed data. Journal of Econometrics 144 , 81–117. 31 Mondal, O. and R. W ang (2024). Partial identiﬁcation of binary choice models with misreported outcomes. arXiv pr eprint arXiv:2401.17137 . Nabi, R., N. S. Hejazi, M. J. van der Laan, and D. Benkeser (2024). Statistical learning for constrained functional parameters in inﬁnite-dimensional models with applications in fair machine learning. arXiv pr eprint arXiv:2404.09847 . Nabi, R. and I. Shpitser (2018). Fair inference on outcomes. In Pr oceedings of the AAAI Conference on Artiﬁcial Intelligence , V olume 32. Nature Editorials (2016). More accountability for big-data algorithms. Natur e 537 , 449. Nguimkeu, P ., A. Denteh, and R. Tchernis (2019). On the estimation of treatment ef fects with en- dogenous misreporting. J ournal of Econometrics 208 , 487–506. Pessach, D., G. Singer , D. A vrahami, H. C. Ben-Gal, E. Shmueli, and I. Ben-Gal (2020). Employees recruitment: A prescriptiv e analytics approach via machine learning and mathematical program- ming. Decision Support Systems 134 , 113290. Rawls, J. (1971). A Theory of J ustice . Harv ard University Press. Robins, J., L. Li, E. Tchetgen Tchetgen, and A. v an der V aart (2008). Higher order inﬂuence functions and minimax estimation of nonlinear functionals. In D. Nolan and T . Speed (Eds.), Pr obability and Statistics: Essays in Honor of David A. F r eedman , V olume 2, pp. 335–421. Beachwood, Ohio: Institute of Mathematical Statistics. Sadurski, W . (1985). Giving Desert Its Due: Social J ustice and Le gal Theory . Springer Science & Business Media. Shehab, M., L. Abualigah, Q. Shambour , M. A. Abu-Hashem, M. K. Y . Shambour , A. I. Alsalibi, and A. H. Gandomi (2022). Machine learning in medical applications: A revie w of state-of-the-art methods. Computers in Biolo gy and Medicine 145 , 105458. Thomas, L., J. Crook, and D. Edelman (2017). Cr edit Scoring and Its Applications . Society for Industrial and Applied Mathematics. T rain, K. E. (2009). Discr ete Choice Methods with Simulation . Cambridge Univ ersity Press. Tsiatis, A. (2006). Semipar ametric Theory and Missing Data . Ne w Y ork: Springer . Tsybako v , A. B. (2009). Intr oduction to Nonparametric Estimation . Springer . 32 v an der V aart, A. and J. A. W ellner (1996). W eak Con ver gence and Empirical Pr ocesses: with Appli- cations to Statistics . Springer Science & Business Media. V anderford, R. (2022). AI hiring tools can violate disability protections, go vernment w arns. The W all Str eet Journal . W ainwright, M. J. (2019). High-Dimensional Statistics: A Non-Asymptotic V iewpoint . Cambridge Uni versity Press. W orld Trade Organization (2001). Doha ministerial declaration. https://www.wto.org/ english/thewto_e/minist_e/min01_e/mindecl_e.htm . Zafar , M. B., I. V alera, M. Gomez-Rodriguez, and K. P . Gummadi (2019). Fairness constraints: A ﬂexible approach for f air classiﬁcation. J ournal of Machine Learning Resear ch 20 , 1–42. Zhang, L., Y . W u, and X. W u (2017). A causal frame work for discovering and removing direct and indirect discrimination. In Pr oceedings of the 26th International J oint Confer ence on Artiﬁcial Intelligence , pp. 3929–3935. Zhao, H. and G. J. Gordon (2022). Inherent tradeof fs in learning fair representations. Journal of Machine Learning Resear ch 23 , 1–26. 33

Identifying the desert decision rule to assess and achieve fairness

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