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Vote-counting: election-system analysis

 
 
 
I decided to do the 3-candidate case more generally, because this example was not a very good one. The Condorcet matrix:
[table="class:grid"]
[tr][td]0[/td][td]c12[/td][td]c13[/td][/tr]
[tr][td]c21[/td][td]0[/td][td]c23[/td][/tr]
[tr][td]c31[/td][td]c32[/td][td]0[/td][/tr]
[/table]
Differences: d1 = c23-c32, d2 = c31-c13, d3 = d12-c21
Circular preferences d1, d2, d3 > 0 or else < 0 -- will use the > 0 case

Maximal lotteries: weights are (d1,d2,d3)/(d1+d2+d3)

Copeland's method: 0-0-0 -- 3-way tie

Minimax: minimum of
  • Winning: c31, c12, c23
  • Margins: d2, d3, d1
  • Opposition: max(c21,c31), max(c12,c32), max(c13,c23)

Kemeny-Young: maximum of
  • 123: c12+c13+c23 = c12+c23+c31 - d1
  • 132: c12+c13+c32 = c12+c23+c31 - d1 - d2
  • 213: c21+c13+c23 = c12+c23+c31 - d2 - d3
  • 231: c21+c23+c31 = c12+c23+c31 - d3
  • 312: c12+c31+c32 = c12+c23+c31 - d2
  • 321: c21+c31+c32 = c12+c23+c31 - d1 - d3
 
Schulze beatpath:
Paths with nonzero strengths:
  • 12: c12
  • 23: c23
  • 31: c31
  • 123: min(c12,c23)
  • 231: min(c23,c31)
  • 312: min(c31,c12)
If c23 > c31 > c12, then 123: c12, 231: c31, 312: c12
D(2,1) > D(1,2) -- c31, c12
D(3,2) < D(2,3) -- c12, c23
D(1,3) < D(3,1) -- c12, c31
Summary: 21 > 12, 23 > 32, 31 > 13
Winning order: 231


Ranked pairs:
All the pairs, unsorted:
(12,c12,c21) (21,c21,c12) (13,c13,c31) (31,c31,c13) (23,c23,c32) (32,c32,c23)

Using c23 > c31 > c12, the pairs can be partially sorted:
(23,c23,c32) (32,c32,c23) / (31,c31,c13) (13,c13,c31) / (12,c12,c21) (21,c21,c12)
Further sorting depends on additional assumptions about the ordering of the c values.

The first one in the beatlist is (23,c23,c32), since c23 is larger than all the others.

Of the others, (32,c32,c23) produces a cycle, and is not added. The others have c31 < c23, c13 < c31, c21, c12 < c31 and we thus add (31,c31,c13)

This gives us enough to specify the winning order: 231


So in the absence of a Condorcet winner, different Condorcet methods will give different winners and orderings.
 
First-past-the-post = plurality voting = single-choice voting: choose one candidate and that's it for each voter. Very simple, but also the worst.

Rated voting = range voting = score voting. Approval voting is a special case of rated voting, with only lowest and highest ratings.

Runoff methods: top-two, sequential. In general, a method with multiple rounds of counting where each round is a single-choice election with all but the last round having multiple winners who move on to the next round. Top-two: two rounds, with the first round having two winners. Sequential: every round's winners is all but the candidate who did the worst.

Runoff methods with ranked or preference votes: contingent (top-two), instant runoff (sequential)


I decided to try to research these criteria in more detail.

 Comparison of electoral systems - Note b:
Condorcet, Smith and Independence of Smith-dominated alternatives criteria are incompatible with Independence of irrelevant alternatives, Consistency, Participation, Later-no-harm, Later-no-help, and Favorite betrayal[clarification needed] criteria.
Click to expand...

From an earlier post, I came up with this list of mutually-exclusive sets of criteria:
  • (Cond, ISDA)
  • (IIA, NFB) and/or (Cons, Part)
  • (LNH)

 Later-no-harm criterion,  Later-no-help criterion - giving some additional candidates some ranks or positive ratings will not affect one's higher-ranked or higher-rated candidates.

Satisfying later-no-harm: runoff methods, the pairwise-opposition variant of minimax Condorcet (use the appropriate Condorcet matrix element)

Satisfying later-no-help: runoff methods, approval and rated voting, Borda count, majority judgment, Bucklin voting

Borda count = ranked-to-rated voting. Majority judgment = rated voting with medians. Bucklin voting = preference votes, with 2nd, 3rd, ... added to 1st until a candidate gets a majority.

Later-no-harm is incompatible with the Condorcet criterion: if a Condorcet winner is present, then that winner is the method's winner.
 
Advocates IRV:
How is RCV better than Approval, Score or Condorcet voting methods? - FairVote
FairVote.org | Single-winner Voting Method Comparison Chart

Later-No-Harm Criterion | The Center for Election Science

Advocates rated voting:
RangeVoting.org - Later no harm
"Later no harm" – an actually-silly (& massively overhyped by "FairVote") voting system criterion
Click to expand...

Claims that later-no-harm is not very good:
Farewell to Pass/Fail - STAR Voting


I've found this nice site: Electowiki, the election methods wiki with some additional methods.
 
Descending Solid Coalitions - Electowiki
For ranked ballots. Find every subset of the candidates, count how many ballots have every member of that subset beating every candidate outside that that subset.

Then sort the subsets by number of voters, then go from the maximum downward. Starting with all the candidates, remove all those that not in each subset, skipping over those that remove all the remaining ones. Whichever candidate survives is the winner.

Descending Acquiescing Coalitions - Electowiki Like the above, but with each subset's candidates also having ties with candidates outside that subset.


The site mentions some additional method criteria.

The Plurality Criterion: "If the number of ballots ranking A as the first preference is greater than the number of ballots on which another candidate B is given any preference, then B must not be elected." Approval voting and IRV satisfy it, as do some Condorcet methods.

"Mono-add-top is a relative criterion which requires that if candidate X is the winner of the election, then adding additional ballots which rank X strictly above all other candidates must not reduce the probability that the winner is still X."

Satisfying participation implies satisfying mono-add-top, though it is not possible to satisfy all three of Condorcet, plurality, and mono-add-top.


Maximize Affirmed Majorities - Electowiki - much like Tideman's Ranked Pairs, but it's hard for me to distinguish them from their descriptions. I'd have to see pseudocode for both methods to be sure.
 
[1804.02973] The Schulze Method of Voting
At Arxiv > Computer Science > Computer Science and Game Theory
Big paper describing it, giving examples of it, and proving theorems about it.

Looking at the public polls of CIVS, I found that most of them had what may be called Condorcet sequences: every member is a Condorcet winner with respect to the later members. I will now consider how a Condorcet matrix with a Condorcet sequence fares in various Condorcet methods.

Copeland's method is easy. Out of n candidates, the first one has score (n-1) - 0 = n-1, the second one (n-2) - 1 = n-3, and the kth one n-2k+1. That method's scores are in sequence.

In the Schulze method, the only beatpaths are from earlier to later members of the sequence. For members i and j, their strength P(i,j) > 0 for i < j and P(i,j) = 0 for i > j. This sorts the members into their sequence order.

In ranked pairs, each right-order pair will be ranked above its corresponding wrong-order pair, so it will get added. Thus getting the sequence order.

In the Kemeny-Young method, the score for the sequence is sum over i,j of D(i,j) where j > i. Departure from the sequence replaces at least some of the D(i,j)'s with D(j,i)'s, which are less. Thus, the K-Y method gives the sequence order.
 
I will work out the minimax method for a 3*3 matrix D:

Winning votes (wins):
  • 0
  • D12
  • max(D13, D23)

Margins (marg):
  • max(D21-D12, D31-D13)
  • max(D12-D21, D32-D23)
  • max(D13-D31, D23-D32)

Pairwise opposition (oppo):
  • max(D21,D31)
  • max(D12,D32)
  • max(D13,D23)

It looks difficult to proceed, so I will have to find constraints on the D's from the ordering. This I will do by finding their values in terms of counts w of the ballots.

D12 = w123 + w132 + w312 + w1x23 + w13x2
D21 = w213 + w231 + w321 + w2x13 + w23x1
D13 = w123 + w132 + w213 + w1x23 + w12x3
D31 = w321 + w312 + w231 + w3x12 + w23x1
D23 = w123 + w213 + w231 + w2x13 + w12x3
D32 = w132 + w312 + w321 + w3x12 + w13x2

w123: 1 then 2 then 3, w1x23: 1 then (tie of 2 and 3), w12x3: (tie of 1 and 2) then 3

It looks like a general proof will be difficult, so I will look for special cases to see if I can find counterexamples.
 
I first consider only one order in the ballots: 1 > 2 > 3: w123 is the only nonzero w. This gives us D12 = D13 = D23 = w123 = w and D21 = D31 = D32 = 0.

The scores of each variant:
  • Wins: 0, w, w
  • Marg: -w, w, w
  • Oppo: 0, w, w
All three variants find the Condorcet winner, the first one, but the remaining two are tied.
 
Some more names for voting methods.
  • Cardinal - rated, score, range
  • Ordinal - ranked, preference
Cardinal and ordinal as in cardinal and ordinal numbers: how many of something and where in sequence.

Not sure about whether I want to implement Benham's method. That's doing a sequential runoff and finding the Condorcet winner at each step, if there is one. Instead of stopping at a majority, it stops at a Condorcet winner.

Condorcet = one-on-one = virtual round robin

My implementations of vote-counting methods: VotingAlgs.zip - for preference voting and proportional allocation. Rated voting is easy to count, though majority judgment (best median) is somewhat tricky to implement. It finds the median rating, and I worked out how to do that with arbitrary vote weights. Sort (weight,rating) by rating, then add up the weights from the lowest rating until one has found 1/2 of the total. That gives the median rating.

Also on GitHub: lkpetrich/Preference-Voting: For counting votes in preference or ranked-choice voting. Large number of algorithms implemented.

I've found several other voting-algorithm implementations on GitHub.
 
I'm thinking some of these vote-counting systems seem complex, difficult to comprehend, abstruse.

And, given that, I'm thinking that if the WIKID* party loses an election determined by a vote-counting system they don't understand, then they'll believe any explanation their dishonest leaders give them.

If their leaders say the election was stolen, they'll believe it.

We hold elections to work the will of the the majority. But we also hold elections to get buy-in from the defeated team, so they can say, "Well, we tried, but we were outvoted, so now we should cooperate with our fairly-elected leaders until the next election."

If the vote-counting system is abstruse, we may lose the advantage of having people believe that elections are fair.

So -- and I don't know whether this is apropos to what I just wrote -- I like the idea of sequentially eliminating losers until there is an eventual winner.

I thought I'd try that on this data:


[TD="class: xl68, width: 89"]How Many[/TD]
[TD="class: xl68, width: 89"]1st[/TD]
[TD="class: xl68, width: 89"]2nd[/TD]
[TD="class: xl68, width: 89"]3rd[/TD]
[TD="class: xl68, width: 89"]4th[/TD]
[TD="class: xl68, width: 89"]5th[/TD]

[TD="class: xl66, width: 89"]18[/TD]
[TD="class: xl65, width: 89"]Sausage[/TD]
[TD="class: xl65, width: 89"]Artichoke[/TD]
[TD="class: xl71, width: 89"] Mushrooms
[/TD]
[TD="class: xl65, width: 89"]Peppers[/TD]
[TD="class: xl65, width: 89"]Anchovies[/TD]

[TD="class: xl66, width: 89"]12[/TD]
[TD="class: xl65, width: 89"]Anchovies[/TD]
[TD="class: xl71, width: 89"] Mushrooms
[/TD]
[TD="class: xl65, width: 89"]Artichoke[/TD]
[TD="class: xl65, width: 89"]Peppers[/TD]
[TD="class: xl65, width: 89"]Sausage[/TD]

[TD="class: xl66, width: 89"]10[/TD]
[TD="class: xl65, width: 89"]Peppers[/TD]
[TD="class: xl65, width: 89"]Anchovies[/TD]
[TD="class: xl71, width: 89"] Mushrooms
 [/TD]
[TD="class: xl65, width: 89"]Artichoke[/TD]
[TD="class: xl65, width: 89"]Sausage[/TD]

[TD="class: xl66, width: 89"]9[/TD]
[TD="class: xl65, width: 89"]Artichoke[/TD]
[TD="class: xl65, width: 89"]Peppers[/TD]
[TD="class: xl71, width: 89"] Mushrooms [/TD]
[TD="class: xl65, width: 89"]Anchovies[/TD]
[TD="class: xl65, width: 89"]Sausage[/TD]

[TD="class: xl70, width: 89"]4
[/TD]
[TD="class: xl71, width: 89"] Mushrooms
[/TD]
[TD="class: xl72, width: 89"]Anchovies[/TD]
[TD="class: xl65, width: 89"]Artichoke[/TD]
[TD="class: xl65, width: 89"]Peppers[/TD]
[TD="class: xl65, width: 89"]Sausage[/TD]

[TD="class: xl70, width: 89"]2[/TD]
[TD="class: xl71, width: 89"] Mushrooms
[/TD]
[TD="class: xl72, width: 89"]Peppers[/TD]
[TD="class: xl65, width: 89"]Artichoke[/TD]
[TD="class: xl65, width: 89"]Anchovies[/TD]
[TD="class: xl65, width: 89"]Sausage[/TD]

Mushrooms got the fewest (6) votes for first place, so we declare mushrooms the loser, and eliminate them from consideration, which leaves us this:

[TABLE="width: 534"]
[TR]
[TD]How Many[/TD]
[TD]1st[/TD]
[TD]2nd[/TD]
[TD]3rd[/TD]
[TD]4th[/TD]
[TD]5th[/TD]
[/TR]
[TR]
[TD]18[/TD]
[TD]Sausage[/TD]
[TD]Artichoke
[/TD]
[TD]Peppers[/TD]
[TD]Anchovies[/TD]
[TD][/TD]
[/TR]
[TR]
[TD]12[/TD]
[TD]Anchovies
[/TD]
[TD]Artichoke
[/TD]
[TD]Peppers[/TD]
[TD]Sausage[/TD]
[TD][/TD]
[/TR]
[TR]
[TD]10[/TD]
[TD]Peppers[/TD]
[TD]Anchovies[/TD]
[TD]Artichoke
[/TD]
[TD]Sausage[/TD]
[TD][/TD]
[/TR]
[TR]
[TD]9[/TD]
[TD]Artichoke
[/TD]
[TD]Peppers[/TD]
[TD]Anchovies[/TD]
[TD]Sausage[/TD]
[TD][/TD]
[/TR]
[TR]
[TD]4[/TD]
[TD]Anchovies[/TD]
[TD]Artichoke
[/TD]
[TD]Peppers[/TD]
[TD]Sausage[/TD]
[TD][/TD]
[/TR]
[TR]
[TD]2[/TD]
[TD]Peppers[/TD]
[TD]Artichoke
[/TD]
[TD]Anchovies[/TD]
[TD]Sausage[/TD]
[TD][/TD]
[/TR]
[/TABLE]

This gives these new totals for first place:
18 Sausage
16 Anchovies
12 Peppers
9 Artichoke

Artichoke has the fewest votes, so we eliminate that:

[TD="class: xl68, width: 89"]How Many[/TD]
[TD="class: xl68, width: 89"]1st[/TD]
[TD="class: xl68, width: 89"]2nd[/TD]
[TD="class: xl68, width: 89"]3rd[/TD]
[TD="class: xl68, width: 89"]4th[/TD]
[TD="class: xl68, width: 89"]5th[/TD]

[TD="class: xl66, width: 89"]18[/TD]
[TD="class: xl65, width: 89"]Sausage[/TD]
[TD="class: xl65, width: 89"]Peppers[/TD]
[TD="class: xl65, width: 89"] Anchovies
[/TD]

[TD="class: xl66, width: 89"]12[/TD]
[TD="class: xl65, width: 89"] Anchovies
[/TD]
[TD="class: xl65, width: 89"]Peppers[/TD]
[TD="class: xl65, width: 89"]Sausage[/TD]

[TD="class: xl66, width: 89"]10[/TD]
[TD="class: xl65, width: 89"]Peppers[/TD]
[TD="class: xl65, width: 89"] Anchovies
[/TD]
[TD="class: xl65, width: 89"]Sausage[/TD]

[TD="class: xl66, width: 89"]9[/TD]
[TD="class: xl65, width: 89"]Peppers[/TD]
[TD="class: xl65, width: 89"] Anchovies
[/TD]
[TD="class: xl65, width: 89"]Sausage[/TD]

[TD="class: xl66, width: 89"]4[/TD]
[TD="class: xl65, width: 89"] Anchovies
[/TD]
[TD="class: xl65, width: 89"]Peppers[/TD]
[TD="class: xl65, width: 89"]Sausage[/TD]

[TD="class: xl66, width: 89"]2[/TD]
[TD="class: xl65, width: 89"]Peppers
[/TD]
[TD="class: xl65, width: 89"] Anchovies
[/TD]
[TD="class: xl65, width: 89"]Sausage[/TD]

The first place totals are now as follows:

21 Peppers
18 Sausage
16 Anchovies

Anchovies has the fewest votes for 1st place, and is therefore eliminated. Thus:


[TD="class: xl68, width: 89"]How Many[/TD]
[TD="class: xl68, width: 89"]1st[/TD]
[TD="class: xl68, width: 89"]2nd[/TD]
[TD="class: xl68, width: 89"]3rd[/TD]
[TD="class: xl68, width: 89"]4th[/TD]
[TD="class: xl68, width: 89"]5th[/TD]

[TD="class: xl66, width: 89"]18[/TD]
[TD="class: xl65, width: 89"]Sausage[/TD]
[TD="class: xl65, width: 89"]Peppers[/TD]

[TD="class: xl66, width: 89"]12[/TD]
[TD="class: xl65, width: 89"]Peppers[/TD]
[TD="class: xl65, width: 89"]Sausage[/TD]

[TD="class: xl66, width: 89"]10[/TD]
[TD="class: xl65, width: 89"]Peppers[/TD]
[TD="class: xl65, width: 89"]Sausage[/TD]

[TD="class: xl66, width: 89"]9[/TD]
[TD="class: xl65, width: 89"]Peppers[/TD]
[TD="class: xl65, width: 89"]Sausage[/TD]

[TD="class: xl66, width: 89"]4[/TD]
[TD="class: xl65, width: 89"]Peppers[/TD]
[TD="class: xl65, width: 89"]Sausage[/TD]

[TD="class: xl66, width: 89"]2[/TD]
[TD="class: xl65, width: 89"]Peppers[/TD]
[TD="class: xl65, width: 89"]Sausage[/TD]

That gives us
37 votes for Peppers, and
18 votes for sausage.

Peppers wins!

And it's true that, in this case, most people preferred peppers to sausage. So maybe that outcome is fair.

I don't know whether we'd be able to persuade the proponents of sausage that it was fair.

Maybe we could. The system is fairly simple, and, in this case, 37 people ranked sausage as their absolute last choice.

--

*WIKID: Whichever Icky Kind I Despise
 
I've done some social-media poll sleuthing. What kinds of polls do social-media sites support?
  • Facebook: Yes. Multiple.
  • Instagram: Yes. (2 options)
  • Twitter: Yes. (max 4 options)
  • Reddit: No.
Rather meager.

I once proposed to help someone write a preference-vote add-on for vBulletin: Preference voting in poll - vBulletin.org Forum Nobody was interested.

I looked again in add-ons for vBulletin, XenForo, phpBB, and Simple Machines, and they usually don't add anything more than administrative options and sometimes the maximum number of options that one can choose in multiple-choice polls.

BTW, I recall getting into a squabble with someone who liked to post single-choice polls where multiple-choice ones may be more appropriate, like favorite ice-cream flavors.

Administrative options include:
  • How long people can vote in it
  • Whether a voter can change their vote
  • Whether a voter can add a poll option
But these are peripheral to my discussion, because they are not about vote-counting algorithms and their features.

I've found on Reddit: Get Rid of First Past the Post Voting System
 
First, it must be noted that alternatives to FPTP depend on casting more than one subvote in one's vote -- more than one ranking or rating. Giving only one candidate a ranking or a rating is "bullet voting", and it is essentially degeneration into FPTP.


I've found this site: The Center for Election Science | Fairer, More Representative Elections - it advocates approval voting.

An Assessment Of Six Single-Winner Voting Methods with these criteria:

1. Simplicity
2. Ease of Administration
3. Elects a Good Winner
4. Represents Candidates’ Support Accurately
5. Tractability - how easy or hard it may be to be adopted
6. Overall Assessment

The methods:

1. Plurality Voting - FPTP - single choice
10, 10, 0, 0, 10 -- 0

2. Condorcet Methods
0, 0, 6, 7, 0 -- 2

3. Ranked Choice Voting (RCV) -- instant runoff (ranking + sequential runoff)
2, 3, 4.5, 1, 7 -- 3.5

4. Range Voting -- rated or score voting
7, 6, 10, 10, 5 -- 7

5. Novel Cardinal Methods (STAR Voting Focus) -- rated vote + top-two runoff
3.5, 4, 10, 10, 2 -- 4

6. Approval Voting
10, 10, 9, 9, 8 -- 10
 
RangeVoting.org - Center for Range Voting - front page

RangeVoting.org - Donald Trump exploits plurality voting system flaws to try to become US president written in July 2015
Plurality voting can suffer a severe pathology we call "fame-based failure." Whenever a famous, media-philic candidate X runs against a large number of lesser-known and comparatively-similar rivals, with care X is almost assured victory almost regardless of what the voters think of the rivals' versus his quality. We explain why that happens, and give three important recent elections in which it did happen: Arnold Schwarzenegger 2003, Hamid Karzai 2009, and Donald Trump (ongoing) 2015 Republican primary (July polls). We then demonstrate how in the first and last of these three examples, approval or score voting would have cured the disease. ("Demonstrate" means "based on poll data.") Finally we consider the proposal by Sam Wang that IRV – Instant Runoff Voting/polling – would be another cure, and demonstrate that idea is dubious at best, and disproven at worst. IRV is highly unsuited for use by pollsters in elections with many candidates.
Click to expand...
Donald Trump won the Republican nomination a year later, and the Presidency a year and a half later.

The Equal Vote Coalition - advocates STAR voting

Compare STAR and IRV - Equal Vote Coalition has this comparison of FPTP, STAR, and IRV:
  • Honesty: F, A, B
  • Equality: D, A+, C
  • Accuracy: D, A, C
  • Simplicity: A, B, C
  • Expressiveness: D, A, B
Compare Top Two - Equal Vote Coalition has this comparison of FPTP, T2 Runoff, and STAR:
  • Equality: Poor, Poor, Good
  • Honesty: Poor, OK, Good
  • Accuracy: Poor, OK, Great!
  • Simplicity: Super!, Poor, Good
  • Expressiveness: Poor, OK, Great!
The two evaluations don't quite agree on FPTP and STAR.
 
STAR Voting - Score Then Automatic Runoff

The purpose of the second step is to provide an incentive for partial rankings, since if one's favorite candidate does not win, then one may have a partial vote for a partial favorite.

From the site, I've found a graphic that compares several methods. Red = F, orange = D, yellow = C, yellow-green = B, green = A
  • Spoiler Effect / Vote Splitting
    • FPTP: F: YES
    • IRV: D: YES
    • App: A: NO
    • Rat: A: NO
    • STAR: A: NO
  • Gives an advantage to some types of candidates
    • FPTP: F: Favors polarizing candidates who are "viable"
    • IRV: D: Strong underdog candidates are at a disadvantage
    • App: C: Favors candidates seen as more "viable"
    • Rat: B: Favors "viable" consensus candidates over polarized majority
    • STAR: A: NO
  • Wasted Votes and Exhausted Ballots
    • FPTP: D: Not voting for a front-runner is a wasted vote
    • IRV: F: Exhausted Ballots are not counted in the final round
    • App: C: Not voting for a front-runner is a wasted vote
    • Rat: B: Scoring viable candidates low can make your vote less powerful
    • STAR: A: Even if your favorites can't win, your vote helps prevent your worst-case scenario
  • Ballots can be tabulated locally?
    • FPTP: A: YES
    • IRV: F: NO
    • App: A: YES
    • Rat: A: YES
    • STAR: A: YES
  • Tabulation Complexity
    • FPTP: D: Basic Addition. 2 Elections Recommended
    • IRV: D: Algebra required. Multiple rounds
    • App: A: Basic Addition. One Round
    • Rat: A: Basic Addition. One Round
    • STAR: B: Basic Addition. 2 rounds of tabulation
  • Accuracy (VSE i.e. Voter Satisfaction Efficiency)
    • FPTP: FL 72 - 86%
    • IRV: C: 80 - 91%
    • App: B: 84 - 95%
    • Rat: B: 84 - 97%
    • STAR: A: 91 - 98%
  • Strategy Resistance Factor (VSE)
    • FPTP: F: 18:1
    • IRV: B: 3:1
    • App: B: 3:1
    • Rat: C: 4:1
    • STAR: A: 1:1
App = approval, Rat = rated / score / range
 
I'll give my assessment of complexity.

Ballots:
  • Approval -- make any number of votes that one wants
  • FPTP, TopTwo -- make only one vote
  • Limited approval (some maximum number > 1) -- must remember that number
  • Rated / score / range -- one rating for each candidate
  • Preference / ranked -- must remember which other candidates one's ranked as one ranks the candidates
A big objection to preference-vote elections is that they would be excessively confusing to many voters. But experience with such elections shows that that is not the case -- majorities of voters use preference-vote ballots correctly without mistakes.

Vote counting:
  • Approval, FPTP, TopTwo, Rated -- one count
  • Preference with Borda (rankings to ratings) -- one count
  • Preference with IRV -- repeated counts
  • Preference with Condorcet methods (make one-on-one matrix) -- one count, then some algorithm for getting the winner out of the Condorcet matrix
That last one can get complicated. While finding a Condorcet winner is fairly easy, one needs an algorithm for finding a fallback winner in case there is no Condorcet winner. I don't think that I want to try to explain the Schulze beatpath algorithm to anyone without mathematical sophistication.

Vote summability: what is the size of data structures of partial sums of votes for n candidates?
  • O(n) -- FPTP, TopTwo, Approval, Rated, Borda
  • O(n^2) -- Condorcet methods
  • O(n!) -- full IRV
Full IRV loses badly, but IRV is often implemented with some maximum number of choices: 2, 3, 4, 5. Using p choices, the size of the summation data structure is n!/(n-p)! ~ n^p.
 
fromderinside said:
After all that we find that votes are either up or down. Bayes basics. Differences are either in the weeds or in the bean counting, of interest to those who play such as chess, but of little use otherwise.
Click to expand...
Seems like nonsense to me.
 
That's also a measure of decline and corruption.

When one votes some will recognize rigged votes don't lead to voter's desired results even though their votes resulted in approval of what one was seeking.

What matters is one sees results one seeks when voting in sufficient number to retain confidence in their voting demonstrating their voice in governing.

So while method are nice, some might even be more efficient than others, unless voters see votes as generating results they agree with often enough to bring them back to the polls. democracy fails.
 
In this thread, I'm mostly interested in the math of vote-counting systems, though some aspects of that math have broader policy implications, like how the system handles electorally similar candidates. Do they help each other? (teams) Hurt each other? (spoilers) Interfere with others' vote-count outcomes? (crowds). If they do such things, then the methods are not cloneproof, they violate "Independence of Clones".

There are other criteria that a method can violate, criteria that often seem like common sense. Like the majority criterion, that a candidate with the majority of top votes must win. That is satisfied by most methods, but there are exceptions, like rated voting and the Borda count. I'd mentioned an example of violation by rated voting, where the top votes were split but the votes with a lower ranking were unified. The Borda count fails that criterion for the same reason, because it includes a ranks-to-ratings step.

A candidate that wins a majority of top votes will be the Condorcet winner, but a Condorcet winner need not win a majority of top votes. That example with a different win for each method is an example of that.
 
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