Corvinus University of Budapest

Axiomatic Districting

Clemens Puppe and Attila Tasnadi

Computational Social Choice Meeting

Maastricht University

April 9, 2014

Introduction Districtings Solutions Axioms Characterization Conclusion

Agenda

1 Introduction

2 Districtings

3 Solutions

4 Axioms

5 Characterization

6 Conclusion

Agenda

1 Introduction

2 Districtings

3 Solutions

4 Axioms

5 Characterization

6 Conclusion

Agenda

1 Introduction

2 Districtings

3 Solutions

4 Axioms

5 Characterization

6 Conclusion

Agenda

1 Introduction

2 Districtings

3 Solutions

4 Axioms

5 Characterization

6 Conclusion

Agenda

1 Introduction

2 Districtings

3 Solutions

4 Axioms

5 Characterization

6 Conclusion

Agenda

1 Introduction

2 Districtings

3 Solutions

4 Axioms

5 Characterization

6 Conclusion

Agenda

1 Introduction

2 Districtings

3 Solutions

4 Axioms

5 Characterization

6 Conclusion

Redistricting and gerrymandering

Redistricting has to be carried out to prevent geographicmalapportionment.

A redistricting can favor a certain party.

If redistricting is manipulated for an electoral advantage, wespeak of gerrymandering.

Districts should be

equally sized in population,

connected and

compact.

Districts should be

connected and

compact.

Districts should be

connected and

compact.

Districts should be

connected and

compact.

Districts should be

connected and

compact.

Districts should be

connected and

compact.

Districts should be

connected and

compact.

Example: Arizona

Example: Illinois

Example: California

Legal issues and court cases.Computer science:

Possibilities: Vickrey (PolSciQ, 1961), Hess et al. (OpRes,1965), Garfinkel and Nemhauser (ManSci, 1970);Constraints: Nagel (Polity, 1972), Altman (RutCLTJ, 1997),Puppe and Tasnadi (MCM, 2008; EL, 2009);Practice: Altman, MacDonald and McDonald (SSCR, 2005),and Chambers and Miller (QJPolSci, 2010).

Political science: Owen and Grofman (PolGeoQ, 1988),Gelman and King (AmPolSciRev, 1994), and Gilligan andMatsusaka (PubChoice, 2006).Economics:

Social welfare: Besley and Preston (QJE, 2007), Coate andKnight (QJE, 2007) and Bracco (JPubEcon);Redistricting games: Friedman and Holden (AER, 2008), andGul and Pesendorfer (AER, 2010);Axiomatic representation: Chambers (GEB, 2008; JET, 2009),and Bervoets and Merlin (IJGT, 2012).

Agenda

1 Introduction

2 Districtings

3 Solutions

4 Axioms

5 Characterization

6 Conclusion

Definition

A districting problem Π is given by (X ,A, µ, µA, µB , t,G ), where

voters are located within a subset X of the plane R2,

A is a σ-algebra on X ,

the distribution of voters is given by a measure µ on (X ,A),

the distributions of party A and party B supporters are givenby measures µA and µB on (X ,A) such that µ = µA + µB ,

t is the number of seats in parliament,

the geography G ⊆ A is a collection of admissible districtssatisfying µ(g) = µ(X )/t and µA(g) 6= µB(g) for all g ∈ Gsuch that

G admits a partitioning of X , i.e.there exist mutually disjointsets g1, . . . , gt ∈ G such that ∪ti=1gi = X .

Definition

A districting for problem Π = (X ,A, µ, µA, µB , t,G ) is a subsetD ⊆ G such that D forms a partition of X and #D = t.

DΠ set of all possible districtings for problem Π.

Definition

A solution F associates to each districting problem Π a non-emptyset of chosen districtings FΠ ⊆ DΠ.

δA(D) number of districts won by party A under D.

δB(D) number of districts won by party B under D.

δA(D) = {δA(D) : D ∈ D} for any D ⊆ DΠ.

δB(D) = {δB(D) : D ∈ D} for any D ⊆ DΠ.

Definition

Agenda

1 Introduction

2 Districtings

3 Solutions

4 Axioms

5 Characterization

6 Conclusion

Optimal Gerrymandering Solutions

Definition

The optimal solution OA for party A determines for alldistricting problems Π the set of those districtings that maximizethe number of winning districts for party A, i.e.

OAΠ = arg max

D∈DΠ

δA(D).

Similarly, the optimal solution OB for party B maximizes thenumber of winning districts for party B.

Optimal Gerrymandering Solutions

Definition

The optimal solution OA for party A determines for alldistricting problems Π the set of those districtings that maximizethe number of winning districts for party A, i.e.

OAΠ = arg max

D∈DΠ

δA(D).

Similarly, the optimal solution OB for party B maximizes thenumber of winning districts for party B.

Most Equal Solution

Definition

The most equal solution ME determines for all districtingproblems Π the set of districtings which are most equal in terms ofwinning districts, i.e.

MEΠ = arg minD∈DΠ

|δA(D)− δB(D)| .

Most Equal Solution

Definition

The most equal solution ME determines for all districtingproblems Π the set of districtings which are most equal in terms ofwinning districts, i.e.

MEΠ = arg minD∈DΠ

|δA(D)− δB(D)| .

Most Unequal Solution

Definition

The most unequal solution MU determines for all districtingproblems Π the set of districtings which are most unequal in termsof winning districts, i.e.

MUΠ = arg maxD∈DΠ

|δA(D)− δB(D)| .

Most Unequal Solution

Definition

The most unequal solution MU determines for all districtingproblems Π the set of districtings which are most unequal in termsof winning districts, i.e.

MUΠ = arg maxD∈DΠ

|δA(D)− δB(D)| .

Least Biased Solution

Definition

The least biased solution LB determines for all districtingproblems Π the set of those districtings that minimize the absolutedifference between shares in winning districts and shares in votes,i.e.

LBΠ = arg minD∈DΠ

∣∣∣∣δA(D)

t− µA(X )

µ(X )

∣∣∣∣ = arg minD∈DΠ

∣∣∣∣δB(D)

t− µB(X )

µ(X )

∣∣∣∣ .

Least Biased Solution

Definition

The least biased solution LB determines for all districtingproblems Π the set of those districtings that minimize the absolutedifference between shares in winning districts and shares in votes,i.e.

LBΠ = arg minD∈DΠ

∣∣∣∣δA(D)

t− µA(X )

µ(X )

∣∣∣∣ = arg minD∈DΠ

∣∣∣∣δB(D)

t− µB(X )

µ(X )

∣∣∣∣ .

Complete Solution

Definition

The complete solution associates to all districting problems Π theset of all possible districtings DΠ.

Complete Solution

Definition

The complete solution associates to all districting problems Π theset of all possible districtings DΠ.

Agenda

1 Introduction

2 Districtings

3 Solutions

4 Axioms

5 Characterization

6 Conclusion

Two-district determinacy

A solution F satisfies two-district determinacy if for all districtingproblems Π with t = 2, the sets δA(FΠ) and δB(FΠ) are singletons.

Two-district determinacy

A solution F satisfies two-district determinacy if for all districtingproblems Π with t = 2, the sets δA(FΠ) and δB(FΠ) are singletons.

Two-district uniformity

A solution F satisfies two-district uniformity if for all districtingproblems Π and Π′ with t = 2:

δA(DΠ) = δA(DΠ′) and δB(DΠ) = δB(DΠ′) impliesδA(FΠ) = δA(FΠ′) and δB(FΠ) = δB(FΠ′).

Two-district uniformity

A solution F satisfies two-district uniformity if for all districtingproblems Π and Π′ with t = 2:

δA(DΠ) = δA(DΠ′) and δB(DΠ) = δB(DΠ′) impliesδA(FΠ) = δA(FΠ′) and δB(FΠ) = δB(FΠ′).

Indifference

A solution F satisfies indifference if for all districting problems Π:

D ∈ FΠ, D ′ ∈ DΠ, δA(D) = δA(D ′) and δB(D) = δB(D ′) impliesD ′ ∈ FΠ.

Indifference

A solution F satisfies indifference if for all districting problems Π:

D ∈ FΠ, D ′ ∈ DΠ, δA(D) = δA(D ′) and δB(D) = δB(D ′) impliesD ′ ∈ FΠ.

Consistency

For all Π, D ∈ DΠ, D ′ ⊆ D, and Y = ∪d∈D′d , denote by Π|Y theinduced subproblem:(

Y ,A|Y , µ|Y , µA|Y , µB |Y ,#D ′,G |Y),

where A|Y := {A ∩ Y : A ∈ A}, G |Y := {g ∈ G : g ⊆ Y }, andµ|Y , µA|Y , µB |Y stand for the restrictions of measures µ, µA, µB to(Y ,A|Y ), respectively.

A solution F satisfies consistency if for all districting problems Π,all D ∈ FΠ, all D ′ ⊆ D and Y = ∪d∈D′d one has D ′ ∈ FΠ|Y .

Consistency

For all Π, D ∈ DΠ, D ′ ⊆ D, and Y = ∪d∈D′d , denote by Π|Y theinduced subproblem:(

Y ,A|Y , µ|Y , µA|Y , µB |Y ,#D ′,G |Y),

where A|Y := {A ∩ Y : A ∈ A}, G |Y := {g ∈ G : g ⊆ Y }, andµ|Y , µA|Y , µB |Y stand for the restrictions of measures µ, µA, µB to(Y ,A|Y ), respectively.

A solution F satisfies consistency if for all districting problems Π,all D ∈ FΠ, all D ′ ⊆ D and Y = ∪d∈D′d one has D ′ ∈ FΠ|Y .

Anonymity

A solution F satisfies anonymity if exchanging the distributions ofparty A and party B voters µA and µB does not change the set ofchosen districtings, i.e.

D ∈ F(X ,A,µ,µA,µB ,t,G) ⇐⇒ D ∈ F(X ,A,µ,µB ,µA,t,G).

Anonymity

A solution F satisfies anonymity if exchanging the distributions ofparty A and party B voters µA and µB does not change the set ofchosen districtings, i.e.

D ∈ F(X ,A,µ,µA,µB ,t,G) ⇐⇒ D ∈ F(X ,A,µ,µB ,µA,t,G).

Agenda

1 Introduction

2 Districtings

3 Solutions

4 Axioms

5 Characterization

6 Conclusion

Linked Geographies

d d t t d td d d t t dd d d t t td d d d t td d t t d td d d t t d@@@@

d d t t d td d d t t dd d d t t td d d d t td d t t d td d d t t d

d d t t d td d d t t dd d d t t td d d d t td d t t d td d d t t dDefinition

The geography G of a problem Π is linked if for any two possibledistrictings D,D ′ ∈ DΠ there exists a sequence D1, . . . ,Dk ofdistrictings such that D = D1, {D2, . . . ,Dk−1} ⊆ DΠ, D ′ = Dk ,and #(Di ∩ Di+1) = t − 2 for all i = 1, . . . , k − 1.

Linked Geographies

d d t t d td d d t t dd d d t t td d d d t td d t t d td d d t t d@@@@

d d t t d td d d t t dd d d t t td d d d t td d t t d td d d t t d

d d t t d td d d t t dd d d t t td d d d t td d t t d td d d t t dDefinition

The geography G of a problem Π is linked if for any two possibledistrictings D,D ′ ∈ DΠ there exists a sequence D1, . . . ,Dk ofdistrictings such that D = D1, {D2, . . . ,Dk−1} ⊆ DΠ, D ′ = Dk ,and #(Di ∩ Di+1) = t − 2 for all i = 1, . . . , k − 1.

A bounded subset A of R2 is strictly connected if its boundary ∂Ais a Jordan curve. A continuous f : X → R is nowhere constant iffor any ∀x ∈ X : ∀N(x) : ∃y ∈ N(x) : f (x) 6= f (y).

Definition (Regular Districting Problems)

Π = (X ,A, µ, µA, µB , t,G ) is called regular if

1 X is a bounded and strictly connected subset of R2,

2 A equals the set of Borel sets on X (i.e. A = B(X )),

3 µ is a finite and absolutely continuous measure on (X ,B(X ))with respect to the Lebesgue measure,

4 G consists of all bounded, strictly connected and µ(X )/t sizedsubsets lying in B(X ) and satisfying the no ties condition,

5 there exists a continuous nowhere constant functionf : X → R such that ∀C ∈ B(X ) : µA(C ) =

∫C f (ω)dµ(ω),

6 µB is given by µB(C ) = µ(C )− µA(C ) for all C ∈ B(X ).

A Characterization and an Impossibility

Theorem

The optimal solutions OA and OB are the only solutions thatsatisfy two-district determinacy, two-district uniformity, indifferenceand consistency on linked geographies.

Corollary

There does not exist a two-district determinate, two-districtuniform, indifferent, consistent and anonymous solution on linkedgeographies.

Theorem

Corollary

Theorem

Corollary

Agenda

1 Introduction

2 Districtings

3 Solutions

4 Axioms

5 Characterization

6 Conclusion

Thank you!

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