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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion

An axiomatic characterization of the prudent order preference function

Claude Lamboray

SMA - University of Luxembourg CODE - Universit´ e Libre de Bruxelles

DIMACS-LAMSADE Workshop on Voting Theory and Preference Modelling

October 27, 2006

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion

The context

Combination of a profile of linear orders into a set of prudent

• rders (Arrow and Raynaud, 1986).

Minimize the strongest opposition

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion

Why characterizing the set of prudent orders?

To discover the particularities of the prudence principle. To build a common axiomatic framework for other prudent ranking rules (e.g. Ranked Pairs Rule, Tideman, 1987). To link the results with other ordinal ranking rules (Barbera, 1988, Fortemps and Pirlot, 2004,...).

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion

Outline

1

Prudent orders Definitions Example Link with the majority relation

2

Characterization of the Prudent Order Preference Function Axioms Results

3

Extended Prudent Order Preference Function Definition Result

4

Conclusion

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Definitions Example Link with the majority relation

Overview

1

Prudent orders Definitions Example Link with the majority relation

2

Characterization of the Prudent Order Preference Function Axioms Results

3

Extended Prudent Order Preference Function Definition Result

4

Conclusion

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Definitions Example Link with the majority relation

Prudent Orders

Basic Notations A profile of q linear orders: u = (O1, O2, . . . , Oq) Majority Margins: Bij = {#k : (ai, aj) ∈ Ok} − {#k : (aj, ai) ∈ Ok} Cut-Relation: (ai, aj) ∈ R>λ ⇐ ⇒ Bij > λ, where λ ∈ {−q, . . . 0, . . . , q}.

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Definitions Example Link with the majority relation

Let β be the smallest value such that the corresponding strict cut relation is acyclic: β = min{λ ∈ {−q, . . . , O, . . . , q} : R>λ is acyclic }. Definition of the prudent order preference function PO(u) = {O ∈ LO : R>β ⊆ O} = E(R>β).

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Definitions Example Link with the majority relation

PO(u) = E(R>β) is a non empty set of linear orders !! Proposition For any acyclic relation R, there exists a profile u of linear orders such that PO(u) = E(R). Proposition The following statements are equivalent: OP is a prudent order OP ∈ arg minO∈LO max(ai,aj)∈O Bij (minimize the strongest

• pposition)

OP ∈ arg maxO∈LO min(ai,aj)∈O Bij (maximize the weakest link)

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Definitions Example Link with the majority relation

Example u = (abcde, adbec, adbec, cdbea, cdbae, bcdae, ecdba) a b c d e a .

• 1
• 1
• 1

3 b 1 . 1

• 1

5 c 1

• 1

. 3 1 d 1 1

• 3

. 5 e

• 3
• 5
• 1
• 5

. β = 1 R>β = R>1 = {(a, e), (b, e), (c, d), (d, e)}. (but B(c, d) ≥ 1 and B(d, b) ≥ 1 and B(b, e) ≥ 1 !) 1 acbde 7 cadbe 2 abcde 8 bcade 3 cabde 9 cdabe 4 acdbe 10 cbdae 5 cbade 11 cdbae 6 bacde 12 bcdae

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Definitions Example Link with the majority relation

Link with the majority relation

M is the strict majority relation : (ai, aj) ∈ M ⇐ ⇒ Bij > 0. Let u be a given profile. Let O be any linear order. kO : the smallest number of times that one has to add O to u such that M(u + kOO) = O Let kMIN = minO∈LO kO. Theorem (Debord, 1986) Let u be a profile such that the strict majority relation is not a linear order. O ∈ PO(u) if and only if M(u + kminO) = O.

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Axioms Results

Overview

1

Prudent orders Definitions Example Link with the majority relation

2

Characterization of the Prudent Order Preference Function Axioms Results

3

Extended Prudent Order Preference Function Definition Result

4

Conclusion

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Axioms Results

Characterization of the Prudent Order Preference Function

A preference function f : f : Oq → P(O) \ ∅ u → f (u).

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Axioms Results

Condorcet Consistency (CC) If M is acyclic, then: f (u) ⊆ E(M) Strong Condorcet Consistency (SCC) If M is acyclic, then: f (u) = E(M) SCC implies CC.

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Axioms Results

E-invariance (EI) Let uE be a profile such that Bij = 0∀i, j. Then: f (u + uE) = f (u) Weak homogeneity (WH) If q is odd, then: f (u) ⊆ f (u + u) Homogeneity (H) If q is odd, then: f (u) = f (u + u) H implies WH

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Axioms Results

Definition Let us consider a linear order O and an ordered pair (ai, aj). We say that the linear order O′ is an update of O in favor of pair (ai, aj) if O = (...ajai...) is such that aj directly precedes ai and O′ = (...aiaj...) is obtained by reversing aj and ai in O. Example: Update in favor of (b, a): O = abcde − → O′ = bacde.

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Axioms Results

Update procedure Let u be a profile. For every pair {ai, aj} do the following: If Bij > 0, then we do an update in favor of pair (ai, aj) of a linear order Ok of profile u. If Bij = 0, then

Do nothing. OR We do an update in favor of pair (ai, aj) of a linear order Ok of profile u. OR We do an update in favor of pair (aj, ai) of a linear order Ok of profile u.

Let uupdate be the profile obtained at the end of this procedure

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Axioms Results

Example u update in favor of uupdate O1 dcba (a, b) dcab O2 cabd (a, c) acbd O3 cbda (b, c) bcda O4 dbac (b, d) bdac O5 dcab (c, d) cdab O6 adbc (d, a) dabc O7 abcd n.a. abcd O8 abcd n.a abcd O9 abcd n.a abcd O10 bcda n.a bcda O11 cbda n.a cbda O12 dabc n.a dabc O13 dabc n.a dabc O14 cbda n.a cbda O15 bacd n.a. bacd B(a, b) = 1; B(a, c) = 1; B(b, c) = 3; B(b, d) = 3; B(c, d) = 3; B(d, a) = 3

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Axioms Results

Majority Oriented Profile Convergence (MOPC) Let u be a profile and let uupdate be the profile obtained using the majority oriented update procedure. Then: f (uupdate) ⊆ f (u). Majority Oriented Profile Invariance (MOPI) Let u be a profile and let uupdate be the profile obtained using the majority oriented update procedure. If the strict majority relation

• f uupdate contains at least one cycle, then:

f (uupdate) = f (u). MOPC does not imply MOPI MOPI does not imply MOPC

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Axioms Results

Theorem The prudent order preference function is the largest preference function (in the sense of the inclusion) which verifies Condorcet Consistency, E-Invariance, Weak Homogeneity and Majority Oriented Profile Convergence. Theorem The prudent order preference function is the only preference function which verifies Strong Condorcet Consistency, E-Invariance, Homogeneity and Majority Oriented Profile Invariance.

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Axioms Results

Theorem The prudent order preference function is the largest preference function (in the sense of the inclusion) which verifies Condorcet Consistency, E-Invariance, Weak Homogeneity and Majority Oriented Profile Convergence. Theorem The prudent order preference function is the only preference function which verifies Strong Condorcet Consistency, E-Invariance, Homogeneity and Majority Oriented Profile Invariance.

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Definition Result

Overview

1

Prudent orders Definitions Example Link with the majority relation

2

Characterization of the Prudent Order Preference Function Axioms Results

3

Extended Prudent Order Preference Function Definition Result

4

Conclusion

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Definition Result

Let us come back to the first example: e a c d b Top-cycle partition = successive application of the top-cycle choice function A1 = {b, c, d}, A2 = {a}, A3 = {e}

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Definition Result

Not every prudent order is consistent with the top cycle partition: A1 = {b, c, d}, A2 = {a}, A3 = {e} 1 acbde 7 cadbe 2 abcde 8 bcade 3 cabde 9 cdabe 4 acdbe 10 cbdae 5 cbade 11 cdbae 6 bacde 12 bcdae

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Definition Result

Extended prudent order preference function ∀u XPO(u) = E(R>β ∪ T), where (ai, aj) ∈ T ⇐ ⇒ ai ∈ Ak and aj ∈ Al and k < l. Extended prudent orders are a refinement of the prudent

• rders: XPO ⊆ PO(u) = E(R>β).

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Definition Result

Extended Condorcet Criterion (XCC) (Truchon, 1998) Let us suppose that the strict majority relation of profile u contains at least one cycle and let A1, A2, . . . , Ap be the top-cycle-partition

• f this profile u. We say that a preference function f verifies the

Extended Condorcet criterion if: ∀ai ∈ Ak, aj ∈ Al, k < l : (ai, aj) ∈ O ∀O ∈ f (u).

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion Definition Result

Theorem The extended prudent order preference function is the largest preference function (in the sense of the inclusion) that verifies Condorcet Consistency, E-Invariance, Weak Homogeneity, Majority Oriented Profile Convergence and the Extended Condorcet Criterion.

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion

Overview

1

Prudent orders Definitions Example Link with the majority relation

2

Characterization of the Prudent Order Preference Function Axioms Results

3

Extended Prudent Order Preference Function Definition Result

4

Conclusion

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Prudent orders Characterization of the Prudent Order Preference Function Extended Prudent Order Preference Function Conclusion

Conclusion

A first characterization of the prudent order preference function. Axioms such as MOPC and MOPI are the most specific of the prudent approach. Other prudent ranking rules (e.g. Ranked Pairs Rule) could be characterized in the same axiomatic framework.

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