Estimating the Margin of Victory for Instant-Runoff Voting* - - PowerPoint PPT Presentation

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Estimating the Margin of Victory for Instant-Runoff Voting* - - PowerPoint PPT Presentation

Oct 04, 2022 222 likes •685 views

Estimating the Margin of Victory for Instant-Runoff Voting* David Cary * also known as Ranked-Choice Voting, preferential voting, and the alternative vote 1 v7 Overview Why estimate? What are we talking about? Estimates

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David Cary

Estimating the Margin of Victory for Instant-Runoff Voting*

* also known as Ranked-Choice Voting, preferential voting, and the alternative vote

v7

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Overview

  • Why estimate?
  • What are we talking about?
  • Estimates
  • Worst-case accuracy
  • Real elections
  • Conclusions
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Why Estimate?

Trustworthy Elections Risk-limiting audits Trustworthy Elections Margin of Victory

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Why Estimate?

IRV Risk-Limiting Audits IRV Trustworthy Elections IRV Margin of Victory (not feasible)

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Why Estimate?

IRV Risk-Limiting Audits IRV Trustworthy Elections ? IRV Margin of Victory (not feasible)

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Why Estimate?

IRV Risk-Limiting Audits IRV Trustworthy Elections IRV Margin of Victory (not feasible sometimes) IRV Margin of Victory Lower Bound

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Proposals for IRV Risk-Limiting Audits

Sarwate, A., Checkoway, S., and Shacham, H.

  • Tech. Rep. CS2011-0967, UC San Diego, June 2011

https://cseweb.ucsd.edu/~hovav/dist/irv.pdf

Risk-limiting audits for nonplurality elections

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Overview

  • Why estimate? because we can;

to do risk-limiting audits

  • What are we talking about?
  • What is Instant-Runoff Voting?
  • What is a margin of victory?
  • Estimates
  • Worst-case accuracy
  • Real elections
  • Conclusions
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Model of Instant-Runoff Voting

  • Single winner
  • Ballot ranks candidates in order of preference.
  • Votes are counted and candidates are eliminated in a

sequence of rounds.

  • In each round, a ballot counts as one vote for the most

preferred continuing candidate on the ballot, if one exists.

  • In each round, one candidate with the fewest votes is

eliminated for subsequent rounds.

  • Ties for elimination are resolved by lottery.
  • Rounds continue until just one candidate is in the round.

That candidate is the winner.

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Consistent IRV Features

  • Number of candidates ranked on a ballot:
  • require ranking all candidates
  • limit maximum number of ranked candidates
  • can rank any number of candidates
  • Multiple eliminations:
  • required*, not allowed, or discretionary*
  • Early termination:
  • tabulation stops when a winner is identified*

* may require an extended tabulation for auditing purposes

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Defining the Margin of Victory

The margin of victory is the minimum total number* of ballots that must in some combination be added and removed in order for the set of contest winner(s) to change with some positive probability.

* the number of added ballots, plus the number of removed ballots

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Overview

  • Why estimate? because we can;

to do risk-limiting audits

  • What are we talking about?
  • Estimates
  • Worst-case accuracy
  • Real elections
  • Conclusions
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Estimates for the Margin of Victory

Last-Two-Candidates upper bound Winner-Survival upper bound Single-Elimination-Path lower bound Best-Path lower bound Time O(1) O(C) O(C2) O(C2 log C) Space O(1) O(1) O(1) O(C)

(C = number of candidates)

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Example IRV Contest

Wynda Winslow 107 112 114 186 332 Diana Diaz 130 133 134 146 — Charlene Colbert 35 46 84 — — Barney Biddle 40 41 — — — Adrian Adams 20 — — — — Round 1 Round 2 Round 3 Round 4 Round 5

Candidates are in reverse order of elimination, with the winner first.

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Last-Two-Candidates Upper Bound

Wynda Winslow 107 112 114 186 332 Diana Diaz 130 133 134 146 — Charlene Colbert 35 46 84 — — Barney Biddle 40 41 — — — Adrian Adams 20 — — — — 40 Round 1 Round 2 Round 3 Round 4 Round 5

Margin of Survival for Winner in round C – 1, the round with just the last two candidates.

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Winner-Survival Upper Bound

Wynda Winslow 107 112 114 186 332 Diana Diaz 130 133 134 146 — Charlene Colbert 35 46 84 — — Barney Biddle 40 41 — — — Adrian Adams 20 — — — — 87 71 30 40 Round 1 Round 2 Round 3 Round 4 Round 5

Margin of Survival for Winner Smallest Margin of Survival for the Winner in the first C – 1 rounds.

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Vote Totals Not In Sequence By Value

Wynda Winslow 107 112 114 186 332 Diana Diaz 130 133 134 146 — Charlene Colbert 35 46 84 — — Barney Biddle 40 41 — — — Adrian Adams 20 — — — — Round 1 Round 2 Round 3 Round 4 Round 5

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Wynda Winslow 130 133 134 186 332 Diana Diaz 107 112 114 146 — Charlene Colbert 40 46 84 — — Barney Biddle 35 41 — — — Adrian Adams 20 — — — — Round 1 Round 2 Round 3 Round 4 Round 5

Vote Totals Not In Sequence By Value

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Single-Elimination-Path Lower Bound

Smallest Margin of Single Elimination in the first C – 1 rounds.

130 133 134 186 332 107 112 114 146 — 40 46 84 — — 35 41 — — — 20 — — — — 15 5 30 40 Round 1 Round 2 Round 3 Round 4 Round 5

Margin of Single Elimination (MoSE)

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Single Elimination Path

Round 1 Round 2 Round 3 Round 4 Round 5 30 = MoSE(3) 40 = MoSE(4) 5 = MoSE(2) 15 = MoSE(1) candidates {a, b, c, d, w} candidates {c, d, w} candidates {b, c, d, w} candidates {d, w} candidate {w}

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Single-Elimination Path Bottleneck

Round 1 Round 2 Round 3 Round 4 Round 5 30 = MoSE(3) 40 = MoSE(4) 5 = MoSE(2) 15 = MoSE(1) candidates {a, b, c, d, w} candidates {c, d, w} candidates {b, c, d, w} candidates {d, w} candidate {w}

Edge weight = a limited capacity (a bottleneck) for tolerating additions and removals of ballots, while still staying

  • n the path.
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Exceeding a Bottleneck

Round 1 Round 2 Round 3 Round 4 Round 5 30 = MoSE(3) 40 = MoSE(4) 5 = MoSE(2) 15 = MoSE(1) candidates {a, b, c, d, w} candidates {c, d, w} candidates {b, c, d, w} candidates {d, w} candidate {w}

Different Winner ? ? Different Winner

Easy guarantee

  • f same winner:

Stay on the single-elimination path

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Path Bottleneck

Round 1 Round 2 Round 3 Round 4 Round 5 30 = MoSE(3) 40 = MoSE(4) 5 = MoSE(2) 15 = MoSE(1) candidates {a, b, c, d, w} candidates {c, d, w} candidates {b, c, d, w} candidates {d, w} candidate {w}

Path Bottleneck is the smallest individual bottleneck

  • n the path

=

Single- Elimination- Path lower bound

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Multiple Elimination as a Detour

Round 1 Round 2 Round 3 Round 4 Round 5 30 = MoSE(3) 40 = MoSE(4) 5 = MoSE(2) 15 = MoSE(1) candidates {a, b, c, d, w} candidates {c, d, w} candidates {b, c, d, w} candidates {d, w} candidate {w}

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130 133 134 186 332 107 112 114 146 — 40 46 84 — — 35 41 — — — 20 — — — — Round 1 Round 2 Round 3 Round 4 Round 5

Multiple Elimination of k Candidates

+

87 A usable multiple eliminaton, if combined vote total is still the smallest

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130 133 134 186 332 107 112 114 146 — 40 46 84 — — 35 41 — — — 20 — — — — Round 1 Round 2 Round 3 Round 4 Round 5

Margin of Multiple Elimination

MoME(2, 2) = 112 – (46 + 41) = 112 – 87 = 25

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Multiple Elimination as a Detour

Round 1 Round 2 Round 3 Round 4 Round 5 30 = MoSE(3) 40 = MoSE(4) 5 = MoSE(2) 15 = MoSE(1) candidates {a, b, c, d, w} candidates {c, d, w} candidates {b, c, d, w} candidates {d, w} candidate {w} MoME(2, 2) = 25

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12 119 25 196

Usable Multiple Eliminations

Round 1 Round 2 Round 3 Round 4 Round 5 30 40 5 15 264 87 53 19 Which path has the largest path bottleneck? 237 351 23 137

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Best-Path Lower Bound

  • The largest path bottleneck ...
  • of all paths from round 1 to round C ...
  • that consist of usable multiple eliminations.
  • A best path:
  • Guarantees the same winner
  • Maximizes tolerance for additions and removals

among usable multiple elimination paths

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Best-Path Lower Bound Algorithms

  • O(C2 log C) time to sort the vote totals

within each round.

  • The best path can be found in O(C2) time.
  • Using a bottleneck algorithm, which is …
  • A longest path algorithm for a weighted directed

acyclic graph, but calculating the length as the minimum of its component parts, instead of the sum.

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Estimate Relations

Single-Elimination-Path lower bound ≤ Best-Path lower bound ≤ margin of victory ≤ Winner-Survival upper bound ≤ Last-Two-Candidates upper bound

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Early-Termination Estimates

  • For tabulations that stop before C-1 rounds
  • when a candidate has a majority of the

continuing votes

  • more than two candidates are in the round
  • Accuracy is degraded
  • must allow for possible extreme behavior in the

missing rounds of the tabulation.

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Overview

  • Why estimate? to do risk-limiting audits
  • What are we talking about?
  • Estimates – quick: O(C2 log C) time
  • Worst-case accuracy
  • Real elections
  • Conclusions
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  • Ratio with margin of victory is unbounded.
  • No estimate can do better if based only on

tabulation vote totals.

Asymptotic Worst-Case Accuracy

Winner-Survival Upper Bound Margin of Victory Margin of Victory Best-Path Lower Bound

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Asymptotic Worst-Case Example

Best Path Lower Bound Winner-Survival Upper Bound 1 2C-3 contest 1 contest 2 1 2C-3 Margin

  • f

Victory

? ?

Identical Tabulation Vote Totals

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Asymptotic Worst-Case Example

Best Path Lower Bound Winner-Survival Upper Bound 1 2C-3 contest 1 contest 2 1 2C-3 Margin

  • f

Victory Ballots Show Different Margins of Victory

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Overview

  • Why estimate? to do risk-limiting audits
  • What are we talking about?
  • Estimates – quick: O(C2 log C) time
  • Worst-case accuracy – unbounded ratios
  • Real elections
  • Conclusions
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Estimates for Real Elections

  • Australia elections, 2010
  • national House of Representatives
  • 150 contests
  • All California IRV contests since 2004
  • local, non-partisan elections
  • 53 contests
  • 36 from San Francisco, 2004-2011
  • 12 using early termination estimates
  • 17 from Alameda county, 2010: Berkeley,

Oakland, and San Leandro

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Evaluating Estimates

  • There are many ways to analyze the data.
  • What are relevant metrics?
  • A full evaluation requires a context of:
  • specific risk-limiting audit protocols
  • profiles of audit differences.
  • Look at:
  • best available lower bound and upper bound,
  • as a percentage of first-round votes.
  • What is the distribution of estimates?
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Selected Stats

Assessment Total Contests Contests with LB > 10% Contests with LB < 5% Contests with LB < 1% Contests with LB=MoV=UB and LB < 5% and LB < 1% Contests with UB/LB > 2 and LB < 5% and LB < 1% 150 85 28 2 71 21 1 34 7 1 100% 57% 19% 1% 47% 14% 1% 23% 5% 1% Australia 53 35 14 7 16 4 10 7 7 100% 66% 26% 13% 30% 8% 0% 19% 13% 13% California

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Australia Elections

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California Elections

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Overview

  • Why estimate? to do risk-limiting audits
  • What are we talking about?
  • Estimates – quick: O(C2 log C) time
  • Worst-case accuracy – unbounded ratios
  • Real elections – some estimates useful,

some need improvement

  • Conclusions
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Conclusions

  • Risk-limiting audits can use lower bounds

for the margin of victory.

  • Estimates can be quickly calculated from

tabulation vote totals.

  • Worst-case ratios with the margin of victory

are unbounded.

  • The Best-Path lower bound can be used for

some risk-limiting audits, but some contests will need better estimates.

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Thanks

  • Members and associates of Californians for

Electoral Reform (CfER)

  • especially Jonathan Lundell
  • San Francisco Voting System Task Force
  • especially Jim Soper
  • anonymous reviewers
  • for many suggestions for improving the paper
  • especially for the idea of the Winner-Survival

upper bound