Survey of Voting Methods that avoid Favorite-Betrayal

By Mike Ossipoff & Warren D. Smith

Executive summary

The Favorite Betrayal Criterion (FBC): Voters should have no incentive to vote someone else over their favorite.

By "incentive" we mean, "increases expected utility of the election winner versus if you'd voted in a way with your favorite top."

FBC is a very basic criterion to make democracy work well. Instant Runoff fails FBC, as do many other systems, including every Condorcet system and also all Condorcet systems where equal-rankings are permitted and even if based on "winning votes" not "margins." Also every point runoff system disobeys FBC, and so does every weighted positional rank-order ballot system except for anti-plurality voting.

That won't do. We need to be able to assure voters that they simply never need to vote their sleazy lesser-evil over their favorite in order to fully protect the lesser-evil. Ever.

So we need an election method that complies with FBC. We shall now survey them.

1. Approval Voting:

  1. Voters may mark, on their ballots, the names of as many candidates as they want to.
  2. The winner is the candidate who is marked on the most ballots.

Approval meets FBC.

Approval is remarkably good, for such a briefly-defined method. (Incidentally, we make the obvious, though significant, remark that Approval Voting causes the election of the candidate who is "ok" in the eyes of the maximum number of voters, provided "ok" means they approved of him.) And Approval is, of course, the most modest change from Plurality. Approval is "Plurality done right." What would it take to change from Plurality to Approval?: On the ballot, the words "Vote for 1" would have to be changed to "Vote for 1 or more." Two new words on the ballot. That's it. Cost of changing from Plurality to Approval? Zero.

Approval could be called "Set Voting", because the voter can vote any set of candidates over any other set of candidates. The voter has complete freedom of what sets, and what size sets. Nothing else equals its elegant simplicity.

But some object to Approval because of a confused and fallacious use of "1-person-1-vote". Of course 1-person-1-vote just means that everyone should have the same voting opportunity, counted the same way. That fallacious objection can be avoided by introducing Approval as Set Voting.

It can also be avoided by introducing Approval as a variation of "the Score System". Let me define the Score System, which, in voting system circles, is known as "Range Voting":

2. Range Voting:

  1. Each voter may give to any candidate any number of points, within a range (such as "0 to 10") pre-specified by the election rules.
  2. The winner is the candidate with the largest average score.

For instance, one popular version of Range Voting lets every voter give to any candidate any number of points from 0 to 10. This is basically what Olympics judges use, except they really are doing 0 to 100 range voting since they work on 0.1-point increments.

That method, 0-to-10 Range Voting, is probably the most winnable and popular one. We abbreviate Range Voting "RV".

So, Approval could be introduced and offered by first defining RV, and then proposing the {0, 1} version of RV – in which a voter may give to any candidate either 1 vote or 0 votes.

But the really familiar and popular RV version is the 0-to-10 RV version. 0-to-10 RV might therefore be the most winnable voting system reform proposal, due to its head start in familiarity and popularity.

For mathematicians we note that range voting could be done using median score instead of average score, or indeed a candidate's "total score" could be any function of his ordered set of scores monotonically increasing and continuous in each argument separately. Also, we could restrict the permitted scores to almost an subset of the real interval [0,1]. The result of all this freedom is a tremendously infinite set of FBC-obeying voting methods.

By the way, all of the RV versions are strategically equivalent to Approval: You maximize your expectation in the election if you give maximum points to every candidate whom you'd vote for if it were an Approval election, and give minimum points to the other candidates. This is not quite true – there are examples where it is strategically best to give some candidates intermediate scores.

For that reason, one might say that there isn't a great practical difference between RV and Approval. But in fact, there are several:

  1. As we said, RV is more familiar and popular. Approval is a new idea, and, at least when not presented properly, is subject to the fallacious "1-person-1-vote" objection.
  2. In Approval, the Lo2E voters who now vote for Kerry instead of Nader would feel compelled to vote for Kerry in addition to voting for Nader. But in RV, many of them may very well give to Kerry slightly fewer points than they give to Nader. That's a greater degree of honesty. RV, then, could let voters vote a little more honestly, showing some preference for their favorite over their lesser-evil.
  3. Additionally, some genuine preferrers of Kerry might give some points to Nader, which is more honest for them than zero.

Experimental polling has shown a pronounced effect of that kind. In the 2004 election Nader would have gotten 83 times as many votes as he did (relative to Bush) if RV had been used, and Cobb would have gotten 63 times more. But Cobb would have only gotten 25 times more with Approval – experimentally, RV always seems to give candidates not from the top-2 parties more votes than either Approval or Plurality, usually far more. This has been called the "nursery effect" since it "coddles" small "infant" third party candidates in a protective shell of voter honesty, allowing them experimentally to get much higher vote counts than with the all-or-nothing Approval system. That effect may be key in allowing third parties to grow to adulthood rather than dying of infant mortality.

An infinite number of variants of Range and Approval voting can be defined which also satisfy FBC, for example range voting with integer scores in {1,2,3,4,5}, and "procrustean" range voting where your vote must score your top-most candidate over 90% of the maximum possible, and must score your bottom-most candidate below 10%, etc.

3. Rank Methods:

Some prefer rank balloting methods. Instant Runoff is the one that we hear about the most. As we said, IRV fails FBC. It also has other problems. For instance, in IRV you can make someone lose by voting him higher. ("Non-monotonicity.") In fact, in IRV, voters can make someone lose by moving him from last place to first place on their ballot. Few or no serious participants in the discussion of voting systems prefer IRV.

But rank balloting has great promise, and some rank methods meet FBC. Many of them were invented by Kevin Venzke.

In Approval or RV, to fully protect your lesser-evil against your greater-evil, you have to vote your lesser-evil equal to your favorite. That's what rank methods can improve on. I'm going to describe one or more methods that accomplish that improvement. Methods that meet FBC, but which, in addition, let you fully protect your lesser-evil against your greater-evil by merely ranking your lesser-evil below your favorite, and above your greater-evil.

3a. Majority-Defeat-Disqualification Approval (MDDA):

  1. Voters may rank, in order of preference, as many or as few candidates as they want to. A voter may rank 2 or more candidates equally at the same rank position if s/he wants to. Ranking X but not Y counts as a way of ranking X over Y. For example a legal vote might be "Nader=Gore>Buchanan>Bush" where Phillips and Moorehead are not ranked.
  2. Candidates are disqualified if another candidate is ranked over them on more than half of the ballots. (Unless that rule would disqualify all the candidates, in which case no one is disqualified.) For example, among the three votes "Nader=Bush", "Gore>Bush", and "Nader>Gore>Bush", Bush is not disqualified by Nader since only 1 out of the 3 ballots (not a majority) rank Nader above Bush. However Bush here is disqualified by Gore.
  3. The winner is the un-disqualified candidate who is ranked on the most ballots.

It is also possible to consider a different version of MDDA where voters rank all candidates but also include a "threshold" in their vote such that all candidates above the threshold are considered "approved." That has been called "deluxe MDDA." It also avoids favorite betrayal, but it is felt by MDDA's inventors to be inferior to plain MDDA for reasons both of simplicity and strategic exploitation.

As we said, MDDA meets FBC. Very few rank methods do.

Additionally, MDDA enforces majority wishes well enough so that when you rank Compromise below Favorite, but above Worst, you are fully and reliably helping Compromise against Worst. Because ranking Worst below Compromise helps Compromise to majority-disqualify Worst, and also presumably you will leave Worst "unranked" to maximally decrease Worst's "approvals" (where ranking someone counts as "approving" them in MDDA).

That would let Nader voters safely rank Nader in 1st place and Kerry in 2nd place.

Let me be more specific, and define some of the particular ways in which MDDA enforces majority wishes:

A preliminary definition:

"CW" stands for "Condorcet winner", a candidate who, when compared separately to each one of the other candidates, is preferred to him/her by more voters than vice-versa.

There is usually a CW. When there isn't, it's difficult to say who should win.

Strategy-Free Criterion (SFC): If no one falsifies a preference, and if more than half of the voters prefer the CW to Y, and vote sincerely, then Y shouldn't win.

MDDA meets SFC. MDDA's compliance with SFC means that the members of that majority need do nothing other than vote sincerely, to ensure that greater-evil Y won't win. It describes a common, plausible set of conditions under which voters don't need to use any strategy, i.e. can just rank sincerely.

Strong Defensive Strategy Criterion (SDSC): If more than half of the voters prefer X to Y, then they should have a way of voting that ensures that Y won't win, without their having to reverse a preference, or fail to vote all of their sincere preferences among those candidates whom they vote over other candidates.

MDDA complies with SDSC. (So do Range and Approval.)

With MDDA, and other methods that comply with SDSC, the members of that majority who prefer X to Y can ensure that Y won't win by merely ranking X but not Y.

Thus, MDDA's SDSC compliance gives additional majority-enforcement. SFC and SDSC are a powerful majority-enforcement combination.

Voting in MDDA, one should rank candidates all the way down to the candidate whom they're sure can get a majority against the worse candidates. That candidate is the needed compromise, and typically is the CW.

Because of MDDA's SFC compliance, you can usually safely rank all the candidates if you want to. Certainly there's little danger if you overshoot a bit, and rank lower than you need to.

But of course you don't want to rank your greater-evil(s). Don't rank anyone who is completely unacceptable to you. In that way, you bring SDSC's majority-enforcement power to bear against those unacceptable candidates, in addition to that of SFC.

As we said, MDDA meets FBC, SFC, and SDSC.

Though MDDA seems to me to be the best proposal for a rank method, let me describe some alternatives. "MDDB" differs from MDDA only in how it chooses from among the un-disqualified candidates:

3b. Majority-Defeat-Disqualification Borda (MDDB):

  1. (Same as for MDDA.)
  2. Candidates are disqualified if another candidate is ranked over them by more than half of the voters. (Unless that rule would disqualify all the candidates, in which case no one is disqualified).
  3. The winner is the un-disqualified candidate who has fewest candidates ranked over him/her, as summed over all the rankings. (E.g. for the two votes "Nader>Gore>Bush" and "Gore>Bush", Bush has 3 candidates ranked above him, as summed over both rankings.)

But MDDA seems superior to MDDB because of its immunity to clones. That was assuming all X's clones are ranked co-equal to X on all ballots. But if voters can have slight preferences among the clones, then MDDA is not clone-immune and indeed every MDD-method is not (and nor is Simpson-Kramer min-max[pairwise opposition] nor ICA), because a majority-preference cycle can appear disqualifying every clone of the winner, including that winner. Similarly, ER-Bucklin is clone-immune with clones equality-ranked, but not with preferences among the clones, since cloning the winner can cause all winner-clones to be delayed in acquiring the necessary vote-majority, allowing somebody else to win sooner. MDDB indeed shares a lot of horror-features with Borda voting.

3c. ER-Bucklin:

  1. (Same as for MDDA.)
  2. We proceed in rounds. If a ballot lists n candidates as tied in kth place, count that ballot as a whole point for all n candidates beginning in the kth round.
  3. Note: A candidate is ranked "in kth place" on a given ballot if there are k-1 candidates who are ranked strictly higher. For example, a ballot marked A>B=C=D>E>F=G=H=I>J should be considered to rank A first, B, C, and D tied for second, E fifth, F, G, H, and I tied for sixth, and J tenth. Thus, the ballot would not count in favor of E until the 5th round, and it would not count in favor of J until the 10th round.
  4. The rounds proceed until the candidate with the most points has more points than 50% of the number of voters; that candidate wins.

As just defined, ER-Bucklin unfortunately can fail to deliver any winner, if enough voters truncate their ballots. We can solve that problem by (in that case) just declaring whoever gets the most votes in the first round, to be the winner (in which case ER-Bucklin just becomes approval voting), but that "solution" unfortunately can reward ballot truncation. Actually Bucklin already does reward ballot truncation, because voters soon realize that ranking somebody lower than their favorite can help to defeat that favorite. Bucklin (although not the ER variant with equal-ranks permitted) was employed in several US states to elect governors, but was abandoned (FairVote alleges) for precisely this reason: most voters simply voted plurality-style, i.e. maximally truncating their ballot. We're not convinced that really was the reason, but it indisputably was abandoned. Other "solutions" would be to outlaw truncated ballots or to artificially extend any truncated ballots to rank the remaining candidates coequal last. Those both have disadvantages. Chris Benham suggested another idea: whoever has the most points at the end of the process just wins, even if the threshhold is not reached. (Bucklin also resembles Approval in the sense that if the last Bucklin round is the kth, then the same result would be obtained by a Bucklin election and an Approval election in which each voter "approved" candidates ranked k or better. )

3d. Simpson-Kramer Min-Max(Pairwise Opposition):

  1. (Same as for MDDA.)
  2. Elects the candidate whose greatest opposition from another candidate is minimal. Pairwise wins or losses are not considered; all that matters is the number of votes ranking the opposed candidate over you.

3e. ICA:

  1. (Same as for MDDA.)
  2. Again as in MDDA, a voter implicitly approves every candidate whom he explicitly ranks.
  3. Let v[a,b] signify the number of voters ranking candidate a above candidate b, and let t[a,b] signify the number of voters ranking a and b equally at the top of the ranking (possibly tied with other candidates).
  4. Define a set S of candidates, which contains every candidate x for whom there is no other candidate y such that v[x,y]+t[x,y]<v[y,x].
  5. If S is empty, then let S contain all the candidates.
  6. Elect the candidate in S with the greatest approval.

In other words, every candidate a is disqualified who pairwise loses to some other candidate b, and would still lose to b even when the voters supporting both equally as first preferences are counted in favor of a. If everyone is disqualified, then no one is. Then the most approved candidate who isn't disqualified is elected.

Perhaps also other voting methods may be converted to be FBC-compliant by using Venzke's "tied at the top trick." (I am not quite sure when this works.)

Venzke has also discussed some variant forms of ICA. Venzke also notes: For another way tied-at-the-top can work, consider a (rather bad) method which elects the candidate with the greatest number of rankings over some other candidate. (That is, if the matrix has no value greater than v[x,y], elect x.) By itself, this method doesn't satisfy FBC because it could be that introducing a strict ranking between x and y could cause v[x,y] to be the greatest win and move the win to x from some other candidate.
But we can use the "tied at the top" rule to fix this. Just say that when x and y are "tied at the top" on a given ballot, this vote counts to both v[x,y] and v[y,x]. Then all that can be "gained" by introducing a strict x>y ranking is that v[y,x] is reduced.

MDD-hybrids: It is possible to create new FBC-complying voting methods by including an MDD (Majority-Defeat-Disqualification) step as a preface to some FBC-obeying method. For example, two MDD-range hybrids would be:

MDDR1:

  1. voters contribute range-style votes.
  2. candidates A scored below B by a majority of the voters, are disqualified (unless all would be disqualified, in which case none are)
  3. among the remaining candidates: your "margin of disfavor" is the sum, over all ballots, of the difference between the max-scored candidate's score and yours.
  4. The winner is the candidate with the least margin-of-disfavor.

MDDR2 would be the same, except in step C redefine a candidate's "margin of disfavor" as the sum, over all ballots, over all candidates scored above him, of the difference between that candidate's score and his.

MDDR1 seems superior to MDDR2 because of its immunity to clones. MDDR1 has the severe disadvantages relative to Range Voting that "no opinion" scores for some candidate are easy to permit in range voting but not in MDDR1; also MDDR1 is more complicated to describe.

Another interesting MDD-hybrid method would be MDD-ER-Bucklin which is ER-Bucklin prefaced by a disqualification of majority-defeated candidates.

4. Lottery methods

The simplest lottery methods, introduced by Gibbard, are "random dictator" and "random pair."

Random dictator:

  1. voters contribute rank-orderings as votes (for simplicity, let us disallow equal rankings).
  2. A random voter is chosen and his top choice is declared the winner.

Random Pair:

  1. voters contribute rank-orderings as votes (equalities can be allowed, or not, either works)
  2. A random pair of candidates is chosen, and the winer is whichever among those two candidates wins the head-to-head election got by removing all other candidates from all votes.

These two schemes are remarkable and unique in that sincere and optimally-strategic voting are always the same thing. However, Gibbard dismissed these schemes as having no practical interest in real elections because "they leave too much to chance."

5. AntiPlurality

The "AntiPlurality" voting system (which we do not recommend) is you name your least-favorite candidate as your vote, and the least-named candidate wins. It and some other related systems technically obey FBC.

How to prove FBC

For all of the voting systems above, one can prove FBC-compliance by the following strategy: If betraying favorite F in order to make X win is the plan, and if that plan actually works, then the alternate non-betrayal plan of simply raising X to be co-equal top with F (carried out by the same set of voters who planned to betray F, using the same set of votes they planned on) also works to make X win. Q.E.D.

Specifically, this proof works for MDDA, MDDB, MDDR1, MDDR2, ICA, ER-Bucklin, Min-Max(pairwise-opposition), and MDD-hybrid-X where X is any FBC-obeying system for which that proof strategy worked.

But for Range and Approval voting (and the lottery methods) a simpler FBC proof works, because these systems actually obey a stronger form of FBC. Specifically, raising or lowering your range or approval vote for F in no way affects the relative election chances of the other candidates. This makes it immediately obvious that betraying F is strategically pointless. With MDDA and MDDB, in contrast, betraying F can be strategically useful – it is just that when it is, there is an alternate non-betrayal strategy that also works.

For example in MDDA:

#voters Their Vote
2 F>X
49 F>X>Y
49 Y>X

Y wins. However, if F is maximally-betrayed we get

#voters Their Vote
2 X
49 X>Y
49 Y>X

and X wins, which those voters prefer. The reason MDDA obeys FBC is that, although this betrayal works, it is not the only thing that works – non-betrayal strategies also work.

So FBC is obeyed is a stronger sense by Range Voting than by MDDA. With range voting, if you blow away F by betraying him, then that will not alter the winner (unless F had won). In contrast, in MDDA, betraying F can alter the winner, and sometimes in a way you consider favorable.

So with range voting, favorite-betrayal just doesn't work.

With MDDA, favorite-betrayal can work, but there is always a non-betrayal method of achieving the same strategic goal.

Comparison

MDDA, MDDB, MDDR1, MDDR2, ER-Bucklin, Simpson-Kramer-Min-Max(PO), and every MDD-hybrid method all might suffer in practice from the devastating DH3 pathology, which, however, plain Range and Approval voting are immune to.

MDDA, MDDB, MDDR2, MDDR1, ER-Bucklin, Simpson-Kramer-Min-Max(PO), and every MDD-hybrid method all are vulnerable to candidate-cloning if the voters express slight preferences among the clones. Range and approval (but the latter only under the assumption clones get equal approval ratings) are immune to cloning. MDDA, MDDR1, ER-Bucklin, and Simpson-Kramer-Min-Max(PO) are, however, immune to clones assuming (probably unrealistically) voters always rate clones as equal.

This leads to the very interesting

Conjecture: Continuum Range Voting (and its obvious variants, such as range voting where the top-scored candidate must be given at least 90% of the maximum score) are the only FBC-obeying voting methods immune to candidate-cloning (with voters assumed to have tiny preferences among the clones) where we disregard non-deterministic election methods.

Later Note (Jan 2007): Big progress has been made toward proving this conjecture.

Also, because the MDD-based methods are considerably more complicated than Range Voting, RV is probably the best proposal for public consumption. In particular, both Approval and RV can be handled by all US voting machines. None of rank-ballot FBC-methods (although Simpson-Kramer and ER-Bucklin come substantially closer than usual) can be. Also the experimental "nursery effect" may not be present with the rank-ballot methods, in which case range would be far superior from the point of view of US third parties.

A range voting advocacy group has formed: the "Center For Range Voting" http://www.RangeVoting.org.


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