This possibility, called the Condorcet paradox, was identified in the late 18th century by Marie-Jean-Antoine-Nicholas de Caritat, the Marquis de Condorcet, a colleague and archcritic of Borda. The three rankings—Gore over Bush over Nader, Bush over Nader over Gore, and Nader over Gore over Bush—are collectively called a Condorcet cycle. Our comparison of majority rule and rank-order voting appears to have resulted in a dead heat: majority rule satisfies every principle on our list except transitivity, and rank-order voting satisfies all but neutrality. This conundrum leads us to consider whether some other electoral system exists that satisfies all the principles. Arrow’s celebrated impossibility theorem says no. It holds that any electoral method must sometimes violate at least one principle [see “Rational Collective Choice,” by Douglas H. Blair and Robert A. Pollak; Scientific American, August 1983].
Beyond Impossibility
BUT ARROW’S THEOREM is unduly negative. It requires that an electoral method must satisfy a given axiom, no matter what voters’ rankings turn out to be. Yet some rankings are quite unlikely. In particular, the Condorcet paradox—the bugaboo of majority rule—may not always be a serious problem in practice. After all, voters’ rankings do not come out of thin air. They often derive from ideology.
To see what implications ideology holds for majority rule, think about each candidate’s position on a spectrum ranging from the political left to the right. If we move from left to right, we presumably encounter the 2000 presidential candidates in the order Nader, Gore, Bush, Buchanan. And if ideology drives voters’ views, then any voter who ranks Nader above Gore is likely to rank Gore above Bush and Bush above Buchanan. Similarly, any voter who ranks Bush above Gore can be anticipated to rank Gore above Nader. We would not expect to find a voter with the ranking Bush, Nader, Gore, Buchanan.
In a pioneering paper published in the 1940s, the late Duncan Black of the University College of North Wales showed that if voters’ rankings are ideologically driven in the above manner— or at least if there are not too many nonideological voters— majority rule will satisfy transitivity. This discovery made possible a great deal of work in political science because, by positing ideological rankings of candidates on the part of voters, researchers could circumvent the Condorcet paradox and make clear predictions about the outcome of majority rule.
Of course, voters may not always conform to such a tidy leftright spectrum. But other situations also ensure transitivity. For another example, look again at the French election. Although Chirac and Jospin led the two major parties, it seems fair to say that they did not inspire much passion. It was the extremist candidate, Le Pen, who aroused people’s repugnance or enthusiasm: evidence suggests that a huge majority of voters ranked him third or first among the three top candidates; few ranked him second. One can argue about whether such polarization is good or bad for France. But it is unquestionably good for majority rule. If voters agree that one candidate of three is not ranked second, transitivity is guaranteed. This property, called value restriction, was introduced in 1966 by Amartya Sen of Harvard University.
In our research on voting, we say that a voting system works well for a particular class of rankings if it satisfies the four axioms when all voters’ rankings belong to that class. For instance, majority rule works well when all rankings are ideologically driven. It also works well when all rankings are “value restricted.” Indeed, we have found that whenever any voting system works well, so does majority rule. Furthermore, majority rule works well in some cases in which other systems do not. We call this the majority dominance theorem. To illustrate, we will imagine a three-way race between Gore, Bush and Nader. Suppose that every voter in fact ranks the candidates as either Gore, Bush, Nader or Bush, Gore, Nader.
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Add CommentIf the election is as close as the Bush/Gore election, there obviously is a serious question about who will do a better job. So splitting hairs about who has a tiny bit more support/trust does not seem all that useful. What we really need is an efficient system to recall someone who is not doing a good job. That is one whose performance is highly unpopular with the people who elected him, not the "high crimes and misdemeanors" law we have now. Something like the recall law for the California Governor. I notice that once the new Governor (Schwarsenegger) realized he could be voted out, he changed his behavior to improve his popularity. Perhaps an election should be called if the official's popularity rating falls below something like 50% for six months. (If he cannot even keep half of the people happy, he is not doing what needs to be done). Things happen much faster now than they did 200 years ago. We cannot afford to have an elected official be allowed to mess things up for years before a change is made.
Reply | Report Abuse | Link to thisDoes not Wisdom call?
Reply | Report Abuse | Link to thisThis is not exactly a new thing. Something very much like this (as I understand the essence of the authors thrust) was used in the General Election of 1952, British Columbia, Canada.
Reply | Report Abuse | Link to thisIt appears that people, aware of the risk of their ideological foe being elected under the new system, chose the third choice candidate as a spoiler vote. That way, the third place choice ended being elected.
On the other hand, it could be argued in light of many successive wins, that the electorate was undergoing a transition to a new (to them) political party that was more in line with the electorate's values, an alienation from the long standing traditional expressions of liberalism and conservatism of the day that echoed the national party systems that were perceived as largely uncaring of regional concerns.
Either way, this electoral system appears to have been deemed by the winners to be too unpredictable a method to guarantee a re-election and was not used again.
http://www.nationmaster.com/encyclopedia/British-Columbia-general-election,-1952
It seems clear that when given a two part choice, a partisan electorate will often choose to vote strategically, rather than logically: ie. A=45%, B=45%, C=10% popular. Voting day choices are:
Love A, hate B, C has no chance=vote A,C
Love B, hate A, C has no chance=vote B,C Love C, hate B, hate A less=vote C,A
Love C, hate A, hate B less=vote C,A
Result is C wins
I would interested as to how such a counterproductive tendency might be factored out.
Correction:
Reply | Report Abuse | Link to thisIt seems clear that when given a two part choice, a partisan electorate will often choose to vote strategically, rather than logically: ie. A=45%, B=45%, C=10% popular. Voting day choices are:
Love A, hate B, C has no chance=vote A,C
Love B, hate A, C has no chance=vote B,C
Love C, hate B, hate A less=vote C,A
Love C, hate A, hate B less=vote C,B
Result is C wins
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ps, to the forum master, I find the pale grey font hard to work with. I had to go back and resubmit owing to a typo
How about a system where candidates get rated on a 1-10 scale lets say.
Reply | Report Abuse | Link to thisBy popularity, its A=46%, B=44%, C=10%
However, the people that prefer A, prefer A by a small margin, while the people that prefer B, definitely don't like A.
In essence, in this case, my opinion is that B should win. Such a rating system would allow that.
100 hundred people vote:
The ppl that prefer A (46%), might vote, on average A=7 B=6 C=2
The ppl that prefer B (44%), might vote, on average A=5 B=8 C=2
The ppl that prefer C (10%), might vote, on average A=2 B=2 C=8
Then you have:
A: 7*46% + 5*44% + 2*10% = 5.62
B: 6*46% + 8*44% + 2*10% = 6.48 (B is the winner)
C: 2*46% + 2*44% + 8*10% = 2.6
That makes sense to me! What do you all think? Is my logic wrong anywhere? Are there cases where the result wouldn't be representative of what the people want? (I haven't though deeply about this, just on the whim).
This should work extremely well in bi-partisan case of America, where MANY republicans will vote republican, even though they like the Democrat almost just as much...but they are Republican so they have to go with that. Or vice versa of course. Thus, this system will reveal what the people really want. No?
Hey Richard Campbell,
Reply | Report Abuse | Link to thisTo be able to factor out the tendency to select C from the above example, the correct approach would be to allow for higher weight to be assigned to the preferred choice. For example initially you would have done the following,
Love A, hate B, C has no chance= A:2 and C:1
Love B, hate A, C has no chance= B:2 and C:1
Love C, hate B, hate A less=C:2 and A:1
Love C, hate A, hate B less=C:2 and B:1
Instead if the following approach is utilized a better result is attained.
Love A, hate B, C has no chance= A:3 and C:1
Love B, hate A, C has no chance= B:3 and C:1
Love C, hate B, hate A less=C:3 and A:1
Love C, hate A, hate B less=C:3 and B:1
Now when you tally the votes you get a the winner as either A or B (depending on the proportion split for the 10% who love C). The solution is also scalable, in the case where you have 4 candidates, you would select 5, 3, 1, and 0 and so on. The provides a deterrence for the second and third place votes and justifies a higher weight for the first choice, which accurately accounts for the voter's intent.
Hey minulescu,
I think that in your example, those who prefer C should also end up adding to 15, right now A=2 B=2 and C=8 where 2+2+8=12 but for those who prefer A and B the total sum is 15. Moreover, I think that it would be very difficult to expect voters to come up with the proportional split between A, B and C. This would have to be internal to the algorithm or the system, the voters I believe can only be expected to provide their rankings, asking them to do more would be too hard to track and slow down the voting process.
Faheem Merchant
I have seen much discussion on our electoral system since the Bush/Gore election. One of the points that the articles seem to miss is the what the underlying principle of the electoral system is. The assumption is the articles that I have read always seem to be that the popular majority should pick the winner. I was taught at a young age that the purpose of the electoral system was that no one major constituency could always win the election. Many of the articles I read seem to imply that this is bad. But doesn't this prove to be good in that no major constituency can always reign supreme, thereby disenfranchising smaller groups. It sort of forces our system to be somewhat progressive. Doesn't this have virtue?
Reply | Report Abuse | Link to thisRank Order has potential. However, how it operates is crucial.
Reply | Report Abuse | Link to thisI propose the following.
The first pass determines one 'loser'. That persons vote is eliminated and shifts to the second choice. Then with this 'vote shift' another 'loser' is chosen, again shifting all that are currently aligned to that candidate to the next on the list.
When this process ends with one candidate left, you have the Candidate that is most favored by the most people.
Rank Order has potential. However, how it operates is crucial.
Reply | Report Abuse | Link to thisI propose the following.
The first pass determines one 'loser'. That persons vote is eliminated and shifts to the second choice. Then with this 'vote shift' another 'loser' is chosen, again shifting all that are currently aligned to that candidate to the next on the list.
When this process ends with one candidate left, you have the Candidate that is most favored by the most people.
Once upon a time, Scientific American had an article about the mathematics of how the Electoral College works and why our founding fathers settled on that particular system.
Reply | Report Abuse | Link to thisIt was published in the mid-1990's sometime, but I don't remember exactly when and I no longer have it. I'd sure love to re-read it in light of some of the other articles Scientific American has published since.
To Ian St. John:
Reply | Report Abuse | Link to thisI think the process you describe is the Instant Runoff Voting (IRV) method alluded to briefly in this article. It is in use in Australia and seems quite sensible.
-Blair