In global markets with immense competition, companies strive to provide the most reliable service or product for consumers’ satisfaction. However, optimal service can be costly and economically unjustified. In this work, we aim to construct a framework under which an optimal service can be defined and achieved with respect to the total revenue management of the company. Under this framework, we suggest a novel approach connecting quality of service, risk management, and the total revenue by incorporating a “self-insurance” mechanism. We show that by adding a self-insurance mechanism,

1. Roey Feldhaim
Dr. Gal Zahavi
Prof. Liron Yedidsion
Technion Israel
Insurance for Service Failures -
Utilizing Risk Allocation for Revenue Management
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2. Outline
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 Introduction
 Overview
 Motivation
 Research Problem
 Problem Definition: Assumptions, Notations, etc.
 The Model – Plan1 and Plan2
 Numerical Experiment & Results
 Conclusion
 Quality Improvement Optimization
 Summary

3. Introduction
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 You are a company manager
 Your goal - To become more “efficient”
 How can this be done?
 Tools (existing methods):
 Technologic improvement - Expense reduction.
 Marketing - Sales increasing.
 Scheduling, Reorganization, etc’
 Revenue management (RM)
Increase “Net
Income”
Rising the sales,
Reducing the
expenses

4. Introduction
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 Revenue management
 Wikipedia: “application of disciplined analytics that
predict consumer behavior at the micro-market
level and optimize product availability and price
to maximize revenue growth. The primary aim of
Revenue Management is selling the right product
to the right customer at the right time for the
right price and with the right pack. The essence
of this discipline is in understanding customers'
perception of product value and accurately aligning
product prices, placement and availability with each
customer segment”

5. Overview
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 Our solution:
 Basic idea: Unbundling risk of failure from the service -
Risk Allocation
 Advantages:
 A simple but innovative business model
 Company benefits
 Customers welfare benefits
 In search of optimum – System equilibrium
Selling self insurance on the
provided service

6. Risk allocation
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Delays
!
.......
.......
......
price quality
Min Max
price quality
Max Min
Company Customer

7. Company benefits
Self-Financing mechanism
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Sales
increase
• Insurance
premium
Risk
failure
• Compensation
payments
• Lose money
Risk
decreasing
Company
efficiency
• Quality
improvement
• Self financing
• Revenue growth
• Service
improvement

8. Customers welfare benefits
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 Risk decreasing
 Getting compensation in case of failure
 The insurance premium is a indicator for the service quality
Example: A flight from NY to Washington (www.expedia.com)
Which company should I choose?
Delta
American
Airline
US Airline
249.1 $
249.1 $
249.1 $
price
1h 9m
1h 5m
1h 5m
Time

9. Motivation - Basic example
EL AL
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http://skift.com/2014
EL AL = Every Landing Always
Late

10. Motivation - Basic example
EL AL
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 EL-AL Financial reports:
 Revenue 2015 - About 2.054 Billion $
 Profit 2015 - About 144.6 Million $
 Suppose:
 Ticket price: 500 $
 Risk: delays
 Insurance premium : 5%

11. Motivation - Basic example
Cont.
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As a customer, would you agree to:
 Pay 25$ more?
 In case of “failure”, you will be compensated with about 250$
(50% of the ticket)
 Reduce the risk for delays
Meaning for EL-AL:
 Sales increasing potential :
 Up to 100 Million $ (5%)
 Service quality improvement.

12. Extended warranty (EW)
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 An extra period of warranty (mostly purchased)
 Customers: A feeling of protection
 Company: Revenue increasing, enable them to “keep in
touch” with the customers
 Questions of optimization:
 Optimal price? (Padmanabhan and Rao 1993, Desai
and Padmanabhan 2004)
 Distribution channels? (Desai and Padmanabhan 2004,
Li et al. 2012)
 EW duration? (Li et al. 2012)

13. Our model vs. EW
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 Services are differ from durable products in several aspects:
Refurbishing for resale, time of consuming, presale quality
control, money-back guarantee, etc.
 Desai and Padmanabhan (2004):
 Optimal price
 Money compensation guarantee
 Customer’s risk preference
 Our model is differ from Desai and Padmanabhan model:
 Quality of product/service
 Demand function
 Numerical results
 Service improvement

14. Research problem
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 How to increase the company revenue, such that:
 Maximizing the revenue
 Minimizing the probability for default (increasing service
performance)
 Generally, the revenue is:
service
𝑝𝑟𝑖𝑐𝑒
− 𝑐𝑜𝑠𝑡 ×
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠
𝑤ℎ𝑜 𝑤𝑖𝑙𝑙 𝑏𝑢𝑦 𝑡ℎ𝑒 𝑠𝑒𝑟𝑣𝑖𝑐𝑒

15. Assumptions
Companies and customers views
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Companies view
 Competitive market
Due to poor service , customers have damaged (losses
=>money)
Price of service with no losses (“risk free”) denote by P
The companies’ prices are a function of the risk free price
and are different from one another only by their quality (risk)
 Branding, External features, etc., are ignored
Customers view
 Risk Perception - Risk Averse (denote by 𝑾𝑬)
Irrational decisions under uncertainty - Overestimating
probabilities of risks (Kahneman and Tversky, 1979)
 Customers’ willingness to pay, denote by 𝑽𝒓, is effected by
losses, price, reputation, etc. In our model, 𝑽𝒓 depends
𝑾𝑬 distribution
function

16. Expected customer surplus
16
 Customer decision: buy / don’t buy
 Decision element: positive expected surplus
 Therefore, a necessary condition
𝑉
𝑟 ≥ 𝑃 − 𝐸 𝐿 ∙ 𝑊𝐸
We denote the service price by 𝒑𝑳
𝒑𝑳 = 𝑷 − 𝑬[𝑳] ∙ 𝑾𝑬 = 𝑷 − 𝑬𝑳′
Expectancy
of loss
Risk-free
service price
Customer
willingness to pay
Customer’s
risk-perception

17. Example
Car buying
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 A person who wants to buy a car, has four options
 The prices are differences only by the quality
(reliability)

18. Notations
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Remarks
Meaning
𝜔 ∈ Ω
World event
ω
0 ≤ 𝑄(𝜔) ≤ 1
Default level
𝑄(𝜔)
Realization of ω
Strick for default
k
= 𝑃{𝑄(𝜔): 𝜔 ≥ k}
Probability to failure
𝒒𝒌
Loss, Expected loss
L, E[L]
𝑊𝐸 = 𝑊𝑇𝑃
E[L]
Risk Perception
𝑾𝑬
Willingness to pay for the service
𝑽𝒓
in case of default
Future compensation payment at time t
𝑰𝒕
Insurance policy
Insurance utility function
𝐔𝑰(𝑰)
Insurance operation costs
𝜺
=𝒑~
-𝒑∗
Insurance price
𝑺𝑰

19. Notations – Cont.
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Remarks
Meaning
“Risk free” service price
P
Service price
𝒑𝑳
Optimal service price
(The current price)
𝒑∗
Optimal service price with
insurance
𝒑~
functions
:
Remarks
Meaning
World event distribution function
𝒇(𝝎)
Function of 𝝎
Service quality (for the events)
distribution function
𝒒(𝝎)
h 𝑠𝑒𝑟𝑣𝑖𝑐𝑒 ,
ℎ1(𝐸 𝐿 ), h2 (𝑊𝐸)
WTP distribution functions
𝒉, 𝐡𝟏, 𝐡𝟐

20. World event
𝜔, 𝑓(𝜔)
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0
5
10
15
20
25
Number
of
cases
Flight delay
ω=1
ω=2
ω=3
ω=4
ω=6
ω=7
ω=8
𝝎
𝒇(𝝎)

21. Quality distribution function
Examples
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0 5 10 15 20 25 30 35 40 45 50
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
An exponential function

P
ro
b
a
b
ility
0 5 10 15 20 25 30 35 40 45 50
0
0.005
0.01
0.015

P
ro
b
a
b
ility
A step function

22. Service quality
why we need it?
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 Alternative: using the world event function, 𝑓(𝜔)
 But the events are not comparable
 15 min’ delay, what dose it mean?
 20 min’ Vs. 30 min’ delay?
 𝜔 could be several kinds of service failure
 We need a more general function
The service Quality function:
 A "mapping" function
 Represent the customer’s damage

23. Parametric Definitions
k, 𝑞𝑘, 𝑞 𝜔
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23 𝑞𝑘 = 𝑃{𝑄(𝜔): 𝜔 ≥ k}
Flight delay
𝒒𝒌
𝑞(𝜔)
𝒌
𝐩𝐫𝐨𝐛𝐚𝐛𝐢𝐥𝐢𝐭𝐲

24. Parametric Definitions
𝑞𝑘, {𝑄(𝜔): 𝜔 ≥ k}
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24 𝑞𝑘 = 𝑃{𝑄(𝜔): 𝜔 ≥ k}
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Q(𝜔)
Flight delay
1-𝒒𝒌
𝒌

25. The offered Plans
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 Plan 1 - No protection
 Selling the service as is, without insurance
 Risk is on the customer
 Plan 2 - Protection
 Selling the service with insurance
 Risk is allocated between the customer and the company

26. The model
Plan 1
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 The revenue function is:
𝝅(𝒑𝑳) = 𝒑𝑳 ∙
𝒑𝑳
𝑷
𝒉( 𝒙)𝒅𝒙
 Assuming 𝑷 is given (𝒉 is redundant, why?) and the prices
difference is attributed to 𝑬 𝑳 and 𝑾𝑬 (independent of one
another) :
𝝅(𝒑𝑳) = 𝒑𝑳 ∙
𝟎
∞
𝐡𝟐 𝐖𝐄
𝟎
𝑷−𝒑𝑳
𝐖𝐄
𝐡𝟏 𝐄 𝐋 𝒅𝑬 𝑳 𝒅𝑾𝑬
 The optimal solution 𝑝∗ = 𝑎𝑟𝑔𝑚𝑎𝑥𝑷𝑳
𝝅 𝒑𝑳 will be found by
solving
“Risk free” service price
p
Service price
𝒑𝑳
Optimal service price
𝒑∗

27. The model
Plan 2
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 In this plan we also sell insurance on the provided
service. the revenue function is:
𝜋(𝒑∗, 𝒑𝑳) = 𝒑∗ ∙
𝒑∗
𝒑𝑳
ℎ( 𝑥)𝑑𝑥 + (𝒑𝑳 − 𝐈𝐭 − 𝜺) ∙
𝒑𝑳
𝑃
ℎ( 𝑥)𝑑𝑥
Where:
𝒑∗ is the optimal price from Plan1 and
𝐈𝐭 is the expected compensation,
𝜺 is the operation costs

28. The model
Plan 2
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 Hence the correct revenue function is:
𝜋(𝒑∗
, 𝒑𝑳) = 𝒑∗
∙
𝑾𝑬=𝟎
𝑾𝑬=∞
𝒉𝟐(𝑾𝑬)
𝑬 𝑳 =
𝑷−𝒑𝑳
𝑾𝑬
𝑬 𝑳 =
𝑷−𝒑∗
𝑾𝑬
𝒉𝟏( 𝑬 𝑳 ) 𝒅𝑬 𝑳 𝒅𝑾𝑬
+(𝒑𝑳 − 𝐈𝐭 − 𝜺) ∙
𝑾𝑬=𝟎
𝑾𝑬=∞
𝒉𝟐(𝑾𝑬)
𝑬 𝑳 =𝟎
𝑬 𝑳 =
𝑷−𝒑𝑳
𝑾𝑬
𝒉𝟏( 𝑬 𝑳 ) 𝒅𝑬 𝑳 𝒅𝑾𝑬
 If 𝐈𝐭 and 𝜺 are given, the solution, namely, the optimal service
price, 𝒑~, will be obtained by solving
𝒅𝝅(𝒑∗
, 𝒑𝑳)
𝒅𝒑𝑳
= 𝟎

29. The model
Finding 𝐈𝐭 – function of 𝒌
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 The expected compensation, 𝐈𝐭 , can be determined in advance
from the expected future payments, when in the basic case :
𝑼𝑰(𝐼) =
0 𝑤𝑖𝑡ℎ 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 1 − 𝒒𝒌
𝐼𝑡
1
1 + 𝑅 𝑡
𝑤𝑖𝑡ℎ 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝒒𝒌
,
and
𝟏
𝟏+𝑹 𝒕 is the discount function with interest rate 𝑹.
 Consider the expectation :
𝑬(𝑼𝑰(𝑰)) =
𝟏
𝟏 + 𝑹 𝒕
∙ 𝑰𝒕 ∙ 𝒒𝒌
𝐈𝐭 =
𝟏
𝟏 + 𝑹 𝒕 ∙ 𝑰𝒕 ∙
𝒌
∞
𝒒 𝝎 𝒅𝝎
The company determine the 𝑰𝒕, 𝜺 and
k

30. The distribution functions
𝐡𝟏 𝐄 𝐋 , 𝑞 𝜔
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 In this work, in order to simplify the model and to
exemplify the use of our model, we assume the same
function of the events distribution function.
 Therefore,
ℎ1 𝐸 𝐿 = 𝜆𝑒−𝜆𝐸 𝐿
𝑞 𝜔 = 𝜆𝑒−𝜆𝜔

31. Solution Discretization
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 We could not find a close form solution to the integral.
Thus we find an approximate solution by using
discretization.
 For 𝒉𝟐(𝑾𝑬), there is no solution when 𝑾𝑬 → ∞ or 𝐖𝐄 = 0,
we bound the function’s domain to the interval 0+, 18
as represented in Ganderton et. al (2000), such that
P 𝐖𝐄 = 0 𝑜𝑟 𝐖𝐄 > 18 = 0
𝐖𝐄=0+
𝐖𝐄=18
𝐡𝟐 𝐖𝐄
𝟎
𝑷−𝒑𝑳
𝐖𝐄
𝐡𝟏 𝐄 𝐋 𝒅𝑬 𝑳 ∙ ∆𝑾𝑬

32. Numerical experiment
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 General information:
 Real data was take from http://www.transtats.bts.gov
 The data is for a specific airline company which provides a
particular flight from New York (LGA) to Washington DC (DCA)
 The scheduled flight duration is about one hour
 Additional data
 Suppose the risk free price is 𝑷 = 𝟔𝟎𝟎 $
 𝒉𝟏 𝑬 𝑳 ∶ 0.3032𝑒 −0.3032∙𝐸 𝐿
 𝑞 𝜔 = 0.3032𝑒 −0.3032∙𝜔

33. Concavity of the Revenue
function
𝑰𝒕 = 500$, 𝜀 = 0.1$, 𝑘 = 15 𝑚𝑖𝑛′ (𝝀 = 𝟎. 𝟑𝟎𝟑𝟐)
Plan 1
Plan 2
𝑝∗
≅ 540, 𝑝~
≅ 585
𝜋 𝑝∗
= $507 , 𝜋 𝑝∗
,𝑝 = $534
0 100 200 300 400 500 600
0
100
200
300
400
500
600
Price
Revenue
500 510 520 530 540 550 560 570 580 590 600
460
470
480
490
500
510
520
530
540
Price
Revenue

34. [𝝅 𝒑∗
, 𝝅 𝒑∗
, 𝒑 , 𝝅(𝒑)]
𝒑
𝒑
𝒑∗
λ
[199.3, 199.3, 351.5]
432.7
810.6
402.8
0.01
[364.5,408.23,514.3]
533.6
558.5
472.2
0.05
[428.5,469.84,549.75]
559.6
570.9
500.8
0.1
[487, 522, 573]
577.9
581.6
529.6
0.2
[507, 538, 580]
583.0
585.0
540.0
0.3
[523,551,584]
586.6
587.7
548.0
0.4
[532.1,558,586.63]
588.6
589.3
554.1
0.5
[539.7, 563.6, 588.55]
590.2
590.6
558.6
0.6
[546, 568, 590]
591.4
591.6
562.2
0.7
[550, 571, 591]
592.3
592.4
565.1
0.8
[554, 574, 591.7]
593.0
593.0
567.6
0.9
[557.5, 576, 592.6]
593.6
593.6
569.6
1.0
[568, 583, 594.8]
595.5
595.3
576.6
1.5
[587, 594, 598.2]
598.4
598.2
589.9
5.0
[591, 596, 598.8]
598.9
598.8
592.9
8.0
Results
𝑰𝒕 = 500$, 𝜀 = 0.1$, 𝑘 = 15 𝑚𝑖𝑛
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λ # 𝑒𝑣𝑒𝑛𝑡𝑠
𝑡𝑖𝑚𝑒
is
called the ‘rate
parameter’
𝒑 is the ‘No
𝑊𝐸’ price

35. Results
Graph 1 - Prices
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
400
450
500
550
600
650
700
750
800
850

Price
p*
p~
p-

36. Results
Graph 2 - Revenues
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
150
200
250
300
350
400
450
500
550
600

Revenue
(p*)
(p~
)
(p-
)

37. Results
Graph 3 – Marginal payment
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4

Customer
marginal
payment
(%)

38. Intermediate Conclusions
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 Relative to the chosen functions, people are willing to pay
about 𝟒% − 𝟏𝟖% more, in order to reduce the risk (𝜆
∈ [0.02,1])
 The average additional revenue is about 𝟓. 𝟖% ! (𝜆
∈ [0.02,1])
 Generally, there is no bound for the improvement as a
function of λ
 The revenue for ‘No 𝑊𝐸’ is an upper bound for the
revenue of each λ (Why?)
 The prices of Plan 2 and ‘No 𝑊𝐸’ are very close (for 𝜆

39. Sensitivity analysis of k
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 k is a decision variable
 The meaning: changing the threshold for default
 For the initial data:
 High value of k – high revenue
 But, will customers buy insurance?
 Low value of k – low revenue
 Is it worth the trouble of providing the
service on the provider’s part?
𝝀 = 𝟎. 𝟑 , 𝒑∗
= 𝟓𝟒𝟎
𝝅 𝒑∗
, 𝒑
𝒑
𝒌
NaN
NaN
1
506.43
736.67
*3
506.43
647.62
*5
506.46
599.57
7
517.15
591.5
11
526
588.31
13
534
586
15
535.17
585.65
16
536.4
585.3
18
537.09
585.15
20
537.47
585.05
22
537.68
585
24
537.8
584.97
26
⋯
⋯
⋯

40. Optimal quality improvement
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 We improve quality by reducing the probability for default (by changing
Lambda)
 If the improvement is free – “unlimited” improvement
 Otherwise, denote 𝑪(𝒈(𝝀𝟎), 𝒈(𝝀𝟏)) as the cost for improving the quality
from 𝒈(𝝀𝟎) to 𝒈(𝝀𝟏)
 A necessary condition is positive profit :
𝑝𝑓 𝝀 = [𝝅 𝝀𝟏 − 𝝅(𝝀𝟎)] − 𝑪(𝒈(𝝀𝟎), 𝒈(𝝀𝟏)) > 𝟎
when 𝑝𝑓 𝝀 is the profit function. The optimal 𝝀 is obtain by
𝜕𝑝𝑓 𝝀
𝜕𝝀
= 0

41. Optimal quality improvement – Cont.
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 However, this equation is hard to solve explicitly
 An approximate solution.
 For the previous example, if 𝝀𝟎=0.3 and
𝑪(𝒈(𝝀𝟎), 𝒈(𝝀𝟏)) =
𝟏
𝒈(𝝀𝟏)
−
𝟏
𝒈(𝝀𝟎)
, the optimal 𝝀 is:
𝝀𝟏 ≈ 𝟎. 𝟓𝟖

42. Summary
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 We presented a simple but innovative model for revenue
management
 The model uses “risk allocation” under reasonable assumptions
 To validate our model, we showed a real numerical experiment
 The service provider increases his income and incentivized to
improve his service quality
 The results are consistent with our expectations,

43. ‫כ‬
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44. ‫כ‬
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45. ‫כ‬
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46. Customer WTP for insurance
𝐖𝐄 distribution function
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 WTP for insurance – how dose it look?
 Ganderton et. al (2000) - Experimental results
 Shows the bias which exists in insurance buyers
𝑾𝑬 =
𝑾𝒊𝒍𝒍𝒊𝒏𝒈𝒏𝒆𝒔𝒔 𝒕𝒐 𝒑𝒂𝒚
𝑬𝒙𝒑𝒆𝒄𝒕𝒆𝒅 𝒍𝒐𝒔𝒔𝒆𝒔
0 2 4 6 8 10 12 14 16 18
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
WTP/E[L]
Probability
Data
Fitted curve

47. The model
general schema
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WT
P
𝑬 𝑳
𝑾𝑬
pric
e
WT
P

48. Future work
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 Finding the optimal function of 𝝀
 Using various distribution functions (for quality,
expected losses).
 Implementation the model in real service
company