On the Role of Risk Perceptions in Cyber Insurance Contracts Shutian Liu and Quanyan Zhu

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On the Role of Risk Perceptions in Cyber Insurance

Contracts

Shutian Liu and Quanyan Zhu

Abstract—Risk perceptions are essential in cyber insurance

contracts. With the recent surge of information, human risk

perceptions are exposed to the inﬂuences from both beneﬁcial

knowledge and fake news. In this paper, we study the role of the

risk perceptions of the insurer and the user in cyber insurance

contracts. We formulate the cyber insurance problem into a

principal-agent problem where the insurer designs the contract

containing a premium payment and a coverage plan. The risk

perceptions of the insurer and the user are captured by coherent

risk measures. Our framework extends the cyber insurance

problem containing a risk-neutral insurer and a possibly risk-

averse user, which is often considered in the literature. The

explicit characterizations of both the insurer’s and the user’s

risk perceptions allow us to show that cyber insurance has

the potential to incentivize the user to invest more on system

protection. This possibility to increase cyber security relies on

the facts that the insurer is more risk-averse than the user (in

a minimization setting) and that the insurer’s risk perception

is more sensitive to the changes in the user’s actions than the

user himself. We investigate the properties of feasible contracts

in a case study on the insurance of a computer system against

ransomware.

I. INTRODUCTION

Human risk perception plays an important role in cyber

insurance contracts. On the one hand, individuals who are

risk-averse tend to overreact to severe potential losses that are

not likely to happen. On the other hand, they are eager to

seek additional resources to defend against cyber losses. The

risk-sharing property of cyber insurance contracts that mitigates

the cyber losses of users depends on the user’s risk-aversion.

The cyber insurance market does not exist when the users are

risk-neutral [

]. In an era of information explosion, people

may either intend to adjust their risk attitudes according to

expert advice or be manipulated by fake news to reform they

risk preferences without awareness [

]. The instability of risk

perception can have essential impacts on the insurance market

and the resiliency of cyber systems. Therefore, there is a need

to study how users with different risk perceptions behave when

they face potential cyber losses and how the optimal insurance

plan should change according to the risk attitudes.

Linear contracts involving a prepaid premium and a coverage

plan are among the most considered contract models in the

cyber insurance market. The premium is a money transfer from

the user to the insurer for entering the contract and the coverage

plan describes the proportion of cyber losses covered by the

insurer. In a linear contract model, the insurer is often assumed

to be risk-neutral and evaluate her loss using expectations; the

user is set to exhibit risk-aversion. The risk-aversion of an

individual can be modeled by a nonlinear utility function [3],

[

]. The risk-aversion is captured by the fact that the utility

function is assumed to be increasing and concave in return.

There is a recent trend on the study risk quantiﬁcation

adopting coherent risk measures (CRMs). The axiomatic

deﬁnition of CRMs maintains the generality of the risk

modeling and provides rich insights towards applications.

The dual representation of a CRM shows its robustness to

probabilistic uncertainty. Reformulation techniques [

], [

[

] have also enabled convenient and tractable computation

methods of risks modeled by various CRMs.

Protection Investment x

Coverage !

Premium q

Loss (1-!)"

Insurer

User

Cyber Risk "

Loss !"

Cyber Insurance

Decreased Cyber Risk

Increased

Protection Investment

Enhanced System Security

Risk Perception

Figure 1. Cyber insurance has the potential to enhance system security by

incentivizing the user to invest more on system protection, if the risk perception

of the insurer exhibits more risk-aversion and is more sensitive towards the

distributional shifts of the cyber losses.

We study the role of human risk perceptions in cyber

insurance using a holistic framework which incorporates the

modern risk modeling approach and a linear principal-agent

(P-A) model. In particular, we use CRMs to describe the

risk-aversion of the principal and the agent. The reason is

two-fold. First, CRMs allow us to investigate the probabilistic

distortion to the random cyber losses caused by human risk-

aversion. Second, cyber risks are challenging to quantify due

to the difﬁculty in transforming the cyber losses to monetary

losses. Therefore, the probabilistic robustness that CRMs

possess leads to reliable and safe estimations of cyber risks.

Our framework builds on the hidden-action linear contract

problem to capture the information asymmetry between the

insurer and the user. Speciﬁcally, the principal minimizes her

loss function by designing the contract containing a premium

and a coverage rate subject to the individual rationality (IR)

constraint which guarantees beneﬁcial participation and the

incentive compatibility (IC) constraint which corresponds to the

rationality of the agent. Due to the IC constraint, the contract

problem appears in the form of a bilevel program. Though

a full-information counterpart to this problem where the IC

arXiv:2210.15010v1 [cs.CR] 26 Oct 2022

constraint is absent produces a lower optimal loss, the hidden-

information does not suffer from the moral hazard issue [

]. In

this work, we focus on the inﬂuence of risk measures on the

insurance. Hence, we do not consider additional nonlinearities

on top of the random cyber loss modeled by a random variable

endowed with a parametric distribution. The validity of linear

contracts follows from the monotonicity property of the solution

to general contract problems [8].

Hidden-action contract problems are challenging because

of the bilevel nature. However, leveraging the linearity of the

contract and the ﬁrst-order approach, we can simplify the

problem and derive its optimality conditions. The conditions

allow us to characterize the coverage and the premium in terms

of the derivative of the risk of the user with respect to his

action. By choosing proper risk measures, practitioners can

obtain optimal contracts which satisfy desired properties.

One of the most essential features of the principal-agent

models lies in that the distribution of the random losses is

parametric in the agent’s action. In our framework, how the

risks perceived by the insurer and the user change according to

the user’ actions captures how sensitive the insurer and the user

are towards the parameterization. We show that when, compared

to the user’s risk perception, the insurer’s risk perception

exhibits more aversion to random cyber losses and is more

sensitive to the parameterization, cyber insurance can enhance

system security by incentivizing the protection investment of the

agent. These requirements suggest the following characteristics

of the insurer. First, the insurer, who bears the responsibility

in evaluating the system risks, should be able to estimate the

cyber losses more cautiously than the user. Second, aiming

to design an incentive contract, the insurer should possess a

higher level of awareness of how the actions from the user

inﬂuence the system risks stochastically than the user himself.

Our result enriches the literature by introducing the possibility

that cyber insurance can incentivize the user’s system protection

investment and hence enhance the overall system security.

This possibility is not observed in traditional cyber insurance

problems where the risk perceptions are captured by nonlinear

utility functions [1], [9], [10].

The paper is organized as follows. In Section II, we ﬁrst

introduce the risk preference modeling, then we incorporate it

into the cyber insurance contract design problem. Section III

contains the analysis of the game. We discuss the roles of risk

perceptions in shaping the optimal contract and the relation

between risk sensitivity and system security. We use a case

study to further investigate the insurance contracts in Section

IV. Finally, Section V concludes the paper.

II. PROBLEM FORMULATION

In this section, we ﬁrst introduce the deﬁnition of CRMs

and their analytical properties. Then, we introduce the cyber

insurance contract design problem with the risk preferences of

the insurer and the user described by CRMs.

A. Risk Preference Modeling

Consider the probability space

(Ξ,F)

of cyber loss samples

ξ∈Ξ⊂R+

endowed with the reference probability measure

. Let

Z:=Lp(Ξ,F,P)

denotes the space of random losses

Z:Ξ→R

with ﬁnite

-th order moment. The parameter

lives

[1,+∞)

. A risk measure

is a function

Z→R

that assigns

a deterministic value to a random loss. Classic approaches to

risk modeling includes using the expected loss, the standard

deviation of the loss, the value-at-risk, and etc. These risk

metrics can come in handy in many real situations due to their

simplicity and straightforwardness of interpretation. However,

the classic risk metrics are lacking in the following two ways.

First, one risk metric cannot fully characterize the behavior

of a random loss. A simple example would be that using

the expectation to characterize the risk of a Gaussian random

loss has

50%

chance to fail when the randomness is realized.

Second, human risk perceptions are different across individuals.

According to [

], humans tend to distinguish between losses and

gains and are likely to perceive the true probability of random

events with biases. Hence, risk metrics should characterize

human behaviors beyond merely risk-neutrality.

In this paper, we will use CRMs to characterize the risk

sensitivities that the insurer and the user exhibit.

Deﬁnition 1

(Coherent Risk Measures [

])

A function

ρ:

Z→R

is called a Coherent Risk Measure if for

Z,Z0∈Z

satisﬁes

(A1) Monotonicity:

ρ(Z)≥ρ(Z0)

Z(ξ)≥Z0(ξ)

for almost

everywhere ξ∈Ξ.

(A2) Convexity:

ρ(tZ + (1−t)Z0)≤tρ(Z)+(1−t)ρ(Z0)

for

t∈[0,1].

(A3) Translation equivariance: ρ(Z+a) = ρ(Z) + a if a ∈R.

(A4) Positive homogeneity: ρ(tZ) = tρ(Z)if t ≥0.

One deﬁnition of risk-aversion [

] is referred to the fact

that the perceived risk is not smaller than the expectation of the

random loss, i.e.,

ρ[·]≥E[·]

. A convex risk measure captures

the risk-aversion of decision-makers in this sense.

A CRM captures the decision-maker’s robustness consid-

eration to probabilistic uncertainty due to the following dual

representation [11], [13]:

ρ[Z(ξ)] = sup

ζ∈AZΞ

Z(ξ)ζ(ξ)dP(ξ),(1)

where

A⊂Z∗

denotes the dual set associated with the risk

measure

ρθ

and contains probability density functions with

respect to the probability measure

. The set

Z∗

denotes the

dual space of

deﬁned by

Z∗:Lq(Ξ,F,P)

with

p+1

q=1

The optimization problem (1) admits an optimal solution since

the set Ais convex and compact when p∈[1,+∞)[13].

The following is an important property of a risk measure.

Deﬁnition 2.

(Law-invariance.) A risk measure

ρ:Z→R

law-invariant with respect to the reference probability measure

, if

∀Z1,Z2∈Z

such that

P(Z1≤t) = P(Z2≤t)

for all

t∈R

then ρ(Z1) = ρ(Z2).

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OntheRoleofRiskPerceptionsinCyberInsuranceContractsShutianLiuandQuanyanZhuAbstractRiskperceptionsareessentialincyberinsurancecontracts.Withtherecentsurgeofinformation,humanriskperceptionsareexposedtotheinuencesfrombothbenecialknowledgeandfakenews.Inthispaper,westudytheroleoftheriskperceptionsofth...

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On the Role of Risk Perceptions in Cyber Insurance Contracts Shutian Liu and Quanyan Zhu

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