Decomposition-Integral: Unifying Choquet and the Concave Integrals Yaarit Even Tel-Aviv University December 2011

Non-additive integral Decision making under uncertainty Game theory Multi-criteria decision aid (MCDA) Insurance and financial assets pricing

In this presentation 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.

Definitions Let 𝑁 be a finite set, 𝑁 =𝑛. A capacity 𝑣 over 𝑁 is a function 𝑣: 2 𝑁 → 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 𝑖=1 𝑘 𝛼 𝑖 𝕀 𝐴 𝑖 such that: (i) 𝑖=1 𝑘 𝛼 𝑖 𝕀 𝐴 𝑖 ≤𝑋 (ii) 𝛼 𝑖 ≥0 and 𝐴 𝑖 ⊆𝑁 for every 𝑖=1,…,𝑘. If there is an equality in (i), then 𝑖=1 𝑘 𝛼 𝑖 𝕀 𝐴 𝑖 is a decomposition of 𝑋. The value of a decomposition w.r.t. 𝑣 is 𝑖=1 𝑘 𝛼 𝑖 𝑣 𝐴 𝑖 .

𝐷-sub-decompositions and 𝐷-decompositions Let 𝐷 be a set of subsets of 𝑁, 𝐷⊆ 2 𝑁 . 𝑖=1 𝑘 𝛼 𝑖 𝕀 𝐴 𝑖 is a 𝐷-sub-decomposition of 𝑋 if it is a sub-decomposition of 𝑋 and 𝐴 𝑖 ∈𝐷 for every 𝑖=1,…,𝑘. 𝑖=1 𝑘 𝛼 𝑖 𝕀 𝐴 𝑖 is a 𝐷-decomposition of 𝑋 if it is a decomposition of 𝑋 and 𝐴 𝑖 ∈𝐷 for every 𝑖=1,…,𝑘.

Examples Suppose 𝐷= 2 𝑁 and 𝑋= 𝕀 𝑁 . 𝑖=1 𝑛 𝕀 𝑖 and 𝕀 𝑁 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 .

The decomposition-integral A vocabulary ℱ is a set of subsets of 2 𝑁 . 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: ℱ 𝑋𝑑𝑣 = max { 𝐴 𝑖 ∈𝐷 𝛼 𝑖 𝑣 𝐴 𝑖 ; 𝐴 𝑖 ∈𝐷 𝛼 𝑖 𝕀 𝐴 𝑖 is ℱ-allowable sub-decomposition of X}. The sub-decomposition attaining the maximum is called the optimal sub-decomposition of 𝑋.

Examples 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 𝑋𝑑𝑣 = 0∗ 1 3 + 1 2 +1=1 1 2 . 𝑌 has an optimal sub-decomposition: 2∗ 𝕀 23 , and ℱ 𝑌𝑑𝑣 = 2∗ 5 6 =1 2 3 .

The decomposition-integral as a generalization of known integrals Choquet integral The concave integral Riemann integral Shilkret integral And other plausible integration schemes.

The concave integral 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 𝕀 𝑆 ≥𝑣 𝑆 for every 𝑆⊆𝑁. Lemma (Lehrer): 𝑐𝑎𝑣 𝑋𝑑𝑣 = 𝑚𝑎𝑥{ 𝑆⊆𝑁 𝛼 𝑆 𝑣 𝑆 ; 𝑆⊆𝑁 𝛼 𝑆 𝕀 𝑆 =𝑋 , 𝛼 𝑆 ≥0}.

The concave integral as a decomposition-integral Define ℱ 𝑐𝑎𝑣 = 2 𝑁 . 𝑐𝑎𝑣 𝑋𝑑𝑣 = ℱ 𝑐𝑎𝑣 𝑋𝑑𝑣 = 𝑚𝑎𝑥{ 𝐴 𝑖 ∈𝐷 𝛼 𝑖 𝑣 𝐴 𝑖 ; 𝐴 𝑖 ∈𝐷 𝛼 𝑖 𝕀 𝐴 𝑖 is ℱ 𝑐𝑎𝑣 -allowable sub-decomposition of X}. Since 𝑣 is monotonic w.r.t. inclusion, we have: 𝑐𝑎𝑣 𝑋𝑑𝑣 = 𝑚𝑎𝑥 { 𝐴 𝑖 ∈𝐷 𝛼 𝑖 𝑣 𝐴 𝑖 ; 𝐴 𝑖 ∈𝐷 𝛼 𝑖 𝕀 𝐴 𝑖 is ℱ 𝑐𝑎𝑣 -allowable decomposition of X}.

The concave integral Since ℱ 𝑐𝑎𝑣 allows for all decompositions, for every vocabulary ℱ, the following inequality holds: ℱ ∙𝑑𝑣 ≤ ℱ 𝑐𝑎𝑣 ∙𝑑𝑣 , for every 𝑣. Concluding, that of all the decomposition-integrals, the concave integral attains the highest value.

Choquet integral Definition: 𝐶ℎ 𝑋𝑑𝑣= 0 ∞ 𝑣 𝑖∈𝑁 𝑋 𝑖 ≥𝛼 𝑑𝛼 = 𝐶ℎ 𝑋𝑑𝑣= 0 ∞ 𝑣 𝑖∈𝑁 𝑋 𝑖 ≥𝛼 𝑑𝛼 = = 𝑖=1 𝑛 𝑋 𝜎 𝑖 − 𝑋 𝜎 𝑖−1 𝑣 𝐴 𝑖 , where 𝜎 is a permutation over 𝑁 that satisfies 𝑋 𝜎 1 ≤…≤ 𝑋 𝜎 𝑛 and 𝐴 𝑖 = 𝜎 𝑖 ,…,𝜎 𝑛 ( 𝑋 𝜎 0 =0). 𝑋= 𝛼 𝑖 𝕀 𝐴 𝑖 , where 𝛼 𝑖 = 𝑋 𝜎 𝑖 − 𝑋 𝜎 𝑖−1 .

Choquet integral as a decomposition-integral Definitions: Any two subsets 𝐴 and 𝐵 of 𝑁 are nested if either 𝐴⊆𝐵 or 𝐵⊆𝐴. A set 𝐷⊆ 2 𝑁 is called a chain if any two events 𝐴, 𝐵∈𝐷 are nested.

Choquet integral as a decomposition-integral Define ℱ 𝐶ℎ to be the set of all chains. 𝐶ℎ 𝑋𝑑𝑣 = ℱ 𝐶ℎ 𝑋𝑑𝑣 = 𝑚𝑎𝑥 { 𝐴 𝑖 ∈𝐷 𝛼 𝑖 𝑣 𝐴 𝑖 ; 𝐴 𝑖 ∈𝐷 𝛼 𝑖 𝕀 𝐴 𝑖 is ℱ 𝐶ℎ -allowable sub-decomposition of X} = 𝑚𝑎𝑥 { 𝐴 𝑖 ∈𝐷 𝛼 𝑖 𝑣 𝐴 𝑖 ; 𝐴 𝑖 ∈𝐷 𝛼 𝑖 𝕀 𝐴 𝑖 is ℱ 𝐶ℎ -allowable decomposition of X}.

Examples 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 𝑋𝑑𝑣=1∗𝑣 𝑁 +1∗𝑣 12 =1 1 2 𝑐𝑎𝑣 𝑋𝑑𝑣=2∗𝑣 1 +1∗𝑣 2 +1∗𝑣 23 =1 5 6

Riemann integral A partition of 𝑁 is a set 𝐷= 𝐴 1 ,…, 𝐴 𝑘 , such that all 𝐴 𝑖 ’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 .

Shilkret integral Define ℱ 𝑠𝑖𝑛𝑔 = 𝐴 ;𝐴⊆𝑁 . The Shilkret integral can be defined as: ℱ 𝑠𝑖𝑛𝑔 𝑋𝑑𝑣 = 𝑚𝑎𝑥 { 𝐴 𝑖 ∈𝐷 𝛼 𝑖 𝑣 𝐴 𝑖 ; 𝐴 𝑖 ∈𝐷 𝛼 𝑖 𝕀 𝐴 𝑖 is ℱ 𝑠𝑖𝑛𝑔 -allowable sub-decomposition of X} = 𝑚𝑎𝑥 𝛼𝑣 𝐴 ; 𝛼 𝕀 𝐴 ≤𝑋, 𝐴⊆𝑁, 𝛼≥0 = 𝑚𝑎𝑥 𝛼∙𝑣 𝑋≥𝛼 ; 𝛼≥0 .

Properties of the decomposition-integral Positive homogeneity of degree one: for every 𝜆>0, ℱ 𝜆𝑋𝑑𝑣 =𝜆 ℱ 𝑋𝑑𝑣 for every 𝑋, 𝑣 and ℱ. The decomposition-integral and additive capacities: Proposition: Let 𝑃 be a probability and ℱ a vocabulary. Then, 𝔼 𝑃 𝑋 = ℱ 𝑋𝑑𝑃 for every r.v. 𝑋 iff every 𝑋 has a 𝐷-decomposition, 𝐷∈ℱ.

Properties of the decomposition-integral - continuation 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 .

Properties of the decomposition-integral - continuation 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⊆ 2 𝑁 is minimal if the variables 𝕀 𝐴 , 𝐴∈𝐶 are algebraically independent.

Properties of the decomposition-integral - continuation 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+𝑌 𝑑𝑣 ℱ 𝑋+𝑌 𝑑𝑣 . Let ℱ= ℱ 𝐶ℎ . 𝑋 and 𝑌 are comonotone iff their optimal decompositions use the same 𝐷 in ℱ.

Properties of the decomposition-integral - continuation 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 D-decompositions of X and Y use 𝐷= 1 , 2 : ℱ 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

Properties of the decomposition-integral - continuation 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 𝛽 𝐴 𝕀 𝐴 , 𝛽 𝐴 >0, and the optimal decomposition of 𝑋 is 𝐵𝜖 𝐶 ′ 𝛼 𝐵 𝕀 𝐵 , 𝛼 𝐵 >0, and 𝐶⊆𝐶′.

Properties of the decomposition-integral - continuation 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 𝛿 𝐴 𝕀 𝐴 = 𝐵∈ 𝐷 ′ 𝛾 𝐵 𝕀 𝐵 , there is 𝐷 ′′ 𝜖 ℱ that contains all the 𝐴’s with 𝛿 𝐴 >0 and all the 𝐵’s with 𝛾 𝐵 >0. Then, for every 𝑣 and every 𝑋 and 𝑌, where 𝑌 is leaner than 𝑋, ℱ 𝑋𝑑𝑣 + ℱ 𝑌𝑑𝑣 = ℱ 𝑋+𝑌 𝑑𝑣 .

Desirable properties Concavity Monotonicity w.r.t. stochastic dominance Translation-invariance

Concavity ℱ ∙𝑑𝑣 is concave if for every two r.v. 𝑋 and 𝑌, and 𝛾∈ 0, 1 : ℱ 𝛾𝑋+ 1−𝛾 𝑌 𝑑𝑣 ≥𝛾 ℱ 𝑋𝑑𝑣 + 1−𝛾 ℱ 𝑌𝑑𝑣 Theorem 1: The decomposition-integral ℱ ∙𝑑𝑣 is concave for every 𝑣, iff there exists a vocabulary ℱ′ containing only one 𝐷 𝐷⊆ 2 𝑁 𝐷⊆ 2 𝑁 such that ℱ ∙𝑑𝑣 = ℱ ′ ∙𝑑𝑣 .

A new characterization of the concave integral Corollary 1: A decomposition-integral ℱ ∙𝑑𝑣 satisfies (i) ℱ 𝕀 𝐴 𝑑𝑣 ≥𝑣 𝐴 for every event 𝐴 and capacity 𝑣; and (ii) ℱ ∙𝑑𝑣 is concave, iff ℱ= ℱ 𝑐𝑎𝑣 .

Monotonicity w.r.t. stochastic dominance 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).

Monotonicity w.r.t. stochastic dominance 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 : 𝐶ℎ 𝑋𝑑𝑣=1 1 3 > 𝐶ℎ 𝑌𝑑𝑣 =1 1 6 ℱ= ℱ 𝑐𝑎𝑣 : 𝑐𝑎𝑣 𝑋𝑑𝑣=1 5 6 < 𝑐𝑎𝑣 𝑌𝑑𝑣=2

Translation-invariance ℱ ∙𝑑𝑣 is translation-invariant for every 𝑣, if for every 𝑐>0, ℱ 𝑋+𝑐 𝑑𝑣 = ℱ 𝑋𝑑𝑣 +𝑐, when 𝑣 𝑁 =1. 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 𝕀 𝑁 .

Translation-invariance 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 𝑋𝑑𝑣=1 2 3 and 𝐶ℎ 𝑋+1 𝑑𝑣=2 2 3 . 𝑐𝑎𝑣 𝑋𝑑𝑣=2∗𝑣 12 +1∗𝑣 23 =2 , but - 𝑐𝑎𝑣 𝑋+1 𝑑𝑣=3∗𝑣 12 +2∗𝑣 23 =3 1 3 .

A new characterization of Choquet integral Corollary 2: A decomposition-integral ℱ ∙𝑑𝑣 satisfies (i) ℱ ∙𝑑𝑃 = 𝔼 𝑃 ∙ 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 = 𝔼 𝑃 ∙ for every probability 𝑃; and (ii) it is translation-invariant for every 𝑣 iff ℱ= ℱ 𝐶ℎ .

For Conclusion 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.

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