Where is the problem? Facts and Hypotheses. A general frame and - - PowerPoint PPT Presentation

PREFERENCES ON INTERVALS∗

Alexis Tsouki` as(*), Philippe Vincke(+) (*)LAMSADE-CNRS, Universit´ e Paris Dauphine tsoukias@lamsade.dauphine.fr (+)SMG-ISRO, Universit´ e Libre de Bruxelles pvincke@ulb.ac.be

∗with the help of Meltem ¨

Ozt¨ urk

• Where is the problem?
• Facts and Hypotheses.
• A general frame and some open questions.
• Old and new results.
• Where are we going?

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COMPARING INTERVALS (1)

• P(x, y)

y x

• I(x, y)

y x y x y x

2

COMPARING INTERVALS (2)

• P(x, y)

y x

• Q(x, y)

y x x y

• I(x, y)

y x

3

COMPARING INTERVALS (3)

• P(x, y)

y x

• Q(x, y)

y x

• I(x, y)

x y y x

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What are we looking for?

• Regularities of preference structures:
• Structures;
• Hidden Hypotheses;
• Esthetics.
• Representation Theorems

Given a set A of alternatives and a preference structure on it, which are the necessary and sufficient conditions in or- der to have an appropriate numerical representation through intervals?

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Why these are problems?

• Check coherence of decision maker.
• Check coherence of a numerical representation.
• Show equivalences of numerical representations.
• How to model hesitation?

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Basic References

• Fishburn P.C., Interval Orders and Interval Graphs, J. Wiley,

New York, 1985.

• Trotter W.T., Combinatorics and partially ordered sets, John

Hopkins University Press, Baltimore, 1992.

• Pirlot M., Vincke Ph., Semi Orders, Kluwer Academic, Dor-

drecht, 1997.

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Notation

• A is the set of objects on which preference relations apply;
• x, y, z, w, · · · ∈ A;
• P, Q, I, R, L, · · ·: preference relations;
• P1, P2, · · · Pn: possibly subscribed;
• f, g, h, r, l, · · · : A → R numerical representations;
• f1, f2, · · · fn: possibly subscribed;
• r(x), l(x): respectively right and left extreme of an interval;
• α, β, γ, δ · · ·: constants;
• The usual logical notation applies;
• R.L ⇔ ∀x, y ∃z : R(x, z)∧L(z, y)
• Io(x, y) ⇔ ∀j fj(x) = fj(y) ⇔ x = y

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First Hypotheses I consider the following definition Definition 1 A preference structure is a collection of binary re- lations Pj j = 1, · · · n, partitioning the universe of discourse A×A:

• ∀x, y, j Pj(x, y) → ¬Pi=j(x, y);
• ∀x, y∃j Pj(x, y) ∨ Pj(y, x)

And possibly the following two requirements: RQ1 Each preference relation in a preference structure should be uniquely defined by its properties. RQ2 Each preference structure should be characterised by a unique preference relation.

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Facts and Hypotheses Fact: Any symmetric binary relation can be seen as the union

• f two asymmetric relations, the one being the inverse of the
• ther, and Io.

Remark 1 Separation of symmetric relations helps in under- standing the underlying structure of interval comparison.

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Facts and Hypotheses H1 We consider only intervals of the reals. Therefore there will be no incomparability in the preference structures considered. H2 If necessary we associate to each interval a flat uncertainty distribution. Each point in an interval may equally be the “real value”. H3 Without loss of generality we can consider only asymmetric relations. H4 We consider only discrete sets. Therefore we can consider

• nly strict inequalities.

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A general framework Two different perspectives (A).

• A. The preference structure:
• 1. Only strict preference and indifference.
• 2. An “intermediate” relation between strict preference and in-

difference. 3. Discrete states of preference between strict preference and indifference. 4. A continuous valuation between strict preference and indif- ference. NB: Two interpretations for the “intermediate” relations:

• hesitation due to uncertainty;
• difference states of preference intensity.

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A general framework Two different perspectives (B).

• B. The number of values associated to each interval.
• Two values: a value and a threshold OR a right and a left

extreme point.

• Three values:

a value and two thresholds OR a left, an intermediate and a right extreme point OR an extreme point and an extreme interval.

• Four values: a value and three thresholds OR two extreme

intervals etc..

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A general framework

2 values 3 values > 3 values 2 asymmetric Interval Orders Split Interval Orders Tolerance and relations and Semi Orders and Semi Orders Bi-tolerance

• rders

3 asymmetric PQI Interval Orders Pseudo orders relations and Semi Orders and double

• threshold orders

n asymmetric Multiple relations

• Interval Orders

and Semi Orders valued Valued Preferences relations Fuzzy Interval Orders and Semi Orders Continuous PQI Interval Orders

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A first question Denoting by xi the ”length” of the ith segment of the interval associated to x, how many preference relations is possible to establish,

• for free numerical representations?
• for coherent numerical representations?

∀x, y, j x1 > y1 ⇔ xj > yj

• for weakly monotone numerical representations?

∀x, y, i, j i > j, x1 > y1 ⇔ xi ≥ yj

• for monotone numerical representations?

∀x, y, i i > j, x1 > y1 ⇔ xi ≥ yi ≥ xi−1 ≥ yi−1

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free coherent weak monotone monotone 2 values: 3 2 2 2 3 values: 10 5 4 3 4 values: 35 14 8 4 n values:

(2n)! 2(n!)2 1 n+1(2n n )

?(2n−1)? n

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Is there a structure among such relations? Any such preference relation is defined through a sequence of values of x and y in increasing order. For instance f1(y), · · · fn(y), f1(x), · · · fn(x) identifies the “strongest” preference relation for any structure comparing intervals with n values. We subscribe each relation with an index i ∈ I = {1, · · · m} Definition 2 For any two relations Pl, Pk, l, k ∈ I we write Pl ⊲Pk and we read “relation Pl is stronger than relation Pk” iff relation Pk can be obtained from Pl by a single shift of values of x and y

• r it exists a sequence of Pi such that Pl ⊲ · · · Pi ⊲ · · · Pk.

Relation ⊲ is a partial order defining a complete lattice on the set of possible preference relations.

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For instance, using two values P1(x, y) : f1(y), f1(x), f2(x), f2(y) P2(x, y) : f1(y), f1(x), f2(y), f2(x) P3(x, y) : f1(y), f2(y), f1(x), f2(x) P1

✛

P2

✛

P3 Interval orders: P = P3, I = P1 ∪ P2 ∪ Io ∪ P −1

1

∪ P −1

2

Partial Orders of dimension. 2: P = P3 ∪ P2, I = P1 ∪ Io ∪ P −1

1

Semi Orders: P = P3, I = P2 ∪ Io ∪ P −1

2

, P1 empty PQI Interval orders: P = P3, Q = P2, I = P1 ∪ Io ∪ P −1

1

PQI Semi orders: P = P3, Q = P2, I = Io, P1 empty

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Using three values P1(x, y) : f1(y), f1(x), f2(x), f3(x), f2(y), f3(y) P2(x, y) : f1(y), f1(x), f2(x), f2(y), f3(x), f3(y) P3(x, y) : f1(y), f1(x), f2(y), f2(x), f3(x), f3(y) P4(x, y) : f1(y), f1(x), f2(x), f2(y), f3(y), f3(x) P5(x, y) : f1(y), f1(x), f2(y), f2(x), f3(y), f3(x) P6(x, y) : f1(y), f2(y), f1(x), f2(x), f3(x), f3(y) P7(x, y) : f1(y), f1(x), f2(y), f3(y), f2(x), f3(x) P8(x, y) : f1(y), f2(y), f1(x), f2(x), f3(y), f3(x) P9(x, y) : f1(y), f2(y), f1(x), f3(y), f2(x), f3(x) P10(x, y) : f1(y), f2(y), f3(y), f1(x), f2(x), f3(x)

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P1

✛

P2

• ✠

❅ ❅ ❅ ❅ ❅ ■

P3 P4

✛ ✛

P5 P6

✛ ✛

P7 P8

✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ☛

• ✠

❅ ❅ ❅ ❅ ❅ ■

P9

✛

P10

❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❑

Split Interval orders: P = P10 ∪ P9, I the rest Double Threshold orders: P = P10 Q = P9 ∪ P8 ∪ P6, I the rest Pseudo Orders: P = P10 Q = P9 ∪ P8, I = P5 ∪ P7 ∪ Io ∪ P −1

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∪ P −1

5

, P1, P2, P3, P4, P6 empty Constant thresholds: P = P10 Q = P9, I = P5 ∪ P7 ∪ Io ∪ P −1

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∪ P −1

5

, P1, P2, P3, P4, P6, P8 empty

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Old and new results Definition 3 A P, I preference structure is an interval order iff: ∃l, r : A → R, l(x) < r(x) such that:

• P(x, y) ⇔ l(x) > r(y);
• I(x, y) ⇔ l(x) < r(y) and l(y) < r(x).

Theorem 1 The necessary and sufficient conditions for a P, I preference structure to be an interval order are: P.I.P ⊂ P.

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Old and new results Definition 4 A PQI preference structure on a finite set A is a PQI interval order iff ∃ : l, r : A → R+, such that ∀ x, y ∈ A: i) r(x) ≥ l(x); ii) P(x, y)⇔l(x) > r(y); iii) Q(x, y)⇔r(x) > r(y) ≥ l(x) > l(y); iv) I(x, y)⇔r(x) ≥ r(y) ≥ l(y) ≥ l(x) or r(y) ≥ r(x) ≥ l(x) ≥ l(y). Theorem 2 A P, Q, I preference structure on a finite set A is a PQI interval order iff there exists a partial order Il such that: i) I = Il ∪ Ir ∪ I0 where I0 = {(x, x), x ∈ A} and Ir = I−1

l

; ii) (P ∪ Q ∪ Il).P ⊂ P ; iii) P.(P ∪ Q ∪ Ir) ⊂ P ; iv) (P ∪ Q ∪ Il).Q ⊂ P ∪ Q ∪ Il ; v) Q.(P ∪ Q ∪ Ir) ⊂ P ∪ Q ∪ Ir.

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Old and new results Theorem 3 An interval order is a P2, P1, Io preference struc- ture such that:

• P2P2 ⊆ P2
• P2P1 ⊆ P2
• P −1

1

P2 ⊆ P2

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Old and new results Theorem 4 A PQI interval order is a P3, P2, P1, Io preference structure such that:

• P3P3 ⊆ P3
• P2P3 ⊆ P3
• P3P2 ⊆ P3
• P3P1 ⊆ P3
• P −1

1

P3 ⊆ P3

• P2P2 ⊆ P2 ∪ P3
• P1P2 ⊆ P1 ∪ P2
• P2P −1

1

⊆ P −1 ∪ P2

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Where are we going?

• A general frame for comparing intervals.
• General structure for representation theorems
• Generalisation of positive and negative reasons.
• Data mining and classification.
• New aggregation procedures.

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