1 Introduction

As society becomes increasingly dependent on automated decision-making, algorithmic fairness—especially addressing unwanted bias (see, e.g., Chouldechova & Roth, 2020; Mehrabi et al., 2021)—receives much attention. This is a cross-disciplinary endeavor, where technical AI experts, philosophers, lawyers, behavioral scientists, and many others collaborate and learn from each other. In particular, existing theories from ethics and political philosophy are useful to avoid reinventing the wheel. Thus, it has been pointed out “that attempts to formalise fairness in machine learning contain echoes of these old philosophical debates” (Binns, 2018), and John Rawls has been called “AI’s favorite philosopher” (Procaccia, 2019). In this sense, it is established that algorithmic fairness can learn from philosophy.

The reverse—that insights from the algorithmic fairness literature are fed back into ethics and political philosophy—is less established. However, this short commentary on a recent paper by Baumann and Loi (2023) introduced in Section 2 aims to illustrate this reverse direction of learning, using Nozick ’s (1974) derivation of the minimal state, introduced in Section 3. More precisely, we show in Section 4 that the results from Baumann and Loi have an impact on whether Nozick’s minimal state is redistributive, and in Section 5 that Nozick’s invisble-hand explanation of how a dominant protection agency becomes a minimal state is further complicated by results from the modern algorithmic fairness literature. Section 6 concludes the commentary.

2 Baumann and Loi on Statistical Fairness in Insurance

Baumann and Loi (2023) ask: How should insurers set their premiums in a fair way? While more relevant with increased use of data and machine learning (p. 20), this question is implicit in any insurance scheme. To find fairness candidates, Baumann and Loi turn to the criteria proposed in the algorithmic fairness literature. Importantly, different fairness criteria cannot be simultaneously satisfied—they are “mathematically incompatible” (p. 5) (see Chouldechova, 2017; Kleinberg et al., 2017, for more details).

Baumann and Loi evaluate three algorithmic fairness measures and explain their relation to actuarial fairness—an established perspective in insurance. They argue that one—sufficiency (well-calibration) is morally defensible for insurers to use, whereas independence (statistical parity or demographic parity) and separation (equalized odds) are not. Sufficiency is appropriate because it “allows us to exclude with statistical significance that some group is systematically disadvantaged” (p. 20): testing for sufficiency helps detect systematic unfairness.

3 Nozick’s Derivation of the Minimal State

In Anarchy, State, and Utopia (1974), Nozick famously defends a minimal state—which only protects its citizens’ negative rights—against two contrasting positions: in Part I against individualist anarchists who want no state at all, and in Part II against those who want a larger state that engages in economic redistribution. While most scholarly attention has focused on Part II, especially the counter-arguments against Rawls (1999) in Chapter 7 (pp. 183–231), it is in Part I that we—surprisingly—find the connection to statistical fairness criteria.Footnote 1

The basic problem in Part I is this: All actual states have murky histories of oppression, theft, power-grabs, wars, forced labor, etc.—violations of individual rights, in Nozick’s terminology. Indeed, as we approach the fiftieth anniversary of Anarchy, State, and Utopia, such past sins remain high on the agenda. How should we regard states in light of this? Nozick answers with a hypothetical history of how a minimal state could arise through a series of legitimate, non-rights violating, steps—an invisible-hand explanation (pp. 18–22) of the state. This demonstrates that violating rights is not a necessary feature of states, at least to some extent undermining the anarchist’s case.Footnote 2

Briefly, Nozick’s hypothetical history unfolds as follows in Chapter 2. In the state of nature, everyone must defend their rights themselves. But division of labor leads to the creation of protective associations or agencies,Footnote 3 from which individuals may purchase protection. It is assumed that “each of the agencies attempts in good faith to act within the limits of Locke’s law of nature” (Nozick, 1974, p. 17), i.e., respect negative rights. From market pressures and economies of scale, different protective agencies become dominant in different geographical areas (pp. 15–17).

While dominant, these protective agencies are not states, because they are not monopolies: some independents enforce their rights themselves. May this be prohibited? To answer, Nozick explores issues of prohibition, compensation, and risk in Chapter 4. Why is it ever prohibited to violate others’ rights, rather than allowed if compensation is paid afterwards? And why not prohibit everything not agreed to beforehand? (p. 59) Violations which cannot be compensated for must be prohibited. Other actions can be partly but not fully compensated for: In particular, some things would be feared even if fully compensated afterwards (p. 66), and since compensation (to victims) does does not fully compensate such fear (even among non-victims), compensation is not enough and there is a legitimate public interest in prohibiting acts causing general fear (p. 67), such as allowing epileptics to drive (p. 78). But such prohibitions come with caveats: “when an action of this type is forbidden to someone because it might cause harm to others and is especially dangerous when he does it, then those who forbid in order to gain increased security for themselves must compensate the person forbidden for the disadvantage they place him under.” (p. 81, emphasis in original).

In Chapter 5, such compensation explains why the dominant protection agency in a territory can—indeed must—first become an ultraminimal state—with a monopoly on the use of force, and then a minimal state—which protects everyone. The first step is warranted by the prohibition of risky punishment procedures. If unreliable enforcers of justice impose general fear, such unreliable enforcement of justice may be prohibited (pp. 105–106). “Relatively unreliable procedures” may not be used in punishment, and this stems from epistemic considerations: “If doing act A would violate Q’s rights unless condition C obtained, then someone who does not know that C obtains may not do A.” (p. 106) But when the dominant protection agency thus creates a de facto monopoly on the enforcement of justice, it disadvantages the independents. This warrants the second step: the ultraminimal state must compensate those forbidden to exercise their right to self-defense, and the most cost-effective compensation is to extend protection to the (former) independents (pp. 110–111). Crucially, this is not redistributive as it depends on morally required compensation (p. 114).

4 Statistical Fairness and the Pricing of Protection

Reading Nozick (1974) in the light of Baumann and Loi (2023), we may ask: Is Nozick too quick to dismiss the possibility that the minimal state is redistributive? Nozick discusses the pricing of protection extensively (e.g., on pp. 63–65, pp. 78–87, and pp. 110–113), but always in the context of redistribution from the voluntary customers of the protective agency to those receiving its services as compensation. But as Baumann and Loi (2023) make clear, there may be other kinds of redistribution involved in insurance and insurance-like activities such as the protection discussed by Nozick.

The dominant protective agency employs fair procedures in enforcing justice (“wields this power as well as anyone would expect”, Nozick, 1974, p. 134) and the charitable assumption is that this extends to pricing. But here, the modern concept of algorithmic statistical fairness becomes relevant, for as Baumann and Loi point out, there are several candidates for fair pricing: at least independence, separation, and sufficiency. Following Baumann and Loi , a protective agency pricing using independence will be redistributive in its implementation of risk solidarity—low-risk individuals will subsidize high-risk individuals—not mere chance solidarity (p. 11). By contrast, a protective agency pricing using separation will not even implement chance solidarity. As observed by Baumann and Loi , this goes against the point of insurance (p. 14), and presumably, also against the point of protection agencies. Finally, a protective agency pricing using sufficiency engages only in chance solidarity.

Thus, proving that the minimal state is not redistributive requires more than Nozick explicitly offers—an explanation of why dominant protective agencies are not redistributive in their pricing. A sketch of such an explanation is the following: Protective agencies pricing using independence will attract high-risk clients, but not low-risk ones (as pointed out, in the general insurance context, by Baumann & Loi, 2023, p. 13). Protective agencies pricing using separation will struggle to attract customers at all, since they do not offer chance solidarity. Thus, Nozick’s “market pressures, economies of scale, and rational self-interest” (pp.  16–17) may lead protective agencies to price their services using sufficiency. But this is only a sketch—more attention should be paid to the details.

5 Statistical Fairness and the Prohibition of Risky Enforcement of Justice

The question of redistributive pricing is the most obvious and explicit question raised when reading Nozick (1974) in the light of Baumann and Loi (2023). However, an even larger question is implicitly raised by the modern literature on algorithmic fairness. As we have seen, the prohibition of risky enforcement of justice is what underpins both the monopoly on force and protection for all. But what, exactly, is prohibited, and can there be reasonable disagreement?

Nozick discusses at length that there may be disagreement about which procedures are known to be risky, e.g., that there may be different preferences about false-positive and false-negative rates in criminal justice (p. 96). Such difficulties propel Nozick to discuss procedures not known not to be risky, setting the stage for letting the dominant protection agency cut the Gordian knot: Everyone may prohibit procedures which are actually defective, but the dominant protective agency has the final word (pp. 108–109); it “will publish a list of those procedures it deems fair and reliable (and perhaps of those it deems otherwise); and it would take a brave soul indeed to proceed to apply a known procedure not yet on its approved list” (p. 103).

Again, algorithmic statistical fairness becomes relevant. When the dominant protective agency prohibits the use of tea leafs in enforcement of justice (Nozick, 1974, p. 101), the scope for reasonable disagreement is small. But what if it announces that something like the controversial COMPAS system for classifying probable reoffenders is a fair and reliable procedure? Now the scope of reasonable disagreement becomes substantial, because COMPAS is fair on some notions of fairness and unfair on others (see Dressel & Farid, 2018; Hedden, 2021, for more background). All fairness notions cannot be had simultaneously—as mentioned in Section 2, they are mathematically incompatible.

This need not be problematic. Nozick does not require a single unique solution of exactly which risky procedures are prohibited—“considerations of fear, division of the benefits of exchange, and transaction costs delimit our area, but [...] do not yet triangulate a solution in all its detail” (Nozick, 1974, p. 73). The dominant protective agency only needs to cut the Gordian knot in one way which does not violate rights, not in a unique way.

But the simultaneous incompatibility of different statistical fairness measures may undermine the ability of the dominant protective agency to cut the Gordian knot. Competitors “can enter the market and attempt to wean customers away from the dominant protective agency” (p. 109). While prohibiting tea leafs will not scare many customers away, prohibiting some methods satisfying some statistical fairness measures and mandating other methods satisfying other measures may upset large customer groups—as COMPAS did—and customers of protective agencies who want to ‘defund the police’ can do so by voting with their feet. But worse, the simultaneous incompatibility property suggests that such customer upheaval may result no matter which particular fairness measure is chosen. In a worst-case scenario, this may even create cyclic majorities (among customers, shareholders, or board members) driving the protective agency into a money-pump (see Gustafsson, 2022, for an introduction), e.g., there may be one majority favoring independence over separation, another majority favoring sufficiency over independence, and a third majority favoring separation over sufficiency. Such complications may undermine the dominant protective agency in a way somewhat reminiscent of how “the invisible hand strikes back” as Childs (1977), an early anarchist critic of Nozick’s, suggested.

Arguments that such complications would not undermine the dominant protective agency may take forms similar to Baumann and Loi —one particular fairness definition is morally required in one particular context, or to Hedden (2021)—none but (perhaps) one particular fairness definition is reasonable in any context. If such arguments hold sway, they impose additional moral requirements—beyond compensation—on protective agencies, but even if they do not, this reasoning demonstrates the relevance of algorithmic statistical fairness to Nozick’s argument.

6 Conclusions

It is generally acknowledged that algorithmic fairness has much to learn from ethics and political philosophy. In this short commentary, we have demonstrated that the reverse is also true, by relating the argument about fair insurance pricing from Baumann and Loi (2023) to the derivation of the minimal state offered by Nozick (1974). The fact that this connection is bidirectional suggests that the algorithmic fairness perspective may shed additional light on ethics and political philosophy more generally.