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Democratic Elections in Faulty Distributed Systems Himanshu Chauhan and Vijay K. Garg Parallel and Distributed Systems Lab, Department of Electrical and Computer Engineering, PDSL, UT Austin Democratic Elections - ICDCN13 1 / 33 Outline

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Democratic Elections in Faulty Distributed Systems

Himanshu Chauhan and Vijay K. Garg

Parallel and Distributed Systems Lab, Department of Electrical and Computer Engineering,

PDSL, UT Austin Democratic Elections - ICDCN’13 1 / 33

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Outline

Motivation

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Outline

Motivation Social Choice and Social Welfare

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Outline

Motivation Social Choice and Social Welfare Social Choice with Byzantine Faults

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Outline

Motivation Social Choice and Social Welfare Social Choice with Byzantine Faults Social Welfare with Byzantine Faults

Pruned-Kemeny-Young Scheme for Byzantine Social Welfare

PDSL, UT Austin Democratic Elections - ICDCN’13 2 / 33

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Outline

Motivation Social Choice and Social Welfare Social Choice with Byzantine Faults Social Welfare with Byzantine Faults

Pruned-Kemeny-Young Scheme for Byzantine Social Welfare

Simulation Results

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Outline

Motivation Social Choice and Social Welfare Social Choice with Byzantine Faults Social Welfare with Byzantine Faults

Pruned-Kemeny-Young Scheme for Byzantine Social Welfare

Simulation Results Conclusion

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Motivation – Leader Election

Conventional Problem Node with the highest id should be the leader. All the nodes in the system should agree on the leader.

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Motivation – Leader Election

Conventional Problem Node with the highest id should be the leader. All the nodes in the system should agree on the leader. Philosophers of Ancient Athens would protest!

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Motivation – Leader Election

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Democratic Leader Election

Elect a leader

Each node has individual preferences Conduct an election where every node votes

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Democratic Leader Election

Elect a leader

Each node has individual preferences Conduct an election where every node votes

Use Case: Job processing system Leader distributes work in the system

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Democratic Leader Election

Elect a leader

Each node has individual preferences Conduct an election where every node votes

Use Case: Job processing system Leader distributes work in the system Worker nodes vote, based upon:

Latency of communication with prospective leader Individual work load

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Democratic Leader Election

Elect a leader

Each node has individual preferences Conduct an election where every node votes

Use Case: Job processing system Leader distributes work in the system Worker nodes vote, based upon:

Latency of communication with prospective leader Individual work load

Enter ‘Byzantine’ Voters!

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Why Not Use Top-Choice Approach?

‘Multivalued Byzantine Agreement’, Turpin and Coan 1984, ‘k−set Consensus’, Prisco et al. 1999

Every voter sends her top choice Run Byzantine Agreement

Agree on the choice with most votes

PDSL, UT Austin Democratic Elections - ICDCN’13 5 / 33

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Why Not Use Top-Choice Approach?

‘Multivalued Byzantine Agreement’, Turpin and Coan 1984, ‘k−set Consensus’, Prisco et al. 1999

Every voter sends her top choice Run Byzantine Agreement

Agree on the choice with most votes P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c

PDSL, UT Austin Democratic Elections - ICDCN’13 5 / 33

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Why Not Use Top-Choice Approach?

‘Multivalued Byzantine Agreement’, Turpin and Coan 1984, ‘k−set Consensus’, Prisco et al. 1999

Every voter sends her top choice Run Byzantine Agreement

Agree on the choice with most votes P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c Elect choice with most votes (at top) : c or b

PDSL, UT Austin Democratic Elections - ICDCN’13 5 / 33

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Why Not Use Top-Choice Approach?

‘Multivalued Byzantine Agreement’, Turpin and Coan 1984, ‘k−set Consensus’, Prisco et al. 1999

Every voter sends her top choice Run Byzantine Agreement

Agree on the choice with most votes P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c Elect choice with most votes (at top) : c or b But . . .

PDSL, UT Austin Democratic Elections - ICDCN’13 5 / 33

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Why Not Use Top-Choice Approach?

‘Multivalued Byzantine Agreement’, Turpin and Coan 1984, ‘k−set Consensus’, Prisco et al. 1999

Every voter sends her top choice Run Byzantine Agreement

Agree on the choice with most votes P1 P2 P3 P4 P5 P6 P7 1st choice b b b a 2nd choice a a a a a a b 3rd choice b b b Elect choice with most votes (at top) : c or b But . . . #(a > b) = 4, #(b > a) = 3

PDSL, UT Austin Democratic Elections - ICDCN’13 5 / 33

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Why Not Use Top-Choice Approach?

‘Multivalued Byzantine Agreement’, Turpin and Coan 1984, ‘k−set Consensus’, Prisco et al. 1999

Every voter sends her top choice Run Byzantine Agreement

Agree on the choice with most votes P1 P2 P3 P4 P5 P6 P7 1st choice c c c a 2nd choice a a a a a a 3rd choice c c c c Elect choice with most votes (at top) : c or b But . . . #(a > b) = 4, #(b > a) = 3 and #(a > c) = 4, #(c > a) = 3

PDSL, UT Austin Democratic Elections - ICDCN’13 5 / 33

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Model & Constructs

System n processes (voters) f Byzantine processes (voters) : bad Non-faulty processes (voters) : good f < n/3

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Model & Constructs

System n processes (voters) f Byzantine processes (voters) : bad Non-faulty processes (voters) : good f < n/3 Jargon A: Set of candidates Ranking: Total order over the set of candidates. Vote: A voter’s preference ranking over candidates. Ballot : Collection of all votes. Scheme : Mechanism that takes a ballot as input and outputs a winner.

PDSL, UT Austin Democratic Elections - ICDCN’13 6 / 33

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Conducting Distributed Democratic Elections

Use Interactive Consistency

Agree on everyone’s vote1 Agree on the ballot

Use a scheme to decide the winner

1We use Gradecast based Byzantine Agreement by Ben-Or et al. PDSL, UT Austin Democratic Elections - ICDCN’13 7 / 33

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Byzantine Social Choice

Social Choice Given a ballot, declare a candidate as the winner of the election.

Arrow 1950-51, Buchanan 1954, Graaff 1957

Byzantine Social Choice Given a set of n processes of which at most f are faulty, and a set A of k choices, design a protocol elects one candidate as the social choice, while meeting the ‘protocol requirements’.

PDSL, UT Austin Democratic Elections - ICDCN’13 8 / 33

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Byzantine Social Welfare

Social Welfare Given a ballot, produce a total order over the set of candidate.

Arrow 1950-51, Buchanan 1954, Graaff 1957, Farquharson 1969

Byzantine Social Welfare Given a set of n processes of which at most f are faulty, and a set A of k choices, design a protocol that produces a total order over A, while meeting the ‘protocol requirements’.

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Protocol Requirements

1 Agreement: All good processes decide on the same choice/ranking.

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Protocol Requirements

1 Agreement: All good processes decide on the same choice/ranking. 2 Termination: The protocol terminates in a finite number of rounds.

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Validity Condition

Validity: Requirement on the choice/ranking decided, based upon the votes of good processes.

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Validity Condition

Validity: Requirement on the choice/ranking decided, based upon the votes of good processes. S: If v is the top choice of all good voters, then v must be the winner. S′: If v is the last choice of all good voters, then v must not be the winner. M′: If v is last choice of majority of good voters, then v must not be the winner.

PDSL, UT Austin Democratic Elections - ICDCN’13 11 / 33

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Validity Conditions

P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c Table: Ballot of 7 votes (P6, P7 Byzantine)

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Validity Conditions

P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c Table: Ballot of 7 votes (P6, P7 Byzantine) M (Elect majority of good voters) : elect b

PDSL, UT Austin Democratic Elections - ICDCN’13 12 / 33

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Validity Conditions

P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c Table: Ballot of 7 votes (P6, P7 Byzantine) M (Elect majority of good voters) : elect b P (Do not elect a candidate that is not the top choice of any good voters) : do not elect a

PDSL, UT Austin Democratic Elections - ICDCN’13 12 / 33

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Byzantine Social Choice – Impossibilities

BSC(k, V ) Byzantine Social Choice problem with k candidates, and validity condition/requirement V . BSC(2, M):

PDSL, UT Austin Democratic Elections - ICDCN’13 13 / 33

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Byzantine Social Choice – Impossibilities

BSC(k, V ) Byzantine Social Choice problem with k candidates, and validity condition/requirement V . BSC(2, M): M: elect top choice of majority of good votes

PDSL, UT Austin Democratic Elections - ICDCN’13 13 / 33

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Byzantine Social Choice – Impossibilities

BSC(k, V ) Byzantine Social Choice problem with k candidates, and validity condition/requirement V . BSC(2, M): M: elect top choice of majority of good votes Impossible to solve for f ≥ n/4

PDSL, UT Austin Democratic Elections - ICDCN’13 13 / 33

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Byzantine Social Choice – Impossibilities

BSC(k, V ) Byzantine Social Choice problem with k candidates, and validity condition/requirement V . BSC(2, M): M: elect top choice of majority of good votes Impossible to solve for f ≥ n/4 Reason: f ≥ n/4 ⇒ can not differentiate b/w good and bad votes BSC(2, M′): M′: do not elect the last choice of majority of good votes Impossible to solve for f ≥ n/4

PDSL, UT Austin Democratic Elections - ICDCN’13 13 / 33

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Byzantine Social Choice – Possibilities

BSC(k, S ∧ M′): S: if v is first choice of all good voters, elect v M′: if v′ is last choice of majority of good voters, do not elect v′

PDSL, UT Austin Democratic Elections - ICDCN’13 14 / 33

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Byzantine Social Choice – Possibilities

BSC(k, S ∧ M′): S: if v is first choice of all good voters, elect v M′: if v′ is last choice of majority of good voters, do not elect v′ Solvable for k ≥ 3

PDSL, UT Austin Democratic Elections - ICDCN’13 14 / 33

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Byzantine Social Choice – Possibilities

BSC(k, S ∧ M′): S: if v is first choice of all good voters, elect v M′: if v′ is last choice of majority of good voters, do not elect v′ Solvable for k ≥ 3 Approach:

P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c Round 1 : Agree on last choices of all voters

PDSL, UT Austin Democratic Elections - ICDCN’13 14 / 33

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Byzantine Social Choice – Possibilities

BSC(k, S ∧ M′): S: if v is first choice of all good voters, elect v M′: if v′ is last choice of majority of good voters, do not elect v′ Solvable for k ≥ 3 Approach:

P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c Round 1 : Agree on last choices of all voters Remove any candidates that appears ⌊(n − f )/2 + 1⌋ times or more

PDSL, UT Austin Democratic Elections - ICDCN’13 14 / 33

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Byzantine Social Choice – Possibilities

BSC(k, S ∧ M′): S: if v is first choice of all good voters, elect v M′: if v′ is last choice of majority of good voters, do not elect v′ Solvable for k ≥ 3 Approach:

P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c n = 7, f = 2, ⌊(n − f )/2 + 1⌋ = 3 Round 1 : Agree on last choices of all voters Remove any candidates that appears ⌊(n − f )/2 + 1⌋ times or more f < n/3 ∧ k ≥ 3 ⇒ at least one candidate that would not be removed

PDSL, UT Austin Democratic Elections - ICDCN’13 14 / 33

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Byzantine Social Choice – Possibilities

BSC(k, S ∧ M′): S: if v is first choice of all good voters, elect v M′: if v′ is last choice of majority of good voters, do not elect v′ Solvable for k ≥ 3 Approach:

P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c n = 7, f = 2, ⌊(n − f )/2 + 1⌋ = 3 Round 1 : Agree on last choices of all voters Remove any candidates that appears ⌊(n − f )/2 + 1⌋ times or more f < n/3 ∧ k ≥ 3 ⇒ at least one candidate that would not be removed Round 2 : Use top choices from remaining candidates, agree and decide

PDSL, UT Austin Democratic Elections - ICDCN’13 14 / 33

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BSC(k, V ) Results – Summarized

Requirement Unsolvable Solvable S

  • k ≥ 2

S′

  • k ≥ 2

M f ≥ n/4 ∧ k ≥ 2

  • M′

f ≥ n/4 ∧ k = 2 k ≥ 3 P f ≥ 1 ∧ k ≥ n f < min(n/k, n/3) ∧ 2 ≤ k < n Table: Impossibilities & Possibilities for BSC(k, V )

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Byzantine Social Welfare – Schemes

Given a ballot, produce a total order over the set of candidates

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Byzantine Social Welfare – Schemes

Given a ballot, produce a total order over the set of candidates Place-Plurality Scheme:

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Byzantine Social Welfare – Schemes

Given a ballot, produce a total order over the set of candidates Place-Plurality Scheme: k candidates

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Byzantine Social Welfare – Schemes

Given a ballot, produce a total order over the set of candidates Place-Plurality Scheme: k candidates for 1 ≤ i ≤ k ci = candidate with most votes at position i in ballot result[i] = ci done

PDSL, UT Austin Democratic Elections - ICDCN’13 16 / 33

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Byzantine Social Welfare – Schemes

Given a ballot, produce a total order over the set of candidates Place-Plurality Scheme: k candidates for 1 ≤ i ≤ k ci = candidate with most votes at position i in ballot result[i] = ci done

P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c

PDSL, UT Austin Democratic Elections - ICDCN’13 16 / 33

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Byzantine Social Welfare – Schemes

Given a ballot, produce a total order over the set of candidates Place-Plurality Scheme: k candidates for 1 ≤ i ≤ k ci = candidate with most votes at position i in ballot result[i] = ci done

P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c

PDSL, UT Austin Democratic Elections - ICDCN’13 16 / 33

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Byzantine Social Welfare – Schemes

Given a ballot, produce a total order over the set of candidates Place-Plurality Scheme: k candidates for 1 ≤ i ≤ k ci = candidate with most votes at position i in ballot result[i] = ci done

P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c

PDSL, UT Austin Democratic Elections - ICDCN’13 16 / 33

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Byzantine Social Welfare – Schemes

Given a ballot, produce a total order over the set of candidates Place-Plurality Scheme: k candidates for 1 ≤ i ≤ k ci = candidate with most votes at position i in ballot result[i] = ci done

P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c

Result : b ≻ a ≻ c

PDSL, UT Austin Democratic Elections - ICDCN’13 16 / 33

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Median of a Ballot

Distance (d) between rankings: # of pair-orderings on which rankings differ

Pairwise Comparison, Condorcet, circa 1785

PDSL, UT Austin Democratic Elections - ICDCN’13 17 / 33

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Median of a Ballot

Distance (d) between rankings: # of pair-orderings on which rankings differ

Pairwise Comparison, Condorcet, circa 1785 r r ′ d a b 1 b a – differ on c c (a, b)

PDSL, UT Austin Democratic Elections - ICDCN’13 17 / 33

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Median of a Ballot

Distance (d) between rankings: # of pair-orderings on which rankings differ

Pairwise Comparison, Condorcet, circa 1785 r r ′ d a c 2 b b – differ on c a (a, b) and (b, c)

PDSL, UT Austin Democratic Elections - ICDCN’13 17 / 33

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Median of a Ballot

Distance (d) between rankings: # of pair-orderings on which rankings differ

Pairwise Comparison, Condorcet, circa 1785 r r ′ d a c 2 b b – differ on c a (a, b) and (b, c)

Median (m) of ballot: Ranking that has least distance from overall pair-wise comparisons in the ballot

PDSL, UT Austin Democratic Elections - ICDCN’13 17 / 33

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Kemeny-Young Scheme

(1) J. Kemeny, 1959, (2) H. Young, 1995

Goal: Get as close to the median as possible.

PDSL, UT Austin Democratic Elections - ICDCN’13 18 / 33

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Kemeny-Young Scheme

(1) J. Kemeny, 1959, (2) H. Young, 1995

Goal: Get as close to the median as possible. For ranking r, let Pr := ordered pairs from r.

PDSL, UT Austin Democratic Elections - ICDCN’13 18 / 33

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Kemeny-Young Scheme

(1) J. Kemeny, 1959, (2) H. Young, 1995

Goal: Get as close to the median as possible. For ranking r, let Pr := ordered pairs from r. Example: r = a ≻ b ≻ c then, Pr = {(a, b) (b, c) (a, c)}

PDSL, UT Austin Democratic Elections - ICDCN’13 18 / 33

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Kemeny-Young Scheme

(1) J. Kemeny, 1959, (2) H. Young, 1995

Goal: Get as close to the median as possible. For ranking r, let Pr := ordered pairs from r. Example: r = a ≻ b ≻ c then, Pr = {(a, b) (b, c) (a, c)} For a given ballot B: score(r, B) =

  • p ∈ Pr

(frequency of p in B) Sk: set of all permutations of k candidates (k! permutations) foreach ranking r ∈ Sk do compute scorer = score(r, B) done

PDSL, UT Austin Democratic Elections - ICDCN’13 18 / 33

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Kemeny-Young Scheme

(1) J. Kemeny, 1959, (2) H. Young, 1995

Goal: Get as close to the median as possible. For ranking r, let Pr := ordered pairs from r. Example: r = a ≻ b ≻ c then, Pr = {(a, b) (b, c) (a, c)} For a given ballot B: score(r, B) =

  • p ∈ Pr

(frequency of p in B) Sk: set of all permutations of k candidates (k! permutations) foreach ranking r ∈ Sk do compute scorer = score(r, B) done select ranking with maximum scorer value as the outcome

PDSL, UT Austin Democratic Elections - ICDCN’13 18 / 33

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Kemeny-Young Scheme – Example

Candidates: {a,b,c}

P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c #(a ≻ b) = 4, #(b ≻ a) = 3, #(a ≻ c) = 4, #(c ≻ a) = 3, #(b ≻ c) = 4, #(c ≻ b) = 3

PDSL, UT Austin Democratic Elections - ICDCN’13 19 / 33

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Kemeny-Young Scheme – Example

Candidates: {a,b,c}

P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c #(a ≻ b) = 4, #(b ≻ a) = 3, #(a ≻ c) = 4, #(c ≻ a) = 3, #(b ≻ c) = 4, #(c ≻ b) = 3 Permutations: a a b b c c b c a c a b c b c a b a

PDSL, UT Austin Democratic Elections - ICDCN’13 19 / 33

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Kemeny-Young Scheme – Example

Candidates: {a,b,c}

P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c #(a ≻ b) = 4, #(b ≻ a) = 3, #(a ≻ c) = 4, #(c ≻ a) = 3, #(b ≻ c) = 4, #(c ≻ b) = 3 Permutations: a a b b c c b c a c a b c b c a b a pairs: {(a, b) (b, c) (a, c)}

PDSL, UT Austin Democratic Elections - ICDCN’13 19 / 33

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Kemeny-Young Scheme – Example

Candidates: {a,b,c}

P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c #(a ≻ b) = 4, #(b ≻ a) = 3, #(a ≻ c) = 4, #(c ≻ a) = 3, #(b ≻ c) = 4, #(c ≻ b) = 3 Permutations: a a b b c c b c a c a b c b c a b a 12 pairs: {(a, b) (b, c) (a, c)}

PDSL, UT Austin Democratic Elections - ICDCN’13 19 / 33

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Kemeny-Young Scheme – Example

Candidates: {a,b,c}

P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c #(a ≻ b) = 4, #(b ≻ a) = 3, #(a ≻ c) = 4, #(c ≻ a) = 3, #(b ≻ c) = 4, #(c ≻ b) = 3 Permutations: a a b b c c b c a c a b c b c a b a 12 11 11 10 10 9

PDSL, UT Austin Democratic Elections - ICDCN’13 19 / 33

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Kemeny-Young Scheme – Example

Candidates: {a,b,c}

P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c #(a ≻ b) = 4, #(b ≻ a) = 3, #(a ≻ c) = 4, #(c ≻ a) = 3, #(b ≻ c) = 4, #(c ≻ b) = 3 Permutations: a a b b c c b c a c a b c b c a b a 12 11 11 10 10 9 Kemeny-Young Scheme Result: a ≻ b ≻ c

PDSL, UT Austin Democratic Elections - ICDCN’13 19 / 33

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Pruned-Kemeny-Young Scheme (this paper)

Objective: Minimize the influence of bad voters on the outcome

PDSL, UT Austin Democratic Elections - ICDCN’13 20 / 33

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Pruned-Kemeny-Young Scheme (this paper)

Objective: Minimize the influence of bad voters on the outcome f bad voters (f < n/3) B: Agreed upon ballot; Sk: set of all permutations of k candidates

PDSL, UT Austin Democratic Elections - ICDCN’13 20 / 33

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Pruned-Kemeny-Young Scheme (this paper)

Objective: Minimize the influence of bad voters on the outcome f bad voters (f < n/3) B: Agreed upon ballot; Sk: set of all permutations of k candidates foreach ranking r ∈ Sk do F = f most distant rankings from r in B define B′ = B\F compute scorer = score(r, B′) done

PDSL, UT Austin Democratic Elections - ICDCN’13 20 / 33

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Pruned-Kemeny-Young Scheme (this paper)

Objective: Minimize the influence of bad voters on the outcome f bad voters (f < n/3) B: Agreed upon ballot; Sk: set of all permutations of k candidates foreach ranking r ∈ Sk do F = f most distant rankings from r in B define B′ = B\F compute scorer = score(r, B′) done select ranking with maximum scorer value as the outcome

PDSL, UT Austin Democratic Elections - ICDCN’13 20 / 33

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Pruned-Kemeny-Young – Example

n = 7, f = 2 P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c

PDSL, UT Austin Democratic Elections - ICDCN’13 21 / 33

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Pruned-Kemeny-Young – Example

n = 7, f = 2 P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c a a b b c c b c a c a b c b c a b a

PDSL, UT Austin Democratic Elections - ICDCN’13 21 / 33

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Pruned-Kemeny-Young – Example

n = 7, f = 2 P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c a a b b c c b c a c a b c b c a b a

PDSL, UT Austin Democratic Elections - ICDCN’13 21 / 33

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SLIDE 74

Pruned-Kemeny-Young – Example

n = 7, f = 2 P1 P2 P3 P6 P7 1st choice b b b c a 2nd choice a a a a b 3rd choice c c c b c a a b b c c b c a c a b c b c a b a 11

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SLIDE 75

Pruned-Kemeny-Young – Example

n = 7, f = 2 P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c a a b b c c b c a c a b c b c a b a 9 8 11 6 10 6

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SLIDE 76

Pruned-Kemeny-Young – Example

n = 7, f = 2 P1 P2 P3 P4 P5 P6 P7 1st choice b b b c c c a 2nd choice a a a a a a b 3rd choice c c c b b b c a a b b c c b c a c a b c b c a b a 9 8 11 6 10 6 Pruned-Kemeny Scheme Result: b ≻ a ≻ c

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SLIDE 77

Evaluating Scheme Efficacy

Suppose ω is an ideal ranking over k candidates ω as the election outcome ⇒ maximum social welfare

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SLIDE 78

Evaluating Scheme Efficacy

Suppose ω is an ideal ranking over k candidates ω as the election outcome ⇒ maximum social welfare All good voters in the system favor ω

goodProb: probability of a good voter putting a ≻ b in her vote if a ≻ω b

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SLIDE 79

Evaluating Scheme Efficacy

Suppose ω is an ideal ranking over k candidates ω as the election outcome ⇒ maximum social welfare All good voters in the system favor ω

goodProb: probability of a good voter putting a ≻ b in her vote if a ≻ω b

All bad voters in the system act hostile

try to minimize social welfare by voting against ω badProb: probability of a bad voter putting b ≻ a in her vote if a ≻ω b

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SLIDE 80

Evaluating Scheme Efficacy

Suppose ω is an ideal ranking over k candidates ω as the election outcome ⇒ maximum social welfare All good voters in the system favor ω

goodProb: probability of a good voter putting a ≻ b in her vote if a ≻ω b

All bad voters in the system act hostile

try to minimize social welfare by voting against ω badProb: probability of a bad voter putting b ≻ a in her vote if a ≻ω b

Analyze outcomes generated by schemes

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SLIDE 81

Evaluating Scheme Efficacy

Suppose ω is an ideal ranking over k candidates ω as the election outcome ⇒ maximum social welfare All good voters in the system favor ω

goodProb: probability of a good voter putting a ≻ b in her vote if a ≻ω b

All bad voters in the system act hostile

try to minimize social welfare by voting against ω badProb: probability of a bad voter putting b ≻ a in her vote if a ≻ω b

Analyze outcomes generated by schemes # of voters = 100, # of bad voters = 33, badProb = 0.9

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SLIDE 82

Simulation Results

Average (of 50 ballots) distances of produced outcomes from the ideal ranking

0.5 1 1.5 2 2.5 3 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

  • Avg. Distance from Ideal

goodProb PlacePlurality Kemeny Pruned

(a) # of Candidates = 3

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SLIDE 83

Simulation Results, contd.

Average (of 50 ballots) distances of produced outcomes from the ideal ranking

1 2 3 4 5 6 7 8 9 10 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

  • Avg. Distance from Ideal

goodProb PlacePlurality Kemeny Pruned

(b) # of Candidates = 5

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SLIDE 84

Simulation Results, contd.

Average (of 50 ballots) distances of produced outcomes from the ideal ranking

5 10 15 20 25 30 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

  • Avg. Distance from Ideal

goodProb PlacePlurality Kemeny Pruned

(c) # of Candidates = 8

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SLIDE 85

Conclusion

Introduction of democratic election problem in distributed systems

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SLIDE 86

Conclusion

Introduction of democratic election problem in distributed systems Pruned-Kemeny-Young Scheme for Byzantine Social Welfare problem

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SLIDE 87

Future Work

Pruned-Kemeny-Young (and Kemeny-Young)

NP-Hard

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SLIDE 88

Future Work

Pruned-Kemeny-Young (and Kemeny-Young)

NP-Hard Yet produce ‘better’ results

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SLIDE 89

Future Work

Pruned-Kemeny-Young (and Kemeny-Young)

NP-Hard Yet produce ‘better’ results Explore techniques for finding ‘better’ outcomes in polynomial steps

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SLIDE 90

Thanks!

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SLIDE 91

Backup

0.5 1 1.5 2 2.5 3 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

  • Avg. Distance from Ideal

goodProb PlacePlurality Pairwise Borda Kemeny Pruned

(d) # of Candidates = 3

1 2 3 4 5 6 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

  • Avg. Distance from Ideal

goodProb PlacePlurality Pairwise Borda Kemeny Pruned

(e) # of Candidates = 4

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SLIDE 92

Backup

Average (of 50 ballots) distances of produced outcomes from the ideal ranking

1 2 3 4 5 6 7 8 9 10 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

  • Avg. Distance from Ideal

goodProb PlacePlurality Kemeny Pruned

(f) # of Candidates = 5

2 4 6 8 10 12 14 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

  • Avg. Distance from Ideal

goodProb PlacePlurality Kemeny Pruned

(g) # of Candidates = 6

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SLIDE 93

Backup

1 2 3 4 5 6 7 8 9 10 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

  • Avg. Distance from Ideal

goodProb PlacePlurality Pairwise Borda Kemeny Pruned

(h) # of Candidates = 5

2 4 6 8 10 12 14 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

  • Avg. Distance from Ideal

goodProb PlacePlurality Pairwise Borda Kemeny Pruned

(i) # of Candidates = 6

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SLIDE 94

Backup

2 4 6 8 10 12 14 16 18 20 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

  • Avg. Distance from Ideal

goodProb PlacePlurality Pairwise Borda Kemeny Pruned

(j) # of Candidates = 7

5 10 15 20 25 30 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9

  • Avg. Distance from Ideal

goodProb PlacePlurality Pairwise Borda Kemeny Pruned

(k) # of Candidates = 8

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SLIDE 95

Related Work

Arrow’s Impossibility Theorem, and his work on Social Choice and Welfare Theory

1950, 1951

Pairwise Comparison Schemes, Social Welfare Schemes, Theory of Voting, Welfare Economics

Condorcet circa 1785, Buchanan 1954, Graaff 1957, Kemeny 1959, Farquharson 1969, Ishikawa et al. 1979, Young 1988

Multivalued Byzantine Agreement Schemes, Byzantine Leader Election, k-set Consensus

Turpin and Coan 1984, Ostrovsky et al. 1994, Russell et al. 1998, Kapron et al. 2008, Prisco et al. 1999

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