Yaarit Even
Tel-Aviv University
December 2011
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Decision making under uncertainty
Game theory
Multi-criteria decision aid (MCDA)
Insurance and financial assets pricing
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A new definition for integrals w.r.t. capacities.
Defining Choquet and the concave integrals by
terms of the new integral.
Properties of the new integral.
Desirable properties and the conditions for which
the new integral maintains them.
Let 𝑁 be a finite set, 𝑁 = 𝑛.
 A capacity 𝑣 over 𝑁 is a function 𝑣: 2𝑁 → 0, ∞
satisfying:
(i) 𝑣 𝜙 = 0.
(ii) if 𝑆 ⊆ 𝑇 ⊆ 𝑁 , then 𝑣 𝑆 ≤ 𝑣 𝑇 .
 A random variable (r.v.) 𝑋 over 𝑁 is a function
𝑋: 𝑁 → ℝ.
 A subset of 𝑁 will be called an event.
Sub-decompositions and
decompositions of a random variable
Let 𝑋 be a random variable.
 A sub-decomposition of 𝑋 is a finite summation
𝑘
𝑖=1 𝛼𝑖 𝕀𝐴𝑖 such that:
(i) 𝑘𝑖=1 𝛼𝑖 𝕀𝐴𝑖 ≤ 𝑋
(ii) 𝛼𝑖 ≥ 0 and 𝐴𝑖 ⊆ 𝑁 for every 𝑖 = 1, … , 𝑘.
𝑘
 If there is an equality in (i), then 𝑖=1 𝛼𝑖 𝕀𝐴𝑖 is a
decomposition of 𝑋.
 The value of a decomposition w.r.t. 𝑣 is
𝑘
𝑖=1 𝛼𝑖 𝑣 𝐴𝑖 .
Let 𝐷 be a set of subsets of 𝑁, 𝐷 ⊆ 2𝑁 .
𝑘
 𝑖=1 𝛼𝑖 𝕀𝐴𝑖 is a 𝐷-sub-decomposition of 𝑋 if it is a
sub-decomposition of 𝑋 and 𝐴𝑖 ∈ 𝐷 for every 𝑖
= 1, … , 𝑘.
𝑘
 𝑖=1 𝛼𝑖 𝕀𝐴𝑖 is a 𝐷-decomposition of 𝑋 if it is a
decomposition of 𝑋 and 𝐴𝑖 ∈ 𝐷 for every 𝑖
= 1, … , 𝑘.
Suppose 𝐷 = 2𝑁 and 𝑋 = 𝕀𝑁 .
𝑛
𝑖=1 𝕀 𝑖 and 𝕀𝑁 are both 𝐷-decompositions of 𝑋.

Suppose 𝑁 = 3, 𝐷 = 1 , 12 , 123 , 𝑋
= 2,2,1 and 𝑌 = 1,2,2 .
𝑋 has a 𝐷-decomposition: 0 ∙ 𝕀 1 + 𝕀 12 + 𝕀 123 ,
and 𝑌 has only 𝐷-sub-decompositions, such as:
0 ∗ 𝕀 1 + 0 ∗ 𝕀 12 + 𝕀 123 .




A vocabulary ℱ is a set of subsets of 2𝑁 .
A sub-decomposition of 𝑋 is ℱ-allowable if it is a
𝐷-sub-decomposition of 𝑋 and 𝐷 ∈ ℱ.
The decomposition-integral w.r.t. ℱ is defined:
𝑋𝑑𝑣 = max { 𝐴𝑖 ∈𝐷 𝛼𝑖 𝑣 𝐴𝑖 ; 𝐴𝑖 ∈𝐷 𝛼𝑖 𝕀𝐴𝑖 is ℱallowable sub-decomposition of X}.
ℱ

The sub-decomposition attaining the maximum is
called the optimal sub-decomposition of 𝑋.

Suppose 𝑁 = 3
ℱ = 1 , 12 , 123 , 2 , 12 , 23
𝑣 𝑁 = 1, 𝑣 12 = 𝑣 13 = 1/2, 𝑣 23 = 5/6, 𝑣 1
= 𝑣 2 = 𝑣 3 = 1/3. 𝑋 = 2,2,1 and 𝑌 = 1,2,2 .
𝑋 has an optimal decomposition:
0∙𝕀
1
+𝕀
12
+𝕀
123
, and
ℱ
1
3
𝑌 has an optimal sub-decomposition:
2∗𝕀
23
, and
ℱ
5
6
𝑌𝑑𝑣 = 2 ∗ =
1
2
1
2
𝑋𝑑𝑣 = 0 ∗ + + 1 = 1 .
2
1 .
3




Choquet integral
The concave integral
Riemann integral
Shilkret integral
And other plausible integration schemes.

Definition (Lehrer):
𝑐𝑎𝑣
𝑋𝑑𝑣 = 𝑚𝑖𝑛 𝑓 𝑋 ,
where the minimum is taken over all concave
and homogeneous functions 𝑓: ℝ+ 𝑛 → ℝ, such
that 𝑓 𝕀𝑆 ≥ 𝑣 𝑆 for every 𝑆 ⊆ 𝑁.

Lemma (Lehrer):
𝑚𝑎𝑥{
𝑆⊆𝑁 𝛼𝑆 𝑣
𝑐𝑎𝑣
𝑆 ;
𝑋𝑑𝑣=
𝑆⊆𝑁 𝛼𝑆 𝕀𝑆
= 𝑋 , 𝛼𝑆 ≥ 0}.

Define ℱ 𝑐𝑎𝑣 = 2𝑁 .
𝑐𝑎𝑣
𝑋𝑑𝑣 =
ℱ 𝑐𝑎𝑣
𝑋𝑑𝑣 =
𝑚𝑎𝑥{ 𝐴𝑖 ∈𝐷 𝛼𝑖 𝑣 𝐴𝑖 ; 𝐴𝑖 ∈𝐷 𝛼𝑖 𝕀𝐴𝑖 is ℱ 𝑐𝑎𝑣 -allowable
sub-decomposition of X}.
 Since 𝑣 is monotonic w.r.t. inclusion, we have:
𝑐𝑎𝑣
𝑋𝑑𝑣 = 𝑚𝑎𝑥 { 𝐴𝑖 ∈𝐷 𝛼𝑖 𝑣 𝐴𝑖 ; 𝐴𝑖 ∈𝐷 𝛼𝑖 𝕀𝐴𝑖 is
ℱ 𝑐𝑎𝑣 -allowable decomposition of X}.

ℱ
Since ℱ 𝑐𝑎𝑣 allows for all decompositions, for
every vocabulary ℱ, the following inequality
holds:
∙ 𝑑𝑣 ≤
ℱ 𝑐𝑎𝑣
∙ 𝑑𝑣, for every 𝑣.
Concluding, that of all the decomposition-integrals,
the concave integral attains the highest value.
Definition:
𝐶ℎ
∞
𝑋𝑑𝑣 = 0 𝑣 𝑖 ∈ 𝑁 𝑋 𝑖 ≥ 𝛼 𝑑𝛼 =
= 𝑛𝑖=1 𝑋𝜎 𝑖 − 𝑋𝜎 𝑖−1 𝑣 𝐴𝑖 ,
where 𝜎 is a permutation over 𝑁 that satisfies 𝑋𝜎 1
≤ ⋯ ≤ 𝑋𝜎 𝑛 and 𝐴𝑖 = 𝜎 𝑖 , … , 𝜎 𝑛 (𝑋𝜎 0 = 0).


𝑋=
𝛼𝑖 𝕀𝐴𝑖 , where 𝛼𝑖 = 𝑋𝜎
𝑖
− 𝑋𝜎
𝑖−1
.
Definitions:
 Any two subsets 𝐴 and 𝐵 of 𝑁 are nested if
either 𝐴 ⊆ 𝐵 or 𝐵 ⊆ 𝐴.
 A set 𝐷 ⊆ 2𝑁 is called a chain if any two events
𝐴, 𝐵 ∈ 𝐷 are nested.

Define ℱ 𝐶ℎ to be the set of all chains.
𝐶ℎ
𝑋𝑑𝑣 = ℱ 𝐶ℎ 𝑋𝑑𝑣 =
𝑚𝑎𝑥 { 𝐴𝑖 ∈𝐷 𝛼𝑖 𝑣 𝐴𝑖 ; 𝐴𝑖 ∈𝐷 𝛼𝑖 𝕀𝐴𝑖 is ℱ 𝐶ℎ -allowable
sub-decomposition of X}
= 𝑚𝑎𝑥 { 𝐴𝑖 ∈𝐷 𝛼𝑖 𝑣 𝐴𝑖 ; 𝐴𝑖 ∈𝐷 𝛼𝑖 𝕀𝐴𝑖 is ℱ 𝐶ℎ allowable decomposition of X}.
Suppose 𝑁 = 3
𝑣 𝑁 = 1, 𝑣 12 = 𝑣 13 = 1/2, 𝑣 23 = 5/6,
𝑣 1 = 𝑣 2 = 𝑣 3 = 1/3 and 𝑋 = 2,2,1 .

𝐶ℎ
1
𝑋𝑑𝑣 = 1 ∗ 𝑣 𝑁 + 1 ∗ 𝑣 12 = 1
2
𝑐𝑎𝑣
5
𝑋𝑑𝑣 = 2 ∗ 𝑣 1 + 1 ∗ 𝑣 2 + 1 ∗ 𝑣 23 = 1
6



A partition of 𝑁 is a set 𝐷 = 𝐴1 , … , 𝐴𝑘 , such
that all 𝐴𝑖 ’s are pairwise disjoint and their union
is 𝑁.
Define ℱ 𝑝𝑎𝑟𝑡 to be the set of all partitions of 𝑁.
The Riemann integral can be defined as
ℱ 𝑝𝑎𝑟𝑡
∙ 𝑑𝑣.
• Positive homogeneity of degree one:
for every 𝜆 > 0,
and ℱ.
ℱ
𝜆𝑋𝑑𝑣 = 𝜆
ℱ
𝑋𝑑𝑣 for every 𝑋, 𝑣
• The decomposition-integral and additive
capacities:
Proposition: Let 𝑃 be a probability and ℱ a
vocabulary. Then, 𝔼𝑃 𝑋 = ℱ 𝑋𝑑𝑃 for every r.v. 𝑋
iff every 𝑋 has a 𝐷-decomposition, 𝐷 ∈ ℱ.
Monotonicity:
1. Monotonicity w.r.t. r.v.’s: Fix 𝑣 and ℱ and suppose

𝑋 ≤ 𝑌. Then, ℱ 𝑋𝑑𝑣 ≤ ℱ 𝑌𝑑𝑣.
2. Monotonicity w.r.t. capacities: Fix ℱ. If for every
𝐷 ∈ ℱ and every 𝐴 ∈ 𝐷, 𝑢 𝐴 ≤ 𝑣 𝐴 , then for every
r.v. 𝑋,
ℱ
𝑋𝑑𝑢 ≤
ℱ
𝑋𝑑𝑣.
3. Monotonicity w.r.t. vocabularies: Fix 𝑣 and
suppose ℱ and ℱ′ are two vocabularies.
Proposition: ℱ ∙ 𝑑𝑣 ≤ ℱ ′ ∙ 𝑑𝑣 iff for every 𝐷 ∈ ℱ
and every minimal set 𝐶 ⊆ 𝐷, there is 𝐷′ ∈ ℱ′ such
that 𝐶 ⊆ 𝐷′.

A set 𝐶 ⊆ 2𝑁 is minimal if the variables 𝕀𝐴 , 𝐴 ∈ 𝐶
are algebraically independent.



Additivity:
Two variables 𝑋 and 𝑌 are comonotone if for
every 𝑖, 𝑗 ∈ 𝑁, 𝑋 𝑖 − 𝑋 𝑗 𝑌 𝑖 − 𝑌 𝑗 ≥ 0.
Comonotone additivity means that if X and Y are
comonotone, then:
ℱ
𝑋𝑑𝑣 +
ℱ
𝑌𝑑𝑣 =
ℱ
(𝑋
Example: 𝑣 12 = 1, 𝑣 1 = 𝑣 2 = 1/3
𝑋 = 𝜀, 1 and 𝑌 = 1, 𝜀 . Fix ℱ = ℱ 𝑝𝑎𝑟𝑡 . Suppose ℰ
is small enough so that the optimal Ddecompositions of X and Y use 𝐷 = 1 , 2 :

𝑋𝑑𝑣 = ℱ 𝑌𝑑𝑣 = 1/3 1 + ℰ ,
but for 𝑋 + 𝑌, taking D′ = 12 :
ℱ
𝑋 + 𝑌 𝑑𝑣 = 1 + ℰ >
ℱ
𝑋𝑑𝑣 +
ℱ
𝑌𝑑𝑣 = 2/3 1 + ℰ
ℱ
Fix ℱ and 𝑣. 𝑌 is leaner than 𝑋 if there are
optimal decompositions in which 𝑋 employ
every indicator that 𝑌 employs:
The optimal decomposition of 𝑌 is 𝐴∈𝐶 𝛽𝐴 𝕀𝐴 ,
𝛽𝐴 > 0, and the optimal decomposition of 𝑋 is
𝐵𝜖𝐶 ′ 𝛼𝐵 𝕀𝐵 , 𝛼𝐵 > 0, and 𝐶 ⊆ 𝐶′.


Proposition: Fix a vocabulary ℱ such that every 𝑋
has an optimal decomposition for every 𝑣.
Suppose that for every 𝐷, 𝐷′ 𝜖 ℱ, whenever there
are two different decompositions of the same
variable, 𝐴∈𝐷 𝛿𝐴 𝕀𝐴 = 𝐵∈𝐷′ 𝛾𝐵 𝕀𝐵 , there is
𝐷′′ 𝜖 ℱ that contains all the 𝐴’s with 𝛿𝐴 > 0 and
all the 𝐵’s with 𝛾𝐵 > 0. Then, for every 𝑣 and
every 𝑋 and 𝑌, where 𝑌 is leaner than 𝑋,
+
ℱ
𝑌𝑑𝑣 =
ℱ
𝑋 + 𝑌 𝑑𝑣.
ℱ
𝑋𝑑𝑣
 Concavity
 Monotonicity w.r.t. stochastic dominance
 Translation-invariance

ℱ
∙ 𝑑𝑣 is concave if for every two r.v. 𝑋 and 𝑌,
and 𝛾 ∈ 0, 1 :
≥𝛾

ℱ
ℱ
𝛾𝑋 + 1 − 𝛾 𝑌 𝑑𝑣
𝑋𝑑𝑣 + 1 − 𝛾
ℱ
𝑌𝑑𝑣
Theorem 1: The decomposition-integral
∙ 𝑑𝑣 is concave for every 𝑣, iff there exists a
ℱ
vocabulary ℱ′ containing only one 𝐷 (𝐷

Corollary 1: A decomposition-integral
satisfies (i)
ℱ
ℱ
∙ 𝑑𝑣
𝕀𝐴 𝑑𝑣 ≥ 𝑣 𝐴 for every event 𝐴
and capacity 𝑣; and (ii)
= ℱ 𝑐𝑎𝑣 .
ℱ
∙ 𝑑𝑣 is concave, iff ℱ


𝑋 stochastically dominates 𝑌 w.r.t. 𝑣 𝑋 ≽𝑣 𝑌 if
for every number 𝑡 ∈ ℝ, 𝑣 𝑋 ≥ 𝑡 ≥ 𝑣 𝑌 ≥ 𝑡 .
ℱ
∙ 𝑑𝑣 is monotonic w.r.t. stochastic dominance if
𝑋 ≽𝑣 𝑌 implies

ℱ
𝑋𝑑𝑣 ≥
ℱ
𝑌𝑑𝑣
Theorem 2: The decomposition-integral ℱ ∙ 𝑑𝑣 is
monotonic w.r.t. stochastic dominance iff ℱ is the
collection of all chains of the same size 𝑘 (𝑘 ∈ ℕ).
Example: 𝑁 = 3
𝑣 𝑁 = 𝑣 12 = 𝑣 13 = 1, 𝑣 23 = 5/6, 𝑣 2 = 1
/3, 𝑣 1 = 𝑣 3 = 1/6
𝑋 = 1,2,1 and 𝑌 = 2,1,1 .
Obviously, 𝑋 ≽𝑣 𝑌.

ℱ=ℱ
𝐶ℎ
ℱ=ℱ
𝑐𝑎𝑣
𝐶ℎ
:
:
𝑋𝑑𝑣 =
𝑐𝑎𝑣
1
1
3
𝑋𝑑𝑣 =
>
5
1
6
<
𝐶ℎ
𝑌𝑑𝑣 =
𝑐𝑎𝑣
1
1
6
𝑌𝑑𝑣 = 2

ℱ
∙ 𝑑𝑣 is translation-invariant for every 𝑣, if for
every 𝑐 > 0,
𝑣 𝑁 = 1.

ℱ
𝑋 + 𝑐 𝑑𝑣 =
ℱ
𝑋𝑑𝑣 + 𝑐, when
Theorem 3: The decomposition-integral ℱ ∙ 𝑑𝑣
is translation-invariant for every 𝑣 iff the
vocabulary ℱ is (i) composed of chains; and (ii)
any 𝐷 ∈ ℱ is contained in 𝐷′ ∈ ℱ that includes
𝕀𝑁 .
Example: 𝑁 = 3
𝑣 𝑁 = 1, 𝑣 12 = 𝑣 23 = 2/3,
𝑣 13 = 𝑣 1 = 𝑣 2 = 𝑣 3 = 0
𝑋 = 2,4,1 and 𝑐 = 1.
𝐶ℎ
𝐶ℎ
2
2
𝑋𝑑𝑣 = 1 and
𝑋 + 1 𝑑𝑣 = 2 .

𝑐𝑎𝑣
𝑐𝑎𝑣
3
3
𝑋𝑑𝑣 = 2 ∗ 𝑣 12 + 1 ∗ 𝑣 23 = 2 , but 𝑋 + 1 𝑑𝑣 = 3 ∗ 𝑣 12 + 2 ∗ 𝑣 23 =
1
3 .
3


Corollary 2: A decomposition-integral
ℱ
∙ 𝑑𝑣
satisfies (i) ℱ ∙ 𝑑𝑃 = 𝔼𝑃 ∙ for every probability
𝑃; and (ii) it is monotonic w.r.t. stochastic
dominance for every 𝑣 iff ℱ = ℱ 𝐶ℎ .
Corollary 3: A decomposition-integral ℱ ∙ 𝑑𝑣
satisfies (i) ℱ ∙ 𝑑𝑃 = 𝔼𝑃 ∙ for every probability
𝑃; and (ii) it is translation-invariant for every 𝑣 iff
ℱ = ℱ 𝐶ℎ .




A new definition for integrals w.r.t. capacities.
A new characterization of the concave integral.
Two new characterizations of integral Choquet
(that do not use comonotone additivity).
Finding a trade-off between different desirable
properties.