chambers of arrangements and arrow's impossibility theorem › ~terao ›...

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Chambers of Arrangementsand

Arrow’s Impossibility Theorem

Hiroaki Terao

(Hokkaido University, Sapporo, Japan)

Recent developments on geometrical and algebraic methods in EconomicsHokkaido University, Sapporo

2014.08.22

H. Terao (Hokkaido University) 2014.08.22 1/ 33

Contents

...1 Basic concepts about hyperplane arrangements

...2 Arrow’s Impossibility Theorem (economics version)

...3 Arrow’s Impossibility Theorem (arrangementversion)

...4 Two theorems on arrangements and their chambers

...5 Implications

Contents

...5 Implications

Contents

...5 Implications

Contents

...5 Implications

Contents

...5 Implications

Contents

...5 Implications

1. Basic concepts about hyperplane arrangements

Hyperplane Arrangement

A (central) hyperplane arrangement A is:

A := {H1, . . . ,Hn}

in an ℓ-dimensional vector space V over a field Kdefined by Hi = ker(αi) with αi ∈ V∗(1 ≤ i ≤ n).

A := {H1, . . . ,Hn}

Chambers

When K = R (the real number field), the connectedcomponents of

M(A) := V \n∪

are called chambers.

Chambers

When K = R (the real number field), the connectedcomponents of

M(A) := V \n∪

are called chambers.

Intersection lattice

L(A) = {all intersections of hyperplanes belonging toA}= {∩H∈B

H | B ⊆ A}

and introduce a partial order by X ≥ Y⇔ X ⊆ Y to makeL(A) a partially ordered set.[Agree that L(A) has the minimum V.]Then L(A) is called the intersection lattice.

H | B ⊆ A}

Mobius function

Defineµ : L(A)→ Z

byµ(V) := 1, µ(X) := −

∑Y<X

µ(Y).

Poincare polynomial

Define the Poincare polynomial

π(A, t) :=∑

X∈L(A)

µ(X)(−t)codimX.

Mobius function

byµ(V) := 1, µ(X) := −

∑Y<X

µ(Y).

Poincare polynomial

π(A, t) :=∑

X∈L(A)

µ(X)(−t)codimX.

Mobius function

byµ(V) := 1, µ(X) := −

∑Y<X

µ(Y).

Poincare polynomial

π(A, t) :=∑

X∈L(A)

µ(X)(−t)codimX.

Mobius function

byµ(V) := 1, µ(X) := −

∑Y<X

µ(Y).

Poincare polynomial

π(A, t) :=∑

X∈L(A)

µ(X)(−t)codimX.

Mobius function

byµ(V) := 1, µ(X) := −

∑Y<X

µ(Y).

Poincare polynomial

π(A, t) :=∑

X∈L(A)

µ(X)(−t)codimX.

Factorization TheoremTheorem. (H. T. 1981). Suppose that A is a freearrangement in Cℓ with exponents d1,d2, . . . , dℓ.Then

π(A, t) =ℓ∏

(1+ dit).

Zaslavsky’s Chamber-Counting Formula

Theorem.(Thomas Zaslavsky 1975).

|Chambers| = π(A,1).

If A is a free real arrangement in Rℓ with exponentsd1,d2, . . . , dℓ, then |Chambers| = π(A,1) =

∏ℓi=1(1+ di).

π(A, t) =ℓ∏

(1+ dit).

∏ℓi=1(1+ di).

π(A, t) =ℓ∏

(1+ dit).

∏ℓi=1(1+ di).

π(A, t) =ℓ∏

(1+ dit).

∏ℓi=1(1+ di).

π(A, t) =ℓ∏

(1+ dit).

∏ℓi=1(1+ di).

π(A, t) =ℓ∏

(1+ dit).

∏ℓi=1(1+ di).

Catalan arrangement of type B2 is free with exponents (1,5,7)'

%��

��

��@@

α1 = −1α1 = 0

α1 = 1α2 = −1α2 = 0α2 = 1

α1 + α2 = −1

α1 + α2 = 0

α1 + α2 = 1

2α1 + α2 = −12α1 + α2 = 0

2α1 + α2 = 1

The number of chambers isπ(A,1) = (1+ 1× 1)(1+ 5× 1)(1+ 7× 1) = 96

related to the Edelman-Reiner conjecture (solved by M. Yoshinagain 2004)

%��

��

��@@

α1 = −1α1 = 0

α1 = 1α2 = −1α2 = 0α2 = 1

α1 + α2 = −1

α1 + α2 = 0

α1 + α2 = 1

2α1 + α2 = −12α1 + α2 = 0

2α1 + α2 = 1

%��

��

��@@

α1 = −1α1 = 0

α1 = 1α2 = −1α2 = 0α2 = 1

α1 + α2 = −1

α1 + α2 = 0

α1 + α2 = 1

2α1 + α2 = −12α1 + α2 = 0

2α1 + α2 = 1

%��

��

��@@

α1 = −1α1 = 0

α1 = 1α2 = −1α2 = 0α2 = 1

α1 + α2 = −1

α1 + α2 = 0

α1 + α2 = 1

2α1 + α2 = −12α1 + α2 = 0

2α1 + α2 = 1

%��

��

��@@

α1 = −1α1 = 0

α1 = 1α2 = −1α2 = 0α2 = 1

α1 + α2 = −1

α1 + α2 = 0

α1 + α2 = 1

2α1 + α2 = −12α1 + α2 = 0

2α1 + α2 = 1

2. Arrow’s Impossibility Theorem (economics version)

Assume that a society of m people have ℓ policy optionsand that every individual has his/her own order ofpreferences on the ℓ policy options.

A social welfare function can be interpreted as a votingsystem by which the individual preferences areaggregated into a single societal preference.

We require the following two requirements for areasonable social welfare function:

The Two Requirements

(A) the society prefers the option i to the option j if everyindividual prefers the option i to the option j (Paretoproperty),

(B) whether the society prefers the option i to the optionj only depends which individuals prefer the option i tothe option j (pairwise independence).

Arrow’s Impossibility Theorem (Kenneth Arrow 1950).For ℓ ≥ 3, the only social welfare function satisfying the tworequirements (A) and (B) is a dictatorship, that is, the societalpreference has to be equal to the preference of one particularindividual.

(A) the society prefers the option i to the option j if every individualprefers the option i to the option j (Pareto property),(B) whether the society prefers the option i to the option j onlydepends which individuals prefer the option i to the option j(pairwise independence).

Why is Arrow’s theorem true?

What is the reason behind Arrow’s theorem?Condorcet’s paradox by Marquis Condorcet (1743-94)A,B,C : 3 people, 1,2,3 : 3 optionslists of preferences :A : 1 > 2 > 3,B : 2 > 3 > 1,C : 3 > 1 > 2In this situation it is very hard to decide the societalpreference in a “democratic way” like the majority rule.Roughly speaking, this is the reason why Arrow’sImpossibility Theorem holds.

3. Arrow’s Impossibility Theorem (arrangementversion)

A = {H1,H2, . . . ,Hn} : a real central arrangement in Rℓ

Ch = Ch(A) : the set of chambersHj : defined by αj = 0H+j := {x ∈ Rℓ | αj(x) > 0} : a half-spaceH−j := {x ∈ Rℓ | αj(x) < 0} : the other half-spaceB := {+,−}ϵσj : Ch −→ B are defined by ϵσj (C) = στ if C ⊆ Hτj(σ, τ ∈ B, j = 1, . . . ,n)m : a positive integerChm, Bm : the m-time direct productsϵσj : Chm→ Bm is induced from ϵσj : Ch→ B byϵσj (C1,C2, . . . ,Cm) = (ϵσj (C1), ϵσj (C2), . . . , ϵσj (Cm))

Definition 1.

A map Φ : Chm −→ Ch is called an admissible map if there exists afamily of maps φσj : Bm −→ B (1 ≤ j ≤ n, σ ∈ B = {+,−}) whichsatisfies the following two conditions:(1) φσj (+,+, . . . ,+) = +, and(2) the diagram

ϵσj��

Φ // Chϵσj��

Bmφσj // B

commutes for each j, 1 ≤ j ≤ n, and σ ∈ B = {+,−}.

Definition 1.

ϵσj��

Φ // Chϵσj��

Bmφσj // B

Definition 1.

ϵσj��

Φ // Chϵσj��

Bmφσj // B

Definition 1.

ϵσj��

Φ // Chϵσj��

Bmφσj // B

Definition 1.

ϵσj��

Φ // Chϵσj��

Bmφσj // B

Definition 1 (continuing).

Let AM(A,m) denote the set of all admissible maps determined byA and m.When Φ is an admissible map, a family of maps φσj(1 ≤ j ≤ n, σ ∈ B = {+,−}) satisfying the conditions in Definition 1 isuniquely determined by Φ, A and m.

Definition 2.

For 1 ≤ h ≤ m, let Φ : the projection to the h-th component,φσj : the projection to the h-th component.Then Φ is an admissible map with a family of mapsφσj (1 ≤ j ≤ n, σ ∈ B = {+,−}).We call the admissible maps of this type projective admissiblemaps.

Definition 2.

The Braid Arrangement Case

A : the braid arrangement in Rℓ (ℓ ≥ 3)A = {Hij | 1 ≤ i < j ≤ ℓ} where Hij := ker(xi − xj)H+ij := {(x1, x2, . . . , xℓ) ∈ Rℓ | xi > xj}H−ij = {(x1, x2, . . . , xℓ) ∈ Rℓ | xi < xj}.Sℓ : the permutation group of {1,2, . . . , ℓ}Then Ch(A)↔ Sℓ (One-to-one correspondence) :Each chamber of A can be uniquely expressed as{(x1, x2, . . . , xℓ) ∈ Rℓ | xπ(1) < xπ(2) < · · · < xπ(ℓ)} for a permutationπ ∈ SℓThus the set of orders of preferences↔ Sℓ ↔ Ch(A)

Smℓ ↔ Chm

ϵσj��

Φ // Ch

ϵσj��

↔ Sℓ

Bmφσj // B

Other correspondences are:a social welfare function↔ Φa dictatorship↔ the projection to a component(A) (Pareto property)↔ (1) (φσj (+, . . . ,+) = +)(B) (pairwise independence)↔ (2) (commutativity)φσj ◦ ϵσj = ϵσj ◦ Φ (∀j)

Smℓ ↔ Chm

ϵσj��

Φ // Ch

ϵσj��

↔ Sℓ

Bmφσj // B

Smℓ ↔ Chm

ϵσj��

Φ // Ch

ϵσj��

↔ Sℓ

Bmφσj // B

Smℓ ↔ Chm

ϵσj��

Φ // Ch

ϵσj��

↔ Sℓ

Bmφσj // B

Smℓ ↔ Chm

ϵσj��

Φ // Ch

ϵσj��

↔ Sℓ

Bmφσj // B

Smℓ ↔ Chm

ϵσj��

Φ // Ch

ϵσj��

↔ Sℓ

Bmφσj // B

Smℓ ↔ Chm

ϵσj��

Φ // Ch

ϵσj��

↔ Sℓ

Bmφσj // B

Arrow’s impossibility theorem can be formulated as:Arrow’s Impossibility Theorem (arrangement version)IfA is a braid arrangement with ℓ ≥ 3, then every admissible map isprojective.

Condorcet’s paradox can be interpreted in terms of arrangementsand their chambers:

Braid arrangement in R^3

H13 H23

1=21=3 2=3A

2>1>3 1>2>3

B 2>3>1 1>3>2

3>2>1 3>1>2Lists of preferences

A : 1>2>3 C

B : 2>3>1

C : 3>1>2

(H12)+ ¥ cap (H23)+ ¥ cap (H13)- would satisfy all of the

three, but it is empty. (Condorcet’s paradox)

4. Two theorems on arramgements

Decomposability/Indecomposability of an Arrangement

For a central arrangement A, define the rank of Ar(A) = codimRℓ

∩1≤j≤n Hj

Definition 3. A central arrangement A is said to bedecomposable if there exist nonempty arrangements A1 and A2

such that A = A1 ∪A2 (disjoint) and r(A) = r(A1) + r(A2). In thiscase, write A = A1 ⊎A2

A central arrangement A is said to be indecomposable if it is notdecomposable.

∩1≤j≤n Hj

Remark 1. A = A1 ⊎A2 if and only if the defining polynomials forA1 , ∅ and A2 , ∅ have no common variables after an appropriatelinear coordinate change.

Remark 2. It is also known that A is decomposable if and only if itsPoincare polynomial π(A, t) is divisible by (t + 1)2.

An arrangement of only one hyperplane is always indecomposable.

An arrangement of two hyperplanes is always decomposable.

The Boolean arrangement is always decomposable intoarrangements with only one hyperplane.

Any nonempty real central arrangement A can be uniquely (up toorder) decomposed into nonempty indecomposable arrangements:

A = A1 ⊎A2 ⊎ · · · ⊎ Ar .

The following two theorems completely determine the set AM(A,m)of admissible maps.

Theorem 1. For a nonempty real central arrangement A with thedecomposition

A = A1 ⊎A2 ⊎ · · · ⊎ Ar ,

there exists a natural bijectionAM(A,m) ≃ AM(A1,m) × AM(A2,m) × · · · × AM(Ar ,m)for each positive integer m.

Theorem 1. For a nonempty real central arrangement A with thedecomposition

A = A1 ⊎A2 ⊎ · · · ⊎ Ar ,

there exists a natural bijectionAM(A,m) ≃ AM(A1,m) × AM(A2,m) × · · · × AM(Ar ,m)for each positive integer m.

Theorem 2. Let A be a nonempty indecomposable real centralarrangement and m be a positive integer. Then,(1) if |A| = 1, AM(A,m) = {Φ : Chm→ Ch | Φ(C,C, . . . ,C) = C foreach chamber C},(2) if |A| ≥ 3, every admissible map is projective.

Corollary. Decompose a nonempty real central arrangementAinto nonempty indecomposable arrangements asA = A1 ⊎A2 ⊎ · · · ⊎ Aa ⊎ B1 ⊎ B2 ⊎ · · · ⊎ Bb with|Ap| = 1 (1≤ p ≤ a) and |Bq| ≥ 3 (1≤ q ≤ b).Then, for each positive integer m,

|AM(A,m)| = (2a(2m−2))mb

5. Implications

What do Theorems 1 and 2 imply?

Theorem 1. For a nonempty real central arrangement A with thedecomposition A = A1 ⊎A2 ⊎ · · · ⊎ Ar . there exists a naturalbijectionAM(A,m) ≃ AM(A1,m) × AM(A2,m) × · · · × AM(Ar ,m)for each positive integer m.

5. Implications

hyperplane↔ a political issue

arrangement↔ a set of political issues

A = A1 ⊎A2 ⊎ · · · ⊎ Ar .↔ a set of political issues is grouped intocertain subsets

For each Ai with (|Ai | ≥ 3), there is a “mini-dictator.”

For each Ai with (|Ai | = 1), any voting system (e. g., the simplemajority rule) works as long as unanimous decisions are respected.

5. Implications

This is random thoughts which might mean nothing.However, Theorems mean something mathematically.

5. Implications

This is random thoughts which might mean nothing.However, Theorems mean something mathematically.

5. Implications

This work appeared in Advances in Math. 214 (2007) 366–378

5. Implications

I stop here.Thank you!

5. Implications

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