Up to Elections: Results and Voting systems

Properties, Standards & Criteria

for Evaluating & Comparing Multi-Alternative Single-Winner Social Choice Methods.

And Comparison of Some Methods by Those Standards:

by Mike Ossipoff

To write to Mike write nkklrp before the "@" sign, and then write hotmail.com after the "@" sign.

Single-winner methods are for choosing 1 alternative from among several. Typically, in partisan elections, the 1-Vote Plurality method is used. People vote for 1 alternative (or candidate), and the one with most votes wins. Other methods are proposed as improvements.

Various different single-winner methods, differing greatly from eachother, are proposed. There's no consensus on standards for evaluating them. I'm going to describe some standards for single- winner methods.

It's a question of what one wants from a single-winner method. What don't we like about the single-winner method currently in use, the 1-Vote Plurality method?

The "lesser-of-2-evils" problem. Voters know that if they vote for their favorite, then they won't be able to vote for a more winnable compromise, and won't be helping that compromise beat someone much worse. So they feel compelled to vote for the "lesser-of-2-evils", the more winnable, but less liked, compromise. They have to abandon their favorite. That's why we want better single-winner methods.

So that's the problem. Now I'll list some proposed solutions, and then discuss how well they solve the lesser-of-2-evils problem.

METHODS:

When discussing standards, & properties, I'm going to compare 5 methods by those standards & properties:

1. Approval
2. Condorcet's method
3. Young's method
4. Copeland's method
5. Elimination (also known as "Preferential", "MPV", "the Alternative Vote", "Hare's method")

Definitions of the methods:

1. Approval:

Differs from 1-Vote Pluality in that voters may vote for more than 1 alternative if they so choose, giving a whole vote to each alternative they vote for.

Said to be a widely used alternative to 1-Vote Plurality.

The remaining 4 methods are rank-balloting methods. Voters rank some alternatives in order of preference: 1st choice, 2nd choice, 3rd choice, ... etc.

2. Condorcet's method:

A beats B if more voters have ranked A over B than vice versa. If 1 alternative beats each one of the others, then it wins.

If no 1 alternative beats each one of the others then the winner is the alternative over which fewest voters have ranked any 1 particular alternative that beats it. In other words:

For each alternative, determine which alternative that beats it is ranked over it by the most voters. The number of voters ranking that alterntive over it is the measure of how beaten it is. The winner is the alternative least beaten by that measure.

Condorcet was the French 18th century founder of voting theory. With regard to the lesser-evils problem, Condorcet's method has important properties not possessed by the other methods. They'll be described after the methods are defined.

I should say now that I claim that Condorcet's method is by far the best of these methods, and the one that I recommend.

3. Young's method:

As with Condorcet's method, A beats B if more voters have ranked A over B than vice-versa.

If 1 alternative beats each one of the others, then it wins.

If no alternative beats each one of the others, then the winner is the alternative with the smallest "overall margin" against it.

The "overall margin" against an alternative is the sum of the margins by which various alternatives beat it. The margin by which A beats B is the number of voters ranking A over B minus the number of voters ranking B over A.

4. Copeland's method:

As before, A beats B if more voters rank A over B than vice-versa.

The winner is the alternative for which the number of alternatives it beats, minus the number of alternatives that beat it, is the greatest.

Copeland's method often returns a tie, and must have a tie-breaker method specified with it.

5. Elimination: Repeatedly eliminate from the rankings the alternative occupying highest position in fewest rankings. Very popular. Not very good.

*****

PROPERTIES & STANDARDS:

Lesser-of-2-evils guarantees:

Approval guarantees that if a full majority of all the voters have voted A over B, that automatically defeats B, without anyone having to vote A over any alternative which they prefer to A.

Condorcet's method guarantees that if a full majority of all the voters vote A over B, then there's always a way that they can vote that will ensure the defeat of B, without anyone voting A equal to or over any alternative that they prefer to A.

In fact, under all reasonable & plausible conditions, with Condorcet's method, if a full majority of all the voters vote A over B, that will automatically defeat B, without anyone voting A equal to or over any alternative that they prefer to A.

What do I mean by "reasonable & plausible conditions"?

There is a "Condorcet winner, and order-reversal doesn't occur on a scale sufficient to change the election result.

A "Condorcet winner" is an alternative that would beat each one of the others if everyone sincerely ranked all the alternatives-- in other words, if everyone sincerely voted all their preferences.

"Order-reversal" is the practice of voting a less-liked alternative over a more-liked one. Highly implausible voting behavior, very unlikely to happen on a large scale.

Condorcet's method has an additional property:

Even if everyone mistakenly believes a certain alternative to be the necessary compromise, and everyone includes that alternative in their ranking, and no one includes in their ranking any alternative that they like less than it, that can't give the election away to that alternative if there's a Condorcet winner which a majority of the voters have ranked over it.

This is a re-wording of one of Condorcet's method's lesser-of-2-evils guarantees. It's a very important property. It's quite possible for all the voters to mis-estimate winnability, to underestimate the winnability of some candidate, for instance.

I call this property "invulnerabilty to mis-estimate".

***** Condorcet's method, Young's method, & Copeland's method are members of a class of methods called "Condorcet criterion methods. They meet Condorcet's criterion, which requires the election of an alternative that beats each one of the others.

The election of the Condorcet winner is important. Since it possesses a majority over each one of the other alternatives, meaning it's preferred to each one of the other alternatives by a majority of those voters having a preference between the two, it's the rightful winner, and it can obviously easily be made to beat any other alternative. It's important that members of a majority preferring it to another alternative be able to make it beat that other alternative without abandoning more liked alternatives. As you can tell, this is a re-statement of the desirability of the lesser- of-2-evils guarantees described earlier. In this section I'll approach the subject from a different starting point.

The desirability of electing the Condorcet winner is perhaps the only proposition approaching a consensus, on the subject of voting systems.

The Condorcet criterion methods, as I said, always elect an alternative that beats each of the others. The Condorcet winner is the alternative that would beat each of the others if everyone voted a sincere ranking of all the alternatives.

In a Condorcet criterion method, there are 2 things that can defeat a Condorcet winner: trunction & order-reversal.

Truncation is the voting of a short ranking that doesn't include all the alternatives (it can result in the Condrocet winner's defeat if it doesn't include the Condorcet winner). Order-reversal, as I said earlier, is the ranking of a less-liked alternative over a more-liked one.

So then, since it's generally agreed that we should elect the Condorcet winner, and there are only 2 things that can defeat a Condorcet winner in a Condorcet criterion methods, it's of interest how to prevent those things from defeating a Condorcet winner:

A method is "truncation-resistant" if truncation can never gain the election of an alternative over which a full majority of all the voters have ranked the Condorcet winner.

Condorcet's method is the only truncation-resistant method, of the 3 Condorcet criterion methods compared in this article.

A method is "order-reversal resistant" if:

If a full majority of all the voters have ranked the Condorcet winner over a certain other alternative, then, even if order-reversal is used, there's always a way that members of that majority can vote to ensure the defeat of that other alternative, while still voting any preferences that they have between alternatives that they prefer to that other alternative.

This is an awkward re-statement of one of Condorcet's methods's lesser-of-2-evils guarantees. The definition of truncation-resistance is based on Condorcet's method's other lesser-of-2-evils guarantees.

The purpose here is to put these properties in terms of protecting the Condorcet winner against the only things that can defeat a Condorcet winner in a Condorcet criterion method.

Condorcet's method is the only one of the 3 Condorcet criterion methods comprared in this article that is order-reversal resistant.

A method is "Condorcet protective" if it is a Condorcet criterion method, and is truncation-resistant, and is order-reversal resistant.

Condorcet's method is the only Condorcet protective method described in this article. ***

Condorcet's method's lesser-of-2-evils guarantee, I should have added in that section, is unmatched by any other method described in this article. Condorcet's method & Approval are the only one of these 5 methods that have such a guarantee.

Something much weaker can be said for Copeland's method & Young's method: Under the conditions that I called "reasonable & plausible", Copeland & Young guarantee that if a full majority of all the voters rank A over B, there's a way that they can vote that will ensure the defeat of B, without voting A over any alternative they prefer to A. (they might however have to vote A equal to an alternative they prefer to A).

*****

Academic criteria.

The academics have written several criteria. I'll name a few that are most often violated:

1. Condorcet criterion
2. Smith Criterion
3. Condorcet loser criterion
4. Majority criterion

There are a few issues of interst about these criteria. They're sometimes defined according to voters' psychological preferences, and sometimes in terms of actual votes. When defined in terms of psychological preference, they actually aren't met by any methods.

I define them in terms of actual votes, as I did earlier when I defined the Condorcet criterion as requirement that we elect any candidate that beats each one of the others.

But if they're defined that way, in terms of actual votes, in terms of what beats what, then, strictly speaking, they're met by Approval & 1-Vote Plurality, two methods not usually considered to meet these criteria. For instance, Approval or 1-Vote Plurality can be considered a system where one can rank alternatives either 1st or last. In Plurality one can only rank 1 alterntive in 1st place. So then, if in Approval or Plurality, a certain alternative beats each of the others, by being "ranked" over it by more voters than vice-versa, that means it will have more votes than each of the other alterntives, and will win.

If it's felt desirable to define the Condorcet criterion, and the other 4 criteria in that list, so that Approval & Plurality don't meet them, then it might be best to add a requirement, to those criteria, that the method allow as many rank positions as the voter wants to use.

***

Copeland's method meets all 4 of those criteria. Condorcet's method meets the Condorcet criterion & the majority criterion. (any method that meets the Condorcet criterion meets the majority criterion).

To me, those 4 academic criteria, except for the Condorcet criterion, seem less important than the properties & standards described earlier, which deal with the lesser-of-2-evils problem. That is, the academic criteria other than the Condorcet criterion don't address the reason why we want better single-winner methods. So, though Condorcet's method, in its plain form, as I've defined it, doesn't meet all the academic criteria, I claim that the properties that it has are much more important than the properties that it doesn't have. I'll later show how any method, including Condorcet's method can be made to meet all the academic criteria.

Definitions of those 4 academic criteria:

1. I've already defined the Condorcet criterion.

2. The Smith criterion says that if there's a set of alternatives such that every alternative in the set beats every alternative outside the set, then the winner should be chosen from that set.

3. The Condorcet loser criterion says that if an alternative is beaten by every other alternative, then it shouldn't be the winner.

Note that when Condorcet's method fails that criterion, it's when _every_ alternative is beaten, and every other alternative is more beaten by some other alternative than is the winner chosen by Condorcet's method. Also, as I said, I claim that the important properties that I've described for Condorcet's method are more important than these more cosmetic properties that it doesn't have.

4. The majority criterion says that any alternative that is the favorite of a majority should win.

Copeland's method meets all 4 of these criteria. Condorcet's method , in its plain form as I've defined it, meets the Condorcet criterion & the majority criterion, but not the Smith criterion or the Condorcet loser criterion. That's also true of Young's method. Elimination, of these 4 criteria, meets only the majority criterion.

***

But any method can be made to meet all 4 of those criteria by limiting its choice to a _subset_ of the alternatives, called the "Smith set":

The "Smith set" is the smallest set of alternatives such that every alternative in the set beats every alternative outside the set.

I should add that Condorcet's method retains all the properties that I've described for it, whether or not its choice is limited to the Smith set.

Obviously it would be desirable to define Condorcet's method so that its choice is limited to the Smith set, to avoid unnecessary criticism. But when it is defined that way, people tend to say the definition is too complicated. Condorcet's method is well accepted when the Smith set isn't included in its definition. So I'd suggest that Condorcet's method be proposed in its plain form, without any mention of the Smith set. Then, at some later time, the provision to limit its choice to the Smith set could be added, as a separate initiative or proposal, after the adoption of Condorcet's method, or at least after its plain definition is well-known.

If anyone brings up the objection that Condorcet's method doesn't meet Smith's criterion or (especially) the Condorcet loser criterion, that would be a good opening for adding to the proposal the provision to limit the choice to the Smith set.

*****

A note about Elimination: It's popularity makes it necessary to mention some of its problems:

Elimination is the only rank-balloting method that can fail to elect a Condorcet winner even if everyone ranks it 1st or 2nd.

Elimination possesses no lesser-of-2-evils guarantees. Even if a full majority vote A over B, in Elimination, sometimes the only way they can ensure the defeat of B is to vote A by itself in 1st place. (Elimination doesn't allow voting more than 1 alternative in 1st place, or in any other rank position). A, I emphasize, may not be those voters 1st choice, but they must sometimes rank it alone in 1st place to defeat a more disliked alternative, B.

*****

A few notes on other methods:

Runoff-Pairwise shares Condorcet's method's important properties, relating to getting rid of the lesser-of-2-evils problem & protecting the Condorcet winner.

Runoff-Pairwise:

If no 1 alternative beats each one of the others, then simply hold a 2nd balloting between all the alternatives (or all the alternatives in the Smith set), using 1-Vote Plurality or (preferably) Approval.

The results of the 1st balloting, showing what beats what, will make it quite obvious to everyone how far, if at all, they need to compromise in the 2nd balloting.

Only a few close relatives of Condorcet's method share the important properties that it & Runoff-Pairwise have.

***

The commonly used municipal "Runoff" system possesses all the disadvantages & problems of Elimination.

Mike Ossipoff --