Bertsimas et al.

Pricing Algorithmic Insurance

Pricing Algorithmic Insurance

Dimitris Bertsimas Sloan School of Management, Massachusetts Institute of
Technology, MA, USA<EMAIL_ADDRESS>Agni Orfanoudaki Operations Research
Center, Massachusetts Institute of Technology, MA, USA<EMAIL_ADDRESS>
As machine learning algorithms start to get integrated into the decision-
making process of companies and organizations, insurance products will be
developed to protect their owners from risk. We introduce the concept of
algorithmic insurance and present a quantitative framework to enable the
pricing of the derived insurance contracts. We propose an optimization
formulation to estimate the risk exposure and price for a binary
classification model. Our approach outlines how properties of the model, such
as accuracy, interpretability and generalizability, can influence the
insurance contract evaluation. To showcase a practical implementation of the
proposed framework, we present a case study of medical malpractice in the
context of breast cancer detection. Our analysis focuses on measuring the
effect of the model parameters on the expected financial loss and identifying
the aspects of algorithmic performance that predominantly affect the price of
the contract.

Algorithmic Insurance, Machine Learning, Algorithmic Risk

##  1 Introduction

Data-driven analytical models and machine learning (ML) algorithms have
started transforming large facets of the economy, becoming the driver of
innovation in digital marketing, self-driving cars and medical imaging, among
others. While the use of artificial intelligence expands across all segments
of society, algorithms are expected to replace human judgement in many cases
(Coglianese and Lehr 2016). However, there is still a lot of resistance in
employing these tools in practice as different types of concerns are emerging
about algorithmic decision-making (Char et al. 2018). Dietvorst et al. (2015)
defined this phenomenon as algorithm aversion, identifying a wide range of
factors, including ethical issues, related to the use of algorithms that are
not resolved yet (Dietvorst et al. 2018). One of the persistent challenges
related to the implementation of ML models in practice is centered around the
question of responsibility in case of erroneous algorithmic decisions making
(Singh et al. 2016).

This is not the first time that modern societies are called to face such
dilemmas. Similar situations haven gained spotlight in the past, such as when
cars became accessible to the broader consumer audience in urban areas after
World War I. By that time, motor vehicles had become fast and less expensive
and manufacturers were expecting an increasing portion of society to enter the
market. Nevertheless, the vast majority of households were still reluctant to
adopt the technology as it could entail disproportionately high financial
risks. In the absence of insurance policies, injured victims would seldom get
any compensation in an accident, and drivers often faced considerable costs
for damage to their car and property. To ensure that all vehicle owners and
drivers can be protected against the risk of causing injury or death to third
parties, the Road Traffic Act was established in 1930 in the United Kingdom,
introducing the first compulsory car insurance scheme (Senn 1930). Over the
years, legal questions related to responsibility became inextricably linked to
third party liability. In general terms, third party liability insurance
provides protection against claims resulting from physical injuries to people
and/or damage to property (Reed et al. 2016). Nowadays, liability coverage has
become so important that it is often required for automotive insurance
policies, product manufacturers, and anyone who practices medicine or law.

As ML algorithms are expected to replace human decision-making in cases where
their predictive and prescriptive performance yields better outcomes, a new
type of liability insurance could be developed to protect their owners from
risk. Potential examples are self-driving cars (third party liability),
preventive maintenance based on ML (general liability) and image recognition
systems for MRI machines (medical liability). If there were legal contracts
that individuals and organizations could sign to protect themselves from
algorithmic mistakes, the implementation of ML tools would be significantly
faster and less contentious (Bertolini et al. 2016).

To propel the effective use and widespread adoption of ML algorithms, we
introduce in this paper the concept of algorithmic insurance contracts and
provide quantitative tools to evaluate them. Our goal is to propose a
methodology to price insurance contracts in situations where decisions are
made by ML algorithms. Specifically, we address how the properties of ML
models, such as interpretability and generalizability, can influence pricing.

The prerequisites of this type of contracts are, as in all cases of insurance,
(i) an agreement between the parties, (ii) the existence of a risk to the
insured party or potential third parties, (iii) the payment of a premium
(Bertolini 2013). We determine the premium as a function of the assumed risk.
Hence it varies according to both the likelihood of its occurrence
(frequency), and the magnitude of the consequences that may arise once it
materializes (severity). Thus, the main focus of this work is the
determination of an appropriate premium pricing strategy for every policy
given the expected risk exposure.

The pricing of traditional non-life insurance products is usually conducted
using the fundamental principles of actuarial science and asset pricing theory
(Cummins 1990, Embrechts 2000). Leveraging historic observations from prior
realizations of risk, underwriters are able to create probabilistic models
that estimate the potential risk exposure, the relative frequency of adverse
events, and associate an appropriate price.

The key challenge of algorithmic insurance is the absence of historic claims
that could be used to directly apply existing underwriting processes. During
the initial implementation of ML-driven decision-making tools, very limited
information regarding past algorithmic mistakes is available. For example,
since fully autonomous cars have not been deployed yet, it is impossible to
collect information with respect to past accidents they have caused. We
overcome this limitation by centering our approach around the data that
participated in the training and validation process of the ML models. By
leveraging this valuable resource and other properties of the algorithm, we
introduce a series of tools for insurance companies and data scientists that
will allow them to evaluate the litigation risk associated with the
implementation of a ML algorithm under different scenarios.

### Contributions

We present for the first time a data-driven methodology to price insurance
contracts in situations where decisions are made by ML algorithms. Our
contributions can be summarized as follows:

  1. 1.

We introduce an optimization formulation that leverages measures of risk from
the financial literature to simultaneously estimate the risk exposure and
price for a given binary classification model. We extend the formulation using
robust optimization to different types of uncertainty sets around historical
scenarios of loss.

  2. 2.

We propose how to price algorithmic liability based on properties of the
underlying ML method, such as accuracy, interpretability and generalizability.

  3. 3.

We provide a case-study for medical liability and analyze the potential effect
of the model parameters in simulated experiments.

  4. 4.

We show that the most impactful parameter on the risk exposure of the
predictive model is the chosen classification threshold. Our analysis also
illustrates that the convexity of the selected interpretability function
dictates whether a smaller or larger degree of transparency is needed to
significantly reduce the expected financial loss of the contract. Finally, we
demonstrate that the use of Generative Adversarial Networks (GANs) allows
decision makers to measure the effect of distributional changes in the data on
the expected financial loss.

The structure of the paper is as follows. In Section 2, we introduce a case
study of medical liability in the context of breast cancer detection that
serves to define in a concrete way the problem we are addressing. In Section
3, we present the baseline optimization formulation that leads to the
simultaneous estimation of price and risk exposure. In Section 4, we
incorporate the predictive accuracy of a binary classification model into the
estimation process of future expected loss. In Section 5, we demonstrate how
to include a ML algorithm’s interpretability into the pricing strategy.
Section 6 focuses on data generalizability and its effect on the premium
determination. In each of the Sections 4-6, we use the medical liability case
study to showcase a practical implementation of the framework and highlight
the effect of the model parameters. In Section 7, we discuss the key findings
from the computational experiments, the limitations of the framework, and
future applications of the proposed approach. We conclude in Section 8.

##  2 A Case Study of Medical Liability for Malignant Tumor Detection

In this section, we introduce a case study of algorithmic insurance for
medical liability. The case study will illustrate a practical implementation
of the proposed risk evaluation framework.

We delve into breast cancer, a carcinoma that is estimated to affect one out
of eight women at some point in their lifetime (SEER 2020). According to the
American Cancer Society, more than 250,000 women are diagnosed with invasive
breast cancer every year in the US (Siegel et al. 2018). Due to widely
established screening policies and improved therapies, breast cancer has now
one of the lowest mortality rates among carcinomas (Berry et al. 2005).
Nevertheless, medical malpractice related to breast cancer and breast imaging
remains common and costly for both radiologists and healthcare organizations
involved (Lee et al. 2020). In the future, the widespread implementation of
artificial intelligence algorithms is expected to improve the diagnostic
accuracy and reduce diagnostic errors in carcinomas (Kim et al. 2020, Yala et
al. 2019).

Given the transformative role that ML can play in this application area, we
focus on a case study for breast cancer detection to illustrate a practical
implementation of the proposed pricing framework. First, we describe the
dataset that we use to train the underlying predictive model. Subsequently, we
present an overview of historical medical malpractice lawsuits for breast
cancer detection.

### Data Description

For our analysis, we will use the Breast Cancer Wisconsin Diagnostic dataset
from the UCI ML Repository (Dua and Graff 2017). The features of the dataset
represent characteristics of the cell nuclei of a breast mass (Street et al.
1993). This information was acquired from digitized images of fine needle
aspirate (FNA) analyses. FNA biopsies are recommended to women who are
suspected to suffer from breast cancer. During this procedure, a small amount
of breast tissue or fluid is taken from the suspicious area and is checked for
cancer cells. The dataset contains ten features related to the cell nucleus of
each sample, including radius, texture, perimeter, area, smoothness, degree of
compactness, concavity, presence concave points, symmetry, and fractal
dimension. The outcome of interest is whether the sample belongs to a benign
or malignant tumor.

### Medical Malpractice Lawsuits for Breast Cancer

Lawsuits involving breast cancer are the most common cause of medical
malpractice litigation in the United States (Whang et al. 2013). An analysis
from credentialing data of 8401 radiologists, revealed that breast cancer was
the most frequently missed diagnosis, followed by nonvertebral fractures and
spinal fractures (Whang et al. 2013). Breast cancer imaging lawsuits involve
physicians from multiple specialties, radiology being the most common. Lee et
al. (2020) identified 253 cases in the US from 2005 to 2015 that resulted in
plaintiff payment where the average award amount was $978,858. The median
award amount in cases with a verdict was $862,500 with interquartile range
(IQR) ($500,000 to $2,009,460) while, in cases that concluded with a
settlement, it was as high as $1,162,500 with IQR ($17,000 to $2,000,000).

In a separate study conducted from 1995 to 1997, 218 surgical pathology and
FNA claims were reviewed. Breast FNA corresponded to 6% of those records while
breast biopsy accounted for another 14% (Anderson and Troxel 2005). 54% of
breast biopsy claims referred to false-negative diagnoses of breast carcinoma,
whereas 35% were for the false positive diagnosis of cancer, demonstrating the
importance of high sensitivity and specificity for the binary classification
model (Le et al. 2016). Malpractice claims from false positive FNAs are
usually attributed to wrong interpretations by the medical team. The most
common case is when a fibroadenoma is misclassified as carcinoma resulting in
unnecessary mastectomy or axillary node sampling if breast conservation is
elected (Anderson and Troxel 2005). Such cases can result in malpractice
claims of more than $800,000 (Rosen&Perry , Gismondi&Associates 2012).

In the following sections, we analyze the impact of predictive performance,
interpretability, and generalizability on the insurance contract. The goal is
to understand the effect of the model parameters and highlight the algorithmic
aspects that may significantly affect the risk exposure of an insurance
contract.

##  3 Quantifying the Risk Exposure

Our goal in this section is to provide a baseline optimization framework to
quantify the risk exposure of the insurance contract and the associated price.
We present both a nominal and a robust formulation of the problem based on the
expected severity and frequency of the claims.

Our framework estimates risk exposure using tools from the finance literature.
We resort to two well established statistical techniques used to measure the
level of financial risk within a firm or an investment portfolio over a
specific time frame; the Value-at-Risk (VaR) and the Conditional-Value-at-Risk
(CVaR). VaR answers the question of what is the maximum loss with a specified
confidence level (Jorion 2000). CVaR is the expected loss if that confidence
level is ever crossed. Though VaR is a widely accepted and used measure, it is
not coherent as it violates the subadditivity property of coherent measures
(Artzner et al. 1999). VaR is neither convex nor smooth, and it may result to
multiple local extrema (Rockafellar and Uryasev 2000). As a result, optimizing
this measure under different constraints is quite complex. For this reason, we
focus on CVaR, which is a coherent and convex risk measure and can be
optimized using linear optimization (Uryasev 2000, Artzner et al. 1999).

We consider $P$ premium categories, corresponding to the different classes of
claims that we insure. For example, referring to the medical malpractice
problem, we could classify the patients into three age groups
$([0,30),[30,50],[50,120))$, in which case $P=3$. Let
$f(\mathbf{x},\mathbf{z})$ be a loss function that takes as input a decision
vector of premiums $\mathbf{x}=(x_{1},\dots,x_{P})$ and a random vector of
losses $\mathbf{z}=(z_{1},\dots,z_{P})$. The decision vector $\mathbf{x}$
belongs to a feasible set of prices $\mathbf{X}$. The loss function
$f(\mathbf{x},\mathbf{z})$ is equal to the difference between the price and
the future loss in case there is a claim:

$f(\mathbf{x},\mathbf{z})=\sum_{j=1}^{P}\max\\{0,z_{j}-x_{j}\\}.$ (1)

Uryasev (2000) proposed a framework for the simultaneous calculation of VaR
and CVaR as well as the determination of the associated premium based on the
distribution of the losses (see Appendix for an overview). The author also
noticed that in the real-world we do not have the analytical representation of
the density function of $\bm{z}$ to estimate the probability that the loss
function does not exceed a threshold value $\alpha$. However, using past
observations, we have access to scenarios $\mathbf{y}_{j},$ with $j\in[J]$,
sampled from the density $p(\mathbf{z})$. Thus, the following CVaR
approximation was introduced:

$\widetilde{F}_{\beta}(\mathbf{x},\alpha)=\alpha+\nu\sum_{j=1}^{J}(f(\mathbf{x},\mathbf{y}_{j})-\alpha)^{+},$
(2)

where $\beta$ is the a given confidence level, $\nu=((1-\beta)J)^{-1}$ and
$\alpha$ is the variable that represents VaR. If $f(\mathbf{x},\mathbf{y})$ is
convex w.r.t. to $\mathbf{x}$, then $\widetilde{F}_{\beta}(\mathbf{x},\alpha)$
is a convex non-smooth function w.r.t. to $(\mathbf{x},\alpha)$. Thus, linear
optimization can be used to solve the problem of interest. Several studies
have shown that this approach provides a very powerful, fast, and numerically
stable technique which can solve problems with a large number of variables and
past scenarios (McNeil et al. 2015, Markowitz 2010, Bertsimas et al. 2004).

### A Nominal Formulation for Algorithmic Insurance

We adjust the formulation introduced by Uryasev (2000) to the algorithmic
insurance setting. The proposed framework is based on a linear optimization
formulation that leverages data-driven scenarios $\mathbf{y}_{j}$. We present
in the following sections a quantitative process to simulate $J$ scenarios
$\mathbf{y}_{j}$ for a given number of observations $N$. The loss function
$f(\cdot)$ is not linear since it is defined as $f(x,y)=\max\\{0,(y-x)\\}$. We
can also restrict the price vector $\mathbf{x}$ within fixed lower ($l_{p}$)
and upper bounds ($H_{p}$) that reflect the pricing constraints of the insured
party. We let $[P]$ represent the set of premium categories that are included
in the contract. We propose the following baseline formulation:

$\begin{array}[]{ll@{}ll}\text{minimize}&\displaystyle\alpha+\nu\sum_{j=1}^{J}z_{j}&\\\
\text{subject to}&\displaystyle
z_{j}\geq\sum_{p=1}^{P}\max\\{0,(y_{pj}-x_{p})\\}-\alpha,&&j\in[J],\\\
&\displaystyle z_{j}\geq 0,&&j\in[J],\\\ &\displaystyle l_{p}\leq x_{p}\leq
H_{p}&&p\in[P].\end{array}$ (3)

Finally, we can solve Equation (3) using a cutting planes algorithm or by
linearizing the constraints, which results in:

$\begin{array}[]{ll@{}ll}\text{minimize}&\displaystyle\alpha+\nu\sum_{j=1}^{J}z_{j}&\\\
\text{subject to}&\displaystyle
z_{j}\geq\sum_{p=1}^{P}w_{pj}-\alpha,&&j\in[J],\\\ &\displaystyle z_{j}\geq
0,&&j\in[J],\\\ &\displaystyle w_{pj}\geq 0,&&j\in[J],p\in[P],\\\
&\displaystyle w_{pj}\geq y_{pj}-x_{p},&&j\in[J],p\in[P],\\\ &\displaystyle
l_{p}\leq x_{p}\leq H_{p}&&p\in[P].\end{array}$ (4)

### A Robust Formulation for Algorithmic Insurance

Solutions to optimization problems can exhibit high sensitivity to
perturbations in the problem parameters (Ben-Tal and Nemirovski 2000). In the
setting of algorithmic insurance, uncertainty lies at the center of the
problem, gaining even higher significance. In the absence of real-world past
scenarios $\mathbf{y}_{j}$, we propose a data-driven way to generate them in
Sections 4-6, combining the model properties with historical claims of past
cases based on human decisions. Undoubtedly, the presented modeling approach
contains noise in the proposed scenarios $\mathbf{y}_{j}$ which can be
attributed either to the probabilistic assumptions or to the model
performance. Robust optimization offers a solution to this problem proposing
uncertainty models that are not stochastic, but rather deterministic and set-
based. Leveraging these techniques, we introduce two robust formulations.

#### Box Uncertainty with $\Gamma$-Robustness.

We will apply the notion of $\Gamma$-Robustness proposed by (Bertsimas and Sim
2004). We first assume that the scenarios $\mathbf{y}_{j}$ lie in the
uncertainty set:

$\mathcal{U}_{1}=\\{y|\mu_{pj}-\delta_{pj}\gamma_{pj}\leq
y_{pj}\leq\mu_{pj}+\delta_{pj}\gamma_{pj},\|\gamma\|_{\infty}\leq\Gamma\\}.$
(5)

Therefore, in this setting, the formulation of the robust counterpart is the
following:

$\begin{array}[]{ll@{}ll}\text{minimize}&\displaystyle\alpha+\nu\sum_{j=1}^{J}z_{j}&\\\
\text{subject to}&\displaystyle
z_{j}\geq\sum_{p=1}^{P}w_{pj}-\alpha,&&j\in[J],\\\ &\displaystyle z_{j}\geq
0,&&j\in[J],\\\ &\displaystyle w_{pj}\geq 0,&&j\in[J],p\in[P],\\\
&\displaystyle
w_{pj}+x_{p}\geq\mu_{pj}+\delta_{pj}\gamma_{pj},&&j\in[J],p\in[P],\\\
&\displaystyle l_{p}\leq x_{p}\leq H_{p}&&p\in[P].\end{array}$ (6)

#### Polyhedral Uncertainty with $\Gamma$-Robustness.

Here, we assume that the scenarios $\mathbf{y}_{j}$ lie in the uncertainty
set:

$\mathcal{U}_{2}=\\{y|\mu_{pj}-\delta_{pj}\gamma_{pj}\leq
y_{pj}\leq\mu_{pj}+\delta_{pj}\gamma_{pj},\|\gamma\|_{1}\leq\Gamma\\}.$ (7)

Based on this definition, the corresponding robust formulation follows:

$\begin{array}[]{ll@{}ll}\text{minimize}&\displaystyle\alpha+\nu\sum_{j=1}^{J}z_{j}&\\\
\text{subject to}&\displaystyle
z_{j}\geq\sum_{p=1}^{P}w_{pj}-\alpha,&&j\in[J],\\\ &\displaystyle z_{j}\geq
0,&&j\in[J],\\\ &\displaystyle w_{pj}\geq 0,&&j\in[J],p\in[P],\\\
&\displaystyle w_{pj}+x_{p}\geq\mu_{pj}q_{pj}-\mu_{pj}s_{pj}+\Gamma
r,&&j\in[J],p\in[P],\\\ &\displaystyle q_{pj}-s_{pj}=1,&&j\in[J],p\in[P],\\\
&\displaystyle-\delta_{pj}q_{pj}-\delta_{pj}s_{pj}+r\geq
0,&&j\in[J],p\in[P],\\\ &\displaystyle q_{p,j}\geq 0,&&j\in[J],p\in[P],\\\
&\displaystyle s_{p,j}\geq 0,&&j\in[J],p\in[P],\\\ &\displaystyle r\geq 0\\\
&\displaystyle l_{p}\leq x_{p}\leq H_{p}&&p\in[P].\end{array}$ (8)

We refer the reader to the Appendix and the work of Bertsimas and den Hertog
(2021) for the detailed derivation of the robust counterpart from the nominal
problem.

##  4 The Effect of Predictive Performance

The optimization formulations in Section 3 are based on historical scenarios
of loss $\mathbf{y}$. In this section, our goal is to derive these scenarios
of loss in the absence of historical claims based on the predictive
performance of a ML model.

To achieve this objective, we propose a data-driven approach that leverages
available information regarding both the frequency and severity of future
claims. We argue that the claim frequency of an erroneous algorithmic decision
is a function of the model’s predictive performance (e.g., AUC). Higher
sensitivity reduces the probability of a false negative algorithmic mistake
while models with higher specificity are less likely to perform a false
negative error. The expected claims cost (severity), though, will depend on
the nature of the decision and the type of error.

Going back to the case study, suppose that a pathologist receives FNA samples
from which, using their knowledge and experience, determines whether a patient
has a malignant breast tumor or not. Depending on the doctor’s response the
patient will or will not follow cancer treatment. In the case the physician
proposes an erroneous diagnosis, there is an associated cost with this
decision:

  * •

If the patient is diagnosed with cancer but does not actually have it, there
is the additional cost of unnecessary treatment that may even result to a
needless mastectomy. We will assume that this cost is captured by a random
variable $K$ with mean $\mu$ and variance $\sigma_{\mu}$.

  * •

If the patient is not diagnosed with cancer but actually has the disease, the
severity of the outcome for the patient is likely increased, since it is known
that early detection is critical in cancer patients. This increase of severity
is associated with a higher litigation cost, which is captured by a random
variable $L$ with mean $M$ and variance $\sigma_{M}$.

Suppose now that instead of a doctor, a ML model is taking up the task of
deciding, based on the FNA samples, whether the patient has cancer or not.
This is not only a hypothetical example as at the Massachusetts General
Hospital radiology department a ML model is partially responsible for the
screening process of patients with mammograms (Yala et al. 2019). Typically,
the output of such binary classification algorithms is a prediction score. The
model assigns to each input observation an individual risk score that
indicates how likely it is for each sample to be associated with the outcome
of interest (e.g., cancer diagnosis). To map each observation to a crisp class
label, a classification threshold $\tau$ must be defined. For example, if the
pathology department has specified a classification threshold $\tau=0.3$, then
all patients whose FNA outcome has probability of being positive $>0.3$ are
diagnosed with breast cancer. In the same example, all samples for which the
model predicts a score of $\leq 0.3$ are classified as cancer-free.

Let $\mathbf{x}_{i}$ be the feature vector of patient $i$ and $g(\cdot)$ is
the probability that patient $i$ has breast cancer. Then the class of $i$ is
defined as follows:

$\mathrm{class}(i)=\left\\{\begin{array}[]{l}1,\text{ if
}g(\mathbf{x_{i}})>\tau,\\\ 0,\text{ otherwise.}\end{array}\right.$

Depending on the value of this threshold $\tau$, the ability of the algorithm
to identify false positives and false negative cases varies. Higher values of
$\tau$ improve the specificity of the model, avoiding unnecessary alerts to
healthy patients. On the other hand, lower values of $\tau$ improve the
sensitivity of the model, resulting in the timely warning of a higher number
of cancer cases. Both measures are threshold dependent. Thus, we can formally
define them as follows:

  * •

Let $\kappa_{\tau}\in[0,1]$ be the specificity of the ML model for a
classification threshold $\tau$. The probability that a sick patient will be
erroneously classified is then $1-\kappa_{\tau}$.

  * •

Let $\lambda_{\tau}\in[0,1]$ be the sensitivity of the ML model for a
classification threshold $\tau$. Thus, the probability that a healthy patient
will be wrongly classified is $1-\lambda_{\tau}$.

Taking all the above into consideration, the claim cost of a new patient that
is diagnosed by the ML model is captured by the random variable $S$:

$S=(1-\kappa_{\tau})K+(1-\lambda_{\tau})L.$ (9)

Therefore, the expected value of the individual claim cost is equal to:

$\operatorname{\mathbb{E}}(S)=(1-\kappa_{\tau})\mu+(1-\lambda_{\tau})M.$ (10)

The corresponding variance is
$\sigma=(1-\kappa_{\tau})^{2}{\sigma_{\mu}^{2}}+(1-\lambda_{\tau})^{2}{\sigma_{M}^{2}}+2(1-\kappa_{\tau})(1-\lambda_{\tau})\sigma_{\mu}\sigma_{M}\rho(K,L)$.
If we assume that $N$ patients are expected to arrive at the hospital during
the contract period, then the total expected loss of the insurance is:

$\operatorname{\mathbb{E}}(C)=N\operatorname{\mathbb{E}}(S)=N((1-\kappa_{\tau})\mu+(1-\lambda_{\tau})M).$
(11)

### Case Study: Experimental Setup

We perform a series of computational experiments in the case study of
interest. The goal is to evaluate the effect of the model parameters on the
risk appreciation framework. We fix the number of scenarios to $J=1000$ and
assume that $N=100$ patients are served within the contract period. We
hypothesized that the litigation cost variables $K,L$ follow independent
normal distributions. In this setting, we do not distinguish between different
price segments, assuming that $P=1$. We vary the values of
$\mu,\sigma_{\mu},M,\sigma_{M}$ between the lower and upper ranges obtained
from historic medical malpractice cases of breast cancer such as the ones
presented in Section 2. We study the impact of the classification threshold
$\tau$ as well as the confidence level $\beta$. We constrain the contract
premium to $10,000 and $50,000. We quantify the effect of the $\Gamma$
parameter in the robust optimization formulation. We use bootstrapping across
10 random seeds. We report the average performance across all iterations. The
parameter ranges are detailed in Table 1.

Parameter | Range
---|---
$\Gamma$ | 3
$\beta$ | 0.9, 0.95, 0.99
$l_{p}$ | $10,000
$H_{p}$ | $10,000, $50,000
$\mu$ | $100,000 $500,000
$\sigma_{mu}$ | $25,000, $150,000
$M$ | $500,000, $1,000,000
$\sigma_{M}$ | $150,000, $400,000
$\tau$ | 0.01 - 0.75
$J$ | 1,000
$N$ | 100
Table 1: Parameter ranges for the computational experiments.

The data was randomly split into training (75%) and testing sets (25%).
Missing values in each partition were imputed using the MedImpute algorithm
(Bertsimas et al. 2021). We use the Random Forests algorithm to train the
binary classification models (Breiman 2001). We apply 10-fold cross-validation
to set the number of estimators and the maximum depth of the individual tree-
based models. The average AUC of the final model on the testing set is 99.36%.
The statistical analysis was conducted using Python 3.7 and Julia 1.3
(Pedregosa et al. 2011, Bezanson et al. 2017). The codebase for all of the
experiments is available as a Github repository (Orfanoudaki 2021).

### Case Study: The Implementation Framework

The proposed formulation enables decision makers to estimate for a given
confidence level ($\beta$) and a vector of historic claims ($\mathbf{y}$): (i)
the prices ($\mathbf{x}$) for each product class (i.e., age groups, vehicle
types); (ii) the VaR ($\alpha$); (iii) the CVaR which corresponds to the
objective function ($\min\alpha+\nu\sum_{j=1}^{J}z_{j}$).

The input necessary to apply it involves:

  * •

A binary classification model (e.g., image recognition classifier for
mammograms) with a representative testing set $g(\cdot)$;

  * •

Random variables $K,L$ that represent the litigation cost for false negative
and false positive cases with means $\mu,M$, variances
$\sigma_{\mu}^{2},\sigma_{M}^{2}$, and covariance Cov$(K,L)$ respectively;

  * •

The number of patients that the algorithm will serve during the contract, $N$;

  * •

The number of past scenarios for the optimization formulation, $J$;

  * •

Upper and lower bounds for the price, $l_{p},H_{p}$.

If past data from the implemented algorithm with prior cases of litigation
claims were available, we would use the historical observations. In the
absence of such information, we use random variable realizations to get the
cost approximation of false positive and negative cases. Thus, the total cost
for scenario $j$ of a fixed contract period for price segment $p$ can be
computed as follows:

$y_{pj}=\sum_{i=1}^{N}(1-\kappa_{\tau p})K_{pji}+(1-\lambda_{\tau p})L_{pji}.$
(12)

Algorithm 1 summarizes the proposed process that combines all the components
of our approach.

Output: prices ($\mathbf{x}$), VaR ($\alpha$), CVaR
Input: $\beta,g(\cdot),l_{p},H_{p},K,L,J,N,X_{train},X_{test}$

1: Train model $g(\cdot)$ using $X_{train}$

2: Get predicted probabilities for $X_{test}$

3: while $j\leq J$ do

4: while $i\leq N$ do

5: Calculate scenario $\mathbf{y}_{j}$ sampling from $K,L$ for patient $i$.

6: $i=i+1$

7: end while

8: $j=j+1$

9: end while

10: Solve the optimization formulation

Algorithm 1 Framework Implementation Procedure

### Case Study: The Effect of the Classification Threshold $\bm{\tau}$

The first question that we aim to answer is what is the effect of the
classification threshold $\tau$ on the estimated CVaR of the contract. In
Figure 1 we depict two scenarios of a low and a high litigation claims
distribution. The blue curve corresponds to
$\mu=\$100,000,\sigma_{\mu}=\$25,000,M=\$500,000,\sigma_{M}=\$150,000$. The
yellow curve corresponds to
$\mu=\$500,000,\sigma_{\mu}=\$150,000,M=\$1,000,000,\sigma_{M}=\$450,000$. On
the horizontal axis we project the threshold values and on the vertical axis
the CVaR. The findings of this analysis reveal that the most significant
determinant of CVaR is the selected classification threshold $\tau$. The
expected costs for the false positive and false negative case re-scale the
CVaR function. Moreover, the effect of $\mu$ is more prominent for lower
values of $\tau$ that negatively impact the specificity of the model. On the
contrary, $M$ gains more significance for higher values of $\tau$. The average
AUC of the binary classification models is $99.36\%$ and thus for any given
threshold, the probability of a false negative and a false positive is very
low. Notice that the CVaR of both curves in Figure 1 is minimized for
$\tau=0.3$. This is due to the fact that when $\tau\leq 0.3$, the model
sensitivity on the testing set is equal to one. Therefore, for $\tau=0.3$ the
model specificity is maximized for the best possible value of sensitivity.
This analysis demonstrates that the selected classification threshold $\tau$
can dramatically affect the CVaR value even for fixed distributions of
litigation costs.

Figure 1: CVaR as a function of the $\tau$ parameter for two different
combinations of the $K,L$ distributions.

(a) 3-Dimensional Illustration.

(b) 2-Dimensional Illustration
when $M=\$600K$.

(c) 2-Dimensional Illustration
when $\mu=\$500K$.

Figure 2: CVaR as a function of $\mu$ for $\tau=0.3$ and $\tau=0.4$.

### Case Study: The Effect of the Claims Cost Expected Value

The next question that we address relates to quantifying the impact of the
means of the random variables $K,L$ on the contract’s risk exposure. Figure 2
provides three-dimensional and two-dimensional illustrations of CVaR as we
vary the expected values $M$ and $\mu$ for $\tau\in\\{0.3,0.4\\}$. In these
experiments, $\sigma_{\mu}$ and $\sigma_{M}$ correspond to 20% of $\mu$ and
$M$ respectively. Figure 2(a) shows CVaR as a function of both $M$ and $\mu$.
When $\tau=0.3$, the model does not include any litigation cost for false
negative claims and CVaR is a linear function of the model specificity. This
is evident in Figures 2(b)-2(c) too. For a fixed value of $\mu$, any increase
or decrease of $M$ does not affect the contract’s financial risk. On the other
hand, as illustrated in Figures 2(b)-2(c), when $\tau=0.4$ both model
sensitivity and specificity affect the exposed risk of the contract in a
linear fashion. In this case, CVaR depends on the distribution of both $K$ and
$L$ and thus it is a linear function of $\mu$ and $M$. As we decrease the
value of $\tau$ below 0.3, the probability of a false positive claim is
increasing and as a result CVaR is also increasing. For example, when
$\tau=0.25$ the model specificity is 97.24% while when $\tau=0.15$ the
specificity drops to 94.49%. These results highlight that CVaR is a linear
function of both $M$ and $\mu$.

### Case Study: The Effect of Robust Optimization, the Premium and the
Confidence Level $\bm{\beta}$

Subsequently, we investigate the role of the the premium, the type of the
formulation, and the confidence level $\beta$ in the determination of CVaR. We
summarize our findings in Table 2. The premium value $x_{p}$ is set in the
vast majority of the experiments by the upper bound $H_{p}$. Only in cases
where both $M$ and $\mu$ are below $\$50,000$, this constraint is not binding.
Naturally, higher premium values result to lower VaR and CVaR for the
contract. The $\beta$ parameter also has a linear effect on the final risk
estimation as it is incorporated as a scalar in the objective function of the
formulation. In addition, the results in Table 2 reveal the benefit of the
robust optimization approach. As we expected, the box uncertainty sets are
more conservative than the polyhedral uncertainty approach. The latter yields
very similar results to the nominal problem while also accounting for
uncertainty in the scenarios $\mathbf{y}$.

Premium | Formulation | $\bm{\beta}=0.9$ | $\bm{\beta}=0.95$ | $\bm{\beta}=0.99$
---|---|---|---|---
$ 10,000 | box | $ 300,536 | $ 302,959 | $ 306,714
$ 10,000 | nominal | $ 276,117 | $ 278,071 | $ 281,631
$ 10,000 | polyhedral | $ 279,611 | $ 282,084 | $ 285,251
$ 50,000 | box | $ 260,536 | $ 262,959 | $ 266,714
$ 50,000 | nominal | $ 236,117 | $ 238,071 | $ 241,631
$ 50,000 | polyhedral | $ 239,611 | $ 242,084 | $ 245,251
Table 2: Average CVaR as we vary the premium price, the type of formulation,
and the confidence level $\beta$. These results correspond to
$\tau=0.3,\mu=\$100,000,\sigma_{\mu}=\$25,000,\Gamma=3$.

##  5 The Effect of Interpretability

In the effort to price the risk of algorithmic decision-making,
interpretability may play a crucial role. In this section, we incorporate the
degree of a ML model’s transparency in the proposed pricing framework. We
outline a class of functions that capture the impact of interpretability on
algorithmic risk evaluation and showcase a practical application in the case
study of medical liability.

Consequential decision-making up until recently has been strictly controlled
by humans. In this setting, the outcome of any decision can be associated with
reasoning that would justify the action. Thus, a human decision can be
evaluated based on the logic followed and, subsequently, the decision agent
can be held accountable for their judgement. Supervised learning algorithms do
not necessarily provide a reason why a given observation should receive a
specific label. They can only state that certain inputs are correlated with
that label. As a result, interpretability has remained an ill-defined term of
ML (Lipton 2018).

In the context of algorithmic liability, one could argue that the
interpretability of a model is a measure of how much human input could be
involved in the risk estimation process. We will focus on the setting of
complete automation where human input is possible only prior to the model
implementation in practice. In this context, experts may be called to review
and approve the algorithm prior to its integration to avoid erroneous decision
rules in the learner. Consider the case of a fully interpretable, tree-based
model for malignant tumor detection. The physician in charge can easily review
the algorithm’s recommendations based on their own knowledge and experience
prior to its implementation. The level of algorithmic transparency directly
affects the degree to which human judgement can be involved. Interpretable
models allow for synergies between the ML algorithm and the experts’ input.
Therefore, we argue that the combination of artificial and human intelligence
is likely to lead to more accurate estimations and may improve the risk
exposure of the contract.

Our goal is to quantify this effect and provide measures of how
interpretability can impact algorithmic risk evaluation. Suppose that $c_{h}$
is the risk exposure for an insurance contract when a human expert is making
all the decisions. Notice that $c_{h}$ is known from historical claims.
$c_{ml}$ is the risk exposure for the same contract when a ML model is the
sole decision maker. We assume that $c_{h}>c_{ml}$ to ensure that there are
financial incentives from the use of the ML model. We let $\theta\in[0,1]$ be
the interpretability parameter that measures the degree of algorithmic
transparency and assume that a model’s risk exposure $c(\theta)$ is a function
of the interpretability parameter.

When $\theta=0$, the ML model is treated as a “black-box” and thus a human
agent is unable to provide additional input that may improve the model’s
performance ($c(0)=c_{ml}$). To the contrary, when $\theta=1$, the model is
intuitive and explainable for the decision maker and as a result there are
synergies between the ML model and the human agent, resulting in a lower cost
$c(1)=\xi c_{ml}$, $\xi\in(0,1)$.

The determination of the $\xi$ parameter depends on the ML model, the
application and the problem under consideration. We assume that the relative
benefit of interpretability directly depends on the relative ratio of
$\frac{c_{ml}}{c_{h}}\in(0,1)$. The latter ratio captures the relative
improvement of a ML model over human judgement in economic terms. In cases
where the edge of algorithmic decision making is small $(c_{ml}\sim{c_{h}})$,
the value of interpretability is high, since an expert’s opinion can yield
equivalent results to an algorithm. In this setting, the synergies between the
decision maker and the machine are stronger, correspondingly decreasing the
expected risk exposure. On the other hand, when $c_{h}$ is significantly
higher than $c_{ml}$, interpretability gains less importance as human input
might not be as informative. Based on these assumptions, a potential value for
$\xi$ is $(1-\frac{c_{ml}}{c_{h}})$, capturing the synergies between the two
types of decision makers. It follows that when $\theta=1$, the cost is equal
to $c=c_{ml}(1-\frac{c_{ml}}{c_{h}})$. If we model the contractual risk
exposure $c$ for all values of $\theta\in(0,1)$ as the linear interpolation of
these two scenarios (see Figure 3), then
$c=-\frac{c_{ml}^{2}}{c_{h}}\theta+c_{ml}.$

Figure 3: The linear interpolation of the $c$ function for $\theta\in[0,1]$
when $\xi=(1-\frac{c_{ml}}{c_{h}})$.

However, interpretability does not necessarily have a linear effect on risk
exposure and model performance. One can hypothesize that the effect of
interpretability on the cost is non-linear. If $c$ is concave function, the
positive effect of interpretability on the risk exposure is more prominent for
higher values of $\theta$. To the contrary, if we assume that $c$ is convex,
even with small degrees of interpretability, we can observe significant
reductions in the expected risk. Though it remains challenging to fully
characterize the interpretability effect, our approach provides flexible
options to decision makers to measure its impact as a function of the $\theta$
parameter.

Identifying a single value for this parameter and determining a specific
definition or degree of interpretability has been a major challenge in the ML
field (Carvalho et al. 2019). Most definitions involve human input in the
evaluation process which impedes systematic quantitative analysis (Lipton
2016). Bertsimas et al. (2019) have recently introduced a quantitative
approach to specify the price of interpretability as the tradeoff with
predictive accuracy for a given model. Alternative approaches that could be
directly incorporated in the pricing framework include the work of Schmidt and
Biessmann (2019) and Ribeiro et al. (2016). We expect that an increasing
number of interpretability definitions will be available in the future
considering the importance of understanding a model’s proposed associations
between the input variables and the output labels.

### Case Study: The Effect of Interpretability

In Figure 4, we provide concrete examples of functions that model the effect
of interpretability in the risk estimation process for the case study of
medical liability. On the horizontal axis we project the $\theta$ parameter
and on the vertical axis the CVaR. Each graph corresponds to a different
function $c$, including concave, convex, and linear examples. In this setting,
we assume that $\xi=(1-\frac{c_{ml}}{c_{h}})$, $c_{ml}=\$500K$, and consider
four distinct scenarios of $c_{h}$. Notice when $c$ is convex, such as in
Figures 4(b) and 4(d), even with low degrees interpretability, we can derive
effective synergies between the model and the human agent that can
significantly reduce the risk exposure. Respectively, when modeling the effect
of interpretability with a concave function, like the ones presented in
Figures 4(a),4(c), the majority of the risk reductions will only be observed
for higher values of $\theta$. This intuition can guide the decision for the
determination of the interpretability function in future applications of the
framework.

(a) $c=-\frac{c_{ml}^{2}}{c_{h}}(\tan(\frac{\pi}{4}\theta))+c_{ml}$.

(b) $c=-\frac{c_{ml}^{2}}{c_{h}}(\sin(\frac{\pi}{2}\theta))+c_{ml}$.

(c) $c=-\frac{c_{ml}^{2}}{c_{h}}{\theta}^{2}+c_{ml}$.

(d) $c=-\frac{c_{ml}^{2}}{c_{h}}\sqrt{\theta}+c_{ml}$.

(e) $c=-\frac{c_{ml}^{2}}{c_{h}}\theta+c_{ml}$.

Figure 4: Risk Exposure $c$ as a function of the interpretability parameter
$\theta$ and risk exposure $c_{h}$ for a fixed value of $c_{ml}=\$500$K.

##  6 The Effect of Model Generalizability

In this section, the goal is to quantify the effect of algorithmic
generalizability on the pricing framework. We propose the generation of
synthetic data using GANs to measure the impact on the performance of the ML
model and apply our approach in the case study of interest.

Generalizability refers to a model’s ability to properly adapt to new,
previously unseen data, drawn from the same or a different distribution as the
one used to create the model. In the context of algorithmic insurance,
generalizability becomes pertinent when the model is applied to a new
population whose data have not been tested by the algorithm in the past. Going
back to the case study of interest, let the binary classification model for
malignant tumor detection be exclusively trained on samples drawn from a
Caucasian population at an academic hospital in Massachusetts. Suppose that a
community hospital in Louisiana is now interested to integrate the system in
its electronic health records database which started collecting medical data
only one year ago. What is the appropriate pricing strategy for this center?
If the hospital had at its disposal data from the patient population it serves
throughout a span of multiple years, we would directly be able to externally
validate its performance. However, in the absence of this valuable resource,
an alternative approach is needed to adjust the pricing strategy of the
contract.

We propose testing the predictive performance of the algorithm on synthetic
observations that resemble the original training set, controlling the degree
of similarity to get estimations of risk variability. Multiple approaches have
been proposed in the literature to generate synthetic data. A quite common
approach is to induce noise with Gaussian random variables in the existing
dataset. Nevertheless, this simple technique directly affects the existing
correlations and associates between the covariates distorting the true
geometry of the feature space. Another well established technique for data
augmentation is referred to as the Synthetic Minority Oversampling Technique
(SMOTE) which is very frequently applied in classification datasets that have
a severe class imbalance (Chawla et al. 2002). However, this approach does not
let the user directly control the degree of similarity between the newly
created instances and the original data distribution.

The proposed framework requires the use of parameters that effectively control
the degree of similarity between the real and the synthetic data. Our
objective is to introduce a mechanism that would allow decision makers to
quantitatively compare the additional cost of the insurance contract with
respect to the ability of the model to generalize to datasets with different
levels of variability from the original training and testing sets. Though it
is not possible to provide theoretical guarantees about the predictive
performance of a given classifier in a new unknown distribution, we can
conduct extensive simulations that test the discrimination capability of the
learner in adverse scenarios.

### Generative Adversarial Networks to Generate Synthetic Data

To perform this analysis we resort to GANs. This neural network architecture
was first introduced in 2014 with the goal of synthesizing artificial images
that are indistinguishable from authentic figures (Goodfellow et al. 2014).
Since then, GANs have become the predominant method of data augmentation for
images and text (Wang et al. 2017). They are used to increase the amount of
data by adding slightly modified copies of already existing samples or newly
created synthetic data from existing observations (Radford et al. 2015,
Goodfellow 2017, Zhang et al. 2017).

GANs involve a unique architecture in which a pair of networks are trained
simultaneously and in competition with each other. Training a robust GAN
architecture is a non-trivial task due to problems like vanishing gradients
and mode-collapse which may result in poor discrimination performance and
synthetic samples with limited diversity (Radford et al. 2015, Salimans et al.
2016). The Wasserstein GANs (WGAN) architecture was introduced to remedy these
issues using the Wasserstein distance as the loss function (Arjovsky et al.
2017). The Conditional GAN (CGAN) is a specific class of GANs that involves
the conditional generation of images by a generator model, resulting in new
data observations with associated class labels (Mirza and Osindero 2014). This
modification to the GAN architecture permits learning the distributions
specific to each class label, producing samples for both labels with higher
quality.

Recently, researchers showed that WCGANs can lead to very promising results in
synthesizing tabular data, comprising only densely connected layers (McKeever
and Singh Walia 2020, Somorjit and Verma 2020). These efforts highlighted the
potential impact that GANs may have in structured data sources, providing
researchers with higher flexibility and control in the data generation
process. This technique also offers a promising alternative to solve the
problem of interest; creating synthetic data while explicitly controlling the
degree of similarity. By varying the number of epochs, we will control the
differences in the distributions between the synthetic and the real-world
data. This parameter constitutes the number of complete passes through the
training process. A higher number of epochs gives the opportunity to the
algorithm to converge, minimizing the loss function. Thus, we can directly
compare the discriminator loss function to the number of epochs, adjusting the
degree of dissimilarity between the synthetic samples and the real training
set.

Due to their complicated structure and despite large strides in terms of
theoretical progress, evaluating and comparing GANs remains a challenging
task. While several metrics have been proposed, there is no consensus in the
scientific community as to which measure provides a more holistic and
objective model evaluation (Alqahtani et al. 2019, Borji 2019). We propose the
use of the GAN quality index (GQI) for our analysis (Ye et al. 2018). To
compute this metric, we will compare the performance of the generator $G$ and
the model we are seeking to price $C_{\mathrm{real}}$. First, we generate the
synthetic samples with the associated class labels using the CGAN
architecture. A second classifier, called the GAN-induced classifier
$C_{\mathrm{GAN}}$ is trained on the generated data. The GQI is defined as the
ratio of the accuracies (or AUCs) of the two classifiers when applied to the
real test data:

$GQI=\frac{ACC(C_{\mathrm{GAN}})}{ACC(C_{\mathrm{REAL}})}.$

Higher GQI means that the GAN distribution better matches the real data
distribution.

### Case Study: The Effect of Generalizability

We apply the proposed approach to the case study of medical liability using a
GAN architecture. The goal is to investigate the effect of generalizability on
the contract’s risk exposure using synthetic samples while controlling the
degree of similarity to the original dataset. The GAN network was built
leveraging the GAN-Sandbox package and was implemented in Python using the
Keras library with a Tensorflow backend (Dietz 2017, Chollet et al. 2015,
Abadi et al. 2016).

(a) Number of Epochs $=0$.

(b) Number of Epochs $=20$.

(c) Number of Epochs $=40$.

(d) Number of Epochs $=60$.

(e) Number of Epochs $=150$.

(f) Real Data.

Figure 5: The derived distributions from the GAN model of two of the most
predictive features. Cell Shape Uniformity is depicted on the vertical axis
and Clump Thickness is illustrated on the horizontal axis. All values have
been normalized to [0,1]. Each graph corresponds to the output of the GAN
model for a given number of epochs. Figure 5(f) corresponds to the real
distribution of the features.

In Figure 5, we present the changes in the features’ distribution as a
function of the number of epochs parameter in a WCGAN architecture. Following
the architecture presented by Vega-Márquez et al. (2019), the GAN model
comprises a generator network with one input layer and three dense layers and
an adversarial network with one input layer and four dense layers. We measure
the similarity of the derived distributions between the synthetic (GAN-
Generated) and the real data for a different number of epochs in the training
process. An epoch is defined as one cycle through the training process of the
network, corresponding to the number of training iterations between the
generator and the adversarial network. The hypothesis in Section 6 is
validated, as we observe that a higher number of epochs results in features
distributions that better resemble the real data distribution (Figure 5). This
effect is directly present in quantitative metrics, such as the GQI index,
which is also positively correlated with the number of epochs in the model.

In Figure 6, we project CVaR as a function of the number of epochs parameter.
The underlying binary classification model is the same as the one used in
Figure 1 where
$\mu=\$100,000,\sigma_{\mu}=\$25,000,M=\$500,000,\sigma_{M}=\$150,000$,
derived on the training sample of the original Breast Cancer Wisconsin
Diagnostic dataset. The CVaR is then measured with respect to the
discrimination performance of the classifier on the GAN-Generated synthetic
data. The output of CGAN architectures includes both independent features and
associated labels that are subsequently used to compute the sensitivity and
specificity of the model for different values of $\tau$. In Figure 6, we only
project the best CVaR value across all potential thresholds $\tau$. This graph
reveals a linear relationship between the number of epochs and the CVaR. This
finding highlights to decision makers the cost effect of applying a pre-
trained learner in datasets with varied degrees of distribution similarity to
the original training population.

Figure 6: CVaR as a function of the number of epochs parameter. The color
indicates the GQI metric for the underlying binary classification model.

##  7 Discussion

In this paper, we construct a framework for pricing algorithmic risk and
provide a comprehensive case study for its implementation. Our work
constitutes the first attempt to quantify the litigation risk resulting from
erroneous algorithmic decision-making in the context of binary classification
models. The framework is agnostic to the type of learner and application area.
It can be easily extended in other fields, such as predictive maintenance and
autonomous vehicles among others. The proposed models are data-driven and
parametric, since such contracts have not been implemented by the industry yet
and extensive experimentation is needed prior to launching them.

Classification Threshold $\bm{\tau}$: Our work reveals that, given fixed
distributions for litigation costs, the choice of the classification threshold
$\tau$ plays the most significant role in the determination of the contract’s
financial risk. This finding provides decision makers with a new way of
assessing the balance between sensitivity and specificity. The direct
association of risk exposure to the threshold $\tau$ leads to more informed
decisions regarding the financial implementation of a classifier in practice.
Our findings validate the assumption that when a false positive and a false
negative have disproportionate implications for the organization the choice of
the $\tau$ threshold becomes even more critical.

Interpretability: The proposed framework incorporates the critical aspect of
model interpretability. We argue that in cases where the risk of human
decision making is comparable to algorithm-driven decision rules,
interpretability gains greater value. We also show that the convexity of the
interpretability function indicates whether a smaller or a higher degree of
model transparency is needed to achieve significant synergies between human
agents and the ML algorithm.

Generalizability: We consider the use of GANs to evaluate the effect of model
generalizability in the risk evaluation process. We illustrate a linear
relationship between CVaR and the number of training epochs of the GAN
network. The latter is also positively correlated with similarity metrics
between the generated and the true data distribution. Thus, we allow decision
makers to adjust the contract pricing to settings where the classifier is
applied to a new population that was not part of the original training and
validation set.

Limitations: Central to the limitations of this study is the absence of
historic claims records from real-world litigation cases of malpractice. We
assume that insurance companies have in their possession this kind of
information from which the scenarios $\mathbf{y}$ could be directly
constructed. Our analysis takes a conservative view over the litigation
process assuming that every erroneously classified sample will resort to a
litigation claim. In reality, only a portion of misdiagnosed patients file a
malpractice lawsuit and only a subset of those are successful. In this paper,
the computational experiments have been based on normally distributed random
variables but other distributions could also be explored in future
investigations. We would also like to note that medical malpractice is a
particularly challenging field and the implementation of ML models such as the
one in Section 2 faces a lot of regulation constraints. As a result, it is
very likely that such models will continue to be validated by medical experts
in the coming years. Finally, this work does not account for dynamic decision-
making processes, such as the triple FNA strategy, that usually results in
improved clinical outcomes for the patients and significantly reduces the
amount of claims (Anderson and Troxel 2005).

##  8 Conclusions

Our work aims to set the foundations in the novel area of algorithmic
insurance. We propose quantitative tools that allow decision makers, modelers,
and insurance companies to estimate the litigation risk of binary
classification models. This approach takes into consideration the predictive
performance of the classifier accounting also for uncertainty in the data. We
incorporate measures of interpretability and generalizability to provide a
holistic appreciation of the model. We believe that this framework can serve
as the basis of a new research area that will expedite the adoption and
implementation of ML models in practice. The authors would like to thank Dr.
Luca Marighetti for insightful discussions regarding algorithmic insurance.

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## Appendix

### A. Background for the Simultaneous Minimization of CVaR and Calculation of
VaR

Following the notation proposed by Uryasev (2000), we let:

  * •

$p(\mathbf{y})$ the probability density function of $\mathbf{y}$;

  * •

$\Psi(\mathbf{x},\alpha)$ the probability that the loss function
$f(\mathbf{x},\mathbf{y})$ does not exceed some threshold value $\alpha$;

  * •

$\alpha(\mathbf{x},\beta)$ the VaR function; the percentile of the loss
distribution with confidence level $\beta$. Based on this definition,
$\Psi(\mathbf{x},\alpha(\mathbf{x},\beta))=\beta$.

  * •

$\phi_{\beta}(\mathbf{x})$ is the CVaR or the conditional expected loss
defined as:

$\phi_{\beta}(\mathbf{x})=(1-\beta)^{-1}\int_{f(\mathbf{x},\mathbf{y})>\alpha(\mathbf{x},\beta)}f(\mathbf{x},\mathbf{y})p(\mathbf{y})d\mathbf{y}.$
(13)

Equation (13) is complex due to the $\alpha(\mathbf{x},\beta)$ function
involved in its definition. For this reason, the authors of Uryasev (2000)
introduce a simpler function:

$F_{\beta}(\mathbf{x},\alpha)=\alpha+(1-\beta)^{-1}\int_{f(\mathbf{x},\mathbf{y})>\alpha}(f(\mathbf{x},\mathbf{y})-\alpha)p(\mathbf{y})d\mathbf{y},$
(14)

which can be used instead of CVaR. By minimizing this function w.r.t $\alpha$
we can directly derive CVaR.

$\phi_{\beta}(\mathbf{x})=F_{\beta}(\mathbf{x},\alpha(\mathbf{x},\beta))=\mathrm{min}_{\alpha}F_{\beta}(\mathbf{x},\alpha).$
(15)

It also follows that $F_{\beta}(\mathbf{x},\alpha)$ is convex w.r.t. $\alpha$
and that its minimum point is the VaR measure. This finding allows us to
leverage the function $F_{\beta}(\mathbf{x},\alpha)$ for optimizing CVaR and
simultaneously calculating VaR (Uryasev 2000):

$\mathrm{min}_{\mathbf{x}\in\mathbf{X}}\phi_{\beta}(\mathbf{x})=\mathrm{min}_{\mathbf{x}\in\mathbf{X},{\alpha}}F_{\beta}(\mathbf{x},\alpha).$
(16)

In this way, we can optimize CVaR and find VaR in tandem. Under general
conditions (see (Uryasev 1995)) and if $\mathbf{X}$ is also convex, this
problem is equivalent to solving a smooth convex optimization problem.

### B. Robust Formulations

#### Box Uncertainty with $\Gamma$-Robustness.

First, we investigate the problem of
$\max_{y\in\mathcal{U}_{1}}w_{pj}+x_{p}\geq y_{pj}$. That is equivalent to

$\begin{array}[]{ll@{}ll}\text{max}&\displaystyle y_{pj}&\\\ \text{subject
to}&\displaystyle\mu_{pj}-\delta_{pj}\gamma_{pj}\leq
y_{pj},&&j\in[J],p\in[P],\\\ &\displaystyle
y_{pj}\leq\mu_{pj}+\delta_{pj}\gamma_{pj},&&j\in[J],p\in[P],\\\
&\displaystyle\max_{p,j}\gamma_{p,j}\leq\Gamma,&&j\in[J],p\in[P],\\\
&\displaystyle\gamma_{p,j}\geq 0,&&j\in[J],p\in[P].\\\ \end{array}$ (17)

Notice that the maximum of $\gamma_{pj}$ is less than $\Gamma$ if and only if
all $\gamma_{p}j$ are less than $\Gamma$. To maximize over $y_{pj}$, we set
all $y_{pj}$ to their upper bound $y_{pj}=\mu_{pj}+\delta_{pj}\gamma_{pj}$.
Hence, the robust counterpart for this uncertainty set is:

$\begin{array}[]{ll@{}ll}\text{minimize}&\displaystyle\alpha+\nu\sum_{j=1}^{J}z_{j}&\\\
\text{subject to}&\displaystyle
z_{j}\geq\sum_{p=1}^{P}w_{pj}-\alpha,&&j\in[J],\\\ &\displaystyle z_{j}\geq
0,&&j\in[J],\\\ &\displaystyle w_{pj}\geq 0,&&j\in[J],p\in[P],\\\
&\displaystyle
w_{pj}+x_{p}\geq\mu_{pj}+\delta_{pj}\gamma_{pj},&&j\in[J],p\in[P],\\\
&\displaystyle l_{p}\leq x_{p}\leq H_{p}&&p\in[P].\end{array}$ (18)

#### Polyhedral Uncertainty with $\Gamma$-Robustness.

In this setting, we study the problem of
$\max_{y\in\mathcal{U}_{2}}w_{pj}+x_{p}\geq y_{pj}$. That is equivalent to the
formulation:

$\begin{array}[]{ll@{}ll}\text{max}&\displaystyle y_{pj}&\\\ \text{subject
to}&\displaystyle\mu_{pj}-\delta_{pj}\gamma_{pj}\leq
y_{pj},&&j\in[J],p\in[P],\\\ &\displaystyle
y_{pj}\leq\mu_{pj}+\delta_{pj}\gamma_{pj},&&j\in[J],p\in[P],\\\
&\displaystyle\sum_{p,j}\gamma_{p,j}=\Gamma,&&\\\
&\displaystyle\gamma_{p,j}\geq 0,&&j\in[J],p\in[P].\\\ \end{array}$ (19)

In order to define the robust counterpart, we need to look at the dual of
Equation (19).

$\begin{array}[]{ll@{}ll}\text{min}&\displaystyle\mu_{pj}q_{pj}-\mu_{pj}s_{pj}+\Gamma
r&\\\ \text{subject to}&\displaystyle q_{pj}-s_{pj}=1,&&j\in[J],p\in[P],\\\
&\displaystyle-\delta_{pj}q_{pj}-\delta_{pj}s_{pj}+r\geq
0,&&j\in[J],p\in[P],\\\ &\displaystyle q_{p,j}\geq 0,&&j\in[J],p\in[P],\\\
&\displaystyle s_{p,j}\geq 0,&&j\in[J],p\in[P],\\\ &\displaystyle r\geq 0.\\\
\end{array}$ (20)

Due to strong duality, the solution to this problem can be used to derive the
robust counterpart.

$\begin{array}[]{ll@{}ll}\text{minimize}&\displaystyle\alpha+\nu\sum_{j=1}^{J}z_{j}&\\\
\text{subject to}&\displaystyle
z_{j}\geq\sum_{p=1}^{P}w_{pj}-\alpha,&&j\in[J],\\\ &\displaystyle z_{j}\geq
0,&&j\in[J],\\\ &\displaystyle w_{pj}\geq 0,&&j\in[J],p\in[P],\\\
&\displaystyle w_{pj}+x_{p}\geq\mu_{pj}q_{pj}-\mu_{pj}s_{pj}+\Gamma
r,&&j\in[J],p\in[P],\\\ &\displaystyle q_{pj}-s_{pj}=1,&&j\in[J],p\in[P],\\\
&\displaystyle-\delta_{pj}q_{pj}-\delta_{pj}s_{pj}+r\geq
0,&&j\in[J],p\in[P],\\\ &\displaystyle q_{p,j}\geq 0,&&j\in[J],p\in[P],\\\
&\displaystyle s_{p,j}\geq 0,&&j\in[J],p\in[P],\\\ &\displaystyle r\geq 0\\\
&\displaystyle l_{p}\leq x_{p}\leq H_{p}&&p\in[P].\end{array}$ (21)