The "problem" with circularity (was Re: Reply to Blake Cretney)

[Steve Eppley]

Blake Cretney has been doing a fine job replying to Donald Davison's 
musings about circularity, but I have a few words I'd like to add:

Donald Davison wrote:
>      It would be nice if Condorcet did not have those circular 
> ties. Maybe something can be done to eliminate them.
-snip-

The circular ties in some Condorcetian methods (but not in plain 
Condorcet!) correspond exactly to the voters' circular preferences 
which they expressed in their preference order ballots.  They are an 
attribute of the voters, not of the voting method.  One can't 
eliminate voters' preferences (unless we lobotomize voters)!  All one 
can do is ignore their preferences, which wouldn't be as democratic 
as tallying them.

I advise Blake to use the terminology "voters' circular preferences" 
rather than Donald's "circular ties" to make it clearer that 
circularity, when it occurs, is an attribute of the voters and not an 
artifact of some methods.

Most people who have considered these issues have concluded that the 
winner should always be one of the candidates in the top cycle.  (See 
definition below.)  Whenever an organization uses the voting method  
recommended by Robert's Rules of Order or something similar, one of 
the top cycle is elected.  Since most voting is done with such 
methods, it's fair to say that most voting already satisfies the top 
cycle criterion and what we're trying to do is modernize our 
primitive public election methods so they also satisfy it.  

Better methods, satisfying the top cycle criterion, have been used in 
assemblies and not in public elections because given primitive 
technology they are practical in assemblies but not in large public 
elections.  The marginal cost of repeated rounds of voting is small 
in assemblies but large in public elections.  What has now changed is 
that the invention of machine-readable ballots--punch cards, etc.--
have made better methods practical in large public elections.  (The 
most important criterion is Feasibility, and what is feasible changes 
with technological advances.)

   Top Cycle: the smallest, non-empty subset of alternatives such
   that every alternative in the subset "defeats pairwise" every
   alternative not in the subset.  (It is sometimes called the 
   Smith set, and it could be called the "dominant candidates.")  

   "Defeats Pairwise":  Alternative X defeats pairwise alternative Y
   when more voters ranked Y ahead of X than ranked X ahead of Y.

Given that the winner ought to be one of the top cycle, then when the 
top cycle includes only one candidate--not really a cycle--that one 
should be the winner.  When the top cycle includes only one, that one 
is often called the "Condorcet winner."

The "problem" to which Donald alluded is to decide which one of the 
top cycle should be elected when the top cycle has more than one, 
which happens when the voters have circular preferences regarding 
three or more of the dominant candidates.  (In theory non-circular 
ties can also happen when vote counts match exactly, but as with 
existing methods that kind of tie would be extremely rare.)

I think it's much less important, and not really valid, to try to 
identify which one of the top cycle is "best" as long as one of them 
gets elected.  (Recall the discussion about the merits of the 
Smith//Random method.)  It's far more important to use a voting 
method which minimizes incentives for voters to strategize, for 
potential candidates to not compete, and for parties to want to 
nominate only one candidate.  If it minimizes those incentives, it 
will also elect one of the top cycle.

Of course, such a method also needs to be understandable to curious 
voters.  *That* is our real problem, given the poor state of our 
nation's education system.  In my opinion, "pairwise comparison 
principles" should be part of the public school curriculum; then it 
would be easy to adopt the best election methods.

> We are justified in reducing the lower choices because the voters 
> themselves value the lower choices less than their most preferred choice.
-snip-

Donald has apparently confused "utility" with "utility differences."  
(An "absolute" with a "relative.")  A preference is a comparison 
between two alternatives.  The expression "I like Ike" really means 
"I prefer Ike to Adlai."  Support and opposition are relative.

Just because a voter prefers a favorite (A) more than a compromise  
(B) does NOT mean his/her relative preference for B rather than C is 
in some way weaker than other voters' preference for B rather than C. 
It's possible for the "B>C" preference of an ABC voter to be just as 
strong as, or stronger than, the "B>C" preference of a BAC voter.  
Here's an example to illustrate, where for convenience the voters' 
utilities are expressed in dollars:

   Voter 1's evaluations of the candidates' utilities:
      A = +$1.5 billion
      B = +$1 billion
      C = -$1 billion

   Voter 2's evaluations of the candidates' utilities:
      A = +$0.5 billion
      B = +$1 billion
      C = -$1 billion

   Note that both voters evaluate B and C identically.  The
   difference between utilities is what matters; that difference 
   is B-C = $2 billion.  Therefore both voters will have identical
   B>C preference intensities.  (It would have been just as easy 
   to provide an example where the BAC voter's B>C preference is
   counter-intuitively smaller than the ABC voter's B>C preference,
   simply by having voter 2 rate C = -$0.5 billion.)

Information about the strength of voters' preferences is entirely 
lacking from preference order ballots.  One should avoid the urge to 
jump to "intuitive" conclusions about other meanings "inferrable" 
from an order of preference.  Preferences are relative, not absolute, 
and rankings contain no information about preference intensities.
(Sadly, methods which ask voters to express their intensities or 
utility differences invariably create strong incentives for voters to 
exaggerate.)

One thing we can say for sure is that a voter whose preference order 
is ABC wants to be able to vote ABC and have that counted as a full 
strength vote for B over C when that's needed to defeat C.  What 
happens if a system (like plain IRV) doesn't allow the voters to 
reliably do that?  It will deter candidates from competing, it will 
cause each party to want to nominate only one candidate, and it will 
cause voters like the above to insincerely rank B ahead of A whenever 
it appears that B and C are both electable, A is probably not, and B 
could be knocked out of contention if they fail to rank B ahead of A. 
It will often backfire when the voters and potential candidates lack 
the info about which ones are electable, as is the case in U.S. local 
non-partisan elections where plain Instant Runoff is currently being 
promoted.

So the method needs to count those relative preferences about 
"lesser" candidates fully, when the voter needs them counted fully.

We already know that given a "select only one candidate" method, most 
voters are willing to cast their vote for the "compromise" B in order 
to defeat a "greater evil" C.  The problem people have with 
compromising is that they hate to compromise when compromise is 
unnecessary.  They have no problem compromising when they see that's 
needed to prevent an even worse outcome.  Flawed systems make it hard 
or impossible to know when and to what extent compromise is 
necessary, and thus facilitate special interest minorities getting 
their way when a majority prefers something else.  Good systems make 
compromise (and majority electoral coalitions) automatic.

As Blake pointed out, IRV advocates make a big deal about the fact 
that IRV guarantees that ranking a second choice can never help 
defeat one's first choice.  They gloss over the fact that ranking a 
first choice ahead of one's second can easily help elect one's third 
or worse choice, and that to compensate for this flaw the two big 
parties will need to continue nominating only one candidate.  The two 
parties will continue to nominate using a wealth-driven primary 
system, with a tendency to nominate off-center candidates due to the 
higher primary turnout of party activists and the disenfranchisement 
of independent voters in most states' primaries.  That means the 
outcomes will be the same with IRV as we already have in our partisan 
elections, and the two big parties will remain entrenched.  It will 
offer (bogus) evidence that there's no need for more than two 
parties.  As Gary Cox pointed out in his book _Making Elections 
Count_, Instant Runoff obeys Duverger's Law (which says that the two-
party system is caused by the voting method).

Perhaps Donald no longer advocates Instant Runoff, which is suggested 
by his search for a noncircular method which satisfies the Condorcet 
criterion.  It should be pointed out that what Instant Runoff does 
with the voters' less-preferred candidates isn't consistent: 
sometimes it gives zero weight to them, then it may give full weight 
to them, as if magically the voter switched from zero support to full 
support.  One wonders what they mean by "support" when they claim 
that Instant Runoff elects candidates with more support.

>      Condorcet has as many as one third circular ties.
-snip-

Actually, they can occur more often than that.  The more candidates 
competing to be in the center (which is where they'd try to be if 
they want to be elected, given a system which elects centrists and 
defeats minority wings) the greater is the chance that voters' 
preferences will be circular.  Also, the more voters there are, the 
greater is the chance their preferences will be circular.  

Peter Ordeshook's book _Game Theory and Political Theory_ provides a 
table showing the probability of circularity as a function of the 
number of voters and the number of alternatives.  He notes that the 
probability apparently goes to 100% as the number of voters and 
alternatives both go to infinity.

What does it matter if the voters' preferences are frequently 
circular?  It's actually a good sign, in my opinion, because it 
indicates that a lot of electable alternatives are competing and 
therefore give the voters an opportunity to express their most 
important relative preferences.

>      If this theory proves to be true, the gain for the pairwise
> people would be great - no circular ties. 
-snip-

More precisely: We would gain by not having to market to poorly 
educated voters a procedure which is defined using language which is 
potentially more complex (but not necessarily more complex) than some 
other methods.

We've seen several reasonably good methods (better than Instant 
Runoff) which don't use the language of circularity.  Plain Condorcet 
is one, Regular-Champion is another.  Both of these are mentioned in 
greater detail below.  (Borda is yet another method free of the 
language of circularity, but given Borda voters would learn to use a 
"turkey-raising" offensive reversal strategy and would also have an 
incentive to defensively downrank their favorites.)

My humble offering is to move the complexity out of the method, into 
the laps of the dominant or near-dominant candidates (or the laps of 
the sponsors of dominant ballot propositions, when we vote on rival 
propositions).  Simply precede Instant Runoff (or plain Condorcet) 
with two simple steps:

   1. After the voters cast their preference order ballots, publish
      the preference orders (electronically, on the internet).

   2. After step 1 is performed, allow candidates a short period of
      time (perhaps a week) to voluntarily withdraw from contention
      before the final result is tallied.  (No candidate can be
      forced to be a spoiler.)

   3. Tally the preference orders using Instant Runoff (or plain
      Condorcet), ignoring rankings of the withdrawn candidates.

To use Bruce Anderson's syntax, this method is named JITW//IRV (or 
JITW//Condorcet).  JITW stands for "just-in-time withdrawal."  

I mention JITW//IRV before JITW//Condorcet because JITW//IRV is a 
reasonable compromise which can be struck with IRV advocates.  It 
satisfies the criteria stated by IRV advocates at least as well as 
plain IRV, and satisfies the criteria of pairwise advocates, without 
adding significantly to the complexity of the IRV description.

I don't think there can be any serious argument against step 1, 
regardless of the overall system, wherever ballots are machine-
readable.  It would allow the performance of the system to be better 
analyzed.  It would provide more information about the voters' 
preferences, which is the point of having people vote.  The omission 
of this step is a significant oversight in the IRV initiatives 
currently being promoted.  In the context of JITW, the ballot 
information would help candidates decide whether or not to withdraw, 
and would help voters decide whether to encourage their favorite 
candidates to withdraw.

We've seen many examples posted here showing how, given plain Instant 
Runoff, sincere votes such as ABC can backfire and elect C.  (C could 
have been defeated if those voters had instead ranked B first.)  
JITW//IRV would allow candidate A to respect the ABC voters' intense 
preference for B over C, whenever they are intense enough, by 
withdrawing from contention, without forcing those voters to 
insincerely rank B first.

France already uses a related system.  They use a runoff system in 
which any candidate with at least 12.5% of the vote in the primary  
qualifies for a runoff.  They permit candidates to voluntarily 
withdraw before the runoff, and that is exactly what most candidates 
do when they predict they'd otherwise be spoilers.

Some people may argue that JITW// would create the possibility of 
backroom deals among the candidates which would defeat the will of 
the voters.  It should be pointed out that most candidates would be 
forcibly eliminated by the //IRV (or //Condorcet) anyway; only 
"dominant" candidates would be able to improve the outcome by 
withdrawing.  

Also, the proportional representation systems (and non-proportional 
multiparty systems) which many people advocate have an even worse 
backroom deal problem, after elections when a majority coalition is 
being cobbled together to run the legislature and (maybe) select the 
prime minister: In that situation each legislator has a full vote to 
sell as s/he pleases, whereas in JITW// systems a withdrawing 
candidate is merely stepping out of the way of voters' preferences.

JITW guarantees that no candidate can be made into a spoiler against 
his/her will.  (If I correctly understand the Gibbard-Satterthwaite 
"manipulability" theorem, only methods which permit just-in-time 
withdrawal rigorously satisfy this "no spoiling" criterion.  One 
might go further and say that JITW moots the theorem, since the order-
reversal tactic backfires by giving some power to one's "greater 
evil": the power to undo the effect of the reversal.)  So given JITW 
there is no deterrent against competing and no need for a party to 
nominate only one candidate.  (Hence partisan primaries would be of 
little importance.  They'd still be of utmost importance given plain 
IRV.)

JITW also allows any voter to vote sincerely, with no strategic 
dilemma, whenever she trusts her more preferred candidate(s) to 
withdraw when necessary to defeat her least preferred candidate(s).
(Given JITW//IRV or JITW//Condorcet, she can also safely rank her 
more preferred candidates ahead of the others whenever she believes 
they would surely lose anyway given IRV or Condorcet.)  Presumably, 
questions about the circumstances in which candidates would withdraw 
would be asked during the pre-election campaigns, and voters would 
tend to downrank candidates whose answers are troubling.  

JITW//Plurality, clearly the simplest of the plausible methods, would 
also work well whenever supporters can trust their more-preferred 
candidates to withdraw when needed to defeat their less-preferred 
candidates.

* * 

As mentioned above, another solution to Donald's problem is plain 
Condorcet, since its definition doesn't include any language about 
circularity.  It could on occasion elect a candidate which is not in 
the top cycle, in fact, due to that omission from its definition, but 
since it would create competition to be in the center we can expect 
that even in that case the winner would be reasonably close to the 
top cycle.  (That's no worse than the "reasonably close" argument 
which is used by some conservative political scientists to justify 
keeping the two-party system.)  JITW//Condorcet would solve that 
minor problem, since one or more members of the top cycle would have 
the incentive to withdraw in order to defeat that slightly worse 
candidate and elect another member of the top cycle.  I believe that 
JITW//Condorcet would in practice perform as well as Smith//Condorcet 
(which led the 1996 EM poll, just ahead of plain Condorcet).  Perhaps 
even better, because candidates would need to avoid slinging mud at 
others whom they may later need to coax to withdraw.

Yet another solution to Donald's problem is the method advocated by 
Bruce Anderson, which Bruce calls "Regular-Champion" (RC).  It's also 
known as Copeland//Plurality.  RC calculates each candidate's "won-
loss" record:  a pair-defeat is a loss and a pair-win is a win.  The 
candidate with the best won-loss record is elected (Copeland).  If 
there's a tie for best won-loss record (which may or may not be due 
to circularity; the definition doesn't need to mention circularity) 
then the tie is broken by looking only at each voter's first choice 
(the //Plurality tie-breaking component of RC).

Since Donald's problem is really about how to explain methods, not 
with the technical merits of methods, RC probably has an advantage 
there over methods which explicitly refer to the top cycle.  It 
doesn't need to use language which explains circularity, since the 
candidate(s) with the best won-loss record must be in the top cycle 
anyway.  (That's easy to prove.)  RC seems less meritorious than some 
other better pairwise methods, though... recall our discussions about 
its "rich party problem" and about its greater susceptibility to 
order-reversal and truncation.  Also Copeland is highly prone to "win-
loss" ties, even when voters don't strategize, so the //Plurality tie-
breaker procedure would often be invoked.

JITW//RC is probably an excellent method, and I hope Bruce Anderson 
will examine it.  It might be easier to explain than JITW//IRV or 
JITW//Condorcet, but this is difficult to predict. 

I haven't yet read Blake's description of his "Path Voting" method, 
so I don't know whether it needs to include the language of 
circularity.  If it gives each candidate a score and elects the one 
with the best score, as do plain Condorcet and RC, then it probably 
doesn't refer to the top cycle.

>      As more and more voters decide to only make one choice the
> Condorcet method will be forced to function using lower and lower scales.
-snip-

This assertion that voters would "learn" to make only one choice 
(radical truncation) is one which Donald has been repeating for a 
long time, but to my knowledge he has never provided a convincing 
reason.  As far as I know, he has never posted even one example which 
would illustrate why, given the best pairwise methods, voters would 
have an incentive to truncate.

Why is it more likely voters would foolishly "learn" to truncate 
given the best pairwise methods than that voters would learn, given 
plain Instant Runoff, to rank the compromise ahead of a more 
preferred candidate?  Many examples have been posted showing how 
failing to rank the compromise ahead of one's favorite can result in 
the election of one's "greater evil", and we know empirically that 
voters are willing to compromise when necessary.  Those examples 
indicate a clear incentive in Instant Runoff for voters to order-
reverse, abandoning their desire to express their preference for 
their more preferred choices.

Perhaps Donald didn't understand the "truncation resistance" property 
of the Condorcet and Smith//Condorcet methods which led the 1996 poll 
in the EM maillist.  Some other variations, such as measuring the 
size of a candidate's pairloss by taking the margin of difference in 
that pairing (instead of by counting the voters who ranked the 
pairwinner ahead of the pairloser), would be more susceptible to 
truncation.  Truncation might be done strategically given methods 
which aren't truncation-resistant, or innocently because a voter 
wanted to finish voting early.  Truncation resistance means that the 
voter does not have a strategic incentive to rank only one, since 
that strategy won't help the voter.  And it means that innocent 
truncation won't affect outcomes either (unless a lot of voters 
foolishly truncate the compromise candidate who needed their votes).

Pairwise methods treat the ranking of only one candidate as if the 
voter had ranked all other candidates equally last.  It's tallied as 
indifference in the pairings between those other candidates.  A voter 
whose sincere preference order is "ABC", if he voted just "A", would 
have the direct effect of making his "greater evil" C less beaten in 
the "B vs. C" pairing, and maybe change that pairing from "B beat C" 
to "C beat B".  Why does Donald think voters would learn to do that, 
when it so clearly and perversely helps their greater evil?  That 
dubious strategy would have no effect on the "A vs. B" pairing, since 
by voting "ABC" the voter is counted just as fully for A in the "A 
vs. B" and the "A vs. C" pairings as if he'd voted just "A".

The closest I can come to showing how voter behavior related to 
what's asserted by Donald might sometimes occur is this:  It is 
possible to show how supporters of a candidate whom they expect would 
beat all others pairwise, if all voters voted sincerely, would have 
an incentive to downrank the candidate of a faction they feared might 
successfully attempt massive order-reversal.  If the example includes 
only three candidates, then that downranking could take the form of 
truncation:  B>A>C becomes B>A=C which can be expediently voted as 
"B" (since unranked candidates are treated as if they'd been ranked 
equally last).  (In important elections we can expect more than three 
candidates so the downranking would not be the same as ranking only 
one; it would be changing A>B>C>D>E to A>B=C>D>E.)  

By pre-announcing that defensive downranking strategy and 
demonstrating their will to execute it when voting, the supporters of 
the compromise candidate would deter order-reversal.  

That strategy would not be useful to most voters.  It would only be 
useful to the supporters of the (centrist compromise) candidate who 
would be elected anyway if all voters voted sincerely.  There's no 
reason to believe that other voters would behave that way, especially 
since if they did they'd eliminate the only reason (i.e., the threat 
of offensive order-reversal by a minority wing) for anyone to ever do 
it.

Offensive truncation is a useless strategy in methods which are 
truncation-resistant.  Order-reversal could be effective.  Defensive 
downranking could only be useful for certain voters, the supporters 
of the compromise.  If truncation (instead of reversal) were 
routinely used by other voters, as Donald claims it would, then there 
wouldn't be a reason for any voter to defensively downrank!

Example 1:  Resistance to truncation

   46: A      <-- These truncated their ABC preference.  (But why?)
   10: BAC    <-- sincere, no incentive to truncate or reverse
   10: BCA    <-- sincere, no incentive to truncate or reverse
   34: CBA    <-- sincere, no incentive to truncate or reverse

       Pairing          x>A    x>B    x>C
       --------        ----   ----   ----
       A vs. B?         54L    46
       A vs. C?         44            56L
       B vs. C?                34L    20
                       ----   ----   ----
       Largest loss:    54     34     56 
     
   B (34) is still elected, so there's no incentive to truncate.

Example 2:  Defensive downranking by a few deters massive reversal.

   23: ABC    <-- sincere
   23: ACB    <-- Most of these offensively reversed their ABC.
   10: B      <-- These defensively downranked A from B>A>C to B>A=C.
   10: BCA    <-- sincere, no incentive to misrepresent
   34: CBA    <-- sincere, no incentive to misrepresent

       Pairing          x>A    x>B    x>C
       --------        ----   ----   ----
       A vs. B?         54L    46
       A vs. C?         44            46L
       B vs. C?                80L    20
                       ----   ----   ----
       Largest loss:    54     80     46

   C (46) is elected, deterring reversal by making it backfire.  
   The 10 BAC voters easily win this game of "chicken" played 
   with the devious ABC voters who threaten reversal.

Note how, in both examples, the CBA voters have a very strong 
disincentive against truncating.  If any of them truncate, that would 
reduce the size of A's loss in the "A vs. B" pairing from 54 to 
something smaller, which doesn't help C and could easily help elect A 
(their "greater evil").  If they did what Donald claims they would, 
they'd have to be pretty silly.


---Steve     (Steve Eppley    seppley at alumni.caltech.edu)