2.
Discrete
Random
Variables
Part
III:
St
Petersburg
Paradox
ECE
302
Fall
2009
TR
3‐4:15pm
Purdue
University,
School
of
ECE
Prof.
Ilya
Pollak
Problem
2.21:
St
Petersburg
Paradox
•  A
coin
is
flipped
repeatedly
unPl
the
first
H.
•  If
the
first
H
appears
on
the
n‐th
toss,
you
get
n
paid
$2 .
•  How
much
should
a
casino
charge
you
for
playing
this
game?
Problem
2.21:
St
Petersburg
Paradox
•  A
coin
is
flipped
repeatedly
unPl
the
first
H.
•  If
the
first
H
appears
on
the
n‐th
toss,
you
get
n
paid
$2 .
•  How
much
should
a
casino
charge
you
for
playing
this
game?
•  Probability
mass
funcPon
of
your
winnings
1/2
1/4
1/8
2
4
8
1/16
16
…
Problem
2.21:
St
Petersburg
Paradox
•  A
coin
is
flipped
repeatedly
unPl
the
first
H.
•  If
the
first
H
appears
on
the
n‐th
toss,
you
get
n
paid
$2 .
•  How
much
should
a
casino
charge
you
for
playing
this
game?
1
1
1
•  Your
expected
winnings:
2 ⋅ 2 +  2  ⋅ 2 +  2  ⋅ 2 + … = ∞
•  Probability
mass
funcPon
of
your
winnings
2
3
2
1/2
€
1/4
1/8
2
4
8
1/16
16
…
3
How
to
price
the
coin‐flipping
game?
•  The
customer’s
expected
winnings
per
game
are
infinite.
•  The
casino
should
probably
not
offer
this
game!
What
if
the
casino
does
offer
this
game,
for
a
very
large
price?
•  Will
you
accept
any
amount
the
casino
wants
to
charge
in
order
to
play
this
game?
E.g.,
what
if
the
casino
charges
$10,000?
•  Your
expected
winnings
are
sPll
∞
−
10,000
=
∞
What
if
the
casino
does
offer
this
game,
for
a
very
large
price?
•  Will
you
accept
any
amount
the
casino
wants
to
charge
in
order
to
play
this
game?
E.g.,
what
if
the
casino
charges
$10,000?
•  Your
expected
winnings
are
sPll
∞
−
10,000
=
∞
•  But
most
people
people
won’t
pay
more
than
$20‐30
to
play
this.
What
if
the
casino
does
offer
this
game,
for
a
very
large
price?
•  Will
you
accept
any
amount
the
casino
wants
to
charge
in
order
to
play
this
game?
E.g.,
what
if
the
casino
charges
$10,000?
•  Your
expected
winnings
are
sPll
∞
−
10,000
=
∞
•  But
most
people
people
won’t
pay
more
than
$20‐30
to
play
this.
•  Paradox:
the
disconnect
between
the
infinite
expected
profit
and
the
unwillingness
of
most
people
to
pay
much
for
this
game.
ExplanaPon
1:
Expected
UPlity
Hypothesis
(Gabriel
Cramer,
1728
and
Daniel
Bernoulli,
1738)
•  $1000
is
worth
more
to
a
person
whose
total
wealth
is
$1
than
to
a
person
whose
total
wealth
is
$1,000,000.
ExplanaPon
1:
Expected
UPlity
Hypothesis
(Gabriel
Cramer,
1728
and
Daniel
Bernoulli,
1738)
•  $1000
is
worth
more
to
a
person
whose
total
wealth
is
$1
than
to
a
person
whose
total
wealth
is
$1,000,000.
•  Diminishing
marginal
uPlity.
ExplanaPon
1:
Expected
UPlity
Hypothesis
(Gabriel
Cramer,
1728
and
Daniel
Bernoulli,
1738)
uPlity
of
wealth
•  $1000
is
worth
more
to
a
person
whose
total
wealth
is
$1
than
to
a
person
whose
total
wealth
is
$1,000,000.
•  Diminishing
marginal
uPlity.
•  Hence,
people’s
uPlity
funcPons
are
concave.
diminishing
marginal
uPlity:
each
addiPonal
dollar
is
less
valuable
than
the
previous
dollar
wealth
ExplanaPon
1:
Expected
UPlity
Hypothesis
(Gabriel
Cramer,
1728
and
Daniel
Bernoulli,
1738)
•  $1000
is
worth
more
to
a
person
whose
total
wealth
is
$1
than
to
a
person
whose
total
wealth
is
$1,000,000.
•  Diminishing
marginal
uPlity.
•  Hence,
people’s
uPlity
funcPons
are
concave.
uPlity
of
wealth
constant
marginal
uPlity
diminishing
marginal
uPlity:
each
addiPonal
dollar
is
less
valuable
than
the
previous
dollar
wealth
ExplanaPon
1:
Expected
UPlity
Hypothesis
(Gabriel
Cramer,
1728
and
Daniel
Bernoulli,
1738)
•  $1000
is
worth
more
to
a
person
whose
total
wealth
is
$1
than
to
a
person
whose
total
wealth
is
$1,000,000.
•  Diminishing
marginal
uPlity.
•  Hence,
people’s
uPlity
funcPons
are
concave.
uPlity
of
wealth
constant
marginal
uPlity
diminishing
marginal
uPlity:
each
addiPonal
dollar
is
less
valuable
than
the
previous
dollar
wealth
•  People
try
to
maximize
their
expected
uPlity.
ExplanaPon
1:
Expected
UPlity
Hypothesis
(Gabriel
Cramer,
1728
and
Daniel
Bernoulli,
1738)
•  $1000
is
worth
more
to
a
person
whose
total
wealth
is
$1
than
to
a
person
whose
total
wealth
is
$1,000,000.
•  Diminishing
marginal
uPlity.
•  Hence,
people’s
uPlity
funcPons
are
concave.
uPlity
of
wealth
constant
marginal
uPlity
diminishing
marginal
uPlity:
each
addiPonal
dollar
is
less
valuable
than
the
previous
dollar
wealth
•  People
try
to
maximize
their
expected
uPlity.
•  E.g.,
Bernoulli
modeled
people’s
uPlity
funcPon
as
a
log.
(Cramer’s
model
was
a
square
root.)
I.e.,
in
Bernoulli’s
model,
instead
of
E[wealth],
people
try
to
maximize
E[log
of
wealth].
ExpectaPon
of
log‐wealth
when
prior
wealth
is
zero
1/2
PMF
of
winnings
W
1/4
1/8
2
4
1/2
1
2
8
1/16
16
PMF
of
log2W
…
ExpectaPon
of
log‐wealth
when
prior
wealth
is
zero
1/2
PMF
of
winnings
W
1/4
1/8
2
4
1/2
1/4
1
2
8
1/16
16
PMF
of
log2W
…
ExpectaPon
of
log‐wealth
when
prior
wealth
is
zero
1/2
PMF
of
winnings
W
1/4
1/8
2
4
1/2
1/4
1/8
1/16
1
2
3
4
8
1/16
16
PMF
of
log2W
…
…
ExpectaPon
of
log‐wealth
when
prior
wealth
is
zero
1/2
PMF
of
winnings
W
1/4
1/8
2
4
1/2
1/4
1/8
1/16
8
1/16
…
16
PMF
of
log2W
…
1
2
3
4
log2W
is
a
geometric
random
variable
with
p=1/2
ExpectaPon
of
log‐wealth
when
prior
wealth
is
zero
1/2
PMF
of
winnings
W
1/4
1/8
2
4
8
1/2
1/4
1/8
1/16
1/16
…
16
PMF
of
log2W
…
1
2
3
4
log2W
is
a
geometric
random
variable
with
p=1/2
Hence,
E[log2W]
=
2
ExpectaPon
of
log‐wealth
when
prior
wealth
is
zero
1/2
PMF
of
winnings
W
1/4
1/8
2
4
8
1/2
1/4
1/8
1/16
1/16
…
16
PMF
of
log2W
…
1
2
3
4
log2W
is
a
geometric
random
variable
with
p=1/2
Hence,
E[log2W]
=
2
If
a
person’s
uPlity
funcPon
is
log2,
and
if
his
iniPal
wealth
is
zero,
he
will
derive
the
same
uPlity
from
playing
this
game
as
from
being
paid
$4,
because
log24
=
2.
What
if
prior
wealth
is
not
zero?
•  If
the
prior
wealth
is
not
zero,
we
can
ask
what
amount
a
person
would
be
willing
to
pay
for
playing
the
game.
What
if
prior
wealth
is
not
zero?
•  If
the
prior
wealth
is
not
zero,
we
can
ask
what
amount
a
person
would
be
willing
to
pay
for
playing
the
game.
•  For
prior
wealth
w
and
fee
f,
the
expected
uPlity
of
the
wealth
aler
the
game
will
be
E[log2(w
+
W
–
f)].
What
if
prior
wealth
is
not
zero?
•  If
the
prior
wealth
is
not
zero,
we
can
ask
what
amount
a
person
would
be
willing
to
pay
for
playing
the
game.
•  For
prior
wealth
w
and
fee
f,
the
expected
uPlity
of
the
wealth
aler
the
game
will
be
E[log2(w
+
W
–
f)].
–  This
expectaPon
is
finite.
–  A
person
with
iniPal
wealth
w
will
be
willing
to
pay
any
fee
f
that
produces
expected
uPlity
larger
than
log2w.
ExplanaPon
2:
What
about
risk?
•  Since
the
expected
profit
is
infinite,
the
variance
of
the
profit
is
undefined.
ExplanaPon
2:
What
about
risk?
•  Since
the
expected
profit
is
infinite,
the
variance
of
the
profit
is
undefined.
•  Need
some
other
way
of
reasoning
about
the
risk.
ExplanaPon
2:
What
about
risk?
•  Since
the
expected
profit
is
infinite,
the
variance
of
the
profit
is
undefined.
•  Need
some
other
way
of
reasoning
about
the
risk.
•  Note
that
the
E[profit]
is
infinite
only
because
of
humongous
profits
associated
with
extremely
unlikely
events.
ExplanaPon
2:
What
about
risk?
•  Since
the
expected
profit
is
infinite,
the
variance
of
the
profit
is
undefined.
•  Need
some
other
way
of
reasoning
about
the
risk.
•  Note
that
the
E[profit]
is
infinite
only
because
of
humongous
profits
associated
with
extremely
unlikely
events.
•  Suppose
the
casino
charges
$10,000.
Then
–  P(win)
=
P(1st
H
on
14th
toss
or
later)
=
1/214
+
1/215
+
…
=
1/214(1+1/2
+
…
)
=
1/213
≈
0.0001
‐‐‐
i.e.,
you
would
expect
to
lose
in
9999
out
of
every
10,000
games!
ExplanaPon
2:
What
about
risk?
•  Since
the
expected
profit
is
infinite,
the
variance
of
the
profit
is
undefined.
•  Need
some
other
way
of
reasoning
about
the
risk.
•  Note
that
the
E[profit]
is
infinite
only
because
of
humongous
profits
associated
with
extremely
unlikely
events.
•  Suppose
the
casino
charges
$10,000.
Then
–  P(win)
=
P(1st
H
on
14th
toss
or
later)
=
1/214
+
1/215
+
…
=
1/214(1+1/2
+
…
)
=
1/213
≈
0.0001
‐‐‐
i.e.,
you
would
expect
to
lose
in
9999
out
of
every
10,000
games!
–  The
most
likely
outcome
is
a
single
toss,
which
happens
with
probability
1/2
and
leads
to
a
loss
of
$9,998.
More
on
the
risk
•  What
is
the
expected
number
of
tosses
per
game?
More
on
the
risk
•  What
is
the
expected
number
of
tosses
per
game?
•  The
number
of
tosses
is
geometric
with
parameter
p=1/2.
More
on
the
risk
•  What
is
the
expected
number
of
tosses
per
game?
•  The
number
of
tosses
is
geometric
with
parameter
p=1/2.
•  Hence,
its
expected
value
is
2.
More
on
the
risk
•  What
is
the
expected
number
of
tosses
per
game?
•  The
number
of
tosses
is
geometric
with
parameter
p=1/2.
•  Hence,
its
expected
value
is
2.
•  Thus,
in
an
“average”
game,
you
will
win
$4
minus
the
fee.
Any
other
risks
for
the
player?
•  Note
that
the
infinite
expectaPon
is
conPngent
upon
nonzero
probabiliPes
of
arbitrarily
large
payoffs.
Any
other
risks
for
the
player?
•  Note
that
the
infinite
expectaPon
is
conPngent
upon
nonzero
probabiliPes
of
arbitrarily
large
payoffs.
•  But,
if
the
first
tail
comes
on
the
40th
toss,
will
the
casino
actually
be
able
to
pay
you
one
trillion
dollars?
Any
other
risks
for
the
player?
•  Note
that
the
infinite
expectaPon
is
conPngent
upon
the
nonzero
probabiliPes
of
arbitrarily
large
payoffs.
•  But,
if
the
first
tail
comes
on
the
40th
toss,
will
the
casino
actually
be
able
to
pay
you
one
trillion
dollars?
•  Suppose
the
original
condiPons
only
apply
to
the
first
30
tosses.
If
n>30
then
you
only
win
$230.
Any
other
risks
for
the
player?
•  Note
that
the
infinite
expectaPon
is
conPngent
upon
the
nonzero
probabiliPes
of
arbitrarily
large
payoffs.
•  But,
if
the
first
tail
comes
on
the
40th
toss,
will
the
casino
actually
be
able
to
pay
you
one
trillion
dollars?
•  Suppose
the
original
condiPons
only
apply
to
the
first
30
tosses.
If
n>30
then
you
only
win
$230.
•  Then
the
expected
winnings
are
∞
 1 n n
 1  n 30
∑ 2  ⋅ 2 + ∑ 2  ⋅ 2 = 31
n=1
n= 31
30
€
Any
other
risks
for
the
player?
•  Note
that
the
infinite
expectaPon
is
conPngent
upon
the
nonzero
probabiliPes
of
arbitrarily
large
payoffs.
•  But,
if
the
first
tail
comes
on
the
40th
toss,
will
the
casino
actually
be
able
to
pay
you
one
trillion
dollars?
•  Suppose
the
original
condiPons
only
apply
to
the
first
30
tosses.
If
n>30
then
you
only
win
$230.
•  Then
the
expected
winnings
are
∞
 1 n n
 1  n 30
∑ 2  ⋅ 2 + ∑ 2  ⋅ 2 = 31
n=1
n= 31
30
•  The
same
result
is
obtained
by
assuming
that,
beyond
some
very
large
number
($230
in
our
example),
it
makes
no
difference
to
€
people
what
the
winnings
are‐‐‐i.e.,
by
assuming
that
the
uPlity
funcPon
is
perfectly
flat
for
large
winnings
(G.
Cramer,
1728.)