A recent NBC/YouGov opinion survey shows that nearly two-thirds of American adults agree with a pullout from Afghanistan (see question 4). Can we infer that the electorate prefers a pullout to maintaining a small military in Afghanistan or even to a “forever war”? Not at all, as the “Condorcet paradox” (also called the “paradox of voting”) demonstrates. Here is an example.
I am not taking sides on the substantive issues in the Afghan war, but only pointing out that the electorate does not know what “it” wants. In general, we can expect it to prefer both A and non-A; in this case, pullout and non-pullout.
Imagine that the American voters are (or have been or will be) facing three alternatives:
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M: providing Moderate support to an Afghan government keeping the Taliban at bay
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F: waging a Forever war in Afghanistan
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R: Retreating or pulling out from, or never intervening in, Afghanistan
Assume that the preferences of American voters among these three alternatives divide them into three equal groups:
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V1 group (1/3 of voters) where each voter prefers M to F and F to R. Being rational in the sense of having transitive preferences, each voter in this group also prefers M to R. (In the table below, this is symbolically represented by M>F>R, where “>” only means “preferred to.”)
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V2 group (1/3 of voters) where each voter prefers F to R and R to M and, therefore, F to M. (In the table, this is symbolically represented by F>R>M.)
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V3 group (1/3 of voters) where each voter prefers R to M and M to F and, therefore, R to F. (In the table, this is symbolically represented by R>M>F.)
Condorcet Paradox Illustrated
V1 |
M>F>R |
V2 |
F>R>M |
V3 |
R>M>F |
Electorate |
M>F>R>M>F>R>M … |
Suppose the electorate is asked to choose between M and F. Members of V1 and V3, will vote for M, making it the winning alternative with two-thirds of the votes. If the voters are asked instead to choose between F and R, the result of the vote would be F because V1 and V2 vote for F; the winner is F. The electorate thus prefers the moderate solution in Afghanistan (M) to a forever war (F) and a forever war (F) to a retreat (R). If the electorate is rational, it must prefer a moderate solution (M) to a retreat (R).
Now, if the same voters are asked, perhaps at a later election (or opinion poll), to choose between R and M, the majority, V2 and V3, votes for R. Without any voter changing his mind, the majority that preferred a moderate solution (M) to a retreat (R) also prefers the contrary: a retreat (R) to a moderate solution (M). The last line of the table describes the electorate’s intransitive and cycling preferences.
Don’t be surprised if that happens in the real world.
This does not mean that majority votes are never useful, only that we can’t count on the electorate to be rational even if no voter changes his mind. The aggregation of voters’ preferences may give incoherent results and will generally do so as the number of voters or issues increases. For a more thorough discussion of the irrationality of voting and its political implications, see my review of a classic book by William Riker, “Populist Choices Are Meaningless,” Regulation 44:1 (Spring 2021), pp. 54-57; and my article on “The Impossibility of Populism,” The Independent Review 26:1 (Summer 2021), pp. 15-25. Teaser: the equivalent of the Condorcet paradox affects any voting system respecting some basic axioms related to rationality.
READER COMMENTS
Jose Pablo
Aug 30 2021 at 7:39pm
“Without any voter changing his mind” … which is a lot to say.
It should not be difficult to change many American voters mind on this topic, just by playing two different kinds of music before asking the question or by making them watch to different sets of images or by carefully framing the question.
As Caplan argues in The Myth of the Rational Voter, is not that people “opinions” drive their votes, it is people “feelings”. And they are not only irrational by definition, but also much easier to manipulate than “opinions” (which hardly change, no matter what).
Of course, “the electorate does not know what “it” wants”, “it” only feels what “it” feels, and their “feelings” change (or are manipulated) in very little time (and with very limited effort).
Pierre Lemieux
Aug 30 2021 at 9:33pm
Pablo: It is easy to demonstrate that the electorate is irrational if every voter is irrational–if, say, we assume that he flips a coin before voting. The power of the Condorcet-Arrow demonstration is that it shows that the electorate is irrational (has non-transitive preferences) even if the preferences of every voter are rational (in the sense of transitive).
Jose Pablo
Aug 31 2021 at 8:41am
I am not one to be convinced of the beauty of the Impossibility Theorem (preaching to the choir here!) and what it does getting modern democracy” even farther from the mythical image of Greek wise men, sitting in white robes in the agora, deciding on the most relevant issues using elaborate rational arguments.
But by imagining a “rational voter”, Arrow is elaborating on a “unicorn”, a mythological figure that does not exist in the real world. Sure, the “zoology of unicorns” can be very interesting (and have an undeniable beauty), but it is irrelevant since it analyzes a non-existing animal.
Jon Murphy
Aug 31 2021 at 9:42am
But if many people believe the unicorn does exist and bases their own scientific analysis on it, then exploring the zoology of unicorns becomes very relevant.
Many folks across the political spectrum adhere to, and vehemently defend, the unicorn of the rational voter choosing rational outcomes. “The Will of the People” is invoked all the time as justification for political power and action. Voting is often treated as akin to the market process insofar as it captures the “wisdom of the crowd” and transmits that information to decision-makers.
So, yes, the collective organization with a single will is a unicorn. But it is also a common myth and thus bares repeated refutation.
Jens
Aug 31 2021 at 2:56am
Does this follow from formal considerations or is it an empirical result? Does it matter whether it is about the range of issues, the choice of alternatives or the number of voters?
Some points:
Voters rarely decide on concrete alternatives. They pass their decisions on to representatives who also have an influence on the questions to be decided. This can lead to further problems. But it also means that “issues” can be “solved” by “faces”.
The model implies rigid alternatives. It doesn’t have to be like that. Voters can change their decisions, but the questions can also change. In particular, the fact that the original question led to paradoxical/irrational/cyclical results, will be part of the next voting decision. The fact that some parties don’t do that, can be an argument for other parties.
On the one hand, this means that voters change their mind not *in spite of* but *because of* their previous vote. On the other hand, this also means that a different set of alternatives may be considered. It can hardly be prevented that all parties involved (and the world itself) change during this process.
So I think if you want to use these results to embrace deadlocks and to justify them, then that can under certain circumstances be successful to a certain extent (you then have to insist on your position with all severity and with the greatest indignation about everything). But if you look at these results as an element of the political process, it is not an end in itself.
Pierre Lemieux
Aug 31 2021 at 12:42pm
Jens: To answer your first question, the proportion of different “profiles” of individual preferences that give rise to the Condorcet paradox (1 at the limit) is the result of mathematical demonstrations, given certain assumptions of course. See Riker’s Display 5-1 and its sources in a large literature on pp. 273-274 (footnote 3) and 274-275 (footnote 8) even up to 1982). Riker discusses many of your questions including whether the manipulation of the vote (the choice of alternatives by politicians, for example) changes these probabilities (they do).
Also ask yourself whether a voter will find it in his interest to study this (and get a Ph.D. in mathematics) in order to perhaps vote differently next time, given that the probability that his vote will change the result of the election is infinitesimally small. Indeed it never happens.
For the political implications of all that, see my Regulation and my Independent Review articles linked to in my post above.
Jon Murphy
Aug 31 2021 at 11:55am
The model isn’t about changing questions. It shows that, for a single question at a single time period, the electorate (regardless whether it’s voters, representatives, juries, judges, etc) can have intransitive preferences.
By way of example, the example i give in class is choosing between takeout options among friends. Depending on how the vote is arraigned, they could prefer burgers to tacos, tacos to pizza, and pizza to burgers. The fact that they may all want Chinese the next day is irrelevant.
Craig
Aug 31 2021 at 12:46pm
I want pizza, burgers and tacos along with 3% body fat.
Comments are closed.