The effect of information constraints on
decision-making and economic behaviour
Roman V. Belavkin
Middlesex University
April 16, 2010
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
1 / 33
Introduction: Choice under Uncertainty
Paradoxes of Expected Utility
Optimisation of Information Utility
Results
Example: SPB Lottery
References106
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
2 / 33
Introduction: Choice under Uncertainty
Introduction: Choice under Uncertainty
Paradoxes of Expected Utility
Optimisation of Information Utility
Results
Example: SPB Lottery
References106
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
3 / 33
Introduction: Choice under Uncertainty
Learning Systems
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
4 / 33
Introduction: Choice under Uncertainty
Learning Systems
Performance
O
v
t
r
p
t
n
m
k
/
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
4 / 33
Introduction: Choice under Uncertainty
Learning Systems
Performance
O
v
t
r
p
t
n
m
Information
O
k
/
k
m
o
q
t
s
u
w
/
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
4 / 33
Introduction: Choice under Uncertainty
Learning Systems
Performance
O
v
t
r
p
t
n
m
Information
O
k
/
k
m
o
q
t
s
u
w
/
Performance and information have orders, and the relation between
them is monotonic.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
4 / 33
Introduction: Choice under Uncertainty
Learning Systems
Performance
O
v
t
r
p
t
n
m
Information
O
k
/
k
m
o
q
t
s
u
w
/
Performance and information have orders, and the relation between
them is monotonic.
Complete partial orders, domain theory.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
4 / 33
Introduction: Choice under Uncertainty
Learning Systems
Performance
O
v
t
r
p
t
n
m
Information
O
k
/
k
m
o
q
t
s
u
w
/
Performance and information have orders, and the relation between
them is monotonic.
Complete partial orders, domain theory.
Utility theory, information theory
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
4 / 33
Introduction: Choice under Uncertainty
Learning Systems
Performance
O
v
t
r
p
n
m
Information
O
k
/
t
k
m
o
q
t
s
u
w
/
Performance and information have orders, and the relation between
them is monotonic.
Complete partial orders, domain theory.
Utility theory, information theory
Allows for treating both deterministic and non-deterministic case:
x = f (ω) ,
x = f (ω) + rand()
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
4 / 33
Introduction: Choice under Uncertainty
Expected Utility Theory
f : Ω → R a utility function.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
5 / 33
Introduction: Choice under Uncertainty
Expected Utility Theory
f : Ω → R a utility function.
p : R → [0, 1] a probability measure on (Ω, R).
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
5 / 33
Introduction: Choice under Uncertainty
Expected Utility Theory
f : Ω → R a utility function.
p : R → [0, 1] a probability measure on (Ω, R).
The expected utility
Ep {x} :=
X
x(ω)p(ω)
ω∈Ω
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
5 / 33
Introduction: Choice under Uncertainty
Expected Utility Theory
f : Ω → R a utility function.
p : R → [0, 1] a probability measure on (Ω, R).
The expected utility
Z
Ep {x} :=
x(ω) dp(ω)
Ω
Choice under uncertainty
q.p
⇐⇒
Eq {f } ≤ Ep {f }
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
5 / 33
Introduction: Choice under Uncertainty
Expected Utility Theory
f : Ω → R a utility function.
p : R → [0, 1] a probability measure on (Ω, R).
The expected utility
Z
Ep {x} :=
x(ω) dp(ω)
Ω
Choice under uncertainty
q.p
⇐⇒
Eq {f } ≤ Ep {f }
Question (Why expected utility?)
1
Ey {f } = f (ω) if y(Ω0 ) = δω (Ω0 ).
2
x.y
⇐⇒
λx . λy, ∀ λ > 0
3
x.y
⇐⇒
x + z . y + z, ∀ z ∈ Y .
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
5 / 33
Paradoxes of Expected Utility
Introduction: Choice under Uncertainty
Paradoxes of Expected Utility
Optimisation of Information Utility
Results
Example: SPB Lottery
References106
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
6 / 33
Paradoxes of Expected Utility
St. Petersburg lottery
Due to Nicolas Bernoulli (1713)
The lottery is played by tossing a fair coin repeatedly until the first
head appears.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
7 / 33
Paradoxes of Expected Utility
St. Petersburg lottery
Due to Nicolas Bernoulli (1713)
The lottery is played by tossing a fair coin repeatedly until the first
head appears.
If the head appeared on nth toss, then you win £2n .
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
7 / 33
Paradoxes of Expected Utility
St. Petersburg lottery
Due to Nicolas Bernoulli (1713)
The lottery is played by tossing a fair coin repeatedly until the first
head appears.
If the head appeared on nth toss, then you win £2n .
For example
n=3
£23 = £8
n=4
£24 = £16
···
n = 20
£220 = £1, 048, 576
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
7 / 33
Paradoxes of Expected Utility
St. Petersburg lottery
Due to Nicolas Bernoulli (1713)
The lottery is played by tossing a fair coin repeatedly until the first
head appears.
If the head appeared on nth toss, then you win £2n .
For example
n=3
£23 = £8
n=4
£24 = £16
···
n = 20
£220 = £1, 048, 576
To enter the lottery, you must pay a fee of £X
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
7 / 33
Paradoxes of Expected Utility
St. Petersburg lottery
Due to Nicolas Bernoulli (1713)
The lottery is played by tossing a fair coin repeatedly until the first
head appears.
If the head appeared on nth toss, then you win £2n .
For example
n=3
£23 = £8
n=4
£24 = £16
···
n = 20
£220 = £1, 048, 576
To enter the lottery, you must pay a fee of £X
How much is £X?
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
7 / 33
Paradoxes of Expected Utility
Why is it a paradox?
We don’t want to pay more than expect to win:
X ≤ E{win}
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
8 / 33
Paradoxes of Expected Utility
Why is it a paradox?
We don’t want to pay more than expect to win:
X ≤ E{win}
How large is Ep {win}?
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
8 / 33
Paradoxes of Expected Utility
Why is it a paradox?
We don’t want to pay more than expect to win:
X ≤ E{win}
How large is Ep {win}?
Let p(n) be the probability of n ∈ N
head 1
2
3
4
···
win £2 £4 £8 £16 · · ·
1
1
1
p(n) 12
···
4
8
16
n ···
2n · · ·
1
2n · · ·
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
8 / 33
Paradoxes of Expected Utility
Why is it a paradox?
We don’t want to pay more than expect to win:
X ≤ E{win}
How large is Ep {win}?
Let p(n) be the probability of n ∈ N
head 1
2
3
4
···
win £2 £4 £8 £16 · · ·
1
1
1
p(n) 12
···
4
8
16
n ···
2n · · ·
1
2n · · ·
It is easy to see that
∞
Ep {win} = 2 ·
X 2n
1
1
1
1
+ 4 · + 8 · + 16 ·
+ ··· =
=∞
2
4
8
16
2n
n=1
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
8 / 33
Paradoxes of Expected Utility
Why is it a paradox?
We don’t want to pay more than expect to win:
X ≤ E{win}
How large is Ep {win}?
Let p(n) be the probability of n ∈ N
head 1
2
3
4
···
win £2 £4 £8 £16 · · ·
1
1
1
p(n) 12
···
4
8
16
n ···
2n · · ·
1
2n · · ·
It is easy to see that
∞
Ep {win} = 2 ·
X 2n
1
1
1
1
+ 4 · + 8 · + 16 ·
+ ··· =
=∞
2
4
8
16
2n
n=1
One cannot buy what is not for sale.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
8 / 33
Paradoxes of Expected Utility
Classical solutions
Daniel Bernoulli (1738) proposed f (n) = log2 2n = n.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
9 / 33
Paradoxes of Expected Utility
Classical solutions
Daniel Bernoulli (1738) proposed f (n) = log2 2n = n.
Note that for any f (n) we can introduce a lottery p(n) ∝ f −1 (n):
Ep {f } ∝
∞
X
f (n)
n=1
f (n)
=∞
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
9 / 33
Paradoxes of Expected Utility
Classical solutions
Daniel Bernoulli (1738) proposed f (n) = log2 2n = n.
Note that for any f (n) we can introduce a lottery p(n) ∝ f −1 (n):
Ep {f } ∝
∞
X
f (n)
n=1
f (n)
=∞
Some suggest to use only f such that
kf k∞ := sup |f (ω)| < ∞
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic
April 16, behaviour
2010
9 / 33
Paradoxes of Expected Utility
Northern Rock lottery
Due to unknown author (2008)
You can borrow a mortgage of any amount £X
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour10 / 33
Paradoxes of Expected Utility
Northern Rock lottery
Due to unknown author (2008)
You can borrow a mortgage of any amount £X
The amount you repay is decided by tossing a fair coin repeatedly
until the first head appears.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour10 / 33
Paradoxes of Expected Utility
Northern Rock lottery
Due to unknown author (2008)
You can borrow a mortgage of any amount £X
The amount you repay is decided by tossing a fair coin repeatedly
until the first head appears.
If the head appeared on nth toss, then you repay £2n .
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour10 / 33
Paradoxes of Expected Utility
Northern Rock lottery
Due to unknown author (2008)
You can borrow a mortgage of any amount £X
The amount you repay is decided by tossing a fair coin repeatedly
until the first head appears.
If the head appeared on nth toss, then you repay £2n .
For example
n=3
£23 = £8
n=4
£24 = £16
···
n = 20
£220 = £1, 048, 576
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour10 / 33
Paradoxes of Expected Utility
Northern Rock lottery
Due to unknown author (2008)
You can borrow a mortgage of any amount £X
The amount you repay is decided by tossing a fair coin repeatedly
until the first head appears.
If the head appeared on nth toss, then you repay £2n .
For example
n=3
£23 = £8
n=4
£24 = £16
···
n = 20
£220 = £1, 048, 576
How much would you borrow? (£X =?)
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour10 / 33
Paradoxes of Expected Utility
The Allais (1953) paradox
Consider two lotteries:
A : p(£300) =
1
3
(and p(£0) = 23 )
B : p(£100) = 1
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour11 / 33
Paradoxes of Expected Utility
The Allais (1953) paradox
Consider two lotteries:
A : p(£300) =
1
3
(and p(£0) = 23 )
B : p(£100) = 1
Most of the people seem to prefer A . B
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour11 / 33
Paradoxes of Expected Utility
The Allais (1953) paradox
Consider two lotteries:
A : p(£300) =
1
3
(and p(£0) = 23 )
B : p(£100) = 1
Most of the people seem to prefer A . B
Note that
1
2
+ 100 · 0 + 0 · = 100
3
3
EB {x} = 300 · 0 + 100 · 1 + 0 · 0 = 100
EA {x} = 300 ·
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour11 / 33
Paradoxes of Expected Utility
The Allais (1953) paradox
Consider two lotteries:
A : p(£300) =
1
3
(and p(£0) = 23 )
B : p(£100) = 1
Most of the people seem to prefer A . B
Note that
1
2
+ 100 · 0 + 0 · = 100
3
3
EB {x} = 300 · 0 + 100 · 1 + 0 · 0 = 100
EA {x} = 300 ·
Remark
Safety is preferred (i.e. risk averse).
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour11 / 33
Paradoxes of Expected Utility
The Allais (1953) paradox (2)
Consider two lotteries:
C : p(−£300) =
1
3
(and p(£0) = 32 )
D : p(−£100) = 1
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour12 / 33
Paradoxes of Expected Utility
The Allais (1953) paradox (2)
Consider two lotteries:
C : p(−£300) =
1
3
(and p(£0) = 32 )
D : p(−£100) = 1
Most of the people seem to prefer C & D
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour12 / 33
Paradoxes of Expected Utility
The Allais (1953) paradox (2)
Consider two lotteries:
C : p(−£300) =
1
3
(and p(£0) = 32 )
D : p(−£100) = 1
Most of the people seem to prefer C & D
Note that
1
2
− 100 · 0 − 0 · = −100
3
3
ED {x} = −300 · 0 − 100 · 1 − 0 · 0 = −100
EC {x} = −300 ·
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour12 / 33
Paradoxes of Expected Utility
The Allais (1953) paradox (2)
Consider two lotteries:
C : p(−£300) =
1
3
(and p(£0) = 32 )
D : p(−£100) = 1
Most of the people seem to prefer C & D
Note that
1
2
− 100 · 0 − 0 · = −100
3
3
ED {x} = −300 · 0 − 100 · 1 − 0 · 0 = −100
EC {x} = −300 ·
Remark
Risk is preferred (i.e. risk taking).
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour12 / 33
Paradoxes of Expected Utility
Why is it a paradox?
1
U = {0, 1, 2}
E{u} =
P
i
Pi Ui = const
Increasing
utility
P3
E{u} = 21
0
P1
1
Remark
Any linear functional (e.g. Ep {x}) has parallel level sets. If people use
expected utility to make choices, then they are either risk-averse or
risk-taking, but not both.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour13 / 33
Paradoxes of Expected Utility
Prospect theory
Due to Tversky and Kahneman (1981)
It was proposed that the utility is convex, when the choice is among
gains, and concave when the choice is among losses.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour14 / 33
Paradoxes of Expected Utility
Prospect theory
Due to Tversky and Kahneman (1981)
It was proposed that the utility is convex, when the choice is among
gains, and concave when the choice is among losses.
This would make the choice risk averse for gains and risk taking for
losses.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour14 / 33
Paradoxes of Expected Utility
Prospect theory
Due to Tversky and Kahneman (1981)
It was proposed that the utility is convex, when the choice is among
gains, and concave when the choice is among losses.
This would make the choice risk averse for gains and risk taking for
losses.
Remark
This theory is not normative (i.e. it is not derived using rational approach).
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour14 / 33
Paradoxes of Expected Utility
The Ellsberg (1961) paradox
Consider two lotteries:
A : p(£100) =
1
2
(and p(£0) = 12 )
B : p(£100) = unknown
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour15 / 33
Paradoxes of Expected Utility
The Ellsberg (1961) paradox
Consider two lotteries:
A : p(£100) =
1
2
(and p(£0) = 12 )
B : p(£100) = unknown
Most of the people seem to prefer A & B
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour15 / 33
Paradoxes of Expected Utility
The Ellsberg (1961) paradox
Consider two lotteries:
A : p(£100) =
1
2
(and p(£0) = 12 )
B : p(£100) = unknown
Most of the people seem to prefer A & B
Note that
1
1
EA {x} = 100 · + 0 · = 50
2
2
Z 1
EB {x} =
(100 · p + 0 · (1 − p)) dp = 50
0
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour15 / 33
Paradoxes of Expected Utility
The Ellsberg (1961) paradox
Consider two lotteries:
A : p(£100) =
1
2
(and p(£0) = 12 )
B : p(£100) = unknown
Most of the people seem to prefer A & B
Note that
1
1
EA {x} = 100 · + 0 · = 50
2
2
Z 1
EB {x} =
(100 · p + 0 · (1 − p)) dp = 50
0
Remark
More information is preferred.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour15 / 33
Optimisation of Information Utility
Introduction: Choice under Uncertainty
Paradoxes of Expected Utility
Optimisation of Information Utility
Results
Example: SPB Lottery
References106
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour16 / 33
Optimisation of Information Utility
Extreme Value Problems
Unconditional extremum
Maximise f (y):
sup f (y)
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour17 / 33
Optimisation of Information Utility
Extreme Value Problems
Unconditional extremum
Maximise f (y):
sup f (y)
Necessary condition ∂f (ȳ) 3 0.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour17 / 33
Optimisation of Information Utility
Extreme Value Problems
Unconditional extremum
Maximise f (y):
sup f (y)
Necessary condition ∂f (ȳ) 3 0.
Sufficient, if f is concave.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour17 / 33
Optimisation of Information Utility
Extreme Value Problems
Unconditional extremum
Maximise f (y):
sup f (y)
Necessary condition ∂f (ȳ) 3 0.
Sufficient, if f is concave.
Conditional extremum
Maximise f (y) subject to g(y) ≤ λ:
f (λ) := sup{f (y) : g(y) ≤ λ}
.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour17 / 33
Optimisation of Information Utility
Extreme Value Problems
Unconditional extremum
Maximise f (y):
sup f (y)
Necessary condition ∂f (ȳ) 3 0.
Sufficient, if f is concave.
Conditional extremum
Maximise f (y) subject to g(y) ≤ λ:
f (λ) := sup{f (y) : g(y) ≤ λ}
.
Necessary condition ∂f (ȳ) − α∂g(y) 3 0.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour17 / 33
Optimisation of Information Utility
Extreme Value Problems
Unconditional extremum
Maximise f (y):
sup f (y)
Necessary condition ∂f (ȳ) 3 0.
Sufficient, if f is concave.
Conditional extremum
Maximise f (y) subject to g(y) ≤ λ:
f (λ) := sup{f (y) : g(y) ≤ λ}
.
Necessary condition ∂f (ȳ) − α∂g(y) 3 0.
Sufficient if K(y, α) := f (y) + α[λ − g(y)] is concave.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour17 / 33
Optimisation of Information Utility
Representation in Paired Spaces
x ∈ X, y ∈ Y , h·, ·i : X × Y → R
Z
hx, yi :=
x(ω) dy(ω)
Ω
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour18 / 33
Optimisation of Information Utility
Representation in Paired Spaces
x ∈ X, y ∈ Y , h·, ·i : X × Y → R
Z
hx, yi :=
x(ω) dy(ω)
Ω
Separation:
hx, yi = 0 , ∀ x ∈ X ⇒ y = 0 ∈ Y
hx, yi = 0 , ∀ y ∈ Y
⇒ x=0∈X
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour18 / 33
Optimisation of Information Utility
Representation in Paired Spaces
x ∈ X, y ∈ Y , h·, ·i : X × Y → R
Z
hx, yi :=
x(ω) dy(ω)
Ω
Separation:
hx, yi = 0 , ∀ x ∈ X ⇒ y = 0 ∈ Y
hx, yi = 0 , ∀ y ∈ Y
⇒ x=0∈X
Can equip X and Y with kxk∞ and kyk1 .
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour18 / 33
Optimisation of Information Utility
Representation in Paired Spaces
x ∈ X, y ∈ Y , h·, ·i : X × Y → R
Z
hx, yi :=
x(ω) dy(ω)
Ω
Separation:
hx, yi = 0 , ∀ x ∈ X ⇒ y = 0 ∈ Y
hx, yi = 0 , ∀ y ∈ Y
⇒ x=0∈X
Can equip X and Y with kxk∞ and kyk1 .
Statistical manifold:
M := {y ∈ Y : y ≥ 0 , kyk1 = 1}
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour18 / 33
Optimisation of Information Utility
Representation in Paired Spaces
x ∈ X, y ∈ Y , h·, ·i : X × Y → R
Z
hx, yi :=
x(ω) dy(ω)
Ω
Separation:
hx, yi = 0 , ∀ x ∈ X ⇒ y = 0 ∈ Y
hx, yi = 0 , ∀ y ∈ Y
⇒ x=0∈X
Can equip X and Y with kxk∞ and kyk1 .
Statistical manifold:
M := {y ∈ Y : y ≥ 0 , kyk1 = 1}
Expected value
Ep {x} = hx, yi|M
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour18 / 33
Optimisation of Information Utility
Information
Definition (Information resource)
a closed functional F : Y → R ∪ {∞} with inf F = F (z).
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour19 / 33
Optimisation of Information Utility
Information
Definition (Information resource)
a closed functional F : Y → R ∪ {∞} with inf F = F (z).
Example (Relative Information (Belavkin, 2010b))
For z > 0, let
E
 D
y

 ln z , y − h1, y − zi , if y > 0
F (y) :=
h1, zi ,
if y = 0


∞,
if y < 0
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour19 / 33
Optimisation of Information Utility
Information
Definition (Information resource)
a closed functional F : Y → R ∪ {∞} with inf F = F (z).
Example (Relative Information (Belavkin, 2010b))
For z > 0, let
E
 D
y

 ln z , y − h1, y − zi , if y > 0
F (y) :=
h1, zi ,
if y = 0


∞,
if y < 0
∂F (y) = ln yz = x
⇐⇒
y = ex z = ∂F ∗ (x)
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour19 / 33
Optimisation of Information Utility
Information
Definition (Information resource)
a closed functional F : Y → R ∪ {∞} with inf F = F (z).
Example (Relative Information (Belavkin, 2010b))
For z > 0, let
E
 D
y

 ln z , y − h1, y − zi , if y > 0
F (y) :=
h1, zi ,
if y = 0


∞,
if y < 0
∂F (y) = ln yz = x
⇐⇒
y = ex z = ∂F ∗ (x)
The dual F ∗ : X → R ∪ {∞} is
F ∗ (x) := h1, ex zi
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour19 / 33
Optimisation of Information Utility
Utility of Information
If x ∈ X is utility, then the value of event y relative to z is
hx, y − zi = Ey {x} − Ez {x}
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour20 / 33
Optimisation of Information Utility
Utility of Information
If x ∈ X is utility, then the value of event y relative to z is
hx, y − zi = Ey {x} − Ez {x}
Definition (Utility of information)
Ux (I) := sup{hx, yi : F (y) ≤ I}
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour20 / 33
Optimisation of Information Utility
Utility of Information
If x ∈ X is utility, then the value of event y relative to z is
hx, y − zi = Ey {x} − Ez {x}
Definition (Utility of information)
Ux (I) := sup{hx, yi : F (y) ≤ I}
Stratonovich (1965) defined Ux (I) for Shannon information.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour20 / 33
Optimisation of Information Utility
Utility of Information
If x ∈ X is utility, then the value of event y relative to z is
hx, y − zi = Ey {x} − Ez {x}
Definition (Utility of information)
Ux (I) := sup{hx, yi : F (y) ≤ I}
Stratonovich (1965) defined Ux (I) for Shannon information.
Related functions
−U−x (I) := inf{hx, yi : F (y) ≤ I}
Ix (U ) := inf{F (y) : U0 ≤ U ≤ hx, yi}
Ix (U ) := inf{F (y) : hx, yi ≤ U < U0 }
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour20 / 33
Results
Introduction: Choice under Uncertainty
Paradoxes of Expected Utility
Optimisation of Information Utility
Results
Example: SPB Lottery
References106
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour21 / 33
Results
Information Bounded Utility
Definition (Information Bounded Utility)
A function f : Ω → R that admits a solution to the utility of information
problem Uf (I) for I ∈ (inf F, sup F )
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour22 / 33
Results
Information Bounded Utility
Definition (Information Bounded Utility)
A function f : Ω → R that admits a solution to the utility of information
problem Uf (I) for I ∈ (inf F, sup F )
Theorem
A solution to Uf (I) and If (U ) exists if and only if set {x : Fq∗ (x) ≤ I ∗ }
absorbs function f :
∃ β −1 > 0 : Fq∗ (βf ) < ∞
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour22 / 33
Results
Information Bounded Utility
Definition (Information Bounded Utility)
A function f : Ω → R that admits a solution to the utility of information
problem Uf (I) for I ∈ (inf F, sup F )
Theorem
A solution to Uf (I) and If (U ) exists if and only if set {x : Fq∗ (x) ≤ I ∗ }
absorbs function f :
∃ β −1 > 0 : Fq∗ (βf ) < ∞
Remark (Separation of information)
For all I ∈ (inf F, sup F ) there exist β1−1 , β2−1 > 0:
Fq (∂Fq∗ (β1 f )) < I < Fq (∂F ∗ (β2 f ))
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour22 / 33
Results
Information Topology (Belavkin, 2010a)
1
The topology is defined using an
information resource:
inf F = F (y0 )
P3
F : L → R∪{∞} ,
y0
0
1
P1
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour23 / 33
Results
Information Topology (Belavkin, 2010a)
1
The topology is defined using an
information resource:
F : L → R∪{∞} ,
inf F = F (y0 )
P3
Neighbourhoods of y0 :
C := {y : F (y) ≤ I}
y0
0
1
P1
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour23 / 33
Results
Information Topology (Belavkin, 2010a)
1
The topology is defined using an
information resource:
F : L → R∪{∞} ,
inf F = F (y0 )
P3
Neighbourhoods of y0 :
C := {y : F (y) ≤ I}
y0
The topology of information
bounded functions:
0
1
C ◦ := {x : F ∗ (x) ≤ I ∗ }
P1
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour23 / 33
Results
Information Topology (Belavkin, 2010a)
1
The topology is defined using an
information resource:
F : L → R∪{∞} ,
inf F = F (y0 )
P3
Neighbourhoods of y0 :
C := {y : F (y) ≤ I}
y0
The topology of information
bounded functions:
0
1
C ◦ := {x : F ∗ (x) ≤ I ∗ }
P1
Generally, f ∈ C ◦ does not imply
−f ∈ C ◦ .
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour23 / 33
Results
Information Topology (Belavkin, 2010a)
1
The topology is defined using an
information resource:
F : L → R∪{∞} ,
inf F = F (y0 )
P3
Neighbourhoods of y0 :
C := {y : F (y) ≤ I}
y0
The topology of information
bounded functions:
0
1
C ◦ := {x : F ∗ (x) ≤ I ∗ }
P1
Generally, f ∈ C ◦ does not imply
−f ∈ C ◦ .
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour23 / 33
Results
Information Topology (Belavkin, 2010a)
1
The topology is defined using an
information resource:
F : L → R∪{∞} ,
inf F = F (y0 )
P3
Neighbourhoods of y0 :
C := {y : F (y) ≤ I}
y0
The topology of information
bounded functions:
0
1
C ◦ := {x : F ∗ (x) ≤ I ∗ }
P1
Generally, f ∈ C ◦ does not imply
−f ∈ C ◦ .
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour23 / 33
Results
Information Topology (Belavkin, 2010a)
1
The topology is defined using an
information resource:
F : L → R∪{∞} ,
inf F = F (y0 )
P3
Neighbourhoods of y0 :
C := {y : F (y) ≤ I}
y0
The topology of information
bounded functions:
0
1
C ◦ := {x : F ∗ (x) ≤ I ∗ }
P1
Generally, f ∈ C ◦ does not imply
−f ∈ C ◦ .
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour23 / 33
Results
Information Topology (Belavkin, 2010a)
1
The topology is defined using an
information resource:
F : L → R∪{∞} ,
inf F = F (y0 )
P3
Neighbourhoods of y0 :
C := {y : F (y) ≤ I}
y0
The topology of information
bounded functions:
0
1
C ◦ := {x : F ∗ (x) ≤ I ∗ }
P1
Generally, f ∈ C ◦ does not imply
−f ∈ C ◦ .
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour23 / 33
Results
Information Topology (Belavkin, 2010a)
1
The topology is defined using an
information resource:
F : L → R∪{∞} ,
inf F = F (y0 )
P3
Neighbourhoods of y0 :
C := {y : F (y) ≤ I}
y0
The topology of information
bounded functions:
0
1
C ◦ := {x : F ∗ (x) ≤ I ∗ }
P1
Generally, f ∈ C ◦ does not imply
−f ∈ C ◦ .
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour23 / 33
Results
Information Topology (Belavkin, 2010a)
1
The topology is defined using an
information resource:
F : L → R∪{∞} ,
inf F = F (y0 )
P3
Neighbourhoods of y0 :
C := {y : F (y) ≤ I}
y0
The topology of information
bounded functions:
0
1
C ◦ := {x : F ∗ (x) ≤ I ∗ }
P1
Generally, f ∈ C ◦ does not imply
−f ∈ C ◦ .
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour23 / 33
Results
Information Topology (Belavkin, 2010a)
1
The topology is defined using an
information resource:
F : L → R∪{∞} ,
inf F = F (y0 )
P3
Neighbourhoods of y0 :
yn
C := {y : F (y) ≤ I}
y0
The topology of information
bounded functions:
0
1
C ◦ := {x : F ∗ (x) ≤ I ∗ }
P1
Generally, f ∈ C ◦ does not imply
−f ∈ C ◦ .
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour23 / 33
Results
Information Topology (Belavkin, 2010a)
1
The topology is defined using an
information resource:
F : L → R∪{∞} ,
inf F = F (y0 )
P3
Neighbourhoods of y0 :
yn
C := {y : F (y) ≤ I}
y0
The topology of information
bounded functions:
0
1
C ◦ := {x : F ∗ (x) ≤ I ∗ }
P1
Generally, f ∈ C ◦ does not imply
−f ∈ C ◦ .
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour23 / 33
Results
Parametrisation by the Expected Utility
hx, ŷi = U (β)
1
0
Binary set
Uncountable set
−4
−3
−2
−1
0
1
2
3
4
−1
β
Let F (y) be negative entropy (i.e. F (y) is minimised at y0 (ω) = const)
x : Ω → {c − d, c + d}
U (β) = Ψ0 (β) = c + d tanh(β d)
x : Ω → [c − d, c + d]
U (β) = Ψ0 (β) = c + d coth(β d) − β −1
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour24 / 33
Results
Parametrisation by Information
F (ŷ) = I(β)
2
Binary set
Uncountable set
1
−4
−3
−2
−1
0
1
2
3
4
0
β
I = Φ0 (β −1 ) ,
I = β Ψ0 (β) − Ψ(β)
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour25 / 33
Results
F (ȳ) = I
Parametric Dependency
Binary set
Uncountable set
1
−1
−0.5
0
0.5
1
0
hx, ȳi = U
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour26 / 33
Example: SPB Lottery
Introduction: Choice under Uncertainty
Paradoxes of Expected Utility
Optimisation of Information Utility
Results
Example: SPB Lottery
References106
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour27 / 33
Example: SPB Lottery
A Solution of the SPB Paradox
f : N → R is information bounded iff for some β −1 > 0:
∗
F (βf ) =
∞
X
q(n)eβf (n) < ∞
n=1
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour28 / 33
Example: SPB Lottery
A Solution of the SPB Paradox
f : N → R is information bounded iff for some β −1 > 0:
∗
F (βf ) =
∞
X
q(n)eβf (n) < ∞
n=1
Using
P∞
n=1
rn
<∞
∃ β −1 > 0 :
βf (n) < − ln z(n) ,
∀n ∈ N
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour28 / 33
Example: SPB Lottery
A Solution of the SPB Paradox
f : N → R is information bounded iff for some β −1 > 0:
∗
F (βf ) =
∞
X
q(n)eβf (n) < ∞
n=1
Using
P∞
n=1
rn
<∞
∃ β −1 > 0 :
P
where z(n) = q(n) q(n).
βf (n) < − ln z(n) ,
∀n ∈ N
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour28 / 33
Example: SPB Lottery
A Solution of the SPB Paradox
f : N → R is information bounded iff for some β −1 > 0:
∗
F (βf ) =
∞
X
q(n)eβf (n) < ∞
n=1
Using
P∞
n=1
rn
<∞
∃ β −1 > 0 :
P
where z(n) = q(n) q(n).
βf (n) < − ln z(n) ,
∀n ∈ N
Let q(n) = (e − 1)e−n (i.e. 2−n ).
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour28 / 33
Example: SPB Lottery
A Solution of the SPB Paradox
f : N → R is information bounded iff for some β −1 > 0:
∗
F (βf ) = (e − 1)
∞
X
eβf (n)−n < ∞
n=1
Using
P∞
n=1
rn
<∞
∃ β −1 > 0 :
P
where z(n) = q(n) q(n).
βf (n) < − ln z(n) ,
∀n ∈ N
Let q(n) = (e − 1)e−n (i.e. 2−n ).
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour28 / 33
Example: SPB Lottery
A Solution of the SPB Paradox
f : N → R is information bounded iff for some β −1 > 0:
∗
F (βf ) = (e − 1)
∞
X
eβf (n)−n < ∞
n=1
Using
P∞
n=1
rn
<∞
∃ β −1 > 0 :
P
where z(n) = q(n) q(n).
βf (n) < n ,
∀n ∈ N
Let q(n) = (e − 1)e−n (i.e. 2−n ).
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour28 / 33
Example: SPB Lottery
A Solution of the SPB Paradox
f : N → R is information bounded iff for some β −1 > 0:
∗
F (βf ) = (e − 1)
∞
X
eβf (n)−n < ∞
n=1
Using
P∞
n=1
rn
<∞
∃ β −1 > 0 :
P
where z(n) = q(n) q(n).
βf (n) < n ,
∀n ∈ N
Let q(n) = (e − 1)e−n (i.e. 2−n ).
For f (n) = n, we have β < 1.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour28 / 33
Example: SPB Lottery
A Solution of the SPB Paradox
Using U = Ψ0f (β) obtain
U=
1
,
1 − eβ−1
U0 =
1
1 − e−1
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour29 / 33
Example: SPB Lottery
A Solution of the SPB Paradox
Using U = Ψ0f (β) obtain
1
1
,
U0 =
1 − e−1
1 − eβ−1
The inverse of function U (β) is β = 1 + ln(1 − U −1 ).
U=
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour29 / 33
Example: SPB Lottery
A Solution of the SPB Paradox
Using U = Ψ0f (β) obtain
1
1
,
U0 =
1 − e−1
1 − eβ−1
The inverse of function U (β) is β = 1 + ln(1 − U −1 ).
U=
Using I = β(ln Ψ(β))0 − ln Ψ(β):
If (U ) = (1 + ln(1 − U −1 ))U − ln(e − 1) − ln(U − 1)
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour29 / 33
Example: SPB Lottery
A Solution of the SPB Paradox
Using U = Ψ0f (β) obtain
1
1
,
U0 =
1 − e−1
1 − eβ−1
The inverse of function U (β) is β = 1 + ln(1 − U −1 ).
U=
Using I = β(ln Ψ(β))0 − ln Ψ(β):
If (U ) = (1 + ln(1 − U −1 ))U − ln(e − 1) − ln(U − 1)
Change e to 2 (ln to log2 ).
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour29 / 33
Example: SPB Lottery
A Solution of the SPB Paradox
Using U = Ψ0f (β) obtain
1
,
U0 = 2
1 − 2β−1
The inverse of function U (β) is β = 1 + ln(1 − U −1 ).
U=
Using I = β(ln Ψ(β))0 − ln Ψ(β):
If (U ) = (1 + log2 (1 − U −1 ))U − log2 (U − 1)
Change e to 2 (ln to log2 ).
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour29 / 33
Example: SPB Lottery
A Solution of the SPB Paradox
Using U = Ψ0f (β) obtain
1
,
U0 = 2
1 − 2β−1
The inverse of function U (β) is β = 1 + ln(1 − U −1 ).
U=
Using I = β(ln Ψ(β))0 − ln Ψ(β):
If (U ) = (1 + log2 (1 − U −1 ))U − log2 (U − 1)
Change e to 2 (ln to log2 ).
For the information amount of 0 bits, the optimal entrance fee is
c ≤ U0 = 2.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour29 / 33
Example: SPB Lottery
Expected utility, U = hf, p̄i, (n)
SPB Lottery
−5
2
1
−4
−3
−2
−1
0
1
Parameter, β
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour30 / 33
Example: SPB Lottery
SPB Lottery
Information, I = Fq (p̄), (bits)
2
−5
1
−4
−3
−2
Parameter, β
−1
0
1
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour31 / 33
Example: SPB Lottery
Information, I = Fq (p̄) (bits)
SPB Lottery
1
1
2
3
4
Expected utility, U = hf, p̄i, (n)
5
6
0
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour32 / 33
References
Introduction: Choice under Uncertainty
Paradoxes of Expected Utility
Optimisation of Information Utility
Results
Example: SPB Lottery
References106
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and April
economic
16, 2010
behaviour33 / 33
Example: SPB Lottery
Allais, M. (1953). Le comportement de l’homme rationnel devant le
risque: Critique des postulats et axiomes de l’École americaine.
Econometrica, 21, 503–546.
Belavkin, R. V. (2010a). Information trajectory of optimal learning. In
M. J. Hirsch, P. M. Pardalos, & R. Murphey (Eds.), Dynamics of
information systems: Theory and applications (Vol. 40). Springer.
Belavkin, R. V. (2010b). Utility and value of information in cognitive
science, biology and quantum theory. In L. Accardi, W. Freudenberg,
& M. Ohya (Eds.), Quantum Bio-Informatics III (Vol. 26). World
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