This document describes a social dilemma between two individuals, Robert and Stuart, who must decide how much effort to contribute to a joint project. It presents a game theory model to analyze their strategic situation. The Nash equilibrium is identified as both choosing 1 unit of effort, but their rational self-interest leads them to contribute less than the overall optimal outcome of both choosing the highest effort level. This demonstrates how independent actions in social dilemmas can result in suboptimal collective outcomes.

1.
EC403:
Microeconomics
4
Assessed
Essay
Katherine
Anne
Gilbert
(201135988)
Word
Count:
1992
|
17th
November
2014

2. 2
This
essay
focuses
on
the
theme
of
social
dilemmas
where
the
rational
behavior
of
an
individual
leads
to
a
suboptimal
outcome
from
the
collective
standpoint
(Kollock,
1988).
Robyn
Dawes
(1980)
describes
the
theory
as
having
two
unique
properties:
1)
an
individual
will
receive
a
higher
payoff
if
they
defect
against
the
cooperative
choice
and
2)
all
individuals
will
receive
a
higher
payoff
if
they
all
choose
to
cooperate
than
if
they
all
choose
to
defect.
This
theme
will
be
discussed
through
the
example
of
Robert
and
Stuart
where
a
model
will
be
adopted
to
analyze
their
strategic
situation
through
a
simple
yet
insight-­‐rich
process.
The
game
is
played
regarding
the
level
of
effort
Robert
and
Stuart
both
choose
to
exert
into
a
joint
project.
Game
theory
is
defined
as
the
interaction
between
people
in
society
where
interdependent
behaviors
cause
the
action
of
one
person
to
affect
another
person’s
well
being,
either
positively
or
negatively
(Watson,
2013).
In
this
example,
the
effort
level
that
Robert
chooses
to
exert
will
have
an
impact
on
the
value
of
the
project
which
in
turn
affects
the
payoff
to
Stuart
and
vice
versa.
This
interdependence
shows
a
game
is
being
played.
When
acting
independently,
individuals
are
assumed
to
act
rationally
with
the
incentive
to
maximize
their
own
payout.
Rational
behavior
has
two
unique
properties:
1)
from
experience
or
knowledge,
a
belief
is
formed
regarding
the
expected
strategies
of
opponents
2)
given
this
belief;
a
strategy
is
selected
to
maximize
the
expected
payoff
(Watson,
2013).
When
analyzing
mixed
strategies,
the
beliefs
that
are
formed
are
a
probability
distribution
over
opponent’s
possible
effort
level.
As
this
example
is
a
non-­‐
cooperative
simultaneous-­‐move
game,
players
do
not
fully
internalize
their
chosen
value
of
effort
and
individuals
are
unsure
what
their
opponents
will
choose
therefore
beliefs
are
central
to
individuals
decision
making
process.
The
characteristics
of
this
game
show
there
is
information
asymmetry
in
the
game,
as
players
cannot
observe
opponents
decision
before
taking
theirs.
This
game
will
begin
through
a
discrete
example
of
Robert
and
Stuart
selecting
from
a
finite
choice
of
either
1
or
2
units
of
effort
to
exert
into
the
joint
project.
The
project
value
is
calculated
by
subtracting
the
individual
cost
function
of
Robert
and
Stuart
from
the
value
function,
which
measures
the
cost
minus
the
benefit
from
exerting
a
particular
unit
of
effort.
The
x
value
measures
the
effort
expended
by
Robert
and
the
y
value
measures
the
effort
expended
by
Stuart.
Project
value:
V
(x,
y)
=
5(x
+
y)
+
xy
Project
value
depends
on
the
level
of
effort
exerted
by
Robert
(x)
and
Stuart
(y).
As
Robert’s
cost
function
is
3x2
à
Payoff
function
=
5(x
+
y)
+
xy
–
3x2
As
Stuart’s
cost
function
is
4y2
à
Payoff
function
=
5(x
+
y)
+
xy
–
4y2

3. 3
When
calculating
the
payoffs
to
each
player
from
each
chosen
effort
combination,
the
effort
values
are
substituted
into
the
individual’s
cost
function:
For
V
(1,
1)
!
5(1
+
1)
+
(1
x
1)
=
11
For
V
(2,
1)
!
5(2
+
1)
+
(2
x
1)
=
17
For
V
(1,
2)
!
5(1
+
2)
+
(1
x
2)
=
17
For
V
(2,
2)
!
5(2
+
2)
+
(2
x
2)
=
24
Constructing
the
normal
form
for
this
game
summarizes
the
players
in
the
game,
strategies
available
and
payoff
received
to
each
individual
based
on
the
strategy
combination
the
player’s
choose
(Gibbons,
1992).
The
columns
correspond
to
the
strategies
of
Stuart
and
the
rows
correspond
to
the
strategies
of
Robert.
Normal
Form:
The
underlining
method
identifies
the
rational
best
response
made
by
individuals
if
their
opponent
chooses
a
particular
effort
level.
By
constructing
the
normal
form,
the
Nash
equilibrium
can
be
identified:
Robert
chooses
1
unit
of
effort
and
Stuart
chooses
1
unit
of
effort
at
V(1,
1).
The
Nash
equilibrium
represents
a
point
in
the
game
where
players
have
no
incentive
to
change
their
strategy
and
have
mutually
consistent
best
responses.
For
both
players,
strategy
1
is
the
dominant
strategy
therefore
we
can
predict,
due
to
rational
behavior,
that
neither
player
will
select
strategy
2
(Gibbons,
1992).
The
identification
of
the
best
response
in
this
game
shows
that
both
players
have
the
incentive
to
underperform
and
reduce
their
individual
effort
to
gain
a
higher
payout.
When
analyzing
the
Nash
equilibrium
and
normal
form
of
this
game,
it
is
apparent
that
the
rational
behavior
of
Robert
and
Stuart
does
not
necessarily
imply
coordinated
behavior,
as
the
Nash
equilibrium
is
inefficient
with
both
players
able
to
gain
a
higher
payoff
by
playing
differently.
The
players
realize
they
are
jointly
better
off
if
they
select
1
unit
of
effort,
however
individual
incentives
result
in
individuals
defecting
against
this
choice:
Robert
can
gain
a
payoff
of
13
instead
of
7
and
Stuart
can
gain
a
payoff
of
14
instead
of
8.
In
this
game,
individuals
gain
from
being
non-­‐cooperative
and
have
the
incentive
to
free
ride
on
the
contributions
of
others.
R/S
1
2
1
8,
7
14,
1
2
5,
13
12,
8
V
(1,
1)
à
Robert
=
11
–
3
(1)2
=
8
Stuart
=
11
–
4
(1)2
=
7
V
(1,
2)
à
Robert
=
17
–
3
(1)2
=
14
Stuart
=
17
–
4
(2)2
=
1
V
(2,
1)
à
Robert
=
17
–
3
(2)2
=
5
Stuart
=
17
–
4
(1)2
=
13
V
(2,
2)
à
Robert
=
24
–
3
(2)2
=
12
Stuart
=
24
–
4
(2)2
=
8

4. 4
This
result
is
similar
to
the
Prisoner’s
dilemma
where
individual
incentives
interfere
with
the
interests
of
the
group
(Rapoport,
1965)
resulting
in
group
loss
–
this
concept
will
be
analysed
later
in
the
game.
When
opponents
have
complete
freedom
over
their
effort
choices,
the
players
no
longer
choose
from
a
finite
set
of
strategies
and
can
undertake
a
continuous
strategy.
As
the
strategy
spaces
are
continuous
in
this
game,
the
game
can
be
analysed
by
calculating
best
responses
as
opposed
to
through
a
payoff
matrix.
This
is
calculated
by
differentiating
the
individual’s
payoff
functions
and
re-­‐
arranging
for
x
and
y
respectively.
The
derivative
is
set
to
0
to
find
where
the
slope
of
this
function
is
maximized
at
zero,
which
is
the
best
response
for
each
individual.
Substituting
the
best
response
function
for
y
into
the
x
function
will
find
the
payoff
for
Robert
and
substituting
the
best
response
function
for
x
into
the
y
function
will
find
the
payoff
for
Stuart.
The
Nash
equilibrium
is
at
point
(
𝟒𝟓
𝟒𝟕
,
𝟑𝟓
𝟒𝟕
),
which
represents
the
set
of
effort
choices
for
Robert
and
Stuart
where
both
players
are
maximizing
their
payoff
given
the
actions
of
the
other
player
(Varian,
1987).
Under
this
equilibrium,
Robert
will
exert
!"
!"
units
of
effort
and
Stuart
will
exert
!"
!"
units
of
effort.
Robert:
f(x)
=
5(x
+
y)
+
xy
–
3x2
f(x)
=
5x
+
5y
+
xy
–
3x2
Differentiate
with
respect
to
x:
f’(x)
=
5
+
y
–
6x
Set
f’(x)
=
0
to
find
optimal
point:
0
=
5
+
y
–
6x
6x
=
5
+
y
x
=
𝟓!𝒚
𝟔
BRr(x)
=
𝟓!𝒚
𝟔
Stuart:
f(y)
=
5(x
+
y)
+
xy
–
4y2
f(y)
=
5x
+
5y
+
xy
–
4y2
Differentiate
with
respect
to
y:
f’(y)
=
5
+
x
–
8y
Set
f’(y)
=
0
to
find
optimal
point:
0
=
5
+
x
–
8y
8y
=
5
+
x
y
=
𝟓!𝒙
𝟖
BRs(y)
=
𝟓!𝒙
𝟖
Substitute
y
into
equation
x:
X
=
(5
+
!!!
!
!
)
6x
=
5
+
!!!
!
48x
=
40
+
5
+
x
47x
=
45
x
=
𝟒𝟓
𝟒𝟕
Substitute
x
into
equation
y:
Y
=
!!
𝟒𝟓
𝟒𝟕
!
8y
=
5
+
𝟒𝟓
𝟒𝟕
376y
=
235
+
45
376y
=
280
y
=
!"#
!"#
y
=
𝟑𝟓
𝟒𝟕

5. 5
Substituting
the
x
and
y
values
into
individual
payoff
functions
and
subtracting
the
effort
costs,
each
partner
receives
the
following
payoff:
From
the
reaction
functions
calculated,
the
relationship
can
be
diagrammatically
presented
by
finding
points
on
Robert
and
Stuart’s
reaction
function.
This
is
calculated
by
setting
the
x
and
y
values
of
Robert
and
Stuart’s
reaction
function
to
0
as
follows:
The
following
graph
depicts
the
best-­‐response
functions
of
the
two
players.
When
calculating
the
equilibrium,
the
dominated
strategies
are
removed
and
the
lower
and
upper
bounds
converge
to
the
point
where
the
players’
best
response
functions
cross,
which
is
where
the
equilibrium
point
lies.
These
response
functions
are
positively
sloped
as
they
are
a
complementary
strategy
set.
Robert:
x
=
𝟓!𝒚
𝟔
To
calculate
reaction
function:
When
y
=
0,
x
=
!!!
!
=
!
!
When
x
=
0,
0
=
!!!
!
=
-­‐5
Points
on
reaction
function:
(5/6,
0)
and
(0,
-­‐5)
Stuart:
y
=
𝟓!𝒙
𝟖
To
calculate
reaction
function:
When
y
=
0,
x
=
!!!
!
=
-­‐5
When
x
=
0,
y=
!!!
!
=
!
!
Points
on
reaction
function:
(-­‐5,
0)
and
(0,
5/8)
Payoff
Robert:
5x
+
5y
+
xy
-­‐
3x2
5(
𝟒𝟓
𝟒𝟕
)
+
5(
𝟑𝟓
𝟒𝟕
)
+
(
𝟒𝟓
𝟒𝟕
)(
𝟑𝟓
𝟒𝟕
)
–
3(
𝟒𝟓
𝟒𝟕
)2
=
6.47
Payoff
Stuart:
5x
+
5y
+
xy
–
4y2
5(
𝟒𝟓
𝟒𝟕
)
+
5(
𝟑𝟓
𝟒𝟕
)
+
(
𝟒𝟓
𝟒𝟕
)(
𝟑𝟓
𝟒𝟕
)
–
4(
𝟑𝟓
𝟒𝟕
)2
=
7.00
(
𝟒𝟓
𝟒𝟕
,
𝟑𝟓
𝟒𝟕
)
Robert’s
Effort
Level
(x)
Stuart’s
Effort
Level
(y)

6. 6
Game-­‐theoretic
analysis
generally
assumes
that
each
player
behaves
rationally
according
to
his
preferences
however;
it
does
not
take
into
account
other
motivations
such
as
that
of
the
altruistic
player.
Tony
will
be
introduced
to
this
example
as
the
social
planner
who
has
the
main
objective
of
maximizing
the
total
payoff
of
the
project,
which
is
then
split
between
Robert
and
Stuart.
The
function
used
to
consider
this
game
of
social
welfare
is
the
project
value
function
of
both
Robert
and
Stuart
(5(x
+
y)
+
xy
+
5(x
+
y)
+
xy)
minus
the
two
cost
functions
of
Robert
(3x2)
and
Stuart
(4y2)
to
calculate
the
best
responses
to
maximize
the
total
payoffs
from
the
game.
!
V
(x,
y)
=
5(x
+
y)
+
xy
+
5(x
+
y)
+
xy
–
3x2
–
4y2
!
V
(x,
y)
=
10(x
+
y)
+
2xy
-­‐
3x2
–
4y2
The
x
and
y
effort
values
are
calculated
by
substituting
the
x
and
y
figures
into
one
another
as
before.
Robert:
f(x)
=
10(x
+
y)
+
2xy
–
3x2
–
4y2
f(x)
=
10x
+
10y
+
2xy
–
3x2
–
4y2
Differentiate
with
respect
to
x:
f’(x)
=
10
+
2y
–
6x
Set
f’(x)
=
0
to
find
optimal
point:
0
=
10
+
2y
–
6x
6x
=
10
+
2y
x
=
𝟏𝟎!𝟐𝒚
𝟔
Stuart:
f(y)
=
10(x
+
y)
+
2xy
–
3x2
-­‐
4y2
f(y)
=
10x
+
10y
+
2xy
–
3x2
-­‐
4y2
Differentiate
with
respect
to
y:
f’(y)
=
10
+
2x
–
8y
Set
f’(y)
=
0
to
find
optimal
point:
0
=
10
+
2x
–
8y
8y
=
10
+
2x
y
=
𝟏𝟎!𝟐𝒙
𝟖
Substitute
y
into
equation
x:
X
=
!" ! !
!"!!!
!
!
6x
=
10
+
2(
!"!!!
!
)
48x
=
80
+
20
+
4x
44x
=
100
x
=
!""
!!
x
=
𝟐𝟓
𝟏𝟏
Substitute
x
into
equation
y:
Y
=
!" ! !
!"!!!
!
!
8y
=
10
+
2(
!"!!!
!
)
48y
=
60
+
20
+
4y
48y
=
80
+
4y
44y
=
80
y
=
!"
!!
y
=
𝟐𝟎
𝟏𝟏

7. 7
Payoff
Robert:
5x
+
5y
+
xy
-­‐
3x2
5(
𝟐𝟓
𝟏𝟏
)
+
5(
𝟐𝟎
𝟏𝟏
)
+
(
𝟐𝟓
𝟏𝟏
)(
𝟐𝟎
𝟏𝟏
)
–
3(
𝟐𝟓
𝟏𝟏
)2
=
9.09
Payoff
Stuart:
5x
+
5y
+
xy
–
4y2
5(
𝟐𝟓
𝟏𝟏
)
+
5(
𝟐𝟎
𝟏𝟏
)
+
(
𝟐𝟓
𝟏𝟏
)(
𝟐𝟎
𝟏𝟏
)
–
4(
𝟐𝟎
𝟏𝟏
)2
=
11.36
Acting
independently,
Robert
can
gain
a
payoff
of
6.47
and
Stuart
can
gain
a
payoff
of
7,
however,
when
adding
an
altruistic
player
to
maximize
social
welfare,
Robert
can
gain
a
larger
payoff
of
9.09
and
Stuart
can
gain
a
larger
payoff
of
11.36.
This
would
suggest
that
Robert
and
Stuart
should
optimally
act
together
to
ensure
they
both
gain
larger
payoffs
than
if
they
were
acting
independently.
When
comparing
the
effort
levels
of
each
player
to
the
Nash
equilibrium
calculated
when
acting
independently,
the
social
planner
states
the
players
should
exert
more
than
double
the
effort
that
the
players
would
exert
when
acting
independently.
Analysing
in
terms
of
the
effort
to
payoff
ratio,
the
individuals
gain
more
from
acting
independently
than
if
they
were
to
exert
effort
at
the
social
planners
ideal
value:
Robert
exerting
1
effort
unit:
1.67
payoff
acting
independently
and
exerting
1
effort
unit:
4
payoff
acting
socially
optimally;
Stuart
exerting
1
effort
unit:
9.5
payoff
acting
independently
and
exerting
1
effort
unit:
6.3
payoff
acting
socially
optimally.

8. 8
It
is
evident
that
the
addition
of
the
social
planner
to
the
game
can
result
in
effort
levels
chosen
that
will
increase
the
total
payoff
function
for
the
players.
The
choice
for
players
is
whether
to
cooperate
with
the
social
planners
ideal
outcome
or
whether
to
defect
with
the
belief
that
their
opponent
will
cooperate.
As
Kollock
(1998)
identified,
the
best
outcome
of
a
Prisoner’s
Dilemma
is
unilateral
defection
of
the
first
person,
followed
by
mutual
cooperation,
mutual
defection,
and
the
worst
outcome
is
the
first
person’s
unilateral
cooperation.
The
following
analysis
calculates
the
payoffs
of
Robert
and
Stuart
if
1)
Robert
cooperates
and
Stuart
defects
and
2)
if
Stuart
cooperates
and
Robert
defects
from
the
social
ideal.
Normal
Form:
The
final
normal
form
presents
the
payoffs
to
each
individual
based
on
the
range
of
scenarios
presented
throughout
this
example.
The
Nash
equilibrium
is
defection
against
the
social
planners
ideal
as
it
the
dominant
strategy
for
both
players.
R/S
C
D
C
11.4,
9.09
2.5,
14.7
D
13,
3.6
7.0,
6.5
When
Robert
cooperates
with
socially
optimal
input
(
𝟐𝟓
𝟏𝟏
)
and
Stuart
defects:
Y
=
! !
𝟐𝟓
𝟏𝟏
!
Y
=
0.91
Joint
value
of
project:
5((
𝟐𝟓
𝟏𝟏
)
+
(0.91))
+
(
𝟐𝟓
𝟏𝟏
)(0.91)
=
18
Payouts:-­‐
Robert:
18
–
3(
𝟐𝟓
𝟏𝟏
)2
=
2.5
Stuart:
18
–
4(0.91)2
=
14.7
When
Stuart
cooperates
with
socially
optimal
input
(
𝟐𝟎
𝟏𝟏
)
and
Robert
defects:
X
=
! !
𝟐𝟎
𝟏𝟏
!
X
=
1.14
Joint
value
of
project:
5((1.14
+
(
𝟐𝟎
𝟏𝟏
))
+
(1.14)(
𝟐𝟎
𝟏𝟏
)
=
16.9
Payouts:
Robert:
16.9
–
3(1.14)2
=
13
Stuart:
16.9
–
4(
!"
!!
)2
=
3.6

9. 9
This
example
is
similar
to
the
prisoners’
dilemma.
The
dilemma
is
clear
when
noticing
that
mutual
cooperation
is
superior
and
yields
a
better
outcome
than
mutual
defection
of
both
players.
This
results
in
mutual
defection
not
being
a
rational
outcome
because
the
choice
to
cooperate
is
more
rational
from
a
self-­‐
interested
point
of
view.
The
value
of
the
project
suffers
as
partners
only
care
about
their
own
costs
and
benefits
and
they
do
not
consider
the
benefit
of
their
labor
to
the
rest
of
the
firm.
The
success
of
the
enterprise
requires
cooperation
and
shared
responsibility
of
the
individuals.
In
this
example,
there
are
limits
on
external
enforcement
as
the
partners
cannot
write
a
contract
and
have
no
way
of
verifying
the
levels
of
effort
exerted
by
each
individual.
Even
if
the
players
had
the
ability
to
communicate
before
the
game
and
decide
on
a
collective
decision,
there
is
no
incentive
for
a
rational
player
to
follow
through
on
the
agreement
as
individuals
could
maximize
by
defecting.
Players
understand
that
increasing
effort
and
therefore
the
value
of
the
joint
project
will
result
in
them
only
gaining
a
fraction
of
the
joint
benefit
therefore
the
players
are
less
willing
to
provide
effort.
The
individuals
therefore
expend
less
effort
than
is
best
from
the
social
point
of
view
and
because
both
players
do
so,
they
both
end
up
worse
off.
As
this
example
only
considered
a
single
play
game,
choosing
to
cooperate
is
not
optimal
for
players.
If
the
example
incorporated
repeated
game
analysis,
including
the
punishment
of
the
opponent
for
defecting
would
reduce
the
payoffs.
This
could
result
in
players
choosing
to
cooperate
if
costs
>
benefits
by
defecting.

10. 10
As
individuals’
indifference
curves
depend
on
other
people
(Buchanan,
1962),
this
game
results
in
the
players
imposing
external
effects
on
their
opponent
based
on
the
decision
they
choose.
Externalities
are
defined
as
an
indirect
consequence
of
an
activity,
which
affect
agents
other
than
the
originator
without
intension
(Laffont,
2008).
In
this
example,
Robert
and
Stuart’s
payoff
depends
on
the
effort
exerted
by
the
other
player,
which
is
an
example
of
a
positive
externality
or
external
benefit
to
the
other
player.
As
players
increase
their
effort,
the
value
of
the
overall
project
increases
which
increases
the
payoffs
to
both
players.
When
calculating
the
effort
to
payoff
ratio,
there
is
a
2.8
payoff
difference
when
Robert
and
Stuart
exert
one
unit
of
effort:
Robert
gains
6.7
payoffs
and
Stuart
gains
9.5
payoffs.
It
is
clear
that
Stuart
benefits
more
from
exerting
one
unit
of
effort
and
therefore
Robert
exerts
a
higher
positive
externality
on
Stuart.
The
existence
of
externalities
can
cause
market
failure
if
the
price
mechanism
does
not
take
into
account
the
full
social
costs
and
benefits
of
production
and
consumption.
In
this
game,
there
is
a
difference
between
the
marginal
private
benefit
and
marginal
social
benefit
resulting
in
the
MSB
curve
lying
above
MPB.
The
effort
exerted
and
therefore
the
market
output
is
less
than
the
socially
optimal
output
as
shown
in
Appendix
1
-­‐
society
could
be
better
off
and
welfare
increased
by
encouraging
increased
provision.
There
is
therefore
community
pressure
to
make
individuals
pull
their
weight
to
bring
the
effort
levels
to
the
socially
optimal
ideal.
A
method
of
creating
this
pressure
is
to
provide
a
subsidy
to
the
individuals
or
reward
the
individuals
to
provide
at
this
point
[Appendix
2]
bringing
MSB
=
MPB.

11. 11
REFERENCES
Buchanan,
J.
&
Stubblebine,
W.
(1962).
Externality.
Economica.
29
(116),
371-­‐384.
Dawes,
R.
(1980).
Social
Dilemmas.
Annual
Review
of
Psychology.
31,
169-­‐193.
Gibbons,
R
(1992).
A
Primer
in
Game
Theory.
London:
Pearson
Education
Limited.
Kollock,
P.
(1998).
Social
Dilemmas:
The
Anatomy
of
Cooperation.
Annual
Review
of
Sociology.
24
(1),
183-­‐214.
Laffont,
J.
(2008).
Externalities.
The
New
Palgrave
Dictionary
of
Economics.
2.
Rapoport,
A
(1965).
Prisoner's
Dilemma:
A
Study
in
Conflict
and
Cooperation.
USA:
The
University
of
Michigan
Press.
Varian,
H.;
1987;
“Intermediate
Microeconomics
–
A
Modern
Approach;
Norton,
Chapter
27
pp.
508
Vicarick.
(2011).
Externalities
Graphs:
How
I
Understand
Them.
Available:
http://www.slideshare.net/vicarick/externalities-­‐graphs-­‐how-­‐i-­‐understand-­‐
them.
Last
accessed
14th
Nov
2014.
Watson,
J
(2013).
Strategy:
An
Introduction
to
Game
Theory.
3rd
ed.
New
York:
W.
W.
NORTON
&
COMPANY.
ADDITIONAL
BIBLIOGRAPHY
Carmichael,
F
(2005).
A
Guide
to
Game
Theory.
ENGLAND:
Pearson
Education
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Dixit,
A.,
Skeath,
S.
&
Reiley,
D
(2009).
Games
of
Strategy.
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New
York:
W.
W.
Norton
&
Co.
Osborne,
M
(2004).
An
introduction
to
game
theory.
New
York:
Oxford
University
Press.

12. 12
APPENDICES
Appendix
1
(Vicarick,
2011)
Appendix
2
(Vicarick,
2011)