Chair Insurance and
Social Systems
I
I.
Introduction to decision theory
II.
The basic model of decision theory under risk
r
III
III.
Classical decision principles
IV.
Theory of expected utility (Bernoulli-Princip
ple)
V
V.
D bt on expected
Doubts
t d utility
tilit theory
th
and
d th
the ““non-expected
t d utility”
tilit ” approach
h
VI.
Introduction to Game Theory
19th October 2009
Modelling an
nd Decision Making
1
Chair Insurance and
Social Systems
II
II.
The basic model of decision theory underr risk

Depiction

P b biliti
Probabilities

-
Axioms of probability calculation
-
I
Interpretation
i off probabilities
b bili i
-
Excursion: models without probabilities
Dominance principles
-
State-by-state dominance
-
First-order stochastic dominance
19th October 2009
Modelling an
nd Decision Making
2
II. The basic model of decision theory under risk
Chair Insurance and
Social Systems
Basic model of decision theory
Model parameters:
 Action space (A), consisting of the set of availa
able actions/acts (a).
 State space (), consisting of every state of the
e world () that the decision maker considers
possible and are relevant for the decision.
 Outcome space (X), consisting of the set of ressults/outcomes (x) considered possible.
 Result function g : A  Θ  X which assig
gns to each pair (a,) an explicit result x
with x  g(a,
g(a θ) .
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Modelling an
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II. The basic model of decision theory under risk
Chair Insurance and
Social Systems
Basic model of decision theory
θ1
θ2
...
θj
...
a1
x11
x12
...
x1j
...
x1n
a2
x21
x22
...
x2j
...
x2n
...
...
...
...
...
...
...
ai
xi1
xi2
...
xij
...
xin
...
...
...
...
...
...
...
am
xm1
xm2
...
xmj
...
xmn
19th October 2009
Modelling an
nd Decision Making
θn
4
II. The basic model of decision theory under risk
Chair Insurance and
Social Systems
Axioms of probability calculation
Axioms of probability theory (Kolmogorov):
 A function p, that assigns a real number to everyy event E  Θ is called probability measure and
owing axioms hold true:
p(E) is the probability of the event E, if the follo
Axiom 1: p(E)  0,1 E
Axiom 2: p (Θ)  1
Axiom 3: If E1, E2, … are mutually exclusive evvents with E k  E j  ,  k  j then
 

p  E j   p E1  E 2  ...  
 j1 
19th October 2009

 pE 
j 1
Modelling an
nd Decision Making
j
5
II. The basic model of decision theory under risk
Chair Insurance and
Social Systems
Axioms of probability calculation
Calculation rules for p
probabilities:
p(  )  0,
p(  )  1
p( E )  p(  E )  1
p( E1  E 2 )  p( E1 )  p( E 2 ) - p( E1  E 2 )
Conditional probabilities:
For any two events E1 and E2 the conditio
onal probability of E2 given E1 is
p(E 2 E1 ) 
p 1  E2 )
p(E
p(E 1 )
I d
Independent
d t Events:
E
t
Two events E1 and E2 are stochasticallyy independent,
if
p
p(E 2 E1 )  p(E 2 ) or equivalen
ntly p(E 1  E 2 )  p(E 1 )  p(E 2 )
19th October 2009
Modelling an
nd Decision Making
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II. The basic model of decision theory under risk
Chair Insurance and
Social Systems
Axioms of probability calculation
Bayes‘ rule
Bayes
For two mutually exclusive events E and F
p E F  
19th October 2009
p E   p F E 
p E   p F E 
p E  F 


p F 
p E   p F E   p  E   p F  E 
p (F )
Modelling an
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II. The basic model of decision theory under risk
Chair Insurance and
Social Systems
The philosophy of probability
 logical (or objective a priori) probabilities
→ principle of insufficient reason
 frequency (or objective a posteriori) probabilitie
es
→ probability as the marginal value of relativve frequency
 subjective probabilities (probabilities as subjecttive figures of credibility)
19th October 2009
Modelling an
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II. The basic model of decision theory under risk
Excursion: models without probabilities
Chair Insurance and
Social Systems
Initial situation
 The decision maker is able to identify the differe
ent states of the world, but she cannot figure out
with which probabilities they are going to occur.
→ The approach is methodically not convinccing and will not be discussed any further.
Reasons:
 In general, decision theory is based on the subje
ective probability notion.
 If there are no indications for individuals that the
e credibility (occurrence probability) of a certain
state of the world is greater than the credibility of
o another state, the individuals can allocate the
same probability
b bilit tto every state
t t off the
th world.
ld
→ Principle of insufficient reason (principle of
o indifference)
19th October 2009
Modelling an
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II. The basic model of decision theory under risk
Chair Insurance and
Social Systems
State-by-state
State
by state dominance
An alternative a1 dominates another alternative a2 according to state-by-state dominance,
if a1 leads in no state of the world to a worse re
esult but at least in one state of the
esult,
world to a better result than a2.
x 1 j  x 2 j  j andd  j : x 1 j  x 2 j
Implications:
 Intuitively plausible, because an individual should never choose an alternative that is dominated
according to state-by-state dominance.
 State-by-state dominance usually only leads to an
a exclusion of certain alternatives, but not to the
d t
determination
i ti off the
th optimal
ti l alternative.
lt
ti
19th October 2009
Modelling an
nd Decision Making
10
II. The basic model of decision theory under risk
Chair Insurance and
Social Systems
State-by-state
State
by state dominance
Example
θ1
p θ1   0,4
θ2
p θ 2   0,4
θ3
p θ 3   0,2
a1
10
5
10
a2
10
10
10
a3
0
0
11
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Modelling an
nd Decision Making
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II. The basic model of decision theory under risk
Chair Insurance and
Social Systems
First-order
First
order stochastic dominance
An alternative a1 dominates an alternative a2 acccording to first-order stochastic dominance,
if for every outcome x the a1-probability to attain
n a outcome greater than x, is not smaller and
for at least one result greater than the a2-proba
ability.
1  F1 x   1  F2 x   x
 
 
an
nd  x̂ : 1  F1 x̂  1  F2 x̂
Implications
 First-order
First order stochastic dominance is a less dema
anding concept as state
state-by-state
by state dominance
dominance.
 It is intuitively plausible as resulting decisions arre based on probability distributions.
→ First-order stochastic dominance also onlyy leads to a preselection.
19th October 2009
Modelling an
nd Decision Making
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II. The basic model of decision theory under risk
Chair Insurance and
Social Systems
First-order
First
order stochastic dominance
Example
θ1
θ2
θ3
p θ1   0,4
p θ 2   0,4
p θ 3   0,2
a1
10
20
10
a2
20
10
20
a3
0
0
25
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Modelling an
nd Decision Making
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II. The basic model of decision theory under risk
Chair Insurance and
Social Systems
First-order
First
order stochastic dominance
 Example (cumulative distribution function)
Fi (x)
1
0,8
0,4
10
19th October 2009
20
25
5
Modelling an
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x
14
II. The basic model of decision theory under risk
Chair Insurance and
Social Systems
First-order
First
order stochastic dominance
Cumulative distribution function
P b bilit density
Probability
d
it function
f
ti
f x 
1
f2 x 
f1 x 
F2 x 
0
19th October 2009
x
0
Modelling an
nd Decision Making
F1 x 
x
15