the effect of information constraints on decision …...the effect of information constraints on...
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The effect of information constraints ondecision-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 behaviourApril 16, 2010 1 / 33
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Introduction: Choice under Uncertainty
Paradoxes of Expected Utility
Optimisation of Information Utility
Results
Example: SPB LotteryReferences106
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic behaviourApril 16, 2010 2 / 33
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Introduction: Choice under Uncertainty
Introduction: Choice under Uncertainty
Paradoxes of Expected Utility
Optimisation of Information Utility
Results
Example: SPB LotteryReferences106
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic behaviourApril 16, 2010 3 / 33
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Introduction: Choice under Uncertainty
Learning Systems
PerformanceOO
t//
vt
rp
nm
k InformationOO
t//k
mo
qs
uw
Performance and information have orders, and the relation betweenthem 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 behaviourApril 16, 2010 4 / 33
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Introduction: Choice under Uncertainty
Learning Systems
PerformanceOO
t//
vt
rp
nm
k
InformationOO
t//k
mo
qs
uw
Performance and information have orders, and the relation betweenthem 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 behaviourApril 16, 2010 4 / 33
![Page 6: The effect of information constraints on decision …...The effect of information constraints on decision-making and economic behaviour Roman V. Belavkin Middlesex University April](https://reader034.vdocuments.us/reader034/viewer/2022042205/5ea77abcc18735323e4550ad/html5/thumbnails/6.jpg)
Introduction: Choice under Uncertainty
Learning Systems
PerformanceOO
t//
vt
rp
nm
k InformationOO
t//k
mo
qs
uw
Performance and information have orders, and the relation betweenthem 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 behaviourApril 16, 2010 4 / 33
![Page 7: The effect of information constraints on decision …...The effect of information constraints on decision-making and economic behaviour Roman V. Belavkin Middlesex University April](https://reader034.vdocuments.us/reader034/viewer/2022042205/5ea77abcc18735323e4550ad/html5/thumbnails/7.jpg)
Introduction: Choice under Uncertainty
Learning Systems
PerformanceOO
t//
vt
rp
nm
k InformationOO
t//k
mo
qs
uw
Performance and information have orders, and the relation betweenthem 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 behaviourApril 16, 2010 4 / 33
![Page 8: The effect of information constraints on decision …...The effect of information constraints on decision-making and economic behaviour Roman V. Belavkin Middlesex University April](https://reader034.vdocuments.us/reader034/viewer/2022042205/5ea77abcc18735323e4550ad/html5/thumbnails/8.jpg)
Introduction: Choice under Uncertainty
Learning Systems
PerformanceOO
t//
vt
rp
nm
k InformationOO
t//k
mo
qs
uw
Performance and information have orders, and the relation betweenthem 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 behaviourApril 16, 2010 4 / 33
![Page 9: The effect of information constraints on decision …...The effect of information constraints on decision-making and economic behaviour Roman V. Belavkin Middlesex University April](https://reader034.vdocuments.us/reader034/viewer/2022042205/5ea77abcc18735323e4550ad/html5/thumbnails/9.jpg)
Introduction: Choice under Uncertainty
Learning Systems
PerformanceOO
t//
vt
rp
nm
k InformationOO
t//k
mo
qs
uw
Performance and information have orders, and the relation betweenthem 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 behaviourApril 16, 2010 4 / 33
![Page 10: The effect of information constraints on decision …...The effect of information constraints on decision-making and economic behaviour Roman V. Belavkin Middlesex University April](https://reader034.vdocuments.us/reader034/viewer/2022042205/5ea77abcc18735323e4550ad/html5/thumbnails/10.jpg)
Introduction: Choice under Uncertainty
Learning Systems
PerformanceOO
t//
vt
rp
nm
k InformationOO
t//k
mo
qs
uw
Performance and information have orders, and the relation betweenthem 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 behaviourApril 16, 2010 4 / 33
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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
Choice under uncertainty
q . p ⇐⇒ Eqf ≤ Epf
Question (Why expected utility?)
1 Eyf = f(ω) if y(Ω′) = δω(Ω′).
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 behaviourApril 16, 2010 5 / 33
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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
Choice under uncertainty
q . p ⇐⇒ Eqf ≤ Epf
Question (Why expected utility?)
1 Eyf = f(ω) if y(Ω′) = δω(Ω′).
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 behaviourApril 16, 2010 5 / 33
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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
Epx :=∑ω∈Ω
x(ω)p(ω)
Choice under uncertainty
q . p ⇐⇒ Eqf ≤ Epf
Question (Why expected utility?)
1 Eyf = f(ω) if y(Ω′) = δω(Ω′).
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 behaviourApril 16, 2010 5 / 33
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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
Epx :=
∫Ω
x(ω) dp(ω)
Choice under uncertainty
q . p ⇐⇒ Eqf ≤ Epf
Question (Why expected utility?)
1 Eyf = f(ω) if y(Ω′) = δω(Ω′).
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 behaviourApril 16, 2010 5 / 33
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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
Epx :=
∫Ω
x(ω) dp(ω)
Choice under uncertainty
q . p ⇐⇒ Eqf ≤ Epf
Question (Why expected utility?)
1 Eyf = f(ω) if y(Ω′) = δω(Ω′).
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 behaviourApril 16, 2010 5 / 33
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Paradoxes of Expected Utility
Introduction: Choice under Uncertainty
Paradoxes of Expected Utility
Optimisation of Information Utility
Results
Example: SPB LotteryReferences106
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic behaviourApril 16, 2010 6 / 33
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Paradoxes of Expected Utility
St. Petersburg lottery
Due to Nicolas Bernoulli (1713)
The lottery is played by tossing a fair coin repeatedly until the firsthead 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 behaviourApril 16, 2010 7 / 33
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Paradoxes of Expected Utility
St. Petersburg lottery
Due to Nicolas Bernoulli (1713)
The lottery is played by tossing a fair coin repeatedly until the firsthead 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 behaviourApril 16, 2010 7 / 33
![Page 19: The effect of information constraints on decision …...The effect of information constraints on decision-making and economic behaviour Roman V. Belavkin Middlesex University April](https://reader034.vdocuments.us/reader034/viewer/2022042205/5ea77abcc18735323e4550ad/html5/thumbnails/19.jpg)
Paradoxes of Expected Utility
St. Petersburg lottery
Due to Nicolas Bernoulli (1713)
The lottery is played by tossing a fair coin repeatedly until the firsthead 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 behaviourApril 16, 2010 7 / 33
![Page 20: The effect of information constraints on decision …...The effect of information constraints on decision-making and economic behaviour Roman V. Belavkin Middlesex University April](https://reader034.vdocuments.us/reader034/viewer/2022042205/5ea77abcc18735323e4550ad/html5/thumbnails/20.jpg)
Paradoxes of Expected Utility
St. Petersburg lottery
Due to Nicolas Bernoulli (1713)
The lottery is played by tossing a fair coin repeatedly until the firsthead 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 behaviourApril 16, 2010 7 / 33
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Paradoxes of Expected Utility
St. Petersburg lottery
Due to Nicolas Bernoulli (1713)
The lottery is played by tossing a fair coin repeatedly until the firsthead 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 behaviourApril 16, 2010 7 / 33
![Page 22: The effect of information constraints on decision …...The effect of information constraints on decision-making and economic behaviour Roman V. Belavkin Middlesex University April](https://reader034.vdocuments.us/reader034/viewer/2022042205/5ea77abcc18735323e4550ad/html5/thumbnails/22.jpg)
Paradoxes of Expected Utility
Why is it a paradox?
We don’t want to pay more than expect to win:
X ≤ Ewin
How large is Epwin?Let p(n) be the probability of n ∈ N
head 1 2 3 4 · · · n · · ·win £2 £4 £8 £16 · · · 2n · · ·p(n) 1
214
18
116 · · · 1
2n · · ·
It is easy to see that
Epwin = 2 · 1
2+ 4 · 1
4+ 8 · 1
8+ 16 · 1
16+ · · · =
∞∑n=1
2n
2n= ∞
One cannot buy what is not for sale.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic behaviourApril 16, 2010 8 / 33
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Paradoxes of Expected Utility
Why is it a paradox?
We don’t want to pay more than expect to win:
X ≤ Ewin
How large is Epwin?
Let p(n) be the probability of n ∈ N
head 1 2 3 4 · · · n · · ·win £2 £4 £8 £16 · · · 2n · · ·p(n) 1
214
18
116 · · · 1
2n · · ·
It is easy to see that
Epwin = 2 · 1
2+ 4 · 1
4+ 8 · 1
8+ 16 · 1
16+ · · · =
∞∑n=1
2n
2n= ∞
One cannot buy what is not for sale.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic behaviourApril 16, 2010 8 / 33
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Paradoxes of Expected Utility
Why is it a paradox?
We don’t want to pay more than expect to win:
X ≤ Ewin
How large is Epwin?Let p(n) be the probability of n ∈ N
head 1 2 3 4 · · · n · · ·win £2 £4 £8 £16 · · · 2n · · ·p(n) 1
214
18
116 · · · 1
2n · · ·
It is easy to see that
Epwin = 2 · 1
2+ 4 · 1
4+ 8 · 1
8+ 16 · 1
16+ · · · =
∞∑n=1
2n
2n= ∞
One cannot buy what is not for sale.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic behaviourApril 16, 2010 8 / 33
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Paradoxes of Expected Utility
Why is it a paradox?
We don’t want to pay more than expect to win:
X ≤ Ewin
How large is Epwin?Let p(n) be the probability of n ∈ N
head 1 2 3 4 · · · n · · ·win £2 £4 £8 £16 · · · 2n · · ·p(n) 1
214
18
116 · · · 1
2n · · ·
It is easy to see that
Epwin = 2 · 1
2+ 4 · 1
4+ 8 · 1
8+ 16 · 1
16+ · · · =
∞∑n=1
2n
2n= ∞
One cannot buy what is not for sale.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic behaviourApril 16, 2010 8 / 33
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Paradoxes of Expected Utility
Why is it a paradox?
We don’t want to pay more than expect to win:
X ≤ Ewin
How large is Epwin?Let p(n) be the probability of n ∈ N
head 1 2 3 4 · · · n · · ·win £2 £4 £8 £16 · · · 2n · · ·p(n) 1
214
18
116 · · · 1
2n · · ·
It is easy to see that
Epwin = 2 · 1
2+ 4 · 1
4+ 8 · 1
8+ 16 · 1
16+ · · · =
∞∑n=1
2n
2n= ∞
One cannot buy what is not for sale.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic behaviourApril 16, 2010 8 / 33
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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):
Epf ∝∞∑
n=1
f(n)
f(n)= ∞
Some suggest to use only f such that
‖f‖∞ := sup |f(ω)| < ∞
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic behaviourApril 16, 2010 9 / 33
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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):
Epf ∝∞∑
n=1
f(n)
f(n)= ∞
Some suggest to use only f such that
‖f‖∞ := sup |f(ω)| < ∞
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic behaviourApril 16, 2010 9 / 33
![Page 29: The effect of information constraints on decision …...The effect of information constraints on decision-making and economic behaviour Roman V. Belavkin Middlesex University April](https://reader034.vdocuments.us/reader034/viewer/2022042205/5ea77abcc18735323e4550ad/html5/thumbnails/29.jpg)
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):
Epf ∝∞∑
n=1
f(n)
f(n)= ∞
Some suggest to use only f such that
‖f‖∞ := sup |f(ω)| < ∞
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic behaviourApril 16, 2010 9 / 33
![Page 30: The effect of information constraints on decision …...The effect of information constraints on decision-making and economic behaviour Roman V. Belavkin Middlesex University April](https://reader034.vdocuments.us/reader034/viewer/2022042205/5ea77abcc18735323e4550ad/html5/thumbnails/30.jpg)
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 repeatedlyuntil 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 economic behaviourApril 16, 2010 10 / 33
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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 repeatedlyuntil 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 economic behaviourApril 16, 2010 10 / 33
![Page 32: The effect of information constraints on decision …...The effect of information constraints on decision-making and economic behaviour Roman V. Belavkin Middlesex University April](https://reader034.vdocuments.us/reader034/viewer/2022042205/5ea77abcc18735323e4550ad/html5/thumbnails/32.jpg)
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 repeatedlyuntil 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 economic behaviourApril 16, 2010 10 / 33
![Page 33: The effect of information constraints on decision …...The effect of information constraints on decision-making and economic behaviour Roman V. Belavkin Middlesex University April](https://reader034.vdocuments.us/reader034/viewer/2022042205/5ea77abcc18735323e4550ad/html5/thumbnails/33.jpg)
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 repeatedlyuntil 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 =?)
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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 repeatedlyuntil 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 =?)
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Paradoxes of Expected Utility
The Allais (1953) paradox
Consider two lotteries:
A : p(£300) = 13 (and p(£0) = 2
3)
B : p(£100) = 1
Most of the people seem to prefer A . B
Note that
EAx = 300 · 1
3+ 100 · 0 + 0 · 2
3= 100
EBx = 300 · 0 + 100 · 1 + 0 · 0 = 100
Remark
Safety is preferred (i.e. risk averse).
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Paradoxes of Expected Utility
The Allais (1953) paradox
Consider two lotteries:
A : p(£300) = 13 (and p(£0) = 2
3)
B : p(£100) = 1
Most of the people seem to prefer A . B
Note that
EAx = 300 · 1
3+ 100 · 0 + 0 · 2
3= 100
EBx = 300 · 0 + 100 · 1 + 0 · 0 = 100
Remark
Safety is preferred (i.e. risk averse).
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Paradoxes of Expected Utility
The Allais (1953) paradox
Consider two lotteries:
A : p(£300) = 13 (and p(£0) = 2
3)
B : p(£100) = 1
Most of the people seem to prefer A . B
Note that
EAx = 300 · 1
3+ 100 · 0 + 0 · 2
3= 100
EBx = 300 · 0 + 100 · 1 + 0 · 0 = 100
Remark
Safety is preferred (i.e. risk averse).
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Paradoxes of Expected Utility
The Allais (1953) paradox
Consider two lotteries:
A : p(£300) = 13 (and p(£0) = 2
3)
B : p(£100) = 1
Most of the people seem to prefer A . B
Note that
EAx = 300 · 1
3+ 100 · 0 + 0 · 2
3= 100
EBx = 300 · 0 + 100 · 1 + 0 · 0 = 100
Remark
Safety is preferred (i.e. risk averse).
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Paradoxes of Expected Utility
The Allais (1953) paradox (2)
Consider two lotteries:
C : p(−£300) = 13 (and p(£0) = 2
3)
D : p(−£100) = 1
Most of the people seem to prefer C & D
Note that
ECx = −300 · 1
3− 100 · 0− 0 · 2
3= −100
EDx = −300 · 0− 100 · 1− 0 · 0 = −100
Remark
Risk is preferred (i.e. risk taking).
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Paradoxes of Expected Utility
The Allais (1953) paradox (2)
Consider two lotteries:
C : p(−£300) = 13 (and p(£0) = 2
3)
D : p(−£100) = 1
Most of the people seem to prefer C & D
Note that
ECx = −300 · 1
3− 100 · 0− 0 · 2
3= −100
EDx = −300 · 0− 100 · 1− 0 · 0 = −100
Remark
Risk is preferred (i.e. risk taking).
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Paradoxes of Expected Utility
The Allais (1953) paradox (2)
Consider two lotteries:
C : p(−£300) = 13 (and p(£0) = 2
3)
D : p(−£100) = 1
Most of the people seem to prefer C & D
Note that
ECx = −300 · 1
3− 100 · 0− 0 · 2
3= −100
EDx = −300 · 0− 100 · 1− 0 · 0 = −100
Remark
Risk is preferred (i.e. risk taking).
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Paradoxes of Expected Utility
The Allais (1953) paradox (2)
Consider two lotteries:
C : p(−£300) = 13 (and p(£0) = 2
3)
D : p(−£100) = 1
Most of the people seem to prefer C & D
Note that
ECx = −300 · 1
3− 100 · 0− 0 · 2
3= −100
EDx = −300 · 0− 100 · 1− 0 · 0 = −100
Remark
Risk is preferred (i.e. risk taking).
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Paradoxes of Expected Utility
Why is it a paradox?
10
1U = 0, 1, 2
Eu =P
i PiUi = const
Increasing
utility
Eu = 12
P1
P3
Remark
Any linear functional (e.g. Epx) has parallel level sets. If people useexpected utility to make choices, then they are either risk-averse orrisk-taking, but not both.
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Paradoxes of Expected Utility
Prospect theory
Due to Tversky and Kahneman (1981)
It was proposed that the utility is convex, when the choice is amonggains, and concave when the choice is among losses.
This would make the choice risk averse for gains and risk taking forlosses.
Remark
This theory is not normative (i.e. it is not derived using rational approach).
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Paradoxes of Expected Utility
Prospect theory
Due to Tversky and Kahneman (1981)
It was proposed that the utility is convex, when the choice is amonggains, and concave when the choice is among losses.
This would make the choice risk averse for gains and risk taking forlosses.
Remark
This theory is not normative (i.e. it is not derived using rational approach).
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Paradoxes of Expected Utility
Prospect theory
Due to Tversky and Kahneman (1981)
It was proposed that the utility is convex, when the choice is amonggains, and concave when the choice is among losses.
This would make the choice risk averse for gains and risk taking forlosses.
Remark
This theory is not normative (i.e. it is not derived using rational approach).
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Paradoxes of Expected Utility
The Ellsberg (1961) paradox
Consider two lotteries:
A : p(£100) = 12 (and p(£0) = 1
2)
B : p(£100) = unknown
Most of the people seem to prefer A & B
Note that
EAx = 100 · 1
2+ 0 · 1
2= 50
EBx =
∫ 1
0(100 · p + 0 · (1− p)) dp = 50
Remark
More information is preferred.
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Paradoxes of Expected Utility
The Ellsberg (1961) paradox
Consider two lotteries:
A : p(£100) = 12 (and p(£0) = 1
2)
B : p(£100) = unknown
Most of the people seem to prefer A & B
Note that
EAx = 100 · 1
2+ 0 · 1
2= 50
EBx =
∫ 1
0(100 · p + 0 · (1− p)) dp = 50
Remark
More information is preferred.
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Paradoxes of Expected Utility
The Ellsberg (1961) paradox
Consider two lotteries:
A : p(£100) = 12 (and p(£0) = 1
2)
B : p(£100) = unknown
Most of the people seem to prefer A & B
Note that
EAx = 100 · 1
2+ 0 · 1
2= 50
EBx =
∫ 1
0(100 · p + 0 · (1− p)) dp = 50
Remark
More information is preferred.
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Paradoxes of Expected Utility
The Ellsberg (1961) paradox
Consider two lotteries:
A : p(£100) = 12 (and p(£0) = 1
2)
B : p(£100) = unknown
Most of the people seem to prefer A & B
Note that
EAx = 100 · 1
2+ 0 · 1
2= 50
EBx =
∫ 1
0(100 · p + 0 · (1− p)) dp = 50
Remark
More information is preferred.
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Optimisation of Information Utility
Introduction: Choice under Uncertainty
Paradoxes of Expected Utility
Optimisation of Information Utility
Results
Example: SPB LotteryReferences106
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Optimisation of Information Utility
Extreme Value Problems
Unconditional extremum
Maximise f(y):sup f(y)
Necessary condition ∂f(y) 3 0.
Sufficient, if f is concave.
Conditional extremum
Maximise f(y) subject to g(y) ≤ λ:
f(λ) := supf(y) : g(y) ≤ λ
.
Necessary condition ∂f(y)− α∂g(y) 3 0.
Sufficient if K(y, α) := f(y) + α[λ− g(y)] is concave.
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Optimisation of Information Utility
Extreme Value Problems
Unconditional extremum
Maximise f(y):sup f(y)
Necessary condition ∂f(y) 3 0.
Sufficient, if f is concave.
Conditional extremum
Maximise f(y) subject to g(y) ≤ λ:
f(λ) := supf(y) : g(y) ≤ λ
.
Necessary condition ∂f(y)− α∂g(y) 3 0.
Sufficient if K(y, α) := f(y) + α[λ− g(y)] is concave.
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Optimisation of Information Utility
Extreme Value Problems
Unconditional extremum
Maximise f(y):sup f(y)
Necessary condition ∂f(y) 3 0.
Sufficient, if f is concave.
Conditional extremum
Maximise f(y) subject to g(y) ≤ λ:
f(λ) := supf(y) : g(y) ≤ λ
.
Necessary condition ∂f(y)− α∂g(y) 3 0.
Sufficient if K(y, α) := f(y) + α[λ− g(y)] is concave.
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Optimisation of Information Utility
Extreme Value Problems
Unconditional extremum
Maximise f(y):sup f(y)
Necessary condition ∂f(y) 3 0.
Sufficient, if f is concave.
Conditional extremum
Maximise f(y) subject to g(y) ≤ λ:
f(λ) := supf(y) : g(y) ≤ λ
.
Necessary condition ∂f(y)− α∂g(y) 3 0.
Sufficient if K(y, α) := f(y) + α[λ− g(y)] is concave.
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Optimisation of Information Utility
Extreme Value Problems
Unconditional extremum
Maximise f(y):sup f(y)
Necessary condition ∂f(y) 3 0.
Sufficient, if f is concave.
Conditional extremum
Maximise f(y) subject to g(y) ≤ λ:
f(λ) := supf(y) : g(y) ≤ λ
.
Necessary condition ∂f(y)− α∂g(y) 3 0.
Sufficient if K(y, α) := f(y) + α[λ− g(y)] is concave.
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Optimisation of Information Utility
Extreme Value Problems
Unconditional extremum
Maximise f(y):sup f(y)
Necessary condition ∂f(y) 3 0.
Sufficient, if f is concave.
Conditional extremum
Maximise f(y) subject to g(y) ≤ λ:
f(λ) := supf(y) : g(y) ≤ λ
.
Necessary condition ∂f(y)− α∂g(y) 3 0.
Sufficient if K(y, α) := f(y) + α[λ− g(y)] is concave.
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Optimisation of Information Utility
Representation in Paired Spaces
x ∈ X, y ∈ Y , 〈·, ·〉 : X × Y → R
〈x, y〉 :=
∫Ω
x(ω) dy(ω)
Separation:
〈x, y〉 = 0 , ∀x ∈ X ⇒ y = 0 ∈ Y
〈x, y〉 = 0 , ∀ y ∈ Y ⇒ x = 0 ∈ X
Can equip X and Y with ‖x‖∞ and ‖y‖1.
Statistical manifold:
M := y ∈ Y : y ≥ 0 , ‖y‖1 = 1
Expected valueEpx = 〈x, y〉|M
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Optimisation of Information Utility
Representation in Paired Spaces
x ∈ X, y ∈ Y , 〈·, ·〉 : X × Y → R
〈x, y〉 :=
∫Ω
x(ω) dy(ω)
Separation:
〈x, y〉 = 0 , ∀x ∈ X ⇒ y = 0 ∈ Y
〈x, y〉 = 0 , ∀ y ∈ Y ⇒ x = 0 ∈ X
Can equip X and Y with ‖x‖∞ and ‖y‖1.
Statistical manifold:
M := y ∈ Y : y ≥ 0 , ‖y‖1 = 1
Expected valueEpx = 〈x, y〉|M
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Optimisation of Information Utility
Representation in Paired Spaces
x ∈ X, y ∈ Y , 〈·, ·〉 : X × Y → R
〈x, y〉 :=
∫Ω
x(ω) dy(ω)
Separation:
〈x, y〉 = 0 , ∀x ∈ X ⇒ y = 0 ∈ Y
〈x, y〉 = 0 , ∀ y ∈ Y ⇒ x = 0 ∈ X
Can equip X and Y with ‖x‖∞ and ‖y‖1.
Statistical manifold:
M := y ∈ Y : y ≥ 0 , ‖y‖1 = 1
Expected valueEpx = 〈x, y〉|M
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Optimisation of Information Utility
Representation in Paired Spaces
x ∈ X, y ∈ Y , 〈·, ·〉 : X × Y → R
〈x, y〉 :=
∫Ω
x(ω) dy(ω)
Separation:
〈x, y〉 = 0 , ∀x ∈ X ⇒ y = 0 ∈ Y
〈x, y〉 = 0 , ∀ y ∈ Y ⇒ x = 0 ∈ X
Can equip X and Y with ‖x‖∞ and ‖y‖1.
Statistical manifold:
M := y ∈ Y : y ≥ 0 , ‖y‖1 = 1
Expected valueEpx = 〈x, y〉|M
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Optimisation of Information Utility
Representation in Paired Spaces
x ∈ X, y ∈ Y , 〈·, ·〉 : X × Y → R
〈x, y〉 :=
∫Ω
x(ω) dy(ω)
Separation:
〈x, y〉 = 0 , ∀x ∈ X ⇒ y = 0 ∈ Y
〈x, y〉 = 0 , ∀ y ∈ Y ⇒ x = 0 ∈ X
Can equip X and Y with ‖x‖∞ and ‖y‖1.
Statistical manifold:
M := y ∈ Y : y ≥ 0 , ‖y‖1 = 1
Expected valueEpx = 〈x, y〉|M
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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
F (y) :=
⟨ln y
z , y⟩− 〈1, y − z〉 , if y > 0
〈1, z〉 , if y = 0∞ , if y < 0
∂F (y) = ln yz = x ⇐⇒ y = ex z = ∂F ∗(x)
The dual F ∗ : X → R ∪ ∞ is
F ∗(x) := 〈1, ex z〉
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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
F (y) :=
⟨ln y
z , y⟩− 〈1, y − z〉 , if y > 0
〈1, z〉 , if y = 0∞ , if y < 0
∂F (y) = ln yz = x ⇐⇒ y = ex z = ∂F ∗(x)
The dual F ∗ : X → R ∪ ∞ is
F ∗(x) := 〈1, ex z〉
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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
F (y) :=
⟨ln y
z , y⟩− 〈1, y − z〉 , if y > 0
〈1, z〉 , if y = 0∞ , if y < 0
∂F (y) = ln yz = x ⇐⇒ y = ex z = ∂F ∗(x)
The dual F ∗ : X → R ∪ ∞ is
F ∗(x) := 〈1, ex z〉
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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
F (y) :=
⟨ln y
z , y⟩− 〈1, y − z〉 , if y > 0
〈1, z〉 , if y = 0∞ , if y < 0
∂F (y) = ln yz = x ⇐⇒ y = ex z = ∂F ∗(x)
The dual F ∗ : X → R ∪ ∞ is
F ∗(x) := 〈1, ex z〉
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Optimisation of Information Utility
Utility of Information
If x ∈ X is utility, then the value of event y relative to z is
〈x, y − z〉 = Eyx − Ezx
Definition (Utility of information)
Ux(I) := sup〈x, y〉 : F (y) ≤ I
Stratonovich (1965) defined Ux(I) for Shannon information.
Related functions
−U−x(I) := inf〈x, y〉 : F (y) ≤ IIx(U) := infF (y) : U0 ≤ U ≤ 〈x, y〉Ix(U) := infF (y) : 〈x, y〉 ≤ U < U0
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Optimisation of Information Utility
Utility of Information
If x ∈ X is utility, then the value of event y relative to z is
〈x, y − z〉 = Eyx − Ezx
Definition (Utility of information)
Ux(I) := sup〈x, y〉 : F (y) ≤ I
Stratonovich (1965) defined Ux(I) for Shannon information.
Related functions
−U−x(I) := inf〈x, y〉 : F (y) ≤ IIx(U) := infF (y) : U0 ≤ U ≤ 〈x, y〉Ix(U) := infF (y) : 〈x, y〉 ≤ U < U0
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Optimisation of Information Utility
Utility of Information
If x ∈ X is utility, then the value of event y relative to z is
〈x, y − z〉 = Eyx − Ezx
Definition (Utility of information)
Ux(I) := sup〈x, y〉 : F (y) ≤ I
Stratonovich (1965) defined Ux(I) for Shannon information.
Related functions
−U−x(I) := inf〈x, y〉 : F (y) ≤ IIx(U) := infF (y) : U0 ≤ U ≤ 〈x, y〉Ix(U) := infF (y) : 〈x, y〉 ≤ U < U0
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Optimisation of Information Utility
Utility of Information
If x ∈ X is utility, then the value of event y relative to z is
〈x, y − z〉 = Eyx − Ezx
Definition (Utility of information)
Ux(I) := sup〈x, y〉 : F (y) ≤ I
Stratonovich (1965) defined Ux(I) for Shannon information.
Related functions
−U−x(I) := inf〈x, y〉 : F (y) ≤ IIx(U) := infF (y) : U0 ≤ U ≤ 〈x, y〉Ix(U) := infF (y) : 〈x, y〉 ≤ U < U0
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Results
Introduction: Choice under Uncertainty
Paradoxes of Expected Utility
Optimisation of Information Utility
Results
Example: SPB LotteryReferences106
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Results
Information Bounded Utility
Definition (Information Bounded Utility)
A function f : Ω → R that admits a solution to the utility of informationproblem Uf (I) for I ∈ (inf F, supF )
Theorem
A solution to Uf (I) and If (U) exists if and only if set x : F ∗q (x) ≤ I∗
absorbs function f :∃β−1 > 0 : F ∗
q (βf) < ∞
Remark (Separation of information)
For all I ∈ (inf F, supF ) there exist β−11 , β−1
2 > 0:
Fq(∂F ∗q (β1f)) < I < Fq(∂F ∗(β2f))
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Results
Information Bounded Utility
Definition (Information Bounded Utility)
A function f : Ω → R that admits a solution to the utility of informationproblem Uf (I) for I ∈ (inf F, supF )
Theorem
A solution to Uf (I) and If (U) exists if and only if set x : F ∗q (x) ≤ I∗
absorbs function f :∃β−1 > 0 : F ∗
q (βf) < ∞
Remark (Separation of information)
For all I ∈ (inf F, supF ) there exist β−11 , β−1
2 > 0:
Fq(∂F ∗q (β1f)) < I < Fq(∂F ∗(β2f))
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Results
Information Bounded Utility
Definition (Information Bounded Utility)
A function f : Ω → R that admits a solution to the utility of informationproblem Uf (I) for I ∈ (inf F, supF )
Theorem
A solution to Uf (I) and If (U) exists if and only if set x : F ∗q (x) ≤ I∗
absorbs function f :∃β−1 > 0 : F ∗
q (βf) < ∞
Remark (Separation of information)
For all I ∈ (inf F, supF ) there exist β−11 , β−1
2 > 0:
Fq(∂F ∗q (β1f)) < I < Fq(∂F ∗(β2f))
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Results
Information Topology (Belavkin, 2010a)
1
01
P3
P1
y0
The topology is defined using aninformation resource:
F : L → R∪∞ , inf F = F (y0)
Neighbourhoods of y0:
C := y : F (y) ≤ I
The topology of informationbounded functions:
C := x : F ∗(x) ≤ I∗
Generally, f ∈ C does not imply−f ∈ C.
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Results
Information Topology (Belavkin, 2010a)
1
01
P3
P1
y0
The topology is defined using aninformation resource:
F : L → R∪∞ , inf F = F (y0)
Neighbourhoods of y0:
C := y : F (y) ≤ I
The topology of informationbounded functions:
C := x : F ∗(x) ≤ I∗
Generally, f ∈ C does not imply−f ∈ C.
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Results
Information Topology (Belavkin, 2010a)
1
01
P3
P1
y0
The topology is defined using aninformation resource:
F : L → R∪∞ , inf F = F (y0)
Neighbourhoods of y0:
C := y : F (y) ≤ I
The topology of informationbounded functions:
C := x : F ∗(x) ≤ I∗
Generally, f ∈ C does not imply−f ∈ C.
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Results
Information Topology (Belavkin, 2010a)
1
01
P3
P1
y0
The topology is defined using aninformation resource:
F : L → R∪∞ , inf F = F (y0)
Neighbourhoods of y0:
C := y : F (y) ≤ I
The topology of informationbounded functions:
C := x : F ∗(x) ≤ I∗
Generally, f ∈ C does not imply−f ∈ C.
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Results
Information Topology (Belavkin, 2010a)
1
01
P3
P1
y0
The topology is defined using aninformation resource:
F : L → R∪∞ , inf F = F (y0)
Neighbourhoods of y0:
C := y : F (y) ≤ I
The topology of informationbounded functions:
C := x : F ∗(x) ≤ I∗
Generally, f ∈ C does not imply−f ∈ C.
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Results
Information Topology (Belavkin, 2010a)
1
01
P3
P1
y0
The topology is defined using aninformation resource:
F : L → R∪∞ , inf F = F (y0)
Neighbourhoods of y0:
C := y : F (y) ≤ I
The topology of informationbounded functions:
C := x : F ∗(x) ≤ I∗
Generally, f ∈ C does not imply−f ∈ C.
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Results
Information Topology (Belavkin, 2010a)
1
01
P3
P1
y0
The topology is defined using aninformation resource:
F : L → R∪∞ , inf F = F (y0)
Neighbourhoods of y0:
C := y : F (y) ≤ I
The topology of informationbounded functions:
C := x : F ∗(x) ≤ I∗
Generally, f ∈ C does not imply−f ∈ C.
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Results
Information Topology (Belavkin, 2010a)
1
01
P3
P1
y0
The topology is defined using aninformation resource:
F : L → R∪∞ , inf F = F (y0)
Neighbourhoods of y0:
C := y : F (y) ≤ I
The topology of informationbounded functions:
C := x : F ∗(x) ≤ I∗
Generally, f ∈ C does not imply−f ∈ C.
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Results
Information Topology (Belavkin, 2010a)
1
01
P3
P1
y0
The topology is defined using aninformation resource:
F : L → R∪∞ , inf F = F (y0)
Neighbourhoods of y0:
C := y : F (y) ≤ I
The topology of informationbounded functions:
C := x : F ∗(x) ≤ I∗
Generally, f ∈ C does not imply−f ∈ C.
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Results
Information Topology (Belavkin, 2010a)
1
01
P3
P1
y0
The topology is defined using aninformation resource:
F : L → R∪∞ , inf F = F (y0)
Neighbourhoods of y0:
C := y : F (y) ≤ I
The topology of informationbounded functions:
C := x : F ∗(x) ≤ I∗
Generally, f ∈ C does not imply−f ∈ C.
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Results
Information Topology (Belavkin, 2010a)
1
01
P3
P1
y0
yn
The topology is defined using aninformation resource:
F : L → R∪∞ , inf F = F (y0)
Neighbourhoods of y0:
C := y : F (y) ≤ I
The topology of informationbounded functions:
C := x : F ∗(x) ≤ I∗
Generally, f ∈ C does not imply−f ∈ C.
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Results
Information Topology (Belavkin, 2010a)
1
01
P3
P1
y0
yn
The topology is defined using aninformation resource:
F : L → R∪∞ , inf F = F (y0)
Neighbourhoods of y0:
C := y : F (y) ≤ I
The topology of informationbounded functions:
C := x : F ∗(x) ≤ I∗
Generally, f ∈ C does not imply−f ∈ C.
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Results
Parametrisation by the Expected Utility
43210−1−2−3−4
1
0
−1
〈x,y〉
=U
(β)
β
Binary setUncountable set
Let F (y) be negative entropy (i.e. F (y) is minimised at y0(ω) = const)
x : Ω → c− d, c + d U(β) = Ψ′(β) = c + d tanh(β d)
x : Ω → [c− d, c + d] U(β) = Ψ′(β) = c + d coth(β d)− β−1
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Results
Parametrisation by Information
43210−1−2−3−4
2
1
0
F(y
)=
I(β
)
β
Binary setUncountable set
I = Φ′(β−1) , I = β Ψ′(β)−Ψ(β)
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Results
Parametric Dependency
−1 −0.5 0 0.5 1
1
0
F(y
)=
I
〈x, y〉 = U
Binary setUncountable set
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Example: SPB Lottery
Introduction: Choice under Uncertainty
Paradoxes of Expected Utility
Optimisation of Information Utility
Results
Example: SPB LotteryReferences106
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Example: SPB Lottery
A Solution of the SPB Paradox
f : N → R is information bounded iff for some β−1 > 0:
F ∗(βf) =∞∑
n=1
q(n)eβf(n) < ∞
Using∑∞
n=1 rn < ∞
where z(n) = q(n)∑
q(n).
Let q(n) = (e− 1)e−n (i.e. 2−n).
For f(n) = n, we have β < 1.
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Example: SPB Lottery
A Solution of the SPB Paradox
f : N → R is information bounded iff for some β−1 > 0:
F ∗(βf) =∞∑
n=1
q(n)eβf(n) < ∞
Using∑∞
n=1 rn < ∞
∃β−1 > 0 : βf(n) < − ln z(n) , ∀n ∈ N
where z(n) = q(n)∑
q(n).
Let q(n) = (e− 1)e−n (i.e. 2−n).
For f(n) = n, we have β < 1.
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Example: SPB Lottery
A Solution of the SPB Paradox
f : N → R is information bounded iff for some β−1 > 0:
F ∗(βf) =∞∑
n=1
q(n)eβf(n) < ∞
Using∑∞
n=1 rn < ∞
∃β−1 > 0 : βf(n) < − ln z(n) , ∀n ∈ Nwhere z(n) = q(n)
∑q(n).
Let q(n) = (e− 1)e−n (i.e. 2−n).
For f(n) = n, we have β < 1.
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Example: SPB Lottery
A Solution of the SPB Paradox
f : N → R is information bounded iff for some β−1 > 0:
F ∗(βf) =∞∑
n=1
q(n)eβf(n) < ∞
Using∑∞
n=1 rn < ∞
∃β−1 > 0 : βf(n) < − ln z(n) , ∀n ∈ Nwhere z(n) = q(n)
∑q(n).
Let q(n) = (e− 1)e−n (i.e. 2−n).
For f(n) = n, we have β < 1.
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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)∞∑
n=1
eβf(n)−n < ∞
Using∑∞
n=1 rn < ∞
∃β−1 > 0 : βf(n) < − ln z(n) , ∀n ∈ Nwhere z(n) = q(n)
∑q(n).
Let q(n) = (e− 1)e−n (i.e. 2−n).
For f(n) = n, we have β < 1.
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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)∞∑
n=1
eβf(n)−n < ∞
Using∑∞
n=1 rn < ∞
∃β−1 > 0 : βf(n) < n , ∀n ∈ Nwhere z(n) = q(n)
∑q(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 economic behaviourApril 16, 2010 28 / 33
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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)∞∑
n=1
eβf(n)−n < ∞
Using∑∞
n=1 rn < ∞
∃β−1 > 0 : βf(n) < n , ∀n ∈ Nwhere z(n) = q(n)
∑q(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 economic behaviourApril 16, 2010 28 / 33
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Example: SPB Lottery
A Solution of the SPB Paradox
Using U = Ψ′f (β) obtain
U =1
1− eβ−1, U0 =
1
1− e−1
The inverse of function U(β) is β = 1 + ln(1− U−1).
Using I = β(lnΨ(β))′ − lnΨ(β):
Change e to 2 (ln to log2).
For the information amount of 0 bits, the optimal entrance fee isc ≤ U0 = 2.
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Example: SPB Lottery
A Solution of the SPB Paradox
Using U = Ψ′f (β) obtain
U =1
1− eβ−1, U0 =
1
1− e−1
The inverse of function U(β) is β = 1 + ln(1− U−1).
Using I = β(lnΨ(β))′ − lnΨ(β):
Change e to 2 (ln to log2).
For the information amount of 0 bits, the optimal entrance fee isc ≤ U0 = 2.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic behaviourApril 16, 2010 29 / 33
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Example: SPB Lottery
A Solution of the SPB Paradox
Using U = Ψ′f (β) obtain
U =1
1− eβ−1, U0 =
1
1− e−1
The inverse of function U(β) is β = 1 + ln(1− U−1).
Using I = β(lnΨ(β))′ − lnΨ(β):
If (U) = (1 + ln(1− U−1))U − ln(e− 1)− ln(U − 1)
Change e to 2 (ln to log2).
For the information amount of 0 bits, the optimal entrance fee isc ≤ U0 = 2.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic behaviourApril 16, 2010 29 / 33
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Example: SPB Lottery
A Solution of the SPB Paradox
Using U = Ψ′f (β) obtain
U =1
1− eβ−1, U0 =
1
1− e−1
The inverse of function U(β) is β = 1 + ln(1− U−1).
Using I = β(lnΨ(β))′ − lnΨ(β):
If (U) = (1 + ln(1− U−1))U − ln(e− 1)− ln(U − 1)
Change e to 2 (ln to log2).
For the information amount of 0 bits, the optimal entrance fee isc ≤ U0 = 2.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic behaviourApril 16, 2010 29 / 33
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Example: SPB Lottery
A Solution of the SPB Paradox
Using U = Ψ′f (β) obtain
U =1
1− 2β−1, U0 = 2
The inverse of function U(β) is β = 1 + ln(1− U−1).
Using I = β(lnΨ(β))′ − lnΨ(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 isc ≤ U0 = 2.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic behaviourApril 16, 2010 29 / 33
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Example: SPB Lottery
A Solution of the SPB Paradox
Using U = Ψ′f (β) obtain
U =1
1− 2β−1, U0 = 2
The inverse of function U(β) is β = 1 + ln(1− U−1).
Using I = β(lnΨ(β))′ − lnΨ(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 isc ≤ U0 = 2.
Roman V. Belavkin (Middlesex University) The effect of information constraints on decision-making and economic behaviourApril 16, 2010 29 / 33
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Example: SPB Lottery
SPB Lottery
−5 −4 −3 −2 −1 0 1
2
1
Exp
ecte
dutility
,U
=〈f
,p〉,
(n)
Parameter, β
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Example: SPB Lottery
SPB Lottery
−5 −4 −3 −2 −1 0 1
2
1
Info
rmation,I
=F
q(p
),(b
its)
Parameter, β
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Example: SPB Lottery
SPB Lottery
1 2 3 4 5 6
1
0
Info
rmation,I
=F
q(p
)(b
its)
Expected utility, U = 〈f, p〉, (n)
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References
Introduction: Choice under Uncertainty
Paradoxes of Expected Utility
Optimisation of Information Utility
Results
Example: SPB LotteryReferences106
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Example: SPB Lottery
Allais, M. (1953). Le comportement de l’homme rationnel devant lerisque: Critique des postulats et axiomes de l’Ecole americaine.Econometrica, 21, 503–546.
Belavkin, R. V. (2010a). Information trajectory of optimal learning. InM. J. Hirsch, P. M. Pardalos, & R. Murphey (Eds.), Dynamics ofinformation systems: Theory and applications (Vol. 40). Springer.
Belavkin, R. V. (2010b). Utility and value of information in cognitivescience, biology and quantum theory. In L. Accardi, W. Freudenberg,& M. Ohya (Eds.), Quantum Bio-Informatics III (Vol. 26). WorldScientific.
Ellsberg, D. (1961, November). Risk, ambiguity, and the Savage axioms.The Quarterly Journal of Economics, 75(4), 643–669.
Stratonovich, R. L. (1965). On value of information. Izvestiya of USSRAcademy of Sciences, Technical Cybernetics, 5, 3–12. (In Russian)
Tversky, A., & Kahneman, D. (1981). The framing of decisions and thepsychology of choice. Science, 211, 453–458.
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