Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Wreath Products in Algebraic Voting Theory Committed to Committees Ian Calaway 1 Joshua Csapo 2 Dr. Erin McNicholas 3 Eric Samelson 4 1 Macalester College 2 University of Michigan-Flint 3 Faculty Advisor Department of Mathematics Willamette University 4 Willamette University Willamette Mathematics Consortium REU, July 24, 2015 1 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Outline Algebraic Framework 1 Voting for Committees 2 Wreath Products 3 Decomposing a Q S m [ S n ]-module 4 2 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Algebraic Framework Ballot Structure Vote Aggregation Point Allocation (Voting System) Election Results 3 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Ballots Ballots → List of allowable tabloids A A B A B A B C C B C C These are elements of χ (3) , χ (2 , 1) , χ (1 , 2) , and χ (1 , 1 , 1) , respectively. 4 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Ballots Ballots → List of allowable tabloids A A B A B A B C C B C C These are elements of χ (3) , χ (2 , 1) , χ (1 , 2) , and χ (1 , 1 , 1) , respectively. Many elections focus on a single type of ballot structure. An exception to this is approval voting. 4 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Vote Aggregation: The Profile Space The components of the profile vector indicate the number of votes for a given tabloid option on the ballot. 5 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Vote Aggregation: The Profile Space The components of the profile vector indicate the number of votes for a given tabloid option on the ballot. For an election in which voters give full rankings of three candidates (i.e. an election on the tabloids of χ (1 , 1 , 1) ) the following is an example of a profile: 3 ABC 2 ACB 0 BAC p = � 2 BCA 0 CAB 4 CBA 5 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Vote Aggregation: The Profile Space The space of all possible profiles (i.e. all possible vote totals for a given ballot) is called the profile space. 6 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Vote Aggregation: The Profile Space The space of all possible profiles (i.e. all possible vote totals for a given ballot) is called the profile space. These profile spaces form Q S n -modules. 6 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Vote Aggregation: The Profile Space The space of all possible profiles (i.e. all possible vote totals for a given ballot) is called the profile space. These profile spaces form Q S n -modules. Definition An FG-module is a vector space over the field F where there is a representation of the group G on the vector space, and multiplication is understood as the associated group action. 3 ABC 0 ABC 2 ACB 2 ACB 0 BAC 3 BAC → (12) → 2 BCA 2 BCA 0 CAB 4 CAB 4 CBA 0 CBA 6 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Point Allocation: Voting Systems Consider the following two voting systems for a full ranking ( χ (1 , 1 , 1) ) Plurality → (1 , 0 , 0) 3 ABC 2 ACB 5 A 0 BAC → 2 B 2 BCA 4 C 0 CAB 4 CBA Borda → (2 , 1 , 0) 7 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Point Allocation: Voting Systems Consider the following two voting systems for a full ranking ( χ (1 , 1 , 1) ) Plurality → (1 , 0 , 0) Borda → (2 , 1 , 0) 3 ABC 2 ACB 10 A 0 BAC → 11 B 2 BCA 12 C 0 CAB 4 CBA 8 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Voting Systems as Linear Transformations The previous Borda example can be represented as the following linear transformation: ABC ACB BAC BCA CAB CBA � 2 2 1 0 1 0 � A T (2 , 1 , 0) = 1 0 2 2 0 1 B 0 1 0 1 2 2 C 9 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Voting Systems as Linear Transformations The previous Borda example can be represented as the following linear transformation: ABC ACB BAC BCA CAB CBA � 2 2 1 0 1 0 � A T (2 , 1 , 0) = 1 0 2 2 0 1 B 0 1 0 1 2 2 C Any system that gives candidates points based on their position within voters’ rankings can be represented by a similar linear transformation. 9 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Results Space When one of these voting systems is applied to a profile, a results vector is created. A results vector encodes the number of points that every potential outcome receives for the election. T w ( � p ) = � r 10 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Results Space When one of these voting systems is applied to a profile, a results vector is created. A results vector encodes the number of points that every potential outcome receives for the election. T w ( � p ) = � r The results spaces are also Q S n -module. 10 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Schur’s Lemma Both profile spaces and results spaces are Q S n -modules. These positional scoring systems are Q S n -module homomorphisms, which can be proven with the previous fact and the fact that these systems are neutral. 11 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Schur’s Lemma Both profile spaces and results spaces are Q S n -modules. These positional scoring systems are Q S n -module homomorphisms, which can be proven with the previous fact and the fact that these systems are neutral. Theorem Schur’s Lemma: Every nonzero module homomorphism between irreducible modules is an isomorphism. 11 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Schur’s Lemma Both profile spaces and results spaces are Q S n -modules. These positional scoring systems are Q S n -module homomorphisms, which can be proven with the previous fact and the fact that these systems are neutral. Theorem Schur’s Lemma: Every nonzero module homomorphism between irreducible modules is an isomorphism. Schur’s Lemma helps us determine what actually affects the outcome of an election given a specific voting system. 11 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Schur’s Lemma Both profile spaces and results spaces are Q S n -modules. These positional scoring systems are Q S n -module homomorphisms, which can be proven with the previous fact and the fact that these systems are neutral. Theorem Schur’s Lemma: Every nonzero module homomorphism between irreducible modules is an isomorphism. Schur’s Lemma helps us determine what actually affects the outcome of an election given a specific voting system. Definition A submodule is a subspace of a module that is invariant under the group action. 11 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Key Points of Algebraic Voting Theory Domain 12 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Key Points of Algebraic Voting Theory Domain → Maps 12 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Key Points of Algebraic Voting Theory Domain → Maps → Codomain 12 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Voting for Committees What happens if we want to elect a subset of the candidates rather than a single candidate? 13 / 49
Algebraic Framework Voting for Committees Wreath Products Decomposing a Q S m [ S n ]-module References Voting for Committees What happens if we want to elect a subset of the candidates rather than a single candidate? 4 A 2 B 0 C 2 AB → 1 AC 5 BC 3 ABC 13 / 49
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