Corvinus University of Budapest Axiomatic Districting Clemens Puppe and Attila Tasn´ adi Computational Social Choice Meeting Maastricht University April 9, 2014
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
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Agenda
1 Introduction
2 Districtings
3 Solutions
4 Axioms
5 Characterization
6 Conclusion
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Agenda
1 Introduction
2 Districtings
3 Solutions
4 Axioms
5 Characterization
6 Conclusion
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Agenda
1 Introduction
2 Districtings
3 Solutions
4 Axioms
5 Characterization
6 Conclusion
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Agenda
1 Introduction
2 Districtings
3 Solutions
4 Axioms
5 Characterization
6 Conclusion
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Agenda
1 Introduction
2 Districtings
3 Solutions
4 Axioms
5 Characterization
6 Conclusion
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Agenda
1 Introduction
2 Districtings
3 Solutions
4 Axioms
5 Characterization
6 Conclusion
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization 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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization 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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization 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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization 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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization 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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization 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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization 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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Example: Arizona
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Example: Illinois
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Example: California
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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).
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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).
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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).
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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).
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Agenda
1 Introduction
2 Districtings
3 Solutions
4 Axioms
5 Characterization
6 Conclusion
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization 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 .
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Agenda
1 Introduction
2 Districtings
3 Solutions
4 Axioms
5 Characterization
6 Conclusion
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization 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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization 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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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)| .
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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)| .
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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)| .
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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)| .
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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 )
∣∣∣∣ .
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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 )
∣∣∣∣ .
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Complete Solution
Definition
The complete solution associates to all districting problems Π theset of all possible districtings DΠ.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Complete Solution
Definition
The complete solution associates to all districting problems Π theset of all possible districtings DΠ.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Agenda
1 Introduction
2 Districtings
3 Solutions
4 Axioms
5 Characterization
6 Conclusion
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Two-district determinacy
Axiom
A solution F satisfies two-district determinacy if for all districtingproblems Π with t = 2, the sets δA(FΠ) and δB(FΠ) are singletons.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Two-district determinacy
Axiom
A solution F satisfies two-district determinacy if for all districtingproblems Π with t = 2, the sets δA(FΠ) and δB(FΠ) are singletons.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Two-district uniformity
Axiom
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′).
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Two-district uniformity
Axiom
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′).
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Indifference
Axiom
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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Indifference
Axiom
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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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.
Axiom
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 .
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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.
Axiom
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 .
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Anonymity
Axiom
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).
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Anonymity
Axiom
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).
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Agenda
1 Introduction
2 Districtings
3 Solutions
4 Axioms
5 Characterization
6 Conclusion
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization 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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization 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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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 ).
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
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.
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Agenda
1 Introduction
2 Districtings
3 Solutions
4 Axioms
5 Characterization
6 Conclusion
Clemens Puppe and Attila Tasnadi
Axiomatic Districting
Introduction Districtings Solutions Axioms Characterization Conclusion
Thank you!
Clemens Puppe and Attila Tasnadi
Axiomatic Districting