Dynamics of Reputation

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Dynamics of Reputation

• USC Mathematical Modeling of Social and Economic
  Behavior
• Prof. Adele Diederich, Prof. Peter Oswald
• Jacobs University Bremen
• Spring 2008
• Group F Philipp Brandt, Jakob Hensing, Joseph
  Kiprotich, Simon Schmitt, Benjamin Schrauf

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Content

• Motivation
• Introduction
• Case Studies
• Business The fall of Northern Rock
• Politics Bill Clinton and the Lewinsky scandal
• Mathematics The model
• Possible extensions

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Motivation

• Increased importance of reputation due to mass
  media and information society
• Which parameters determine the dynamic of a
  scandal?
• Under which circumstances can actors recover from
  a decay of reputation, under which circumstances
  do they perish?
• Can the dimension of a scandal be predicted?

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Introduction

• Huberman, B.A. Fang Wu (2004). The dynamics of
  reputations. Journal of Statistical Mechanics
  Theory and Experiment, 27 April 2007. doi
  10.1088/1742-5468/2004/04/P04006
• 2 Firms with long horizons
• Varying quality of products ? varying costs
• Quality level ? assignment of reputation by
  customers
• Private information ? Public information
• Outcomes Stable equilibria or persistent
  nonlinear oscillations of reputation

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Case 1 The fall of Northern Rock

• Britains fifth largest mortgage lender
• Indirect involvement in the US subprime crisis
• Institutional lenders withdrew capital, leading
  to a liquidity problem.
• Panic evoked by announcements of Chancellor
  Alistair Darling urging private investors not to
  withdraw money, assuring emergency funds by the
  Bank of England
• Massive backfire, 14.9.2007 queues of private
  investors trying to rescue their money, 1
  billion pound withdrawn on a single day.
• 2008 Nationalization of the bank, suspension of
  shares

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The fall of Northern Rock
After a period of moderate decline, the offer of
emergency funds by the Bank of England led to the
crash of Northern Rock, since it brought the true
extent of Northern Rocks problems to public
attention.
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Case 2 Bill Clinton and the Lewinsky scandal

• July 1995 Monica Lewinsky intern in the White
  House
• 1998 In another trial involving Clinton, it was
  revealed by coincidence that Clinton and Lewinsky
  had a sexual affair during her intern time
• Clinton denied the accusations, later challenging
  the definition of sexual affair

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Clintons approval ratings
Beginning of the affair (NOT PUBLIC)
Revealing of the scandal (PUBLIC)
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A simplified model

• The simplified model applies the principles of
  the original model to one actor

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A simplified model

• The simplified model applies the principles of
  the original model to one actor
• Model desribes inertia of reputation
• - An event changes the image of a public
  figure/organization, resulting in a change of
  public approval.
• - There is a time lag between the event and the
  publics reaction to it, since people only
  reevaluate their preferences with finite
  frequency.

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A simplified model

• The simplified model applies the principles of
  the original model to one actor
• Model desribes inertia of reputation
• - An event changes the image of a public
  figure/organization, resulting in a change of
  public approval.
• - There is a time lag between the event and the
  publics reaction to it, since people only
  reevaluate their preferences with finite
  frequency.
• Given a time series of public approval ratings
  after an image changing event, the model can
  estimate characteristic parameters
• - At what frequency do people reevaluate their
  opinions?

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ApplicationModeling a Presidents approval
ratings

• F(t) approval rate
• Fraction of population that approves of
  President over time t

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ApplicationModeling a Presidents approval
ratings

• F(t) approval rate
• Fraction of population that approves of
  President over time t
• Fo initial approval rate
• Presidents approval rate at t0
• a evaluation frequency
• Gives the number of times an average member
  of the public reevaluates his/her opinion per
  unit time.
• S ?0,1 likeability after image changing event
• Probability with which a member of the
  public approves of the President upon
  reevaluation after the image changing event has
• taken place. In the model, the Presidents
  approval
• rate F(t) will converge to S.

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Structure of the model
approving citizens
F approve
American Public
(1-F) disapprove
disapproving citizens
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Structure of the model
Reevaluation?
a
yes
F approve
no
1-a
American Public
a
yes
(1-F) disapprove
no
1-a
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Structure of the model
Reevaluation?
Result of reevaluation?
approve
a
S
yes
F approve
no
disapprove
S-1
1-a
American Public
approve
a
S
yes
(1-F) disapprove
no
1-a
S-1
disapprove
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Structure of the model
Reevaluation?
Result of reevaluation?
approve
a
S
yes
F approve
no
disapprove
S-1
1-a
Fa(1-S)
American Public
approve
a
S
(1-F)aS
yes
(1-F) disapprove
no
1-a
S-1
disapprove
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Derriving the equation
Goal To describe the time dynamics of the
fraction F(t) of people supporting the president.
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Derriving the equation
Goal To describe the time dynamics of the
fraction F(t) of people supporting the
president. Therefore, we need to know f1
the time rate at which disapproving citizens
become supporters f2 the time rate at which
supportive citizens become detractors
new supporters
new detractors
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Derriving the equation
Reevaluation?
Result of reevaluation?
approve
a
S
yes
F approve
new detractors
no
disapprove
S-1
1-a
f2 Fa(1-S)
American Public
new supporters
approve
a
S
f1 (1-F)aS
yes
(1-F) disapprove
no
1-a
S-1
disapprove
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Derriving the equation
Goal To describe the time dynamics of the
fraction F(t) of people supporting the
president. dF Change of supportive fraction in
time dt
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Derriving the equation
Goal To describe the time dynamics of the
fraction F(t) of people supporting the
president. dF Change of supportive fraction in
time dt dF (new supporters) - (new
detractors) dt f1
- f2
dt Sa(1-F)
- (1-S)aF dt
a(S-F)dt
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Derriving the equation
Goal To describe the time dynamics of the
fraction F(t) of people supporting the
president. dF Change of supportive fraction in
time dt dF (new supporters) - (new
detractors) dt f1
- f2
dt Sa(1-F)
- (1-S)aF dt
a(S-F)dt
dF/dt a(S-F)
differential equation
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Solving the equation
The derived equation is a first order,
linear, inhomogeneous differential equation with
constant coefficients.
dF/dt a(S-F)
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Solving the equation
The derived equation is a first order,
linear, inhomogeneous differential equation with
constant coefficients. The general solution F(t)
of the inhomogeneous equation can be found by
adding a particular solution Fp(t) of the
inhomogeneous equation to the general solution
Fg(t) of the homogeneous equation F(t) Fg(t)
Fp(t)
dF/dt a(S-F)
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Solving the equation
The derived equation is a first order,
linear, inhomogeneous differential equation with
constant coefficients. The general solution F(t)
of the inhomogeneous equation can be found by
adding a particular solution Fp(t) of the
inhomogeneous equation to the general solution
Fg(t) of the homogeneous equation F(t) Fg(t)
Fp(t) The homogeneous differential equation
is dF/dt -aF By trivial integration, the
general solution can be found to be Fg(t)
ce-at Where c is a constant determined by the
problems boundary condition F0, which is the
presidents initial aproval rating
dF/dt a(S-F)
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Solving the equation
dF/dt a(S-F)

• A particular solution to the inhomogeneous
  equation is F(t) S,
• since then
• dF/dt 0 and a(S-F) 0

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Solving the equation
dF/dt a(S-F)

• A particular solution to the inhomogeneous
  equation is F(t) S,
• since then
• dF/dt 0 and a(S-F) 0
• Therefore, the general solution of the
  inhomogeneous equation is
• F(t) Fg(t) Fp(t) ce-at S

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Solving the equation
dF/dt a(S-F)

• A particular solution to the inhomogeneous
  equation is F(t) S,
• since then
• dF/dt 0 and a(S-F) 0
• Therefore, the general solution of the
  inhomogeneous equation is
• F(t) Fg(t) Fp(t) ce-at S
• If we take the presidents approval level to be F0
  at t0, the constant c
• can be determined
• F0 ce-at S at t 0
• c F0 S
• F(t) (F0 S)e-at S

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Modeling the Lewinsky Scandal
By fitting the models function F(t) to Bill
Clintons approval ratings after after
publication of the Lewinsky scandal, the models
parameters can be estimated for the scandal.
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Modeling the Lewinsky Scandal
t0 is set to Jan 21, 1998 On this day, several
news organizations reported on the affair between
Clinton and Lewinsky
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Modeling the Lewinsky Scandal
t0 is set to Jan 21, 1998 On this day, several
news organizations reported on the affair between
Clinton and Lewinsky
Monthly surveys, blue dots indicate the
approval ratings for 8 months immediately after
the scandal was revealed. The red graph is the
best fit of the function F(t) to the data.
approval rating population
F(t) (F0 S)e-at S
time t months
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Modeling the Lewinsky Scandal
t0 is set to Jan 21, 1998 On this day, several
news organizations reported on the affair between
Clinton and Lewinsky
Monthly surveys, blue dots indicate the
approval ratings for 8 months immediately after
the scandal was revealed. The red graph is the
best fit of the function F(t) to the data.
approval rating population
F(t) (F0 S)e-at S
Parameter approximation
F(t) (0.65 0.58)e-2.77t 0.58 R-square
0.79 F0 initial approval rate (0.65 ) S
likeability (0.58 ) a evaluation
frequency (2.77 per month)
time t months
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Modeling the Lewinsky Scandal
Result During the Lewinsky scandal, the average
American citizen reevaluated his/her opinion
about the President 2.77 times per month.
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Problems
The model assumes a sudden jump in likeability
of the President from F0 to S. This assumption is
not completely accurate, since the details of the
scandal became public over a period of
months. The approval ratings to which the curve
was fitted in order to approximate the parameters
has a category for undecided citizens. The model
does not take account of such a category,
assuming that the fraction 1-F of citizens who
are not supporters are all detractors.
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Outlook

• Additional factors must be considered in a more
  comprehensive model
• Memory effects
• Public and private information

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Thank you for your attention!