Equal Opportunity and Luck: Empirical Exploration Using the Canadian Longitudinal Study on Aging

TL;DR

This paper explores how luck affects equality of opportunity in aging populations using data from the Canadian Longitudinal Study on Aging.

Contribution

It introduces the 'playing field framework' to assess whether unexplained variation (luck) reflects unfair inequality in equality of opportunity analyses.

Findings

Distributions of residuals (luck) are not always fair across age-sex groups.

Luck should be explicitly considered in equality of opportunity frameworks.

Ignoring residuals can lead to underestimating unfair inequality.

Abstract

Equality of opportunity (EOp) is a broad category of egalitarian theories that has attracted considerable attention in recent decades. Empirical implementations of EOp primarily focus on the explained component of inequality, classifying determinants of the outcome (e.g., health) into effort—legitimate causes of inequality—and circumstance—illegitimate causes of inequality. Largely overlooked is unexplained variation, which in statistical analysis manifests as residuals and is often ignored as a statistical annoyance. The true random component of residuals is now often referred to as luck. In this paper, we propose the playing field framework that serves as a pragmatic test as to whether residuals signal unfairness in empirical EOp analyses and that enables empirical explorations of roles of luck within the EOp framework. Using a large sample of Canadian older adults, our empirical application of the playing field framework shows that distributions of residuals are not always fair, though there is no consistent pattern of unfairness across age-sex groups. The paper’s three main conclusions are: luck matters; luck should be explicitly incorporated in the EOp framework through the brute luck-effort characterization; and residuals are not just an innocuous statistical annoyance but can represent unfair inequality, and ignoring them can underestimate unfair inequality.

Figures

Extended, fictionalized Gauguin and typologies of luck. D: Dworkin; LPT: Lefranc, Pistolesi, Trannoy; N: Nagel; V: Vallentyne

Frailty and typologies of luck. D: Dworkin; LPT: Lefranc, Pistolesi, Trannoy; N: Nagel; V: Vallentyne

Playing field, within-tranche comparison, and within-type comparison

Examples of distributions of luck by circumstance and effort, 55–64 years female and 75–84 years male. Note: Distributions for 55–64 years old females show covariation of residuals both with social background luck and effort, and distributions for 75–84 years old males show no covariation between residuals and social background luck and between residuals and effort. The statistical assessment of these distributions is in Table [1](#Tab1)

Full text

Introduction

Health inequality is a pressing and persistent concern in many jurisdictions globally. Strict equality in health, however, is rarely identified as the goal, and society typically strives for alleviating unfair health inequality. Egalitarian theories offer diverse accounts as to which health inequality should be considered unfair and why (MacKay & Sreenivasan, 2021). Equality of opportunity (“EOp” henceforth), a broad category of egalitarian theories, has become predominant among egalitarian theories in recent decades (Elford, 2023). EOp states that inequality due to factors beyond individual control is unfair and inequality due to factors within individual control is fair. According to this view, for example, inequality in health resulting from childhood socioeconomic circumstances is unfair, while inequality resulting from a freely chosen commitment to health-enabling behaviours is fair.

In the standard understanding of EOp operationalized for empirical implementation, a person has an opportunity set, exercises freedom, then realizes the outcome (Trannoy, 2016). Early efforts to implement EOp empirically focused on measuring opportunity sets directly. But because opportunity sets are unobservable, measuring them is difficult. As a result, this effort largely died out (Ferreira & Peragine, 2016; Trannoy, 2016). Most empirical EOp work now takes the indirect approach, which builds on the opportunity-freedom-outcome understanding. Two perspectives are possible for the indirect approach. The ex ante perspective examines inequality after the opportunity set is revealed but before freedom is exercised. Alternatively, the ex post perspective examines inequality after both the opportunity set is revealed and freedom is exercised. In now standard terminology, the revealed opportunity set is referred to as circumstance and the exercised freedom as effort (Roemer, 1998). EOp is operationalized by either the compensation or reward principle. The compensation principle states that, from the ex ante perspective, inequality between circumstance types is to be compensated, whereas from the ex post perspective, inequality within each effort tranche (i.e., each effort level) is to be compensated (Ferreira & Peragine, 2016). The reward principle aims to preserve inequality between effort tranches within each circumstance type, and how to meet this aim leads to different versions of the reward principle (Ferreira & Peragine, 2016). EOp does not demand satisfaction of both the compensation and reward principles; in fact, they often conflict, such as, when circumstance and effort correlate (Fleurbaey & Schokkaert, 2011).

Diverse methodological approaches have been developed to implement empirically the compensation and reward principles of EOp. Empirical development has focused primarily on the statistically “explainable” component of inequality (e.g., García-Gómez et al., 2015; Kovacic & Orso, 2022). Researchers have debated extensively which explanatory variables should be classified as circumstance—illegitimate causes of inequality—or effort—legitimate causes of inequality—within the constraints of imperfect real-world data (Jusot & Tubeuf, 2019). They have also examined potential interactions between circumstance and effort (e.g., how family background influences preferences over health behaviours) and their implications for estimates of unfair inequality (Bricard et al., 2013; Jusot et al., 2013). While these issues are important, they are relevant only to the component of inequality that can be explained by circumstance and effort variables. Empirically, however, much variation is left unexplained. Unexplained variation in recent good-quality empirical studies, for example, has accounted for 53–83% of all variation in health (Asada et al., 2018; Jusot et al., 2013), 52–85% in income (Almås et al., 2011; Ferreira & Gignoux, 2008, 2011), and 71–98% in education (Brunori et al., 2010). Unexplained variation in statistical analysis manifests as residuals and is often ignored as a statistical annoyance, largely overlooked in the empirical literature. For example, the increasingly popular empirical implementation of ex ante EOp perspective using the Shapley decomposition method focuses exclusively on the component of inequality explained by circumstance variables, implying residuals are unimportant and uninteresting from the perspective of EOp (Kovacic & Orso, 2022). Whether unexplained variation is classified as fair or unfair, however, drastically changes the estimate of unfair inequality. The estimate of unfair health inequality is 30% greater when classifying unexplained variation as unfair rather than fair (Asada et al., 2015), a much larger impact than the 7% difference observed when the frequently debated ethical status of health behaviours is changed (Asada et al., 2014). How much unfair inequality exists at any given time in society is fundamental information for guiding policy development. The lack of attention to unexplained inequality and its ethical status could misguide social and health policy.

The limited empirical EOp literature has paid attention to unexplained variation through concerns for randomness and residuals. Trannoy (2016) and Ferreira and Peragine (2016) argue that pervasive randomness in our everyday life cannot be ignored in the EOp framework of circumstance and effort. Fleurbaey and Schokkaert (2011), Roemer and Trannoy (2016), and Jusot and Tubeuf (2019), on the other hand, acknowledge the issue as a problem of residuals or unexplained inequality in econometric models. Residuals include both systematic errors, resulting from omitted variables and model misspecification, and true randomness. True randomness is now often referred to as luck (Lefranc et al., 2009; Lefranc & Trannoy, 2017). However, the empirical investigation of luck, and specifically the challenge of identifying the component of unexplained inequality that is luck, is formidable (see, e.g., Bago D’uva et al., 2012; Lefranc et al., 2009).

This paper explores the role of luck in empirical EOp analyses and proposes a pragmatic framework, hereafter referred to as the playing field, to guide such analyses. Because a component of residuals is luck, the playing field framework builds on existing typologies of luck and ethical analysis of different types of luck. The playing field assesses covariations between residuals and the type of luck that can be considered unfair and between residuals and the type of luck that can be considered fair. The playing field, thus, serves as a pragmatic test as to whether residuals signal unfairness in empirical EOp analyses. Furthermore, the assessment of covariations between residuals and different types of luck enables empirical explorations of the role of luck within the EOp framework.

The plan of this paper is as follows. As a theoretical foundation for the playing field framework, Sect. 2 synthesizes three well-developed typologies of luck in economics and philosophy and examines which types of luck are considered fair or unfair. Section 3 introduces the playing field framework. Section 4 empirically illustrates this framework using a population-based health dataset and standard parametric econometric methods. Section 5 reports empirical results, showing how residuals covary with a type of luck that can be considered as unfair at least in some population subgroups. Section 6 concludes the paper with three main points: luck plays important roles within the EOp framework; the empirical operationalization of EOp should be reframed from circumstance and effort to brute luck and effort; and the playing field framework suggests residuals are more than an innocuous annoyance in empirical implementations of EOp but can represent unfair inequality.

Typologies of Luck

We synthesize three prominent typologies of luck and examine the ethical status of luck. The first two—one by Lefranc et al. (2009) and Lefranc and Trannoy (2017) and another by Dworkin (1981a, b)—are familiar in the economics literature. The third, by the philosopher Thomas Nagel, derives from the concept of moral luck (Williams & Nagel, 1976). Despite philosophical inquiry of moral luck over 40 years and its relevance, Nagel’s typology is not widely discussed in the philosophical (but see Lippert-Rasmussen, 2023) and empirical EOp literature. These different typologies share some common conceptions of luck, although different scholars apply different labels to them. All three typologies also align with the opportunity-freedom-outcome understanding central to EOp.

Lefranc et al. (2009) and Lefranc and Trannoy (2017) provide the most comprehensive economics treatment of luck, introducing four types of luck: genetic luck, social circumstance luck, accidental luck, and moral hazard (“LPT typology” henceforth). Luck, they argue, is neither circumstance nor effort, and an examination of types of luck is necessary to determine the ethical status of luck as a cause of inequality. This LPT typology corresponds closely to long-standing philosophical inquiries into luck such as Dworkin’s typology of brute and option luck (1981a, 1981b). Prior to LPT, Dworkin’s typology had arguably been the most frequently relied upon view of randomness in the empirical EOp literature (e.g., Fleurbaey, 2008). The philosopher Bernard Williams offers another influential concept of moral luck (Nelkin, 2021): situations where individuals are morally assessed even though what they are assessed for is beyond their control (Williams & Nagel, 1976). Moral luck threatens the fundamental premise of moral philosophy, since moral assessments of actions taken by individuals apply only to those actions under their control. Responding to Williams, in a third typology, Nagel analyzes moral luck in four ways: resultant, circumstantial, constitutive, and causal luck (Williams & Nagel, 1976).

To illustrate these various types of luck, we extend the case of the somewhat fictionalized life of the painter Gauguin originally discussed by Williams and Nagel (1976). All references to Gauguin below are to our scenario. Gauguin’s life consists of four random events (i.e., luck) and one decision point. Figure 1 depicts his life with circles for random events and a square for the decision point. Imagine Gauguin and the associated first random event, whether or not he is talented for painting with a disposition to pursue it. This is what Lefranc et al. (2009) call genetic luck and Nagel constitutive luck: “the kind of person you are, where this is not just a question of what you deliberately do, but of your inclinations, capacities, and temperament” (Williams & Nagel, 1976, p. 140). Assuming he won the genetic talent lottery, Gaugin then experienced the second random event, whether or not the family he was born into was poor. This random event is referred to as circumstance in the EOp framework. Although circumstance is not explicitly called luck, it is a type of luck, a birth lottery related to family and social background, hence referred to as social background luck by LPT (Lefranc & Trannoy, 2017; Lefranc et al., 2009). Genetic luck and social background luck are intertwined; had Gaugin been born into a family in poverty, a talented Gauguin might have ended his life without even noticing or developing his talent. In contrast, if he won both the genetic and birth lottery, a talented Gauguin from a wealthy family continued on in his life, became a successful stockbroker and picked up painting in his free time. He then had the third random event, Paris stock market crash. This is what Lefranc and Trannoy (2017) call accidental luck and Nagel circumstantial luck: “the kind of problems and situations one faces” (Williams & Nagel, 1976, p. 140). Accidental or circumstantial luck may be a health event such as a disease, a period event such as a pandemic, and serendipity such as meeting a good teacher, but for Gauguin, a stock market crash reduced his income but also contributed to his transition to a full-time painter. For Nagel, circumstantial luck includes circumstance in the standard EOp empirical operationalization, and constitutive and circumstantial luck together represents causal luck, “the product of antecedent circumstances outside of the will’s control” (Williams & Nagel, 1976, p. 146).Fig. 1. Extended, fictionalized Gauguin and typologies of luck. D: Dworkin; LPT: Lefranc, Pistolesi, Trannoy; N: Nagel; V: Vallentyne

After some years, Gauguin faced a choice: whether to go to Tahiti or stay in Paris. This choice or decision is referred to as effort in the standard EOp empirical operationalization. This choice is a deliberate gamble for Gauguin—by staying in Paris, he would likely be an average painter; by going to Tahiti, he would either be a magnificent painter or a miserable painter (miserable in the sense that his success as a painter and associated living standard would be much less than in Paris). This outcome is the last type of luck, the one closely connected to the prior decision. Gauguin’s Tahiti decision changed the probabilities of the potential outcomes—average, magnificent, or miserable painter—hence, between the decision and the outcome, there is a luck that is influenced by the decision, and in turn, influences the outcome. From the ex ante perspective—before making the decision and before knowing how the gamble would turn out—Lefranc and Trannoy (2017) call this luck moral hazard, a favourable (unfavourable) shift of the outcome distribution due to the high (low) effort. From the ex post perspective—after making the decision—subsequent randomness—whether Gauguin becomes an average, magnificent, or miserable painter—reflects what Nagel calls resultant luck: “luck, good or bad, in the way things turn out” (Williams & Nagel, 1976, p. 140) and what Dworkin calls option luck: “a matter of how deliberate and calculated gambles turn out—whether someone gains or loses through accepting an isolated risk he or she should have anticipated and might have declined” (Dworkin, 1981b, p. 293). Dworkin contrasts such option luck with brute luck, which he defines as “a matter of how risks fall out that are not in that sense deliberate gambles” (Dworkin, 1981b, p. 293). In our fictionalized case of Gauguin depicted in Fig. 1, brute luck thus includes the three types of luck (genetic/constitutive luck, circumstance, and accidental/circumstantial luck) that happened before the decision point. Vallentyne (2002) further categorized brute luck into initial brute luck—one that happens before the onset of agency (i.e., adulthood)—and late brute luck—one that happens after the onset of agency. Thus, initial brute luck includes genetic/constitutive luck and social background luck, and late brute luck corresponds to accidental/circumstantial luck.

These three typologies of luck offer diverse views as to which types of luck are legitimate or illegitimate causes of inequality. For example, genetic/constitutive luck is the kind of luck that falls under the skin and becomes constituent of an individual. Thus, libertarians who honour individual self-ownership may consider it as a legitimate cause of inequality and argue for preserving its influence on differential outcomes (but then libertarians are unlikely to be interested in egalitarian theories of EOp). Lefranc and Trannoy (2017) discuss a possibility of asymmetrical ethical assessment of good and bad luck: society might wish to respect inequality due to good genetic luck but to compensate for inequality due to bad genetic luck. Vallentyne (2002) considers “initial opportunities for advantage” (initial brute luck) as an illegitimate cause of inequality and argues that any luck before the onset of agency (e.g., adulthood) should be compensated.

Furthermore, the ethical status of option luck has been richly debated. Dworkin argues against compensating bad option luck as it is part of the package for the choice made (1981a, 1981b). Fleurbaey, however, argues that whether or not to compensate bad option luck depends on a person’s degree of risk preference (2008). For Gauguin, Dworkin would not compensate for ending up a miserable painter should Gauguin become one. Fleurbaey would need to know Gauguin’s risk aversion. If Gauguin turned out to be a miserable painter, regretted his decision, and vowed not to take such a gamble again, Fleurbaey would compensate for Gauguin’s loss as a miserable painter. If Gauguin turned out to be a miserable painter, but did not regret his decision, and would take such a gamble again should there be another opportunity, Fleurbaey would not compensate Gauguin. While Dworkin and Fleurbaey consider choice (or effort) and outcomes of the option luck in a binary manner, Lefranc and Trannoy (2017) introduce a distributional consideration for option luck: the benefit (drawback) of high (low) effort includes a favourable (unfavourable) shift of the outcome distribution. According to Lefranc and Trannoy (2017), moral hazard (or option luck) is a legitimate cause of inequality. More precisely, ex ante, Lefranc and Trannoy would respect Gauguin’s decision to go to Tahiti that would shift the outcome distribution favourably to include a possibility of becoming a magnificent painter. Ex post, they would also respect Gauguin’s decision to go to Tahiti, but, when assessing the success of his painting career due to his Tahiti decision, they would account for the option luck that had fallen on him.

Two observations emerge from this synthesis. First, luck plays a critical role in EOp. Indeed, in Roemer’s pragmatic formulation of EOp that became the standard terminology of circumstance and effort, circumstance is originally defined as causes beyond individual control (1993). If taken literally, circumstance in Roemer could have included all types of luck, or at least all forms of brute luck. In Roemer’s operationalization, however, everything that is not captured by observable and measurable circumstance is considered as effort (Lefranc & Trannoy, 2017; Roemer, 1993). LPT moved luck to the forefront of the standard EOp empirical operationalization and advocated that luck be included as a third element to circumstance and effort. However, given that circumstance is one form of luck, we believe that it is more straightforward to frame the EOp framework in terms of luck and effort rather than circumstance, effort, and luck. Therefore, we concur with the literal interpretation of Roemer’s original formulation of circumstance as causes beyond individual control, whether or not it is observable or measurable.

Second, the luck-effort characterization of the EOp framework follows Dworkin’s distinction between brute and option luck: brute luck (i.e., genetic luck, social background luck, and accidental luck) is unfair while option luck is fair. Further, we advocate for the distributional understanding of option luck (i.e., moral hazard), as opposed to the binary understanding of outcomes of option luck. By positing that effort shifts the outcome distribution, the moral hazard view provides clear support for the legitimacy of option luck. In consideration for this dichotomy of the ethical status of brute and option luck, we restate the EOp framework as brute luck and effort, with the acknowledgment of effort associated with moral hazard.

This brute luck-effort characterization of the EOp framework, together with the ethical assessment of brute luck as unfair and moral hazard as fair, provides a foundation for the playing field framework introduced in the next section.

Playing Field Framework

The playing field framework assesses fairness and unfairness of residuals in empirical implementations of EOp through covariations between residuals and brute luck and residuals and effort (including moral hazard). If residuals covary with brute luck, an unfair source of inequality, then, residuals signal unfairness; if residuals covary with effort (and moral hazard), a fair source of inequality, then, residuals do not signal unfairness. The playing field functions both as a pragmatic test as to whether residuals signal unfairness in empirical EOp analyses and as a framework for empirically exploring the roles of luck within the EOp framework.

Below we first define residuals. We then explain how the playing field framework builds on the existing typologies of luck discussed in Sect. 2. Subsequently, we formally define the playing field and how it considers covariations between residuals and brute luck and between residuals and effort as fair or unfair. Introducing the playing field framework, we use a health outcome, but the framework can be applied to other outcomes such as income and education.

Following our formulation of EOp in terms of brute luck and effort, we assume the observed health of an individual ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h$$\end{document} ) results from some function ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g$$\end{document} ) of luck ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L$$\end{document} ) and effort ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E$$\end{document} ) of the individual: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h = g(L, E)$$\end{document} . Luck of the individual is determined by some function of brute luck ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B$$\end{document} ) and moral hazard ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M$$\end{document} ): \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L = L(B, M)$$\end{document} . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M$$\end{document} are random variables. B is a random variable that does not depend on any other variable. Moral hazard ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M$$\end{document} ), following the characterization by Lefranc and Trannoy (2017), follows the distribution function ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f$$\end{document} ) of a random variable parameterized by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E$$\end{document} is not random and is determined by the individual.

In empirical work, the researcher cannot observe all elements of these variables. Hence, brute luck ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B$$\end{document} ) comprises observed brute luck ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b$$\end{document} ) and unobserved brute luck ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${b}^{*}$$\end{document} ): \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B = B(b, { b}^{*})$$\end{document} , and effort ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E$$\end{document} ) comprises observed effort ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e$$\end{document} ) and unobserved effort ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${e}^{*}$$\end{document} ): \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E(e, {e}^{*})$$\end{document} . Note how \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${e}^{*}$$\end{document} combine to yield \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E$$\end{document} depends on brute luck, hence: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E = E(e, {e}^{*}| b, { b}^{*})$$\end{document} . Following from above, moral hazard follows the distribution function ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f$$\end{document} ) of a random variable parameterized by observed effort ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e$$\end{document} ) and unobserved effort ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${e}^{*}$$\end{document} ): \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M \leadsto f(e, {e}^{*})$$\end{document} . Empirically, therefore, we predict health based on the health regression: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{h}=\widehat{g}\left(b,e\right)+\widehat{\varepsilon }\left({ b}^{*}, e, {e}^{*}\right)$$\end{document} . Residuals ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{\varepsilon })$$\end{document} capture what we cannot observe, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${b}^{*}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${e}^{*}$$\end{document} , as well as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M$$\end{document} , which is why \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{\varepsilon }$$\end{document} is a function of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e$$\end{document} .

To assess fairness and unfairness of the distribution of health residuals, consider a similar schema to the one used for the fictionalized Gauguin case discussed in Sect. 2. Figure 2 depicts the presence of hereditary disease risk (genetic luck), family wealth (social background luck), the presence of major health events under 50 years old (accidental luck), health habits (effort), and option luck associated with health habits to determine the level of frailty of an older person. Recall that brute luck is beyond individual control, thus unfair according to EOp and that option luck, or more precisely, the moral hazard characterization of option luck, is considered fair because it is associated with precedent effort. Because covariation between residuals and brute luck varies by effort, and covariation between residuals and effort varies by brute luck, we ask: do residuals covary with brute luck conditional upon effort, and do residuals covary with effort conditional upon brute luck?Fig. 2. Frailty and typologies of luck. D: Dworkin; LPT: Lefranc, Pistolesi, Trannoy; N: Nagel; V: Vallentyne

Formally, to investigate covariations between residuals and brute luck and between residuals and effort, select one particular type of brute luck (i.e., genetic luck, social background luck, or accidental luck) and one particular effort. Consider a matrix for four subgroups of individuals, two brute luck types (bad vs. good) in rows and two effort tranches (low vs. high) in columns (Fig. 3). We call this 2 × 2 matrix the playing field. The playing field framework uses that matrix to draw the distribution of residuals for each of the four subgroups of individuals and then measure covariations across tranches or types. Similar to the circumstance-effort matrix comparisons for the standard EOp without consideration for luck (Ferreira & Peragine, 2016), we can make brute luck-effort matrix comparisons. To examine the covariation between residuals and brute luck (i.e., do residuals vary across brute luck conditional upon effort?), we can make within-tranche comparisons. To examine the covariation between residuals and effort (i.e., do residuals vary across efforts conditional upon brute luck?), we can make within-type comparisons.Fig. 3. Playing field, within-tranche comparison, and within-type comparison

When comparing distributions, we posit that greater dispersion of residuals worse than less dispersion of residuals because the former suggests greater uncertainty of the outcome than the latter. Because better brute luck and greater effort are generally associated with better outcomes, we further assume better brute luck and greater effort are associated with less uncertainty of the outcome, that is, less dispersion of residuals.

To assess the fairness and unfairness of the distribution of residuals, within-tranche comparison (column) asks whether the distribution of residuals varies by brute luck conditional upon effort. When the distribution of residuals does not differ by brute luck, that is, there is no covariation between residuals and brute luck, we consider the distribution of residuals to be fair. If we see covariation among either one of the two tranches (low effort, high effort) or both, we conclude there is unfairness in the distribution of residuals. In our health context, brute luck on which we focus is social background luck measured by childhood family wealth (i.e., rich (good luck) vs. non-rich (bad luck)) and the effort on which we focus is health habits (i.e., healthy habits (high effort) vs. unhealthy habits (low effort)). Then, unfairness in the case of covariation might be expressed as follows by the non-rich with unhealthy habits, “I did not practice good health behaviour, but it is unfair I also experience greater uncertainty of the outcome than if I were rich with unhealthy habits, just because of my bad social background luck,” and by the non-rich with healthy habits, “I tried my best to practice good health behaviour. It is unfair, just because of my bad social background luck, that I end up with having greater uncertainty than if I were rich with healthy habits.”

The within-type comparison (row) asks whether the distribution of residuals varies by effort conditional upon brute luck. When the distribution of residuals varies by effort in a similar manner across bad and good brute luck—high effort reduces the degree of uncertainty regardless of brute luck—we consider these distributions of residuals to be fair. When the distribution of residuals varies by effort differently across bad and good brute luck, we consider such differential covariation by brute luck as unfair. The situation when the distribution of residuals does not differ by effort (i.e., no covariation) goes counter to the moral hazard characterization of luck in the sense that good health habits are not fully rewarded in the form of reduced uncertainty of the outcome. We consider no covariation neither fair nor unfair. Assume, as before, that social background luck, measured by family wealth is a type of brute luck and that health habits is a kind of effort. Should residuals covary by effort only among the rich, unfairness might be expressed as follows by the non-rich with healthy habits, “It is unfair that only the rich enjoy the benefit of practicing good health behaviours that reduce uncertainty of the outcome.” Should residuals covary by effort only among the non-rich, the unfairness might be expressed as follows by the non-rich with unhealthy habits, “I accept that I did not practice good health behaviour, but it is unfair that I experience greater uncertainty than the rich with unhealthy habits just because of my bad social background luck.”

In summary, the playing field framework offers a pragmatic test to assess fairness and unfairness of the distribution of residuals. This assessment builds on the well-developed literature on typologies of luck that considers brute luck as an unfair source of inequality and moral hazard as a fair source of inequality. The playing field applies to one particular type of brute luck and one particular kind of effort, thus, it enables an inference as to which type of luck contributes to the unfair distribution of residuals.

Empirical Strategy

We now explain the data and methods we used to apply the playing field framework to assess empirically unfairness and fairness of health outcomes among a large sample of older adults in Canada. We examine health because brute and option luck likely play a significant role in determining health outcomes, and because effort, in the form of health behaviours, is observable in health. Health behaviours, after adjustment for social background luck, can be considered as factors within individual control (Jusot & Tubeuf, 2019).

Data: We used the Canadian Longitudinal Study on Aging (CLSA) (Raina et al., 2009, 2019), a population-based, longitudinal study following approximately 50,000 non-institutionalized Canadians aged 45–85 at the time of recruitment in 2011–2015. For our analysis we used only the baseline, cross-sectional data from the CLSA Comprehensive, a subset of 30,097 participants, who under went in-home face-to-face computer-assisted interviews as well as computer-assisted interviews and clinical and physical assessments at data collection sites. Sampling frames for the CLSA Comprehensive are provincial universal health care registration databases and random digit dialing of landlines. To be included in the sampling frames, potential participants must have resided within a 25 or 50 km radius (depending on the population density) of the 11 data collection sites across Canada. Exclusion criteria included: individuals who spoke neither English nor French; the cognitively impaired at the time of recruitment; residents of the three territories; residents of federal First Nations reserves and other First Nations settlements; and full-time members of the Canadian Armed Forces. After removing 1059 observations due to missing information, the sample size for our analysis was 29,038.

Health outcome variable: Our health outcome is frailty, defined as a “decline in functioning across multiple physiological systems, accompanied by an increased vulnerability to stressors” (Hoogendijk et al., 2019) (p. 1365). Frailty is a measure of overall health, well suited to the equal opportunity framework and particularly relevant to studies of older adults. To measure frailty, we constructed the widely-used cumulative deficit based frailty index proposed by Rockwood and Mitnitski (2007). The frailty index counts 30 or more deficits in health, where deficits can be symptoms, signs, diseases, disabilities, or laboratory-assessed abnormalities. We constructed the frailty index using 44 variables available in the CLSA Comprehensive (Appendix 1), ranging from physically assessed performance measures to self-reported chronic conditions. The frailty index is the ratio of the number of deficits present to the total number of deficits considered (e.g., 10/30 = 0.333), where a frailty index of 0.000 indicates the absence of any deficit, and 1.000 indicates the full expression of deficits. Previous studies show that, regardless of the number and specific types of deficits used, the index follows a continuous, gamma distribution; the rate of deficit increase measured by the frailty index is about 0.030 per year; the maximum aggregation of deficits measured by the frailty index is about 0.700 (above which a person dies); and a difference of 0.020 or greater is meaningful as it typically indicates a change in response categories in one of the variables used to construct the frailty index (Fallah et al., 2011; Rockwood & Mitnitski, 2007). To convert the adverse health measure of the frailty index to a positive measure of health, we transformed the frailty index into the “flipped” frailty index (1–frailty index), whereby 0.000 represents the full expression of deficits and 1.000 represents the absence of any deficit. In our analytical sample of 29,038 observations, the flipped frailty index values ranged from 0.353 to 1.000 with the mean value of 0.862 and the standard deviation of 0.089.

Residuals: We first estimated the best fit model for the entire analytical sample of 29,038 individuals using unweighted ordinary least squares (OLS) with the flipped frailty index as the dependent variable. We tried log-transformation of the right-skewed frailty index, but it did not yield marked improvement in model fit. Explanatory variables included in the model were: sex, age, age squared, height, height squared, weight, racial/cultural background, immigration status, education, income, household size, home ownership, nutritional risk, marital status, social support, social participation, alcohol consumption, smoking, fruit and vegetable consumption, sleep, physical activity, province, and rurality (see Appendices 2 and 3 for details). These explanatory variables are known to be determinants of health expected to improve the explanatory power of the model (hence, reducing systematic errors in residuals).

Because the purpose of the present analysis was to examine the utility of the playing field framework, improving the explanatory power of the model drove our selection of the explanatory variables. Further, because the present analysis required only the estimation of residuals and did not assess EOp based on the compensation or reward principle, it did not require that we classify these variables to circumstance and effort as in the standard econometric implementation of the EOp framework. Adjusted R-squared for the final model was 0.509.

Social background luck, household income: We focused on social background luck as a type of brute luck and used household income as its indicator. Household income in reality would be influenced by both social background luck and effort, and our decision to use it as an indicator of social background luck primarily came from limitations of the dataset, which did not have a variable that clearly and solely captures social background luck. In the playing field analysis, we focused on two extreme ends of the income distribution as they would give the strongest signal for potential covariation between residuals and social background luck. Specifically, we created a binary variable for household income based on the five-category self-reported household income variable available in the CLSA: “Rich” is defined as household income greater than $100,000 versus “Non-rich” defined as household income less than$50,000 or income information missing (who typically are low income). Of the analytical sample of 29,038 observations, 10,089 (34.74%) were classified as rich, 9,341 (32.17%) as non-rich, and 9,608 (33.09%) as neither rich nor non-rich (household income ≥ $50,000 and <$100,000). When applying the playing field framework, we focused on the rich (n = 10,089) and the non-rich (n = 9,608).

Effort, health behaviours: As an indicator of effort, we created a binary health-behaviour variable based on smoking, sleeping, and physical activity variables. “Healthy habits” were defined as never smoked, sleeping 6–8 h, and physical activity score in the top 50%; “unhealthy habits” were defined as former or current smoker, sleeping less than 6 h or more than 8 h, and physical activity score in the bottom 50%. Of the analytical sample of 29,038 observations, 6,075 (20.92%) were classified as practicing healthy habits, 1775 (6.11%) as practicing unhealthy habits, and 21,188 (72.97%) as practicing neither healthy nor unhealthy habits. For the playing field analysis, we focused on those with healthy habits (n = 6075) and those with unhealthy habits (n = 1775).

Playing fields: Following the common epidemiological stratifications, we defined the playing field by sex (male vs. female) and age groups (45–54, 55–64, 65–74, and 75–84 years). This approach assumes that it is reasonable to compare distributions of residuals by social background luck, measured by income, and effort, measured by health behaviour, among persons in the same age-sex group. We thus had eight playing fields within each of which we assessed fairness and unfairness of distributions of residuals by social background luck and effort. In addition to the overall model described above, we ran stratified models by age, obtained predicted values, and calculated residuals for individuals in each age-sex category to which we applied the playing field framework.

Analysis: The analysis focused on the 5393 participants who were classified as either rich or non-rich and as practicing either healthy habits or unhealthy habits. In each of the eight age-sex playing fields we applied the within-tranche and the within-type comparisons, as discussed in Sect. 3. To determine if two distributions were different, we used three methods. First, we visually inspected kernel density plots of distributions of residuals. Second, we compared the standard deviations of the distributions. Third, we used the Kolmogorov–Smirnov test of equality of distributions, which is based on cumulative distribution functions, has been used in EOp empirical work on health (Bricard et al., 2020; Rosa Dias, 2009; Trannoy et al., 2009), and is more sensitive than the median-based Kruskal–Wallis test used by Bago d’Uva et al. (2012). Furthermore, given the large number of comparisons in our analysis—two comparisons within tranche and two comparisons within type for each of eight playing fields, resulting in 32 comparisons in total—we applied the Bonferroni correction to reduce false-positive results. Because the primary purpose of our analysis is to illustrate our methodological approach and not to draw inferential conclusions, our analysis was not weighted by survey weights, and variance estimates did not account for the complex survey design of the CLSA.

Empirical Results

Table 1 shows results of the within-tranche and the within-type comparisons of distributions of residuals in eight playing fields. For each of the four subgroups in each of the eight playing fields (e.g., the non-rich with unhealthy habits among 45–54 years old males), the top row number (e.g., 45) is the sample size and the bottom row number (e.g., 0.081) is the standard deviation of the distribution of residuals. The row p-values at the bottom of each playing field (e.g., 0.637 and 0.268) are for the Kolmogorov–Smirnov test for the within-tranche comparison of distributions of residuals (one for the healthy-habit and one for the unhealthy-habit tranche). When one of these row p-values is statistically significant (i.e., p < 0.0125), we consider the distribution of residuals by effort to be unfair.Table 1. Sample size, standard deviation, and p-value for the Kolmogorov–Smirnov test in playing fieldsFor each of the four subgroups in each of the eight playing fields (e.g., the non-rich with unhealthy habits among 45–54 years old males), the top row number (e.g., 45) is the sample size and the bottom row number (e.g., 0.081) is the standard deviation of the distribution of residuals. The row p-values at the bottom of each playing field (e.g., 0.637 and 0.268) are for the Kolmogorov–Smirnov test for the within-tranche comparison of distributions of residuals (one for the healthy-habit and one for the unhealthy-habit tranche). The column p-values at the furthest column of each playing field (e.g., 0.016 and 0.180) are for the Kolmogorov–Smirnov test for the within-type comparison of distributions of residuals (one for the rich and one for the non-rich). We consider p < 0.0125 as statistically significant, written in bold

The column p-values at the furthest column of each playing field (e.g., 0.016 and 0.180) are for the Kolmogorov–Smirnov test for the within-type comparison of distributions of residuals (one for the rich and one for the non-rich). When one of these column p-values is statistically significant but another is not (i.e., residuals covary with effort differentially by social background luck), or when both of the column p-values are statistically not significant (i.e., “effort does not pay” in the sense that it did not alter the distribution of luck), we consider the distribution of residuals by social background luck to be unfair. Applying the Bonferroni correction, we consider p < 0.0125 as statistically significant; significant results are written in bold.

The prevalence of healthy and unhealthy habits across age-sex groups in Table 1 exhibits a noteworthy age-related pattern. Whereas healthy habits are much more prevalent than unhealthy habits among 45–54 and 55–64 years old males and females, unhealthy habits are more prevalent than or almost equally prevalent to healthy habits among 65–74 and 75–84 years old males and females, especially among the non-rich. The reasons for this are not known, but it may be that: (1) older people think investment in healthy habits is of less value than younger people, especially among the non-rich; and/or (2) older people are less able to practice healthy habits due to declining health.

The standard deviation of each subgroup in Table 1 largely confirms our assumption that better social background luck and greater effort are associated with narrower distributions of residuals. Within each playing field, the smallest standard deviation is always for the rich with healthy habits, whereas the largest standard deviation is often for the non-rich with unhealthy habits. The Kolmogorov–Smirnov test results in Table 1 show that residuals covary with social background luck among males with healthy habits in the 55–64 and 65–74 age groups and among females with healthy habits in the 45–54 and 55–64 age groups (within-tranche comparison), thus, residuals are unfairly distributed among them. The top graph in Fig. 4 is an example of such unfair distribution of residuals. Among 55–64 age group females with healthy habits, the distribution of residuals is narrower among the rich than among the non-rich (within-tranche comparison).Fig. 4. Examples of distributions of luck by circumstance and effort, 55–64 years female and 75–84 years male. Note: Distributions for 55–64 years old females show covariation of residuals both with social background luck and effort, and distributions for 75–84 years old males show no covariation between residuals and social background luck and between residuals and effort. The statistical assessment of these distributions is in Table 1

Residuals covary with effort among the rich but not among the non-rich both for males and females in the 55–64 age group, and residuals covary with effort among the non-rich but not among the rich for females in the 45–54 age group (within-social background type comparison), thus, residuals are unfairly distributed among these groups. The same top graph in Fig. 4, as before, is also an example of such unfair distribution of residuals. Among rich 55–64 age group females, the distribution of residuals is narrower among those with healthy habits than among those with unhealthy habits, but such covariation between residuals and effort is not observed among the non-rich. In addition, among males in the 45–54, 65–74, and 75–84 age groups and among females in the 65–74 and 75–84 age groups, residuals do not covary with effort, thus, residuals are neither fairly nor unfairly distributed among these groups. The bottom graph in Fig. 4 is an example of such a distribution of residuals that is neither fair nor unfair. Among rich and non-rich 75–84 age group males, the distribution of residuals is similar among those with healthy habits and among those with unhealthy habits.

Taken together, as further summarized in Table 2, males and females in the 55–64 age group and females in the 45–54 age group have unfair distributions of residuals in terms of covariations with both social background luck and effort, and males in the 65–74 age group have unfair distributions of residuals in terms of covariation with social background luck.Table 2. Summary of unfair distributions of residualsAge groupMaleFemaleCovariation with social background luckCovariation with effortCovariation with social background luckCovariation with effort45–54FairNeitherfair or unfairUnfairUnfair55–64UnfairUnfairUnfairUnfair65–74UnfairNeitherfair or unfairFairNeitherfair or unfair75–84FairNeitherfair or unfairFairNeitherfair or unfair

Discussion

Acknowledging the importance of luck in the EOp framework but faced with the challenges of disentangling luck from residuals in the empirical EOp analyses, we sought to develop a pragmatic framework to assess the ethical status of residuals. Drawing inspiration from the literature on luck in philosophy and economics, we reframed the standard circumstance-effort EOp characterization by Roemer as the brute luck-effort characterization with the acknowledgment of effort associated with moral hazard. We proposed the playing field framework that assesses the unfairness and fairness of residuals in terms of covariation between residuals and brute luck (unfair) and between residuals and effort, which is associated with moral hazard (fair). Using a sample of Canadian older adults, we empirically applied the playing field framework. Both covariations between residuals and brute luck and between residuals and effort suggest that distributions of residuals are not always fair, though there is no consistent pattern of unfairness across age-sex groups.

The playing field framework is primarily a pragmatic test for the ethical status of residuals. Yet, because it makes the ethical assessment based on the typologies of luck and the ethical analysis of different types of luck, it can also serve as a framework that assists empirical exploration of luck. Empirical analysis of luck within the EOp framework to date is limited because it is challenging to identify the component of residuals that is luck and distinguish different types of luck, as in the case for two pioneer studies (Lefranc et al., 2009 and Bago d’Uva et al., 2012). Instead of attempting to estimate luck directly, or assuming that residuals are a reasonable approximation of luck, the playing field explores luck indirectly through covariations between residuals and observable brute luck and between residuals and observable effort. In our empirical implementation, we chose social background luck as an example of brute luck. Should data be available, it would be possible to apply the playing field analysis to other types of brute luck, for example, genetic luck, measured by the existence of hereditary disease in the family or the age of death of parents, and accidental luck, measured by the existence of major health events under the age of 50.

As a proof-of-concept of the playing field framework, our empirical analysis did not extend to the standard EOp analysis. In future work, as a pragmatic test for the ethical status of residuals, the playing field can supplement empirical EOp analyses using the standard parametric econometric methods. The playing field forces analysts to pay attention to residuals and can indicate whether disregarding them in the analysis may underestimate unfairness. Using the playing field to strengthen empirical EOp analyses, future work needs to select carefully variables indicating brute luck. In this study, among different types of brute luck, we focused on social background luck and used household income as an indicator. Household income admittedly is not the best indicator for social background luck as it represents both social background luck and effort. Our choice was limited with the availability of variables in the dataset we used. Variables that capture parental information and/or adverse childhood experience (Chen et al., 2022; Kovacic & Orso, 2022) are a promising candidate to capture social background luck.

This study demonstrated the utility of the playing field framework, but its implementation posed important questions such as the criteria used to compare residual distributions, the use of the age-sex group within which to compare residual distributions, and the size of data to implement the playing field. These issues need to be examined more thoroughly in future work. The question of the goodness of the shape of distributions of residuals deserves further discussion, for example, whether, and if so, how, the skewness of the distribution (as indicated by residual distributions in Fig. 4) should play a role in our assessment of goodness of the shape of distributions, and why we might consider a certain shape of distribution of residuals as worse than others. These questions connect to the discussion regarding asymmetric compensation of genetic luck in Lefranc and Trannoy (2017): we may wish to compensate bad luck, but we may wish to allow people enjoy good luck. Methodologically, there is no single, clearly preferred method for testing equivalence of two distributions. We opted for the commonly used Kolmogorov–Smirnov test, but future work should explore alternative methods.

Furthermore, it is important to consider how to define the population groups for whom it is appropriate to compare distributions of residuals. Our age-sex definition of the playing field has face validity as it follows the common epidemiological stratifications with the understanding that health differences by age and between sexes are to be expected and are acceptable. In addition, if we assume unfair health may manifest differently between males and females, sex stratification of the playing field analyses is appropriate. Implicit in this view is the understanding of age and sex as biological determinants of health, hence, differences by age and sex are uninteresting in the ethical examination of the distribution of health. Age and sex, however, are both biological and social determinants of health, supported by a simple observation of a wide range of health outcomes among people of the same age and a now common distinction between sex (biological determinant) and gender (social determinant). We acknowledge thoughtful discussion should be made to define an appropriate playing field but defer careful discussion on these issues to future work. Finally, it would be useful to determine a reasonable sample size for the playing field analysis. It is desirable to have a large sample size for the stratified distributional analysis the playing field analysis calls for, and it is also preferable to contrast four groups in the 2 × 2 playing fields, as we opted to focus on two extremes of social background luck and effort in our analysis. Taken together, the playing field analysis demands large data sets, but what the minimum sample size is for a reasonable analysis requires future exploration. To sum, the lack of consistent pattern of unfair distributions of residuals across age-sex groups observed in this study calls for future studies applying the playing field framework to richer data that enable the use of more sophisticated methods to distinguish fair and unfair determinants. Only then can we confidently assess potential age-sex patters of unfair distributions of residuals. This study can be considered as a first step aimed at illustrating the potential of the approach.

We conclude this paper with three key points that emerged from the development and empirical application of the playing field. First, luck matters. The existing typologies of luck offer further vocabulary to enrich our understanding and empirical exploration of EOp. Second, the operationalization of the EOp framework can benefit from explicitly incorporating luck. For the development of the playing field, we classified circumstance as a type of luck and reframed the standard circumstance-effort operationalization of EOp (Roemer, 1998) as brute luck-effort with the acknowledgment that effort is associated with option luck, which is characterized as moral hazard (Lefranc & Trannoy, 2017). Finally, the playing field framework is useful, and it showed that residuals are unlikely just an innocuous statistical annoyance but can represent unfair inequality. Given that residuals are often more than 50% of observed inequality, ignoring them can result in vast underestimation of unfair inequality. Our results suggest such underestimation is possible, and the playing field can be used as a method to identify and mitigate potential underestimation.