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Decomposition-Integral: Unifying Choquet and the Concave Integrals

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.

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Decomposition-Integral: Unifying Choquet and the Concave Integrals

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  1. Decomposition-Integral: Unifying Choquet and the Concave Integrals YaaritEven Tel-Aviv University December 2011

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

  3. 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.

  4. Definitions Let be a finite set, . • A capacity over is a function satisfying: (i) . (ii) if , then . • A random variable (r.v.) over is a function . • A subset of will be called an event.

  5. Sub-decompositions and decompositions of a random variable Let be a random variable. • A sub-decomposition of is a finite summation such that: (i) (ii) and for every • If there is an equality in (i), then is a decomposition of . • The value of a decomposition w.r.t. is .

  6. -sub-decompositions and -decompositions Let be a set of subsets of , . • is a -sub-decomposition of if it is a sub-decomposition of and for every • is a -decomposition of if it is a decomposition of and for every

  7. Examples • Suppose and . and are both -decompositions of . • Suppose , , and . has a -decomposition: , and has only -sub-decompositions, such as: .

  8. The decomposition-integral • A vocabulary is a set of subsets of . • A sub-decomposition of is -allowable if it is a -sub-decomposition of and . • The decomposition-integral w.r.t. is defined: = is -allowable sub-decomposition of X}. • The sub-decomposition attaining the maximum is called the optimal sub-decomposition of .

  9. Examples • Suppose , , ,. and . has an optimal decomposition: ,and = . has an optimal sub-decomposition: , and = .

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

  11. The concave integral • Definition (Lehrer): , where the minimum is taken over all concave and homogeneous functions , such that for every . • Lemma (Lehrer): = .

  12. The concave integral as a decomposition-integral • Define . == is -allowable sub-decomposition of X}. • Since is monotonic w.r.t. inclusion, we have: = is -allowable decomposition of X}.

  13. The concave integral • Since allows for all decompositions, for every vocabulary , the following inequality holds: , for every . Concluding, that of all the decomposition-integrals, the concave integral attains the highest value.

  14. Choquet integral • Definition: = = , where is a permutation over that satisfies and . • , where .

  15. Choquet integral as a decomposition-integral Definitions: • Any two subsets and of are nested if either or . • A set is called a chain if any two events are nested.

  16. Choquet integral as a decomposition-integral • Define to be the set of all chains. == is -allowable sub-decomposition of X} = is -allowable decomposition of X}.

  17. Examples • Suppose , , , and .

  18. Riemann integral • A partition of is a set , such that all ’s are pairwise disjoint and their union is . • Define to be the set of all partitions of . • The Riemann integral can be defined as .

  19. Properties of the decomposition-integral • Positive homogeneity of degree one: • for every for every , and . • The decomposition-integral and additive capacities: • Proposition: Let be a probability and a vocabulary. Then, for every r.v. iff every has a -decomposition, .

  20. Properties of the decomposition-integral - continuation • Monotonicity: 1. Monotonicity w.r.t. r.v.’s: Fix and and suppose . Then, . 2. Monotonicity w.r.t. capacities: Fix . If for every and every , , then for every r.v. , .

  21. Properties of the decomposition-integral - continuation 3. Monotonicity w.r.t. vocabularies: Fix and suppose and are two vocabularies. Proposition: iff for every and every minimal set , there is such that . • A set is minimal if the variables are algebraically independent.

  22. Properties of the decomposition-integral - continuation • Additivity: • Two variables and are comonotone if for every , . • Comonotoneadditivitymeans that if X and Y are comonotone, then: . • Let . and are comonotoneiff their optimal decompositions use the same in .

  23. Properties of the decomposition-integral - continuation • Example: , and . Fix . Suppose is small enough so that the optimal D-decompositions of X and Y use : , but for , taking :

  24. Properties of the decomposition-integral - continuation • Fix and . is leaner than if there are optimal decompositions in which employ every indicator that employs: The optimal decomposition of is , , and the optimal decomposition of is , , and .

  25. Properties of the decomposition-integral - continuation • Proposition: Fix a vocabulary such that every has an optimal decomposition for every . Suppose that for every , whenever there are two different decompositions of the same variable, , there is that contains all the ’s with and all the ’s with . Then, for every and every and , where is leaner than , .

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

  27. Concavity • is concave if for every two r.v. and , and : • Theorem 1: The decomposition-integral is concave for every , iff there exists a vocabulary containing only one such that .

  28. A new characterization of the concave integral • Corollary 1: A decomposition-integral satisfies (i) for every event and capacity ; and (ii) is concave, iff.

  29. Monotonicity w.r.t. stochastic dominance • stochastically dominates w.r.t. if for every number , . • is monotonic w.r.t. stochastic dominance if implies • Theorem 2: The decomposition-integral is monotonic w.r.t. stochastic dominance iff is the collection of all chains of the same size ().

  30. Monotonicity w.r.t. stochastic dominance • Example: , , , and . Obviously, . : > : <

  31. Translation-invariance • is translation-invariant for every , if for every , , when . • Theorem 3: The decomposition-integral is translation-invariant for every iff the vocabulary is (i) composed of chains; and (ii) any is contained in that includes .

  32. Translation-invariance • Example: , , and . and . , but -.

  33. A new characterization of Choquet integral • Corollary 2: A decomposition-integral satisfies (i) for every probability ; and (ii) it is monotonic w.r.t. stochastic dominance for every iff. • Corollary 3: A decomposition-integral satisfies (i) for every probability ; and (ii) it is translation-invariant for every iff .

  34. 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 comonotoneadditivity). • Finding a trade-off between different desirable properties.

  35. THE END

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