# Rethinking Fairness for Human-AI Collaboration

Haosen Ge, Wharton School, University of Pennsylvania
Hamsa Bastani, Wharton School, University of Pennsylvania
Osbert Bastani, University of Pennsylvania

###### Abstract.

Existing approaches to algorithmic fairness aim to ensure equitable outcomes
_if_ human decision-makers comply perfectly with algorithmic decisions.
However, perfect compliance with the algorithm is rarely a reality or even a
desirable outcome in human-AI collaboration. Yet, recent studies have shown
that selective compliance with fair algorithms can amplify discrimination
relative to the prior human policy. As a consequence, ensuring equitable
outcomes requires fundamentally different algorithmic design principles that
ensure robustness to the decision-maker’s (a priori unknown) compliance
pattern. We define the notion of compliance-robustly fair algorithmic
recommendations that are guaranteed to (weakly) improve fairness in decisions,
regardless of the human’s compliance pattern. We propose a simple optimization
strategy to identify the best performance-improving compliance-robustly fair
policy. However, we show that it may be infeasible to design algorithmic
recommendations that are simultaneously fair in isolation, compliance-robustly
fair, and more accurate than the human policy; thus, if our goal is to improve
the equity and accuracy of human-AI collaboration, it may not be desirable to
enforce traditional fairness constraints.

††conference: ; ;

##  1\. Introduction

As machine learning algorithms are increasingly deployed in high-stakes
settings (e.g., healthcare, finance, justice), it has become imperative to
understand the fairness implications of algorithmic decision-making on
protected groups. As a consequence, a wealth of work has sought to define
algorithmic fairness (Dwork et al., 2012; Hardt et al., 2016; Corbett-Davies
and Goel, 2018; Kleinberg et al., 2016; Chen et al., 2023) and learn machine
learning-based policies that satisfy fairness constraints to ensure equitable
outcomes across protected groups (Kim et al., 2019; Bastani et al., 2022;
Kearns et al., 2019; Joseph et al., 2016; Basu, 2023).

Most of this work focuses on whether the algorithm makes fair decisions in
isolation. Yet, these algorithms are rarely used in high-stakes settings
without human oversight, since there are still considerable legal and
regulatory challenges to full automation. Moreover, many believe that human-AI
collaboration is superior to full automation because human experts may have
auxiliary information that can help correct the mistakes of algorithms,
producing better decisions than the human or algorithm alone. For example,
while many powerful AI systems have been developed for diagnosing medical
images (Esteva et al., 2017), the Center for Medicare and Medicaid Services
only allows AI systems to assist medical experts with diagnosis (Rajpurkar et
al., 2022).

However, human-AI collaboration introduces new complexities — the overall
outcomes now depend not only on the algorithmic recommendations, but also on
which individuals the human decision-maker complies with the algorithmic
recommendation. Recent case studies have shown mixed results on whether human-
AI collaboration actually improves decision accuracy (Campero et al., 2022) or
fairness (Van Dam, 2019). For instance, a recent experiment examines
diagnostic quality when radiologists are assisted by AI models (Agarwal et
al., 2023). The authors find that, although the AI models are substantially
more accurate than radiologists, access to AI assistance does not improve
diagnostic quality on average; the authors show that this is due to selective
compliance of the algorithmic recommendations by humans, which they
hypothesize is driven by improper Bayesian updating. Similarly, a recent study
evaluates the impact of algorithmic risk assessment on judges’ sentencing
decisions in Virginia courts (Stevenson and Doleac, 2022; Van Dam, 2019).
Although risk assessment promised fairer outcomes (Kleinberg et al., 2018),
the authors find that it brought no detectable benefits in terms of public
safety or reduced incarceration; in fact, racial disparities increased in the
subset of courts where risk assessment appears most influential. Once again,
the mismatch is driven by selective compliance to algorithmic recommendations,
which appears to be at least partly driven by conflicting objectives between
judges and the algorithm (e.g., judges are more lenient towards younger
defendants). Selective compliance has significant fairness implications, e.g.,
in this case, the authors note that “judges were more likely to sentence
leniently for white defendants with high risk scores than for black defendants
with the same score.” These case studies make it clear that ensuring equitable
outcomes in human-AI collaboration requires accounting for humans’ complex and
unexpected compliance patterns. To this end, Morgan and Pass (2019); Gillis et
al. (2021) show that the outcomes of human-AI collaboration can be arbitrarily
less fair than either those of the human alone or the algorithm alone.

To resolve this state of affairs, we introduce the notion of compliance-robust
algorithms — i.e., algorithmic decision policies that are guaranteed to
(weakly) improve fairness in final outcomes, regardless of the human’s
(unknown) compliance pattern. In particular, given a human decision-maker and
her policy (without access to AI assistance), we characterize the class of
algorithmic recommendations that never result in collaborative final outcomes
that are less fair than the pre-existing human policy, even if the decision-
maker’s compliance pattern is adversarial. Next, we prove that there exists
considerable tension between traditional algorithmic fairness and compliance-
robust fairness. Unless the true data-generating process is itself perfectly
fair, it is often infeasible to design an algorithmic policy that is fair in
isolation, compliance-robustly fair, and more accurate than the human-only
policy; this raises the question of whether traditional fairness is even a
desirable constraint to enforce for human-AI collaboration. If the goal is to
improve fairness and accuracy in human-AI collaboration outcomes, it may be
preferable to design an algorithmic policy that is accurate and compliance-
robustly fair, but not fair in isolation. Our last result shows that the
tension between traditional fairness and compliance-robust fairness is
prevalent. Specifically, we prove that for a broad class of fairness
conditions, fair policies are not necessarily compliance-robustly fair,
implying that compliance-robust fairness imposes fundamentally different
constraints compared to traditional fairness.

###  1.1. Related Literature

Our paper contributes to a fast-growing literature that highlights the gap
between traditional fairness and fairness in human-AI collaboration. For
example, Morgan and Pass (2019) shows that a fair algorithmic policy cannot
guard against humans from exploiting its recommendations in a discriminatory
manner. In fact, it is often impossible to design an algorithmic policy that
is both fair and robust to human exploitation. Our results have similar
implications on the tension between traditional fairness and compliance-robust
fairness. However, we differ in that our compliance-robustly fair policy class
still guarantees fairer outcomes, even when humans selectively comply based on
factors other than the algorithmic policy’s output. Gillis et al. (2021) and
McLaughlin and Spiess (2022) study why and how humans deviate from algorithmic
recommendations using a Bayesian persuasion framework. They investigate
potential remedies like disclosing information about protected groups (along
with the recommendation), or withholding recommendations altogether; these
approaches require knowledge of how the human may react to the recommendation,
which is often a priori unknown. Thus, we focus on characterizing algorithmic
policies that guarantee more equitable outcomes even under adversarial
compliance patterns.

We also relate to a broad literature on how algorithmic advice benefits human
decision-makers. For example, Bastani et al. (2021) study the optimal design
of interpretable “tips” that can boost human performance; Noy and Zhang (2023)
show that ChatGPT can significantly increase the productivity of college-
educated professionals; and, Sim et al. (2020) show that algorithms can
substantially increase the diagnostic accuracy of radiologists. Relatedly,
Angelova et al. (2022) investigates when human decision-makers override
algorithmic recommendations, and Athey et al. (2020) investigate when it is
optimal to give agents decision authority.

##  2\. Problem Formulation

#### Human-AI decision making.

Consider a decision making problem where each individual is associated with a
type $x\in\mathcal{X}=[k]=\\{1,...,k\\}$ (e.g., education, prior defaults), a
protected attribute $a\in\mathcal{A}=\\{0,1\\}$ (e.g., gender), and a true
outcome $y\in\mathcal{Y}=\\{0,1\\}$ (e.g., whether they can repay a
loan).111Note $\mathcal{X}$ can encompass multiple categorical features that
are “flattened” into a single dimension. For mathematical simplicity, we
restrict to categorical features (i.e., $\mathcal{X}$ has finite possible
values), a binary protected attribute, and a binary outcome. Our results
straightforwardly extend to protected attributes with multiple classes, but
allowing for continuous features and outcomes requires modifying the primary
fairness definition we use (Hardt et al., 2016) by introducing slack
variables. Let $\mathbb{P}(x,a,y)$ over
$\mathcal{X}\times\mathcal{A}\times\mathcal{Y}$ denote the joint distribution
of the types, protected attributes, and outcomes of individuals for whom
decisions must be made; we will not require knowledge of $\mathbb{P}(x,a,y)$
to construct compliance-robust policies. We make the following assumption that
our population has good “coverage” across variables/outcomes:

###### Assumption 1.

We have $\mathbb{P}(x,a,y)>0$ for all $x\in\mathcal{X}$, $a\in\mathcal{A}$,
and $y\in\mathcal{Y}$.

###### Definition 2.1.

A _decision making policy_ is a mapping
$\pi:\mathcal{X}\times\mathcal{A}\to[0,1]$ that maps each feature-attribute
pair to a probability $\pi(x,a)$; then, the decision is
$\hat{y}\sim\text{Bernoulli}(\pi(x,a))$.222We use the commonly employed
Bernoulli distribution, but it suffices for $\mathbb{P}[\hat{y}=1]$ to simply
be increasing in $\pi(x,a)$.

The algorithm designer’s goal is for the decision to equal the true outcome —
i.e., $\hat{y}=y$ (e.g., we would ideally give each individual a loan if and
only if they will repay the loan).333Note that the human’s objective may vary,
e.g., judges are more lenient towards younger defendants (Van Dam, 2019). We
consider a human decision-maker represented by a policy $\pi_{H}$ (without
access to algorithmic assistance). When given access to recommendations from
an algorithmic policy $\pi_{A}$, the human instead makes decisions according
to a _compliance function_ $c:\mathcal{X}\times\mathcal{A}\mapsto\\{0,1\\}$,
where $c(x,a)=1$ indicates that the human adopts the algorithmic decision for
individuals $(x,a)$. Then, the joint human-AI policy will be

$\displaystyle\pi_{C}(x,a)$ $\displaystyle=\begin{cases}\pi_{A}(x,a)&\text{if
}c(x,a)=1\\\ \pi_{H}(x,a)&\text{otherwise}.\end{cases}$

Note that $\pi_{H}$ can be estimated via supervised learning on historical
decision-making data in the absence of algorithmic recommendations; in
contrast, the compliance function $c$ cannot be learned until an algorithm
$\pi_{A}$ has already been deployed, potentially with poor consequences. Thus,
we assume knowledge of $\pi_{H}$ but study compliance-robust policies that do
not require knowledge of $c$.

#### Fairness.

We primarily analyze the well-studied notion of “equality of opportunity”
(Hardt et al., 2016; Kleinberg et al., 2016), which requires
$\mathbb{P}[\pi=1\mid y=1,a=0]=\mathbb{P}[\pi=1\mid y=1,a=1]$ for any chosen
decision policy $\pi$. In other words, the true positive rate for each
protected group should be equal. In Section 6, we show that the qualitative
challenges that we illustrate in this paper arise for a very general class of
fairness definitions, subsuming demographic parity (Calders et al., 2009;
Zliobaite, 2015) and calibration within groups (Kleinberg et al., 2016) .

To this end, for a policy $\pi$, we denote the average score for subgroup $a$
by

$\displaystyle\overline{\pi}(a)=\sum_{x\in\mathcal{X}}\pi(x,a)\mathbb{P}(x\mid
a,y=1).$

Traditional algorithmic fairness would require the algorithmic policy
$\pi_{A}$ to satisfy $\overline{\pi}_{A}(0)=\overline{\pi}_{A}(1)$, without
accounting for the human policy $\pi_{H}$ or the compliance function $c$.

Next, without loss of generality, we assume Group 1 is better off than Group 0
in terms of “opportunity” under the human-alone policy:

###### Assumption 2.

We have $\overline{\pi}_{H}(1)\geq\overline{\pi}_{H}(0)$.

We now introduce some definitions. Let $\alpha$ be the slack in group fairness
for a policy $\pi$:

$\displaystyle\alpha(\pi)=|\overline{\pi}(1)-\overline{\pi}(0)|.$

###### Definition 2.2.

We say an algorithmic policy $\pi_{A}$ _reduces fairness_ under compliance
function $c$ if the resulting human-AI policy $\pi_{C}$ satisfies
$\alpha(\pi_{C})>\alpha(\pi_{H})$.

To avoid the above, we do not require that the human-AI decisions be perfectly
fair — only that they are (weakly) fairer than the human policy without
algorithmic assistance.

###### Definition 2.3.

Given $\pi_{H}$, an algorithmic policy $\pi_{A}$ is _compliance-robustly fair_
if there does not exist any compliance function $c$ that reduces fairness for
$\pi_{A}$.

Let $\Pi_{\text{fair}}$ be the set of compliance-robustly fair policies; note
that these policies need not be fair in the traditional algorithmic fairness
sense. We will characterize $\Pi_{\text{fair}}$ in the next section.

#### Performance.

Algorithmic assistance often aims to not only improve fairness but also the
accuracy of decisions. Ideally, we would produce compliance-robustly fair
recommendations that improve performance relative to the human policy. To
define performance, we consider a loss function
$\ell:[0,1]\times\mathcal{Y}\to\mathbb{R}$, and define the expected loss

$\displaystyle L(\pi)=\mathbb{E}[\ell(\pi(x,a),y)].$

Let the loss-minimizing (but possibly unfair) optimal policy be

$\displaystyle\pi_{*}=\operatorname*{argmin}_{\pi}L(\pi),$

and the highest performing compliance-robustly fair policy be

$\displaystyle\pi_{0}=\operatorname*{\arg\min}_{\pi\in\Pi_{\text{fair}}}L(\pi).$

For analysis, we impose the following mild assumption on our loss:

###### Definition 2.4.

We say a policy $\pi^{\prime}$ has _higher deviation_ than a second policy
$\pi$ if for all $x\in\mathcal{X},a\in\mathcal{A}$, if
$\pi(x,a)\geq\pi_{*}(x,a)$, then $\pi^{\prime}(x,a)\geq\pi(x,a)$, and if
$\pi(x,a)\leq\pi_{*}(x,a)$, then $\pi^{\prime}(x,a)\leq\pi(x,a)$. We say the
deviation is _strictly higher_ if the inequality is strict for any
$x\in\mathcal{X},a\in\mathcal{A}$.

###### Assumption 3.

For any policies $\pi,\pi^{\prime}$, if $\pi^{\prime}$ has higher deviation
than $\pi$, then $L(\pi^{\prime})\geq L(\pi)$; furthermore, if the deviation
is strictly higher, then $L(\pi^{\prime})>L(\pi)$.

In other words, if $\pi^{\prime}$ always deviates farther from $\pi_{*}$ than
$\pi$, then $\pi^{\prime}$ has higher expected loss. It can be easily checked
that common loss functions (e.g., mean squared error, mean absolute error,
cross entropy) satisfy the above definition. It is worth noting that we do not
assume the loss is symmetric — i.e., if $\pi$ and $\pi^{\prime}$ are on
different sides of $\pi_{*}$ for any $x,a$ pair, this assumption does not say
anything about which one attains a lower loss.

##  3\. Characterization of Compliance-Robust Fairness

$a$$a=0$$a=1$$\overline{\pi}(a)$$\pi_{H}$$\pi_{A}$$\alpha(\pi_{H})$ (a)
Fairness reduced if $c(x,0)=0$ and $c(x,1)=1$

$a$$a=0$$a=1$$\overline{\pi}(a)$$\pi_{H}$$\pi_{A}$$\alpha(\pi_{H})$ (b)
Compliance-robust fairness

Figure 1. Examples with trivial individual types (i.e., $\mathcal{X}=\\{1\\}$)
with the same human policy $\pi_{H}$ and two different algorithmic policies
$\pi_{A}$. The human policy is unfair
($\overline{\pi}_{H}(0)\neq\overline{\pi}_{H}(1)$), but the algorithmic policy
in both cases is fair in isolation
($\overline{\pi}_{A}(0)=\overline{\pi}_{A}(1)$). Left: If the human
selectively complies when $a=1$, fairness is reduced (relative to $\pi_{H}$).
Right: Fairness is never reduced for any compliance $c$, i.e., $\pi_{A}$ is
compliance-robustly fair.

Our first main result characterizes $\Pi_{\text{fair}}$. For intuition,
consider the simple example depicted in Figure 1, where there are no types
(i.e., $\mathcal{X}=\\{1\\}$). The left and right panels consider the same
unfair human policy $\pi_{H}$
($\overline{\pi}_{H}(0)\neq\overline{\pi}_{H}(1)$) but two different fair
algorithmic policies $\pi_{A}$
($\overline{\pi}_{A}(0)=\overline{\pi}_{A}(1)$). On the left, if the human
selectively complies when $a=1$, fairness reduces relative to $\pi_{H}$, i.e.,
$\pi_{A}$ is not compliance-robustly fair. On the right, it is easy to check
that no compliance function reduces fairness, i.e., $\pi_{A}$ is compliance-
robustly fair. In general, as we formalize next, compliance-robustness holds
exactly when $\pi_{A}$ is “sandwiched” between $\pi_{H}(x,0)$ and
$\pi_{H}(x,1)$.

###### Theorem 3.1.

Given $\pi_{H}$, an algorithm policy $\pi_{A}$ is compliance-robustly fair iff

(1) $\displaystyle\alpha(\pi_{A})$ $\displaystyle\leq\alpha(\pi_{H})$ (2)
$\displaystyle\pi_{H}(x,0)$ $\displaystyle\leq\pi_{A}(x,0)\qquad(\forall
x\in\mathcal{X})$ (3) $\displaystyle\pi_{A}(x,1)$
$\displaystyle\leq\pi_{H}(x,1)\qquad(\forall x\in\mathcal{X}).$

###### Proof.

First, we show that (1), (2), and (3) are sufficient. Note that (2) and (3)
imply

(4) $\displaystyle\overline{\pi}_{H}(0)$
$\displaystyle\leq\overline{\pi}_{C}(0)$
$\displaystyle\leq\overline{\pi}_{A}(0)$ (5)
$\displaystyle\overline{\pi}_{A}(1)$ $\displaystyle\leq\overline{\pi}_{C}(1)$
$\displaystyle\leq\overline{\pi}_{H}(1),$

respectively, for any compliance function $c$. Now, we have

$\displaystyle\overline{\pi}_{C}(1)-\overline{\pi}_{C}(0)\leq\overline{\pi}_{H}(1)-\overline{\pi}_{H}(0)\leq\alpha(\pi_{H}),$

where the first inequality follows from (4) and (5). Additionally, we have

$\displaystyle\overline{\pi}_{C}(0)-\overline{\pi}_{C}(1)\leq\overline{\pi}_{A}(0)-\overline{\pi}_{A}(1)\leq\alpha(\pi_{A})\leq\alpha(\pi_{H}).$

where the first inequality follows from (4) and (5), and the third from (1).
The claim follows.

Next, we show that (1), (2), and (3) are necessary. Note that (1) is clearly
necessary, or the compliance function $c(x,a)=1$ for all $x,a$ (i.e., the
human always complies with the algorithmic decision) reduces fairness. To see
that (2) is necessary, suppose to the contrary that
$\pi_{H}(0,x_{0})>\pi_{A}(0,x_{0})$ for some $x\in\mathcal{X}$. Then, consider
the compliance function

$\displaystyle c(x,a)$ $\displaystyle=\begin{cases}1&\text{if }x=x_{0},a=0\\\
0&\text{otherwise}.\end{cases}$

For this $c$, it is easy to see that by Assumption 1,
$\overline{\pi}_{H}(0)>\overline{\pi}_{C}(0)$, whereas
$\overline{\pi}_{C}(1)=\overline{\pi}_{H}(1)$. By Assumption 2, it follows
that $\alpha(\pi_{C})>\alpha(\pi_{H})$, so $c$ reduces fairness. The proof for
(3) is similar. ∎

In general, it is easier to find compliance-robustly fair policies when the
human policy is rather unfair. In fact, the following corollary shows that
there does not exist a performance-improving compliance-robust policy when the
human policy is already fair.

###### Corollary 3.2.

If $\alpha(\pi_{H})=0$, then $\pi_{A}$ is compliance-robustly fair iff
$\pi_{A}(x,a)=\pi_{H}(x,a)$ for all $x\in\mathcal{X}$ and $a\in\mathcal{A}$.

###### Proof.

Consider a compliance-robustly fair policy $\pi_{A}$, and assume to the
contrary that $\pi_{A}(x_{0},a_{0})\neq\pi_{H}(x_{0},a_{0})$ for some
$x_{0}\in\mathcal{X}$ and $a_{0}\in\mathcal{A}$. We assume that $a_{0}=1$; the
case $a_{0}=0$ is similar. By Theorem 3.1, we have
$\pi_{A}(x,1)\leq\pi_{H}(x,1)$ for all $x\in\mathcal{X}$, so
$\pi_{A}(x_{0},1)<\pi_{H}(x_{0},1)$. Then, by Assumption 1, we have
$\overline{\pi}_{A}(1)<\overline{\pi}_{H}(1)$. Also by Theorem 3.1, we have
$\pi_{A}(x,0)\geq\pi_{H}(x,0)$ for all $x\in\mathcal{X}$, so
$\overline{\pi}_{A}(0)\geq\overline{\pi}_{H}(0)$. Thus, we have

$\displaystyle\overline{\pi}_{A}(1)<\overline{\pi}_{H}(1)=\overline{\pi}_{H}(0)\leq\overline{\pi}_{A}(0),$

where the equality holds by our assumption that $\alpha(\pi_{H})=0$. Since
$\overline{\pi}_{A}(1)\neq\overline{\pi}_{A}(0)$, we must have
$\alpha(\pi_{A})>0=\alpha(\pi_{H})$, so by Theorem 3.1, $\overline{\pi}_{A}$
is not compliance-robustly fair, a contradiction. ∎

Theorem 3.1 allows us to write down a simple optimization problem (see Figure
2) to compute a compliance-robustly fair policy $\pi_{0}$ that performs the
best (i.e., minimizes the loss $L$). Note that we have assumed nothing on the
class of compliance functions $c(x,a)$ thus far. If we have a priori knowledge
that $c$ only depends on some subset of the type variables, then we can
trivially remove the constraints in the optimization problem in Figure 2 for
those types, enlarging $\Pi_{\text{fair}}$.

$\displaystyle\pi_{0}=$ $\displaystyle\operatorname*{\arg\min}_{\pi}L(\pi)$
subj. to $\displaystyle\alpha(\pi)\leq\alpha(\pi_{H})$
$\displaystyle\pi_{H}(x,0)\leq\pi(x,0)\qquad(\forall x\in\mathcal{X})$
$\displaystyle\pi(x,1)\leq\pi_{H}(x,0)\qquad(\forall x\in\mathcal{X})$ Figure
2. Algorithm for computing the optimal compliance-robustly fair policy
$\pi_{0}$.

##  4\. Performance of Compliance-Robustly Fair Policies

As defined in Algorithm 2, a strong candidate for algorithmic recommendations
is $\pi_{0}$, the best-performing compliance-robust policy. However, in most
cases, we would only provide algorithmic advice if we think it may perform
better than the current human policy, i.e., $L(\pi_{0})<L(\pi_{H})$. In this
section, we provide simple conditions (that can be verified with knowledge of
$\pi_{H}$ and the loss-minimizing $\pi_{*}$) to see if algorithmic advice is
desirable.

Given the human policy $\pi_{H}$ and the optimal policy $\pi_{*}$, let

$\displaystyle u(a)$
$\displaystyle=\\{x\in\mathcal{X}\mid\pi_{H}(x,a)\geq\pi_{*}(x,a)\\}$
$\displaystyle\ell(a)$
$\displaystyle=\\{x\in\mathcal{X}\mid\pi_{H}(x,a)<\pi_{*}(x,a)\\}.$

We now construct the following policy:

$\displaystyle\pi_{B}(x,a)$ $\displaystyle=\begin{cases}\pi_{H}(x,a)&\text{if
}x\in u(0)\cup\ell(1)\\\ \pi_{*}(x,a)&\text{otherwise}.\end{cases}$

It follows by the construction of $\pi_{B}$ that it satisfies:

###### Lemma 4.1.

We have

$\displaystyle\pi_{B}(x,a)=\begin{cases}\max\\{\pi_{*}(x,a),\pi_{H}(x,a)\\}&\text{if
}a=0\\\ \min\\{\pi_{*}(x,a),\pi_{H}(x,a)\\}&\text{if }a=1.\end{cases}$

As we will see shortly, $\pi_{B}$ provides a constructive upper bound on the
performance of any compliance-robustly fair policy, which will be useful for
examining when the performance of $\pi_{0}$ exceeds that of $\pi_{H}$. We
begin by noting that $\pi_{B}$ itself is compliance-robustly fair if it (in
isolation) does not reduce fairness relative to $\pi_{H}$.

###### Lemma 4.2.

If $\alpha(\pi_{B})\leq\alpha(\pi_{H})$, then $\pi_{B}$ is compliance-robustly
fair.

###### Proof.

By Lemma 4.1, $\pi_{B}$ satisfies conditions (2) and (3) in Theorem 3.1 by
construction. If $\alpha(\pi_{B})\leq\alpha(\pi_{H})$, then (1) also holds, so
by Theorem 3.1, $\pi_{B}$ is compliance-robustly fair. ∎

Furthermore, the next result shows that $\pi_{B}$ performs at least as well as
the optimal compliance-robustly fair policy $\pi_{0}$.

###### Lemma 4.3.

We have $L(\pi_{0})\geq L(\pi_{B})$.

###### Proof.

It suffices to prove that $\pi_{0}$ has higher deviation than $\pi_{B}$, in
which case the claim follows by Assumption 3. We need to show that for all
$x\in\mathcal{X}$ and $a\in\mathcal{A}$, we have

(6) $\displaystyle\begin{cases}\pi_{0}(x,a)\leq\pi_{B}(x,a)&\text{if
}\pi_{B}(x,a)\leq\pi_{*}(x,a)\\\ \pi_{0}(x,a)\geq\pi_{B}(x,a)&\text{if
}\pi_{B}(x,a)\geq\pi_{*}(x,a).\end{cases}$

Now, consider a point $x\in u(0)$; in this case, we have

$\displaystyle\pi_{0}(x,0)\geq\pi_{H}(x,0)\geq\pi_{*}(x,0),$

where the first inequality follows since $\pi_{0}$ is compliance-robustly fair
so it satisfies (2), and the second since $x\in u(0)$. Since
$\pi_{B}(x,0)=\pi_{H}(x,0)$ for $x\in u(0)$, (6) holds. Next, consider a point
$x\in\ell(1)$; in this case, we have

$\displaystyle\pi_{0}(x,1)\leq\pi_{H}(x,1)<\pi_{*}(x,1)$

where the first inequality follows since $\pi_{0}$ satisfies (3), and the
second since $x\in\ell(1)$. Since $\pi_{B}(x,1)=\pi_{H}(x,1)$ for
$x\in\ell(1)$, (6) holds. Finally, if $x\not\in u(0)\cup\ell(1)$, then
$\pi_{B}(x,a)=\pi_{*}(x,a)$ for all $a\in\mathcal{A}$, so (6) holds. Thus,
$\pi_{0}$ has higher deviation than $\pi_{B}$, so the claim follows. ∎

As noted earlier, while $\pi_{B}$ provides an upper bound on the performance
of any compliance-robustly fair policy, this upper bound may not be attainable
since $\pi_{B}$ may not be compliance-robustly fair. Nevertheless, our next
result, Theorem 4.4, shows simple conditions under which the optimal
compliance-robustly fair policy $\pi_{0}$ is worth sharing with the human
(i.e., when $L(\pi_{0})<L(\pi_{H})$). Namely, we require that human policy
$\pi_{H}$ is not perfectly fair (in which case, there is no nontrivial
compliance-robustly fair policy by Corollary 3.2), and $\pi_{H}$ deviates from
the performance-optimal policy $\pi_{*}$ in a direction that we can plausibly
correct with algorithmic advice.

###### Theorem 4.4.

Assume that $\alpha(\pi_{H})\neq 0$, and that either
$\pi_{H}(x,1)\neq\pi_{*}(x,1)$ for some $x\not\in\ell(1)$ or
$\pi_{H}(x,0)\neq\pi_{*}(x,0)$ for some $x\not\in u(0)$. Then, we have
$L(\pi_{0})<L(\pi_{H})$.

###### Proof.

If $\alpha(\pi_{B})\leq\alpha(\pi_{H})$, then by Lemma 4.2, $\pi_{B}$ is
compliance-robustly fair; the assumptions in the theorem statement clearly
imply that $L(\pi_{B})<L(\pi_{H})$, so the claim follows. Otherwise, we must
have $\alpha(\pi_{B})>\alpha(\pi_{H})$. Furthermore, Lemma 4.1 implies that
$\overline{\pi}_{B}(1)\leq\overline{\pi}_{H}(1)$ and
$\overline{\pi}_{B}(0)\geq\overline{\pi}_{H}(0)$. Together with Assumption 2,
these three conditions imply that

$\displaystyle\overline{\pi}_{B}(1)<\overline{\pi}_{B}(0).$

Intuitively, this might happen when the optimal policy satisfies
$\overline{\pi}_{*}(1)<\overline{\pi}_{*}(0)$, but the human policy reverses
this relationship; thus, if the human selectively complies to $\pi_{B}$ for
one group but not the other, they will reduce fairness. To compensate, we can
reduce the performance of $\overline{\pi}_{B}$ to “shrink” the gap between
$\overline{\pi}_{B}(1)$ and $\overline{\pi}_{B}(0)$. In particular, consider
scaling the decisions as follows:

$\displaystyle\pi_{A,\lambda}(x,a)=\begin{cases}\pi_{H}(x,a)&\text{if }x\in
u(0)\cup\ell(1)\\\
(1-\lambda)\pi_{B}(x,a)+\lambda\pi_{H}(x,a)&\text{otherwise}.\end{cases}$

Note that $\pi_{A,0}=\pi_{B}$ and $\pi_{A,1}=\pi_{H}$. In addition, it is easy
to see that $\pi_{A,\lambda}$ has strictly lower deviation than $\pi_{H}$ for
all $\lambda\in[0,1)$ (strictness is due to Assumption 1 and our assumption on
$\pi_{H}$ in the theorem statement). Next, by construction, for all
$\lambda\in[0,1]$, we have $\pi_{A,\lambda}(x,1)\leq\pi_{H}(x,1)$ and
$\pi_{A,\lambda}(x,0)\geq\pi_{H}(x,0)$. Now, consider the function

$\displaystyle g(\lambda)$
$\displaystyle=\overline{\pi}_{A,\lambda}(1)-\overline{\pi}_{A,\lambda}(0).$

By the above, we have

$\displaystyle g(0)$
$\displaystyle=\overline{\pi}_{B}(1)-\overline{\pi}_{B}(0)\leq 0$
$\displaystyle g(1)$
$\displaystyle=\overline{\pi}_{H}(1)-\overline{\pi}_{H}(0)\geq 0.$

Thus, by the intermediate value theorem, there exists $\lambda^{*}\in[0,1]$
such that $g(\lambda^{*})=0$. Since

$\displaystyle
g(1)=\overline{\pi}_{H}(1)-\overline{\pi}_{H}(0)=\alpha(\pi_{H})\neq 0,$

we know that $\lambda^{*}\neq 1$, so $\lambda^{*}\in[0,1)$. Thus,
$\pi_{A,\lambda^{*}}$ satisfies (1), (2), and (3), so by Theorem 3.1, it is
compliance-robustly fair. In addition, since $\lambda_{1}^{*}\in[0,1)$, by the
above, it has strictly lower deviation than $\pi_{H}$, so
$L(\pi_{A,\lambda^{*}})<L(\pi_{H})$. Thus, we have $L(\pi_{0})\leq
L(\pi_{A,\lambda^{*}})<L(\pi_{H})$, as claimed. ∎

##  5\. Compliance-Robust Fairness vs. Traditional Fairness

As shown in the last section, compliance-robust fairness and performance
improvement are often compatible; the same holds for traditional fairness and
performance improvement (Hardt et al., 2016). However, we will show that there
is considerable tension between maintaining both types of fairness
(compliance-robust fairness and traditional algorithmic fairness) while
improving performance.

Building on the mild conditions required for a performance-improving
compliance-robust policy in Theorem 4.4, the next lemma establishes additional
conditions that are necessary and sufficient to find a policy $\pi_{A}$ that
is also traditionally fair (i.e., $\alpha(\pi_{A})=0$).

###### Lemma 5.1.

Assume that $\alpha(\pi_{H})\neq 0$, and that either
$\pi_{H}(x,1)\neq\pi_{*}(x,1)$ for some $x\not\in\ell(1)$ or
$\pi_{H}(x,0)\neq\pi_{*}(x,0)$ for some $x\not\in u(0)$. Then, there exists a
compliance-robustly fair policy $\pi_{A}$ satisfying $\alpha(\pi_{A})=0$ such
that satisfies $L(\pi_{A})<L(\pi_{H})$ iff there exists a policy $\pi$
satisfying

(7) $\displaystyle\overline{\pi}(1)$ $\displaystyle\leq\overline{\pi}(0)$ (8)
$\displaystyle\pi(x,1)$ $\displaystyle\leq\pi_{B}(x,1)\qquad\qquad(\forall
x\in\mathcal{X})$ (9) $\displaystyle\pi(x,0)$
$\displaystyle\geq\pi_{B}(x,0)\qquad\qquad(\forall x\in\mathcal{X})$ (10)
$\displaystyle L(\pi)$ $\displaystyle<L(\pi_{H}).$

###### Proof.

We first show that existence of $\pi_{A}$ implies existence of $\pi$. By
Theorem 3.1, $\pi_{A}$ satisfies

$\displaystyle\overline{\pi}_{A}(1)$ $\displaystyle=\overline{\pi}_{A}(0)$
$\displaystyle\pi_{H}(x,0)$ $\displaystyle\leq\pi_{A}(x,0)\qquad\qquad(\forall
x\in\mathcal{X})$ $\displaystyle\pi_{A}(x,1)$
$\displaystyle\leq\pi_{H}(x,1)\qquad\qquad(\forall x\in\mathcal{X}).$

Now, let

$\displaystyle\pi(x,a)=\begin{cases}\max\\{\pi_{A}(x,0),\pi_{B}(x,0)\\}&\text{if
}a=0\\\ \min\\{\pi_{A}(x,1),\pi_{B}(x,1)\\}&\text{if }a=1.\end{cases}$

By construction, $\pi$ satisfies (8) and (9). Furthermore, we have

$\displaystyle\overline{\pi}(1)\leq\overline{\pi}_{A}(1)=\overline{\pi}_{A}(0)\leq\overline{\pi}(0),$

where the first inequality follows since $\pi(x,1)\leq\pi_{A}(x,1)$ and the
second since $\pi$ satisfies $\pi(x,0)\geq\pi_{A}(x,0)$. Thus, $\pi$ satisfies
(7). Finally, to show that $L(\pi)<L(\pi_{H})$, it suffices to show that $\pi$
has lower or equal deviation compared to $\pi_{A}$, since this implies that
$L(\pi)\leq L(\pi_{A})<L(\pi_{H})$. To this end, recall that for all
$x\in\mathcal{X}$ and $a\in\mathcal{A}$, we have
$\pi_{B}(x,a)\in\\{\pi_{H}(x,a),\pi_{*}(x,a)\\}$. If
$\pi_{A}(x,a)\neq\pi_{H}(x,a)$ and $\pi(x,a)\neq\pi_{A}(x,a)$, then we must
$\pi(x,a)=\pi_{B}(x,a)$, so $\pi(x,a)\in\\{\pi_{H}(x,a),\pi_{*}(x,a)\\}$. In
this case, we cannot have $\pi(x,a)=\pi_{H}(x,a)$, since either $a=0$ and
$\pi(x,0)\geq\pi_{A}(x,0)>\pi_{H}(x,0)$, or $a=1$ and
$\pi(x,1)\leq\pi_{A}(x,1)<\pi_{H}(x,1)$. Thus, we must have
$\pi(x,a)=\pi_{*}(x,a)$. In general, it follows that
$\pi(x,a)\in\\{\pi_{A}(x,a),\pi_{*}(x,a)\\}$, which straightforwardly implies
that $\pi$ has lower or equal deviation compared to $\pi_{A}$. The claim
follows.

Next, we prove that existence of $\pi$ implies the existence of $\pi_{A}$.
First, if $\overline{\pi}_{B}(1)\leq\overline{\pi}_{B}(0)$, then the result
follows from the proof of Theorem 4.4, which shows that if
$\overline{\pi}_{B}(1)\leq\overline{\pi}_{B}(0)$, then there exists a
compliance-robustly fair policy $\pi$ such that $\alpha(\pi)=0$. . Thus, it
suffices to consider the case $\overline{\pi}_{B}(1)>\overline{\pi}_{B}(0)$.
In this case, by (8) and (9), we have

$\displaystyle\pi(x,1)$
$\displaystyle\leq\min\\{\pi_{*}(x,1),\pi_{H}(x,1)\\}=\pi_{B}(x,1)\leq\pi_{*}(x,1)$
$\displaystyle\pi(x,0)$
$\displaystyle\geq\max\\{\pi_{*}(x,0),\pi_{H}(x,0)\\}=\pi_{B}(x,0)\geq\pi_{*}(x,0).$

Thus, $\pi_{B}$ has lower or equal deviation compared to $\pi$, so
$L(\pi_{B})\leq L(\pi)<L(\pi_{H})$. Consider

$\displaystyle\pi_{A,\lambda}(x,a)=\lambda\pi(x,a)+(1-\lambda)\pi_{B}(x,a),$

where $\lambda\in[0,1]$. Note that $\pi_{A,0}=\pi_{B}$ and $\pi_{A,1}=\pi$. It
is easy to see that $\pi_{A,\lambda}$ has lower or equal deviation compared to
$\pi$, so $L(\pi_{A,\lambda})\leq L(\pi)<L(\pi_{H})$ for all $\lambda$. Now,
define

$\displaystyle
g(\lambda)=\overline{\pi}_{A,\lambda}(1)-\overline{\pi}_{A,\lambda}(0),$

so

$\displaystyle g(0)$
$\displaystyle=\overline{\pi}_{B}(1)-\overline{\pi}_{B}(0)>0$ $\displaystyle
g(1)$ $\displaystyle=\overline{\pi}(1)-\overline{\pi}(0)<0.$

By the intermediate value theorem, there exists $\lambda^{*}\in(0,1)$ such
that $g(\lambda^{*})=0$. Then, we have $\alpha(\pi_{A,\lambda^{*}})=0$ and
$L(\pi_{A,\lambda^{*}})<L(\pi_{H})$. It also directly follows from Theorem 3.1
that $\pi_{A,\lambda^{*}}$ is compliance-robustly fair. Thus,
$\pi_{A,\lambda^{*}}$ satisfies our desiderata, so the claim follows. ∎

Next, we show a natural setting where we meet the above conditions — namely,
when the data-generating process is such that the optimal performance-
maximizing policy $\pi_{*}$ is already perfectly fair without any added
constraints (i.e., $\alpha(\pi_{*})=0$).

###### Theorem 5.2.

Assume that $\alpha(\pi_{H})\neq 0$, and that either
$\pi_{H}(x,1)\neq\pi_{*}(x,1)$ for some $x\not\in\ell(1)$ or
$\pi_{H}(x,0)\neq\pi_{*}(x,0)$ for some $x\not\in u(0)$. Then, if $\pi_{*}$ is
fair, there is always a fair $\pi_{A}$ that never reduces fairness and
accuracy.

###### Proof.

Consider $\pi_{B}$. Since $\pi_{*}$ is fair and $\pi_{B}(x,1)\leq\pi_{*}(x,1)$
and $\pi_{B}(x,0)\geq\pi_{*}(x,0)$, it immediately follows that
$\overline{\pi}_{B}(1)\leq\overline{\pi}_{B}(0)$. Thus, the claim follows from
Lemma 5.1 (with $\pi=\pi_{B}$). ∎

Unfortunately, it unlikely that an unconstrained performance-maximizing policy
will be inherently fair; this insight has been the driving force of the
algorithmic fairness literature. Rather, we may have to choose between the
properties of compliance-robust fairness (to avoid disparate harm relative to
the human policy), performance improvement (to ensure that algorithmic
recommendations actually drive improved decisions), and traditional fairness
(to ensure the algorithm is fair in isolation). To this end, we now construct
a simple setting where we can only satisfy one criterion — performance
improvement or traditional fairness — for all compliance-robustly fair
policies.

###### Proposition 5.3.

There exists $\mathcal{X}$, $\mathbb{P}$, $L$, and $\pi_{H}$ satisfying
$\alpha(\pi_{H})\neq 0$ and $\pi_{H}(x,1)\neq\pi_{*}(x,1)$ for some
$x\in\ell(1)$ such that $\Pi_{\text{fair}}\neq\varnothing$, and for all
$\pi\in\Pi_{\text{fair}}$, either $\alpha(\pi)\neq 0$ or $L(\pi)>L(\pi_{H})$.

###### Proof.

Let $\mathcal{X}=\\{1\\}$ be singleton; thus, we can omit it from our
notation. Let

$\displaystyle\mathbb{P}(a,y)=\begin{cases}1-\epsilon&\text{if }a=1\wedge
y=1\\\ \epsilon&\text{if }a=1\wedge y=0\\\ \epsilon&\text{if }a=0\wedge y=1\\\
1-\epsilon&\text{if }a=0\wedge y=0\end{cases}$

for any $\epsilon\in(0,1/4]$. Let the loss be

$\displaystyle
L(\pi)=\mathbb{E}[(\pi(a)-y)^{2}]=(1-\epsilon)(\pi(1)-1)^{2}+\epsilon\pi(1)^{2}+\epsilon(\pi(0)-1)^{2}+(1-\epsilon)\pi(0)^{2}.$

Then, it is easy to check that so that

$\displaystyle\pi_{*}(a)=\begin{cases}1-\epsilon&\text{if }a=1\\\
\epsilon&\text{if }a=0.\end{cases}$

In addition, suppose that the human policy is

$\displaystyle\pi_{H}(a)$ $\displaystyle=\begin{cases}1-\epsilon&\text{if
}a=1\\\ \epsilon/2&\text{if }a=0.\end{cases}$

In this case, $\pi_{B}=\pi_{*}$, and
$\alpha(\pi_{B})=\alpha(\pi_{*})<\alpha(\pi_{H})$, so by Theorem 4.4,
$\pi_{B}$ is compliance-robustly fair; in addition, it strictly improves
performance, though it is itself unfair. Thus,
$\Pi_{\text{fair}}\neq\varnothing$.

Next, we show that for any compliance-robustly fair policy $\pi$, if
$\alpha(\pi)=0$, then $L(\pi)\geq L(\pi_{H})$. Since $\mathcal{X}$ is
singleton, we have $\overline{\pi}(a)=\pi(a)$, so $\alpha(\pi)=0$ implies
$\pi(0)=\pi(1)$. Thus, it suffices to consider a policy $\pi(0)=\pi(1)=\beta$.
For any such policy, the loss is

$\displaystyle
L(\pi)=(1-\epsilon)(\beta-1)^{2}+\epsilon\beta^{2}+\epsilon(\beta-1)^{2}+(1-\epsilon)\beta^{2}=(\beta-1)^{2}+\beta^{2},$

which is minimized when $\beta=1/2$, in which case $L(\pi)=1/2$. In contrast,
we have

$\displaystyle L(\pi_{H})$
$\displaystyle=(1-\epsilon)(\epsilon/2)^{2}+\epsilon(1-\epsilon)^{2}+\epsilon(1-\epsilon)^{2}+(1-\epsilon)(\epsilon/2)^{2}$
$\displaystyle=(1-\epsilon)\left((\epsilon/2)^{2}+2\epsilon(1-\epsilon)\right)$
$\displaystyle<2\epsilon,$

where the inequality is strict since $\epsilon>0$. Since $\epsilon\leq 1/4$,
we have $L(\pi_{H})<1/2\leq L(\pi)$. ∎

Given these results, if the goal is to improve fairness and accuracy in human-
AI collaboration outcomes, we argue that it may be preferable to design an
algorithmic policy that is accurate and compliance-robustly fair, but not fair
in isolation.

##  6\. Compliance Issues for General Fairness Conditions

Thus far, we have focused on a single definition of algorithmic fairness —
equality of opportunity (Hardt et al., 2016). However, the challenges arising
from selective compliance and the resulting trade-offs are endemic to a large
class of fairness definitions.

To this end, we now define a general class of fairness criteria, subsuming
demographic parity (Calders et al., 2009; Zliobaite, 2015) and calibration
within groups (Kleinberg et al., 2016). We then show that, under this general
class, fair policies are not necessarily compliance-robustly fair. Thus, in
all cases, one must optimize separately for performance-improving compliance-
robustly fair policies (as illustrated in Algorithm 2). Note that we abuse
notation in this section by redefining
$\pi_{0},\underline{\pi},\overline{\pi}$ for our general fairness definition.

###### Definition 6.1.

A _fairness condition_ is a function $\varphi:\Pi\mapsto\mathbb{R}_{\geq 0}$,
where $\Pi$ is the space of all policies. Given a human policy $\pi_{H}$ and
an algorithmic policy $\pi_{A}$, we say $\pi_{A}$ is _compliance-robustly
fair_ for $\pi_{H}$ if for every compliance function $c$, the resulting human-
AI policy $\pi_{C}$ satisfies $\varphi(\pi_{C})\leq\varphi(\pi_{H})$.

###### Assumption 4.

Given a fairness condition $\varphi$, there exist policies $\pi_{0}$,
$\underline{\pi}$, and $\overline{\pi}$, and a compliance function $c_{0}$,
such that (i) we have

$\displaystyle\varphi(\underline{\pi})\leq\varphi({\pi_{0}})<\varphi(\overline{\pi}),$

(ii) the human-AI policy

$\displaystyle\pi_{C}(x,a)=\begin{cases}\overline{\pi}(x,a)&\text{if
}c_{0}(x,a)=1\\\ \pi_{0}(x,a)&\text{otherwise}.\end{cases}$

satisfies

$\displaystyle\varphi(\pi_{0})<\varphi(\pi_{C})<\varphi(\overline{\pi}),$

and (iii) the human-AI policy

$\displaystyle\pi^{\prime}_{C}(x,a)=\begin{cases}\pi_{0}(x,a)&\text{if
}c_{0}(x,a)=1\\\ \overline{\pi}(x,a)&\text{otherwise}.\end{cases}$

satisfies

$\displaystyle\varphi(\pi^{\prime}_{C})<\varphi(\overline{\pi}).$

We will show that demographic parity and within-group calibration satisfy
Assumption 4.

###### Theorem 6.2.

For any fairness condition $\varphi$ satisfying Assumption 4, there exists a
human policy $\pi_{H}$ and an algorithmic policy $\pi_{A}$ such that
$\varphi(\pi_{A})\leq\varphi(\pi_{H})$ but $\pi_{A}$ is not compliance-
robustly fair for $\pi_{H}$.

###### Proof.

Let $\pi_{0}$, $\underline{\pi}$, $\overline{\pi}$, and $c_{0}$ be as defined
in Assumption 4, and consider the policy

$\displaystyle\pi_{1}(x,a)=\begin{cases}\overline{\pi}(x,a)&\text{if
}c_{0}(x,a)=1\\\ \underline{\pi}(x,a)&\text{otherwise}.\end{cases}$

First, consider the case $\varphi(\pi_{1})\leq\varphi(\pi_{0})$. Consider the
human-AI policies

$\displaystyle\pi_{C}^{(1)}(x,a)$
$\displaystyle=\begin{cases}\overline{\pi}(x,a)&\text{if }c_{0}(x,a)=1\\\
\pi_{0}(x,a)&\text{otherwise}\end{cases}$ $\displaystyle\pi_{C}^{(2)}(x,a)$
$\displaystyle=\begin{cases}\pi_{1}(x,a)&\text{if }c_{0}(x,a)=1\\\
\pi_{0}(x,a)&\text{otherwise}.\end{cases}$

Note that $\pi_{C}^{(1)}=\pi_{C}^{(2)}$ since
$\pi_{C}^{(2)}(x,a)=\pi_{1}(x,a)=\overline{\pi}(x,a)$ if $c_{0}(x,a)=1$. Thus,
$\varphi(\pi_{C}^{(2)})=\varphi(\pi_{C}^{(1)})>\varphi(\pi_{0})$, where the
inequality follows by Assumption 4. Thus, $\pi_{1}$ is not compliance-robustly
fair for $\pi_{0}$, so the claim follows for $\pi_{A}=\pi_{1}$ and
$\pi_{H}=\pi_{0}$.

Otherwise, consider the case $\varphi(\pi_{1})>\varphi(\pi_{0})$. Consider the
policy as before

$\displaystyle\pi_{C}^{(1)}(x,a)=\begin{cases}\overline{\pi}(x,a)&\text{if
}c_{0}(x,a)=1\\\ \pi_{0}(x,a)&\text{otherwise}.\end{cases}$

As before, $\varphi(\pi_{C}^{(1)})>\varphi(\pi_{0})$. Consider the sub-case
where $\varphi(\pi_{C}^{(1)})>\varphi(\pi_{1})$. Let

$\displaystyle\pi_{C}^{(3)}(x,a)$
$\displaystyle=\begin{cases}\pi_{0}(x,a)&\text{if }\tilde{c}_{0}(x,a)=1\\\
\pi_{1}(x,a)&\text{otherwise},\end{cases}$

where

$\displaystyle\tilde{c}_{0}(x,a)=1-c_{0}(x,a).$

Note that $\pi_{C}^{(3)}=\pi_{C}^{(1)}$ since
$\pi_{C}^{(3)}(x,a)=\pi_{1}(x,a)=\overline{\pi}(x,a)$ when $c_{0}(x,a)=1$, and
$\pi_{C}^{(3)}(x,a)=\pi_{0}(x,a)$ otherwise. Thus,
$\varphi(\pi_{C}^{(3)})=\varphi(\pi_{C}^{(1)})>\varphi(\pi_{1})$. Thus,
$\pi_{0}$ is not compliance-robustly fair for $\pi_{1}$, so the claim follows
taking $\pi_{A}=\pi_{0}$ and $\pi_{H}=\pi_{1}$.

Next, consider the sub-case where $\varphi(\pi_{C}^{(1)})<\varphi(\pi_{1})$.
Let

$\displaystyle\pi_{C}^{(4)}(x,a)=\begin{cases}\underline{\pi}(x,a)&\text{if
}\tilde{c}_{0}(x,a)=1\\\ \pi_{1}(x,a)&\text{otherwise}.\end{cases}$

Note that $\pi_{C}^{(4)}=\pi_{1}$ since $\pi_{C}^{(4)}(x,a)=\pi_{1}(x,a)$ if
$c_{0}(x,a)=1$ and $\pi_{C}^{(4)}(x,a)=\underline{\pi}(x,a)$ otherwise. Thus,
$\varphi(\pi_{C}^{(4)})=\varphi(\pi_{1})>\varphi(\pi_{C}^{(1)})$, so
$\underline{\pi}$ is not compliance-robustly fair for $\pi_{1}$. Also, by
Assumption 4,
$\varphi(\pi_{1})>\varphi(\pi_{C}^{(1)})\geq\varphi(\underline{\pi})$, so the
claim follows taking $\pi_{A}=\underline{\pi}$ and $\pi_{H}=\pi_{1}$.

Finally, consider the sub-case $\varphi(\pi_{C}^{(1)})=\varphi(\pi_{1})$. Let

$\displaystyle\pi_{2}(x,a)=\begin{cases}\pi_{0}(x,a)&\text{if }c_{0}(x,a)=1\\\
\overline{\pi}(x,a)&\text{otherwise}.\end{cases}$

If $\varphi(\pi_{1})\leq\varphi(\pi_{2})$, then consider

$\displaystyle\pi_{C}^{(5)}(x,a)=\begin{cases}\pi_{1}(x,a)&\text{if
}c_{0}(x,a)=1\\\ \pi_{2}(x,a)&\text{otherwise}.\end{cases}$

Note that $\pi_{C}^{(5)}=\overline{\pi}$ since
$\pi_{C}^{(5)}(x,a)=\pi_{1}(x,a)=\overline{\pi}(x,a)$ if $c_{0}(x,a)=1$ and
$\pi_{C}^{(5)}(x,a)=\pi_{2}(x,a)=\overline{\pi}(x,a)$ otherwise. Thus,
$\varphi(\pi_{2})<\varphi(\overline{\pi})=\varphi(\pi_{C}^{(5)})$, where the
inequality follows by Assumption 4. Thus, $\pi_{1}$ is not compliance-robustly
fair for $\pi_{2}$, so the claim follows taking $\pi_{A}=\pi_{1}$ and
$\pi_{H}=\pi_{2}$.

Otherwise, we have $\varphi(\pi_{1})>\varphi(\pi_{2})$. Let

$\displaystyle\pi_{C}^{(6)}(x,a)=\begin{cases}\pi_{2}(x,a)&\text{if
}\tilde{c}_{0}(x,a)=1\\\ \pi_{1}(x,a)&\text{otherwise}.\end{cases}$

Similar to before, we have $\pi_{C}^{(6)}=\overline{\pi}$. Thus,
$\varphi(\pi_{1})=\varphi(\pi_{C}^{(1)})<\varphi(\overline{\pi})=\varphi(\pi_{C}^{(6)})$.
Thus, $\pi_{2}$ is not compliance-robustly fair for $\pi_{1}$, so the claim
follows taking $\pi_{A}=\pi_{2}$ and $\pi_{H}=\pi_{1}$. ∎

#### Fairness condition 1: demographic parity

For demographic parity, we can redefine the average function as:

$\displaystyle\overline{\pi}_{D}(a)=\sum_{x\in\mathcal{X}}\pi(x,a)\mathbb{P}(x|a).$

Next, we show that demographic parity satisfies Assumption 4, implying that it
suffers from compliance-related problems. Let us consider
$\mathcal{X}=\\{1\\}$ be singleton, then we can omit it from our notation.
Next, we construct $\overline{\pi},\pi_{0},\underline{\pi}$:

$\displaystyle\underline{\pi}(a)$ $\displaystyle=0.5$
$\displaystyle\pi_{0}(a)$
$\displaystyle=\begin{cases}0.5+\epsilon\quad\text{if $a=1$}\\\
0.5-\epsilon\quad\text{if $a=0$}\end{cases}$ $\displaystyle\overline{\pi}(a)$
$\displaystyle=\begin{cases}0.5+3\epsilon\quad\text{if $a=1$}\\\
0.5-2\epsilon\quad\text{if $a=0$},\end{cases}$

where $\epsilon\in(\frac{1}{6},\frac{1}{4})$. Now, consider the compliance
function:

$\displaystyle c_{0}(a)=\begin{cases}1\quad\text{if $a=1$}\\\ 0\quad\text{if
$a=0$}\end{cases}$

which implies the following $\pi_{C}$ and $\pi^{\prime}_{C}$:

$\displaystyle\pi_{C}(a)$
$\displaystyle=\begin{cases}0.5+3\epsilon&\begin{aligned} \text{if
}a&=1\end{aligned}\\\ 0.5&\begin{aligned} \text{if
}a&=0\end{aligned}\end{cases}$ $\displaystyle\pi^{\prime}_{C}(a)$
$\displaystyle=\begin{cases}0.5&\begin{aligned} \text{if }a&=1\end{aligned}\\\
0.5-2\epsilon&\begin{aligned} \text{if }a&=0\end{aligned}\end{cases}$

It is easy to see that the three conditions of Assumption 4 are satisfied:

  1. (1)

$\varphi(\underline{\pi})\leq\varphi({\pi_{0}})<\varphi(\overline{\pi})$

  2. (2)

$\varphi(\pi_{0})<\varphi(\pi_{C})<\varphi(\overline{\pi})$

  3. (3)

$\varphi(\pi^{\prime}_{C})<\varphi(\overline{\pi})$.

#### Fairness condition 2: calibration within groups

Another popular fairness condition is within-group calibration. As Kleinberg
et al. (2016) summarize: “if the algorithm identifies a set of people as
having a probability $z$ of constituting positive instances, then
approximately a $z$ fraction of this set should indeed be positive instances.”
We can formally define this fairness condition for each group $a$ as:

$\displaystyle\varphi(\pi)=|\overline{\pi}_{D}(x,a)-\mathbb{P}(y=1|a)|,$

where $\overline{\pi}_{D}(x,a)=\sum_{x\in\mathcal{X}}\pi(x,a)\mathbb{P}(x|a)$.
In other words, a policy $\pi$ is fair for group $a$ iff $\varphi(\pi)=0$ for
group $a$. We show that this fairness definition satisfies Assumption 4,
implying its susceptibility to compliance-related problems.

Again, consider $\mathcal{X}=\\{0,1\\}$. Since this fairness condition is
defined for each group, let us consider $a=0$. Now, suppose
$\mathbb{P}(y=1|a=0)=0.8$ and $\mathbb{P}(x=0|a=0)=\mathbb{P}(x=1|a=0)=0.5$.
Let us define $\underline{\pi},\pi_{0},\overline{\pi}$:

$\displaystyle\underline{\pi}(0,0)$
$\displaystyle=\underline{\pi}(1,0)=0.8-\epsilon$ $\displaystyle\pi_{0}(0,0)$
$\displaystyle=\pi_{0}(1,0)=0.8-2\epsilon$ $\displaystyle\overline{\pi}(0,0)$
$\displaystyle=\overline{\pi}(1,0)=0.8-3\epsilon,$

where $\epsilon\in(0,\frac{4}{15})$.Consider the compliance function:

$\displaystyle c_{0}(x,0)=\begin{cases}1\quad\text{if $x=1$}\\\ 0\quad\text{if
$x=0$},\end{cases}$

which implies that $\pi_{C}$ and $\pi^{\prime}_{C}$ being:

$\displaystyle\pi_{C}(x,a)$
$\displaystyle=\begin{cases}0.8-3\epsilon\quad\text{if $x=1$}\\\
0.8-2\epsilon\quad\text{if $x=0$}\end{cases}$
$\displaystyle\pi^{\prime}_{C}(x,a)$
$\displaystyle=\begin{cases}0.8-2\epsilon\quad\text{if $x=1$}\\\
0.8-3\epsilon\quad\text{if $x=0$}.\end{cases}$

It is easy to see that the three conditions of Assumption 4 are satisfied:

  1. (1)

$\varphi(\underline{\pi})\leq\varphi({\pi_{0}})<\varphi(\overline{\pi})$

  2. (2)

$\varphi(\pi_{0})<\varphi(\pi_{C})<\varphi(\overline{\pi})$

  3. (3)

$\varphi(\pi^{\prime}_{C})<\varphi(\overline{\pi})$.

##  7\. Conclusions and Discussions

This paper illustrates the perils of selective compliance for equitable
outcomes in human-AI collaboration. In particular, even algorithms that
satisfy traditional algorithmic fairness criteria can amplify unfairness in
decisions (relative to the human making decisions in isolation).
Unfortunately, a human decision-maker’s compliance pattern is a priori
unknown, and may even change over time, affecting fairness in outcomes.
Therefore, we introduce the concept of compliance-robust fairness and
demonstrate how to derive algorithmic policies that weakly improve fairness
regardless of the human’s compliance pattern. Naturally, it is also important
that the algorithmic advice achieves better performance than the human alone.
We show that, as long as the human policy is slightly sub-optimal and not
perfectly fair, the best performance-improving compliance-robust policy still
generates improvements over the human in isolation. However, it is not always
the case that we can also achieve the third property of traditional fairness —
we may need to rely on algorithmic policies that are unfair in isolation to
achieve compliance-robustly fair human-AI collaboration. Our findings
contribute to the design of human-AI collaboration systems that are
“compliance-aware,” enhancing rather than diminishing fairness in
collaborative decisions.

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# Extreme topological tunability of Weyl-Kondo semimetal to Zeeman coupling

Sarah E. Grefe<EMAIL_ADDRESS>Department of Physics and Astronomy and Rice
Center for Quantum Materials, Rice University, Houston, Texas 77005, USA
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico
87545, USA  Hsin-Hua Lai  Department of Physics and Astronomy and Rice Center
for Quantum Materials, Rice University, Houston, Texas 77005, USA  Silke
Paschen  Institute of Solid State Physics, Vienna University of Technology,
1040 Vienna, Austria  Qimiao Si  Department of Physics and Astronomy and Rice
Center for Quantum Materials, Rice University, Houston, Texas 77005, USA

###### Abstract

There is considerable interest in the intersection of correlations and
topology, especially in metallic systems. Among the outstanding questions are
how strong correlations drive novel topological states and whether such states
can be readily controlled. Here we study the effect of a Zeeman coupling on a
Weyl-Kondo semimetal in a nonsymmorphic and noncentrosymmetric Kondo-lattice
model. A sequence of distinct and topologically nontrivial semimetal regimes
are found, each containing Kondo-driven and Fermi-energy-bound Weyl nodes. The
nodes annihilate at a magnetic field that is smaller than what it takes to
suppress the Kondo effect. As such, we demonstrate an extreme topological
tunability that is isolated from the tuning of the strong correlations per se.
Our results are important for experiments in strongly correlated systems, and
set the stage for mapping out a global phase diagram for strongly correlated
topology.

Strongly correlated topological phases, heavy-fermion systems, Kondo effect,
Weyl semimetal, topological phases, quantum phase transitions

The application of topological concepts to condensed matter physics has
considerably enriched the landscape of quantum phases in the quantum matter
field. Yet two crucial frontiers of the field remain: the search for tunable
topological materials, and the exploration of topological states in strongly
correlated settings. A plethora of quantum states of matter are driven by
strong correlations Keimer and Moore (2017); Paschen and Si (2020). Here,
large Coulomb repulsion often produces local moments that form a building
block of the low-energy physics. The quantum entanglement between the local
moments determines the nature of the ground state. Of particular interest are
heavy fermion systems, which possess both strong correlations and significant
spin-orbit coupling that provides the band inversion suitable for nontrivial
topology.

In heavy fermion metals, itinerant electrons are quantum entangled with the
local moments as well, giving rise to the celebrated Kondo effect Hewson
(2003) and a rich landscape of metallic quantum phases and quantum critical
points Kirchner  _et al._ (2020); Löhneysen  _et al._ (2007). When spin-orbit
coupling is also strong, topological metallic states may also appear in the
phase diagram. Recent theoretical Lai  _et al._ (2018); Grefe  _et al._
(2020a) and experimental Dzsaber  _et al._ (2017, 2021) studies have shown
that the Kondo effect can drive the emergence of Weyl nodes near the Fermi
energy, leading to a Weyl-Kondo semimetal (WKSM). This development provides a
new opportunity to utilize insights about the strong correlation physics of
the heavy fermion metals and elucidate how strong correlations and topology
intersect in metallic systems.

Strong correlations per se amplify a system’s response to external stimuli.
For example, heavy fermion metals have long been explored by a magnetic field,
which is a non-thermal tuning parameter and its relatively small variation can
trigger phase transitions or help access quantum critical points by perturbing
the $4f$ local moments Heuser  _et al._ (1998). This is highlighted by the
usage of a magnetic field to reveal a jump in the normal Hall effect that
originates from an abrupt change in the Fermi surface at a local quantum
critical point Paschen  _et al._ (2004); Nair  _et al._ (2012); Si and Paschen
(2013); Shishido  _et al._ (2005); Park  _et al._ (2006); Knebel  _et al._
(2008); Si  _et al._ (2001); Coleman  _et al._ (2001); Si (2006). What has
been open is whether strongly correlated topology can be readily tuned and, if
so, what sequence of topological quantum phase transitions can be realized.

In this Letter, we study the effect of a Zeeman coupling in a nonsymmorphic
and noncentrosymmetric Kondo lattice model. We find that the combined time-
reversal and inversion symmetry breaking associated with the Zeeman field and
a sublattice-differentiating potential produces a progression of topological
Lifshitz transitions. The Weyl nodes are annihilated at a Zeeman energy that
is smaller than the Kondo energy scale and, thus, the strongly correlated
topological semimetal retains its heavy fermion character under the magnetic
field before it is quenched into a Kondo insulator. The complete node
annihilation happens at a modest (laboratory-accessible) field. This reflects
the extreme tunability of the strongly correlated topology to the Zeeman
coupling, which is isolated from the tuning of the correlation physics per se.
The extreme topological tunability is also to be contrasted with the case of
weakly correlated Weyl semimetals where the Weyl nodes are hard to tune
Hirschberger  _et al._ (2016); Zhang  _et al._ (2017); Caglieris  _et al._
(2018); Ramshaw  _et al._ (2018). Our results are important for experiments on
heavy fermion and related systems Dzsaber  _et al._ (2019). In addition, our
findings set the stage to map out a theoretical global phase diagram for
strongly correlated topology. The latter is important to the identification of
new correlated topological states in theoretical models, and may guide
systematic experimental explorations in this frontier area.

Figure 1:  (a) Phase diagram of the WKSM model as a function of the Zeeman
field normalized by the zero-field Kondo temperature, $M_{0}=M_{z}/T_{K}^{0}$.
“Phases” labels each WKSM regime by its nodal trajectory in the BZ.
TLT=topological Lifshitz transition, which labels regime crossover and
critical points (filled circles) by where it occurs in the BZ and the type of
dispersion. TRI=time-reversal invariant, QBT=quadratic band touching,
ABT=anisotropic band touching. (b)-(d) shows the path of selected nodes
through the fcc BZ for each corresponding WKSM regime: (b) $X$-$W$ (yellow),
(c) $\Gamma$-$X$ (light blue), and (d) $\Gamma$-$X-\Gamma$-$L$ (orange). Red
(blue) refers to $-1\,(+1)$ monopole Weyl nodes, darker (lighter) color
variants denote nodes below (above) the Fermi energy, white-filled circles
mark double Weyl nodes regime.

Model and Methods: To consider the strong coupling (Kondo) limit, we implement
the Anderson lattice model on a diamond crystal structure with
$\mathcal{H}=\mathcal{H}_{c}+\mathcal{H}_{cd}+\mathcal{H}_{d}$ Grefe  _et al._
(2020a); Lai  _et al._ (2018). The conduction electrons corresponding to the
$spd$-orbitals of heavy fermion systems are represented by a Hamiltonian based
on the Fu-Kane-Mele model Fu  _et al._ (2007); Ojanen (2013); Murakami and
Kuga (2008),

$\displaystyle\mathcal{H}_{c}$ $\displaystyle=t\sum_{\langle
ij\rangle,\sigma}\left(c_{i\sigma}^{\dagger}c_{j\sigma}+\textrm{H.c.}\right)+\sum_{i,\sigma}\left(m(-1)^{i}-\mu\right)n_{i\sigma}^{c}$
$\displaystyle+i\lambda\sum_{\langle\langle
ij\rangle\rangle}\left[c^{\dagger}_{i\sigma}\left(\bm{\sigma}\cdot{\bf
e}_{ij}\right)c_{j\sigma}-\textrm{H.c.}\right],$ (1)

with nearest-neighbor $\langle ij\rangle$ hopping $t$, chemical potential
$\mu$, a Dresselhaus-type spin orbit coupling $\lambda$, and an inversion
symmetry breaking sublattice potential $m$. The electrons are coupled to the
$d$ fermion species (representing physical $4f$ moments) through the
hybridization term,

$\displaystyle\mathcal{H}_{cd}$
$\displaystyle=V\sum_{i,\sigma}\left(d_{i\sigma}^{\dagger}c_{i\sigma}+\textrm{H.c.}\right).$
(2)

The localized $4f$ electrons are represented by

$\displaystyle\mathcal{H}_{d}$
$\displaystyle=\sum_{i}[Un^{d}_{i\uparrow}n^{d}_{i\downarrow}+\sum_{\sigma}(E_{d}\sigma_{0}+M_{z}\sigma_{z})d_{i\sigma}^{\dagger}d_{i\sigma}],$
(3)

where $E_{d}$ is much lower in energy than the conduction electron band edge
Hewson (2003), $\sigma_{0},\sigma_{z}$ are Pauli matrices acting on spin
space, and the Coulomb repulsion $U$ acts between $d$ fermions. The Zeeman
interaction with field $M_{z}$ aligned in the $\hat{z}$ direction, breaks
time-reversal symmetry. In a heavy fermion system subjected to a magnetic
field, the response of the local moments dominates over orbital effects, so we
do not consider Landau levels here.

In the $U\rightarrow\infty$ (Kondo) limit, we use the auxiliary boson method
Hewson (2003), with $d_{i\sigma}=f_{i\sigma}b_{i}^{\dagger}$. Here, double-
occupancy of the $d$-electrons is projected out, and small hole-fluctuations
away from single-occupation are tracked by the bosonic field,
$b_{i}\rightarrow\langle b_{i}\rangle=r$. The occupation constraint is
enforced by a term that replaces the Coulomb repulsion,
$\mathcal{H}_{\ell}=\ell\sum_{i,\sigma}(f^{\dagger}_{i\sigma}f_{i\sigma}+r^{2}-1)$.
The parameters $\ell$ and $r$, along with $\mu$, are determined by solving the
system of saddle point equations $\langle\delta\mathcal{H}/\delta
r\rangle=0,~{}\langle\delta\mathcal{H}/\delta\ell\rangle=0$, together with a
condition for the electron filling. We consider a total quarter filling
$n_{d}+n_{c}=1$, which is $1$ fermion per site; in the strong coupling limit
we have $n_{f}+r^{2}=1$, implying that $n_{c}\sim r^{2}$.

Generically, both the time-reversal and inversion symmetry-breaking will
influence the Weyl nodes Grefe  _et al._ (2020b). We study the extent to which
the Kondo-driven Weyl nodal excitations survive the Zeeman coupling and, if
they do, whether and how they can be controlled by the magnetic field. Our
numerical method to solve the self consistency equations is detailed in the
Supplemental Material zee .

Zeeman-Field Tuning of the Weyl Nodes: In what follows, we describe several
Weyl node configurations in the Brillouin zone (BZ) that we encounter as a
function of the Zeeman field. We label each WKSM regime by the Weyl node
trajectories from one high symmetry point to another, and label topological
Lifshitz transitions (TLTs) between regimes/phases by their dispersion type
and the BZ location, as shown in the phase diagram of Fig. 1(a). The Zeeman
field strength is reported as $M_{0}=M_{z}/T_{K}^{0}$, relative to the Kondo
temperature $T_{K}^{0}$, which is estimated from the bandwidth of the heavy
Weyl bands for $M_{z}=0$, in units of $\mu_{0}=k_{B}=1$; the parameters held
constant are $\\{E_{d},V,\lambda,m,t\\}=\\{-7,9.29,0.5,1,1\\}$. We previously
established the WKSM phase for the time-reversal invariant case Grefe  _et
al._ (2020a); Lai  _et al._ (2018), represented by the black $M_{0}=0$
starting point of Fig. 1(a). Here Weyl nodes can be found along all $X$-$W$
lines on the BZ boundary (red and blue dots of Fig. 1(b)).

In the yellow $X$-$W$ regime, the Weyl nodes move along the yellow arrows in
the BZ diagram in Fig. 1(b). The darker (lighter) blue and red circles of
nodes on the $k_{x},k_{y}=\pm 2\pi$ BZ faces (but off the
$k_{x}$-$k_{y}-$plane) indicate that, in addition to moving in
$\mathbf{k}-$space toward $W$, they shift in energy below (above) the Fermi
energy ($E_{F}=0$), establishing Fermi/hole pockets that evolve with the
field, which disappear in the $\mathit{\Gamma}$-$X-\mathit{\Gamma}$-$L$ regime
(details in Ref. zee ). Remarkably, all the nodes besides these (moving toward
$X$) are pinned to $E_{F}$. In particular, the nodes remaining at $E_{F}$ are
the $k_{x},k_{y}=\pm 2\pi$ BZ face nodes lying in the $k_{x}$-$k_{y}-$plane,
and all four nodes on the $k_{y}=\pm 2\pi$ faces.

The $k_{z}=\pm 2\pi$ face nodes meet uniformly at $X$, simultaneously with the
pinned nodes on the $k_{x},k_{y}=\pm 2\pi$ BZ faces, while the off-$E_{F}$
nodes meet at $W$. At $M_{0}=0.128$ (yellow circle, Fig. 1(a)), the dispersion
forms a non-Kramers’ type quadratic band touching at $X$, dubbed the $X$-QBT
point.

The emergence of double-Weyl nodes: Next, the WKSM enters the
$\mathit{\Gamma}$-$X$ regime (Fig. 1(c)). For $M_{0}$ just beyond the $X$-QBT
point, the dispersion undergoes a TLT, and Weyl nodes form just inside the BZ
boundary from the six $X$ points, and move towards $\mathit{\Gamma}$ along the
blue arrows. On the $k_{x},k_{y}$-plane, double Weyl nodes develop with charge
$+2$ Fang  _et al._ (2012); Yang and Nagaosa (2014); Jian and Yao (2015); Li
_et al._ (2016); Huang  _et al._ (2016); Ezawa (2017a, b); Ahn  _et al._
(2017); Yan and Wang (2017); Zhang  _et al._ (2018), yet along the
$\hat{z}$-axis, the Weyl nodes form a monopole charge-opposite pair.

Figure 2:  $\mathit{\Gamma}$-$X-\mathit{\Gamma}$-$L$ WKSM eigenenergy
dispersion at $M_{0}=0.432$; (a) high symmetry path through the BZ, (b) on the
$[011]$ plane, with origin at $\mathit{\Gamma}$. Here, $D$ is the bare
conduction-electron bandwidth.

At the $L$ points when $M_{0}=0.290$ (blue circles, Fig. 1(a),(d)), a
crossover TLT occurs, called the $L$-ABT point to label an anisotropic band
touching at $L$: the dispersion along the $(\pm 1,\pm 1,\pm 1)$ directions is
a quadratic band touching, but is linear in the perpendicular plane of the
hexagonal BZ boundary. When $M_{0}>0.290$, charge $\pm 1$ Weyl nodes form from
the $L$ points along the $\mathit{\Gamma}$-$L$ directions (see Fig. 2). All
sets of nodes (3 pairs of $\mathit{\Gamma}$-$X$, 4 pairs of
$\mathit{\Gamma}$-$L$) progress towards $\mathit{\Gamma}$, and this
$\mathit{\Gamma}$-$X-\mathit{\Gamma}$-$L$ regime corresponds to the orange
regime and arrows in Fig. 1(a),(d).

Figure 3:  (a) Berry curvature projected onto the $[011]$ plane in the
$\mathit{\Gamma}$-$X-\mathit{\Gamma}$-$L$ regime at $M_{0}=0.432$, where
$k_{0}$ points along the $k_{y}$-$k_{z}-$plane. The orange arrows indicate the
node pairs’ path toward simultaneous annihilation at the $\mathit{\Gamma}$
point for $M_{0}>0.432$. The green line marks the BZ boundary, blue/red
(empty) circles mark $\pm 1\,(+2)$ Weyl nodes, and yellow, blue, and orange
circles correspond to the TLT locations of Fig. 1(a).

The normalized Berry curvature field of the
$\mathit{\Gamma}$-$X-\mathit{\Gamma}$-$L$ WKSM is shown in Fig. 3(a),
projected onto the [011] plane to show both the $\mathit{\Gamma}$-$X$ and
$\mathit{\Gamma}$-$L$ nodes. The $\mathit{\Gamma}$-$L$ nodes (blue and red
points) display sink/source monopole fields, while the $\mathit{\Gamma}$-$X$
nodes have field structure for charge $+2$ double-Weyl fermions Fang  _et al._
(2012); Yang and Nagaosa (2014); Jian and Yao (2015); Huang  _et al._ (2016);
Li  _et al._ (2016); Ezawa (2017a, b). In Fig 3(b)-(c), closer field
configurations of the $\mathit{\Gamma}$-$X$ double Weyl fermions are shown.

The emergence of the double Weyl fermions out of a quadratic band touching is
reminiscent of what happens in diamond lattice $\alpha$-Sn Barfuss  _et al._
(2013); Rogalev  _et al._ (2017); in a $k\cdot p$ model, the application of a
magnetic field to the pre-existing Luttinger semimetal leads to double-Weyl
nodes at different energies Zhang  _et al._ (2018). There, it was found that a
linear inversion symmetry breaking term splits the double-Weyl fermions into
single-Weyl fermions. By contrast, in our model the double-Weyl fermions are
pinned to the Fermi energy, and are present over a range of combined time-
reversal and inversion symmetry breaking couplings. The latter reflects the
robustness of the Kondo effect that underlies the Weyl nodes. We also note
that double Weyl fermions enhance the anomalous Hall effect twofold, and their
Fermi arcs come in pairs Ezawa (2017a).

Zeeman-induced Annihilation of the Weyl Nodes: When the Zeeman coupling
approaches the threshold value $M_{0}=0.611$, all the $\mathit{\Gamma}$-$X$
and $\mathit{\Gamma}$-$L$ nodes meet in the zone center $\mathit{\Gamma}$.
This complete annihilation of the Weyl nodes by the Zeeman coupling is
illustrated in Fig. 1(d) as well as in Fig. 3(a).

At the threshold coupling $M_{0}=0.611$, a quadratic band touching critical
point is formed (labeled $\mathit{\Gamma}$-QBT) (orange circle, Fig. 4). The
$\mathit{\Gamma}$-QBT point is non-Kramers (singly degenerate) due to the
broken time-reversal symmetry. When $M_{0}$ goes beyond this threshold, the
$\mathit{\Gamma}$-QBT bands open a gap, leading to a Kondo insulator phase.
The emergence of the $\mathit{\Gamma}$-QBT point is consistent with a
continuous nature of the zero-temperature topological phase transition at this
threshold $M_{0}$ value.

Figure 4:  Eigenenergy dispersion at $\mathit{\Gamma}$-QBT; (a) high symmetry
path through the BZ, (b) on the $k_{x}$-$k_{y}$ plane, with origin at
$\mathit{\Gamma}$.

Indeed, $\mathit{\Gamma}$-QBT marks a second-order thermodynamic quantum phase
transition. This is evidenced by the parameters $\ell$ and $r$ as a function
of $M_{0}$ being continuous and showing nonanalyticities (pronounced kinks)
across the $\mathit{\Gamma}$-QBT point, as seen in Fig. 5.

Pinning of the Weyl nodes to the Fermi energy: An important finding of our
work is that each topologically nontrivial regime contains Fermi-energy-bound
Weyl nodes. In the presence of time-reversal symmetry, this effect has been
extensively demonstrated. The strong correlation effect underlying the Kondo
effect dictates that the Kondo-driven Weyl nodes lie near the Fermi energy
within the narrow energy range of the Kondo scale; this effect, when combined
with the nonsymmorphic nature of the space group symmetry, pins the Kondo-
driven nodes in our model to the Fermi energy Lai  _et al._ (2018); Grefe  _et
al._ (2020a). The Zeeman field only reduces the symmetry of the spin degrees
of freedom, and so the node-pinning mechanism remains free to operate. The
space-group symmetry constraint is a general phenomenon Cano  _et al._ (2018);
Watanabe  _et al._ (2016); Young  _et al._ (2012). Thus, we expect that the
mechanism we have advanced here, for the Zeeman-field-induced WKSM to Kondo-
insulator transition, will apply to Kondo-lattice systems defined on a variety
of nonsymmorphic crystalline structures.

Indeed, we can see the generality of the Fermi-level-bound nature of the
Kondo-driven Weyl nodes by comparing the Zeeman-tuned WKSM-Kondo insulator
transition to one that occurs as a function of tuning $m$ with time-reversal
preserved Ojanen (2013); Lai  _et al._ (2018); Grefe  _et al._ (2020a) (even
though the latter tuning is experimentally difficult to implement). When
$m=0$, inversion symmetry is preserved, resulting in a Dirac-Kondo semimetal
with degeneracies at $X$. A WKSM forms for $0<m<2$ where the inversion-
symmetry breaking splits the Dirac cone into two Weyl node pairs per square BZ
boundary, with trajectories from $X$ to $W$. At $m=2$, each node meets its
opposite charge partner from the neighboring BZ, becoming an anisotropic band
touching at $W$, before annihilating and forming a gapped Kondo insulator
state for $m>2$. Similar to the Zeeman-tuned WKSM-Kondo insulator transition,
the merging of Weyl nodes at $W$ precipitates their annihilation and gap
formation, which in both cases is accompanied by a kink in the saddle-point
parameters across the critical point. Moreover, the topologically-nontrivial
regime always has the Kondo-driven Weyl nodes at the Fermi energy. However, in
the case of Zeeman tuning at nonzero $m$, the double Weyl nodes emerge, and
the gap opening location occurs at the highly symmetric time-reversal-
invariant momentum $\mathit{\Gamma}$, in contrast to annihilation at the
lower-symmetry time-reversal _non_ -invariant momentum $W$.

Figure 5:  The saddle-point parameters as a function of $M_{0}$. Top: scalar
bosonic condensate $r$. Bottom: Lagrange multiplier $\ell$. The dashed line
marks the kinks at the $\mathit{\Gamma}$-QBT point where $M_{0}=0.611$.

Implications of our results: Our results are highly relevant to the magnetic
field experiments done on the nonmagnetic heavy fermion compound Ce3Bi4Pd3,
where a WKSM-Kondo insulator transition has been observed Dzsaber  _et al._
(2019). The experiments also found a heavy-fermion metal phase at high field
values, which is more typical of Kondo-insulator physics and is not the focus
of the present work. Importantly, we have shown that the WKSM phase and its
TLTs take place at a Zeeman field smaller than the Kondo scale, before the
Kondo effect itself is suppressed by the Zeeman coupling. In other words, the
extreme tuning of the strongly correlated topology happens when the strong
correlation physics per se does not experience a qualitative change.

A Zeeman coupling on the order of the Kondo scale in typical heavy fermion
semimetals corresponds to a magnetic field on the order of 10 T. In weakly-
interacting WSM systems, such a magnetic field would have produced a minute
Zeeman effect; indeed, typically, the orbital effect of the magnetic field
dominates, which smears the Weyl fermions and prevent a well-controlled nodal
annihilation Zhang  _et al._ (2017); Caglieris  _et al._ (2018); Ramshaw  _et
al._ (2018); Liang  _et al._ (2017). We expect that the extreme topological
tunability we have demonstrated also applies to related models based on Kondo
effects Feng  _et al._ (2013, 2016); Ok  _et al._ (2017); Chang and Coleman
(2018); Pixley  _et al._ (2017); Cao  _et al._ (2020), as well as to other
materials, such as CeRu4Sn6 Fuhrman  _et al._ (2020); Paschen  _et al._
(2010); Guritanu  _et al._ (2013); Sundermann  _et al._ (2015); Wissgott and
Held (2016); Xu  _et al._ (2017), YbBiPt Guo  _et al._ (2018); Fisk  _et al._
(1991); Chadov  _et al._ (2010); Mun  _et al._ (2013) and CeSbTe Schoop  _et
al._ (2018), which are considered to be WKSM candidates.

Summary: This work has addressed the effect of Zeeman coupling in a
nonsymmorphic and noncentrosymmetric Kondo lattice model, in which the
cooperation of the Kondo effect and space-group symmetry produces Weyl nodes
near the Fermi energy. We have demonstrated an extreme responsiveness of the
Weyl nodes to the Zeeman coupling. Several topologically-distinct semimetal
regimes are induced by the Zeeman coupling, which involve double Weyl points
that may significantly affect the anomalous magnetotransport properties. We
have shown that a Zeeman coupling that is smaller than the zero-field Kondo
energy scale is adequate to fully annihilate all the Weyl nodes, leading to a
second-order topological quantum phase transition to a Kondo insulator. Our
results provide a proof-of-principle demonstration that the extreme tuning of
strongly correlated topology can be realized without the interference of any
qualitative change to the strong correlation physics per se. Equally
important, our work sets the stage for the exploration of a global phase
diagram for strongly correlated topology, which may be important for
identifying new correlated topological states both theoretically and
experimentally.

###### Acknowledgements.

We thank Pallab Goswami, Sami Dzsaber, Jennifer Cano, Diego Zocco, Mathieu
Taupin for useful discussions. Work at Rice has been supported by the NSF
(DMR-1920740), the Robert A. Welch Foundation (C-1411) and the ARO
(W911NF-14-1-0525). Work at Los Alamos was carried out under the auspices of
the U.S. Department of Energy (DOE) National Nuclear Security Administration
under Contract No. 89233218CNA000001, and was supported by LANL LDRD Program.
Work in Vienna was supported by the Austrian Science Fund (projects P29279 and
P29296) and the European Community (H2020 project 824109).

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