Suppose that, unlike current conditions, there are no disinformational media to lead gullible voters to do seriously misinformed favorite-burial. And suppose that voters and factions aren’t subject to the self-destructive rivalries that lead to the chicken-dilemma. I call that “Ideal majoritarian conditions”. Under those conditions, we’re free to choose a voting system for its ideal majoritarian performance, without concern about the strategy problems of favorite-burial need and chicken dilemma.

Ideal majoritarian conditions don’t currently obtain, for our official public political elections. But they seem to obtain in polls, and in many or most organizations. For polls, and for ideal majoritarian organizations, I recommend an ideal majoritarian method called MAM (defined below, and in my previous article, “On Four Kinds of Voting Conditions, with Recommendations“). It’s assumed that it’s still desired to have majoritarian results, majority-rule. But even when maximizing social utility is, instead, the goal, it turns out that majoritarianism will often be the most reliable way to achieve that, because:

When distance between a candidate and a voter in issue-space is measured by “city-block distance” (the sum of the distances in the various mutually—perpendicular issue-co-ordinate dimensions), the Condorcet Winner (a candidate who pairwise-beats each of the others) is the “social utility maximizer”, the candidate whose summed distances to all the voters is the least. And there are good arguments for why city block distance is the more meaningful distance measure in electoral issue-space.

So the ideal majoritarian voting systems are a good choice even when the electorate’s goals aren’t strictly majoritarian. The ideal majoritarian voting systems are Condorcet voting systems. They’re voting systems that meet the Condorcet Criterion, defined in my article before this one. But two kinds of Condorcet methods do even better than the others, by the various majority-rule criteria: Ranked-Pairs, and Beatpath. They’ll be defined later in this article. Though Ranked-Pairs was defined in the previous article as well, a more complete instruction for MAM will be given here, and a definition of Beatpath (though I recommend MAM instead of Beatpath).

A Few Majoritarian Criteria:

• The Condorcet Criterion (CC)
• The Smith Criterion (SC)
• The Mutual Majority Criterion (MMC)
• The Strong Defensive Strategy Criterion (SDSC)
• The Strategy-Free Criterion (SFC)

The best ideal majoritarian voting-systems meet all of the above-listed criteria. Steve Eppley uses “ballots-only” (they don’t mention preferences) versions of SDSC and SFC, known respectively by the names Minimal Defense, and Truncation-Resistance. The advantage of referring, in criteria, to preferences is that doing so often confers universal applicability on a criterion. Ballots-only criteria are often misleadingly, un-meaningfully and unintentionally passed by Plurality. Therefore, my versions of CC, SC, MMC, SDS C and SFC all refer to preferences.

The name “Strategy-Free Criterion”, is probably misleading, and Eppley’s name “Truncation-Resistance” is probably a much better name. But it’s probably too late for me to change my name for the criterion. I defined CC in the article before this one, and in a few previous ones as well, so I’ll leave its definition out this time. But I’ll state here the definitions of the other above-listed criteria, because, criteria are what distinguish voting-systems.

Smith Criterion:

1. X is socially-preferred to Y if the number of people who prefer X to Y is greater than the number of people who prefer Y to X.
2. The Smith set is the smallest set of candidates such that every candidate in the set is socially-preferred to every candidate outside the set.
3. If everyone votes sincerely, then the winner should come from the Smith set.

(The Smith set defined above is usually called the “Sincere Smith Set” to distinguish it from the Voted Smith Set, the one referred to in the definition of Woodall’s method, a Green scenario method defined in my earlier article, “Voting Systems for the Green Scenario”. )

Every voting-system that meets the Smith Criterion also meets the Condorcet Criterion and the Mutual Majority Criterion.

Mutual-Majority-Criterion (MMC):

A mutual majority is a set of voters who all prefer some same set of candidates to all of the other candidates. Those candidates are a “majority-preferred set” of theirs. Their innermost majority-preferred set is their majority preferred set that doesn’t contain a smaller majority-preferred set of theirs. If a mutual majority vote sincerely, then the winner should be from their innermost majority-preferred set. I suggest that MMC is the most basic and important majority-rule criterion.

Definition of sincere voting (for the purposes of these criteria):

• A voter votes sincerely if s/he doesn’t falsify a preference, or fail to vote a felt preference that the balloting system in use would have allowed hir to vote in addition to the preferences that s/he did vote.
• To vote a preference is to vote one candidate over another.
• To falsify a preference is to vote X over Y and not prefer X to Y.
• To vote a felt preference is to vote X over Y and prefer X to Y.

Strong Defensive Strategy Criterion (SDSC):

If a majority prefer X to Y, then they should have a way of voting that will ensure that Y won’t win, without any member of that majority voting a candidate equal to or over a candidate whom s/he likes more (except if voting them both at bottom). A voter simply puts a candidate at bottom if s/he doesn’t vote that candidate over anyone, and votes someone over that candidate.

SDSC is the opposite of CD (defined in my earlier article “Voting Systems for the Green Scenario”). CD and SDSC are mutually incompatible. CD says that 1-sided coalitions don’t count, but SDSC says that 1-sided coalitions _do_ count, to defeat a majority-disliked candidate. That’s an example of the majoritarian gains that become available when avoidable strategy-problems, such as media-deception-induced favorite-burial, or rivalry-&-distrust-motivated chicken-dilemma are absent.

Strategy-Free Criterion (SFC):

If a majority prefer the CW to candidate Y, and vote sincerely, and if no one falsifies a preference, then Y shouldn’t win. “CW”, here, refers to the “Sincere Condorcet Winner”, a candidate who is socially-preferred to each one of the other candidates.

As mentioned above, Eppley’a name of “Truncation-Resistance” better describes SFC. Ideal majoritarian methods could be defined as methods that meet all of the above-listed majoritarian criteria, which could also be called majority-rule criteria.

Two Ideal Majoritarian Voting-Systems:

The only two ideal majoritarian voting systems that I’m aware of are Ranked-Pairs (particularly in its MAM version) and Beatpath. (Even if Kemeny meets the above-listed criteria too, Kemeny unnecessarily fails Independence From Clones, which, along with its NP-Hard computation-time, disqualifies it from consideration. MAM and Beatpath meet Independence From Clones.)

Steve Eppley and Norman Petry did simulations to compare MAM and Beatpath. They both, independently, found that, in the vast majority of instances where MAM and Beatpath returned different winners, the number of voters preferring MAM’s winner to Beatpath’s winner was greater than the number of voters with the opposite preference.

Therefore it can fairly be said that MAM is the ideal majoritarian voting system qualifying for recommendation. Additionally, RP, exemplified by MAM, has a much more simple, brief, obvious, and natural definition. The various RP versions differ according to how they deal with equal defeats. The best turns out to be the one called MAM, which randomly deals with equal defeats. So I’ll first define RP, and some of its versions, and later will define Beatpath last. (I’ll define Beatpath only because it’s used by a fair number of organizations, probably due to heavy promotion.)

Ranked-Pairs (RP):

1. A “defeat is an instance of one candidate beating another (as defined above in this article).
2. A “cycle” is a cyclical sequence of defeats, such as A beats B beats C beats A.
3. A defeat “contradicts” a set of other defeats if it is in a cycle consisting of only it and them.
4. If X beats Y, then the strength of that defeat is measured by the number of people who rank X over Y. (If, for two defeats being compared, that quantity is equal, then, among those two defeats, the stronger one is the defeat with fewer people ranking the defeated candidate over the defeating one.)
5. A defeat is a discarded defeat if it contradicts a set of not-discarded defeats that are stronger than it is.
6. The winner is the candidate who doesn’t have any not-discarded defeats.

I’ll call that “Basic RP”. That defines RP when there are no equal defeats, where defeat strength is defined as above. But, in organizational elections or family elections, or in polls, with fewer voters, there can be defeats that are equal in strength, making it desirable to add to the above definition, to take equal-defeats into account.

The above RP definition can be, and sometimes is, used for counting RP even when there are very few voters. But, its disadvantage for such small elections is that, when there are equal defeats or pairwise-ties, it can discard insufficiently many defeats, sometimes failing to return a winner, resulting in unnecessary indecisiveness or vulnerability to clone-candidates. The best RP version for small elections is MAM, defined below. It uses randomization to deal with equal defeats and pairwise-ties. There are other RP versions, without randomization, that avoid the problem of not discarding enough defeats—but they can discard _too many_ defeats, again resulting in the same problems.

For family, committee, organization or poll elections, the following change could, and, for best results, should, be added to the above-stated basic RP definition, making a different version of RP, known as Maximize Affirmed Majorities (MAM):

Maximize Affirmed Majorities (MAM):

Paragraph #5 from the basic RP definition is slightly changed, as follows: A defeat is a discarded defeat if it contradicts a set of defeats each of which is either stronger than it, or else equal to it and higher than it in the dominance-order.

1. To determine the dominance-order of the defeats, first randomly choose a ballot.
2. Only one randomly-chosen ballot is needed per election.
3. To determine the defeats’ dominance-order from the randomly-chosen ballot:

Suppose that we’re comparing defeat xy, in which x beats y, and defeat zw, in which z beats w:

xy is above zw in the dominance-order if, in the randomly chosen ballot, w is ranked over y. But if, in that randomly-chosen ballot, if w and y are ranked equally, then xy is above zw in the dominance-order if that ballot ranks x over z.

Other than that modification, MAM’s definition is the same as that of Basic RP. The dominance-order effectively makes it as if there were no equal defeats. When there are few voters, the advantage of MAM is that it doesn’t discard unnecessarily many defeats. That’s desirable, because a defeat is a public decision that a certain candidate (the defeated one) shouldn’t be elected. But MAM also doesn’t indecisively leave unnecessarily-many defeats un-discarded. Thereby, it avoids the above-named problems of non-randomizing RP versions.

When voters are at the same location together, or when they fully trust the people and software doing the count, then any randomizing method, such as drawing a number from a bag, or a computer pseudorandom routine, would be fine. But when voters are not at the same location, as in an Internet vote, and when they don’t necessarily trust the persons and software doing the count, then there’s still a secure and trustworthy way to do the randomization:

Based on some numerical attributes of the set of rankings, maybe with some arithmetic operations being done on certain numbers describing the rankings-set, a pre-announced rule chooses one of the rankings as the basis of the dominance-order for that poll or election.

Determinist RP versions:

There are various RP versions that less-frequently need randomization. They’re often called “deterministic RP versions” But, when there are few voters, those non-randomizing (“deterministic”) RP versions all suffer from the above-described disadvantages, when they discard unnecessarily many, or indecisively-few, defeats. MAM is better than any deterministic RP version. MAM is what I recommend, when there are few enough voters for there to be equal defeats, or when it’s desired to take into account the possibility that there could be.

But, though it isn’t as good as MAM, I’ll describe one “deterministic” (having less often a need for randomization) RP version, which I call Hierarchial Deterministic Ranked-Pairs (HDRP):

Hierarchical Deterministic Ranked-Pairs (HDRP):

1. A defeat, D, is a discarded defeat if it contradicts a set of defeats that includes a not-discarded defeat that’s stronger than D, and doesn’t include any defeat that is weaker than D.
2. Additionally, a defeat, D, is a discarded defeat if it contradicts a set of defeats each of which is 1) equal to D; and 2) not declared discarded in paragraph 1 (immediately preceding this paragraph).

Beatpath:

Another ideal majoritarian method is called “Beatpath”, or “Schulze’s method”. But, in simulations by Steve Eppley and Norman Petry, it was found that, in the vast majority of instances where Beatpath and MAM choose differently, more voters prefer the MAM result to the Beatpath result than vice-versa. Therefore, I don’t recommend Beatpath. I recommend MAM, when an ideal majoritarian method is desired, for ideal majoritarian conditions. But, for completeness, I should define Beatpath in this article.

Beatpath’s beatpath definition:

1. There is a beatpath from X to Y if either X beats Y, or X beats something from which there is a beapath to Y.
2. The beatpath from X to Y consists of the defeats which, based on paragraph 1, make it possible to truly say that there is a beatpath from X to Y.
3. The strength of a beatpath is measured by its weakest defeat.
4. If the strongest beatpath from X to Y is stronger than the strongest beatpath from Y to X, then X beatpath-beats Y.
5. The winner is the candidate who beatpath-beats each one of the other candidates.

Of course, Beatpath’s name derives from this definition. It should be mentioned that the definitions of Beatpath lack MAM’s thorough rules for dealing with the ties that can happen when there are few voters. That could misleadingly make MAM look more complicated than Beatpath. If the strongest beatpath from X to Y is equal to the strongest beatpath from Y to X, then one could solve the tie by saying that the strengths of those two beatpaths shall be compared by looking at their _2nd_ weakest defeats.

Beatpath’s Schwartz-set definition:

Beatpath is equivalent to Cloneproof-Schwartz-Sequential-Dropping (CSSD), which will be defined here. First the Schwartz set’s 2 definitions must be defined:

1. The Schwartz set’s unbeaten set definition:

1. An unbeaten set is a set of alternatives none of which are beaten by anything outside that set.
2. An innermost unbeaten set is an unbeaten set that doesn’t contain a smaller unbeaten set.
3. The Schwartz set is the set of alternatives that are in innermost unbeaten sets.

2. Beatpath definition of Schwartz set:

1. Alternative X is in the Schwartz-set unless, for some Y, there is a beatpath from Y to X, but not from X to Y.

Now, with the Schwartz set defined, here is the definition of CSSD:

Cloneproof-Schwartz-Sequential-Dropping (CSSD):

1. The current Schwartz set is the Schwartz set based only on un-dropped defeats.
2. Repeatedly, drop the weakest defeat among the alternatives in the current Schwartz set.
3. Continue until there are no defeats among the current Schwartz set.
4. The member(s) of that current Schwartz set win(s).

Maybe Beatpath’s beatpath definition is the briefer of Beatpath’s two definitions. But the CSSD definition is more naturally and obviously motivated and justified. I’ve defined Beatpath (and CSSD, which is equivalent to Beatpath) because Beatpath and CSSD are used by many organizations, but MAM is what I recommend for ideal majoritarian conditions, for the above-stated reasons.

As mentioned above, and in my previous article, polls seem to have ideal majoritarian conditions. We can assume that family votes do, as well. The Condorcet Internet Voting Service offers four rank-counts, by which it will automatically count polls at that website. They are:

• Benham (referred to there as Condorcet-IRV)
• MAM
• Basic RP, as defined above (referred to there as CIVS-RP)
• Beatpath (referred to there as Schulze)

(Benham is a Green scenario voting system, defined in my previous article on kinds of voting conditions, and also in various articles of mine before that).

There’s convincing evidence that successful chicken dilemma defection isn’t happening in polls, and that, therefore, polls aren’t having any chicken-dilemma problem. So, evidently, the conditions that obtain in polls are ideal majoritarian conditions. Therefore, for polls, such as those at CIVS, I recommend MAM. That best of all ideal majoritarian voting systems is one of those that is offered at CIVS, for automatically counting its polls.

At CIVS are several polls among political candidates, political parties, and categories of political parties. They’re all public polls, and anyone can vote in them. Likewise, anyone can easily start a new rank-balloting single-winner poll, on any question or issue, political or otherwise, important or trivial.