A state has on its ballot a Proposition 13-style measure to limit property taxes and a state-devised, "moderate" countermeasure. In a pre-election poll, voters are asked to make separate choices between "13" and the moderate measure, the moderate and no change, and "13" and no change. A majority prefers "13" to the moderate measure and the moderate to no change, so "13" would have to win over no change, right? Not necessarily. Without any inconsistency on their part, the same group of respondents can show a majority preferring no change over the Proposition 13 measure.
In the summer of 1976, a sample vote on the upcoming presidential elections was taken in a university class of 22 students. The paired votes among the three leading candidates were: Ford 14, Carter 8; and Carter 12, Reagan 10; but Reagan 13, Ford 9.
Immediately after an election, the "what-if" columns and commentaries begin to appear. In general, they decry the inequities and vicissitudes of the election process. The close Ford-Carter election in 1976 provides an excellent example. If a mere 8,000 voters in both Ohio and Hawaii had voted for Ford rather than Carter, Ford would have won in the electoral college—even though losing the popular vote. And so, the questions: What if there had been no televised presidential debates, or Ford had not made his Eastern European "blooper" in the second debate? Does Carter owe special favors to blacks (he lost the white vote) or concessions to labor (the labor vote was the difference in the industrialized Midwest) or cabinet positions to Baptists or Jews? Was Ford's use of advertising counterproductive in relation to the Carter Playboy interview? Was the vice-presidential selection the deciding factor? Did Watergate play an important role in Ford's defeat?
These questions, and many others concerning the election process, have a common underlying theme. Each is based on the fact that most politicians are elected for reasons other than something called the real issues, that election outcomes swing on minor issues with little or no substance. And this casts a foreboding shadow over the ability of democracy to function in a rational fashion. But, although the questions thus raised about collective decisions are serious enough, there is a more fundamental problem encountered in arriving at group consensus. It has been discussed in literature since the 18th century and is generally known today as Arrow's Impossibility Theorem (after economist and Nobel Laureate Kenneth Arrow) or simply the paradox of voting.
STRANGE RESULTS The academic literature of voting theory consists of a complex system of axioms, conditions, and proofs, but we don't need to know all of this to understand the basic problem involved in the paradox of collective consensus. Under certain conditions—conditions that are present in the pattern of our contemporary political process—irrational or illogical outcomes can and do appear because of an inherent paradox in group decision making. The paradox does not involve any of the commonly heard criticisms of the voting process but results from what mathematicians call intransitivity.
Transitivity, when it comes to choices, amounts to logical consistency of choices or preferences. For instance, suppose you prefer classical music to jazz and jazz to rock. Then it is reasonable that, given a choice between classical and rock, you would show a preference for classical music. This logical consistency of choices is the transitive result. If, however—even after admitting that you like classical better than jazz and jazz better than rock—you choose rock over classical, the result would be inconsistent and would be called intransitive. The heart of the paradox of voting is this: even when each person within a group is logically consistent, or transitive, about his or her preferences, the individuals can, through the voting process, produce a collective result that is intransitive.
A class of students choosing Ford over Carter, Carter over Reagan, then Reagan over Ford is an example of an intransitive group outcome. Common sense and logic would seem to say that if Ford was preferred to Carter and Carter preferred to Reagan, then Ford would naturally be the winner in an election with Reagan. But the mock election showed Reagan the victor by four votes. When confronted with this result, the students' immediate reaction was that some members of the class had been illogical in their choice patterns and switched allegiances when voting different pairs. But this, as the paradox of voting reveals, need not be the case. Each class member can be consistent in his individual voting preference, and yet an inconsistent result can occur when the group attempts to make a collective decision.
To see the point, assume the class has only 3 members rather than 22. Suppose one student prefers Ford over Carter and Carter over Reagan; another student prefers Carter over Reagan and Reagan over Ford; and the third prefers Reagan over Ford and Ford over Carter. So we have:
Student 1 Ford, Carter, Reagan
Student 2 Carter, Reagan, Ford
Student 3 Reagan, Ford, Carter
But if the students vote for the three candidates in pairs, an intransitive result occurs. In the race between Ford and Carter, students one and three vote for Ford and student two for Carter. The winner is Ford: two votes to one. In the Carter-Reagan contest, students one and two vote for Carter and student three for Reagan. The winner is Carter: two votes to one. But in the Ford-Reagan vote—with Ford the rational winner because the three had selected Ford over Carter and then Carter over Reagan—the outcome is the opposite. Student one votes for Ford but students two and three, each remaining consistent in his preferences, vote for Reagan, who thus emerges the winner over Ford.
CONGRESSIONAL PARADOX Computer models have been used to find that when there are three individuals (or motions, amendments, or items) and voters have what are called strong preference orderings—which means they know what they want—the inconsistent, or intransitive, results will exceed eight percent. Examples of these intransitivities have been found in amendment voting in the US Senate and various state legislatures, as well as in such nonpolitical activities as scoring track meets.
In 1955 the Senate passed a five-year $18 billion highway bill offered by Senator Gore of Tennessee—a case that typifies a nonrational outcome. On this issue the Senate was divided into three basic blocks—northern Democrats, southern Democrats, and Republicans. The original bill was offered with the provision that fair-pay standards (the Davis-Bacon Act, 1931) would be included. The Republicans were opposed to the bill, but if it were to pass they wanted Davis-Bacon to apply. The southern Democrats liked the bill without Davis-Bacon and felt no bill was better than one with fair-pay standards. The northern Democrats wanted the bill with the provision but preferred the bill without Davis-Bacon to no bill at all:
Republicans: No bill, bill with Davis-Bacon, bill without Davis-Bacon
Northern Democrats: Bill with Davis-Bacon, bill without Davis-Bacon, no bill
Southern Democrats: Bill without Davis-Bacon, no bill, bill with Davis-Bacon
The Senate would defeat the bill if offered with the Davis-Bacon provision—gaining support from only the Northern Democrats. In voting on the inclusion of Davis-Bacon, the provision would pass with only the Southern Democrats in opposition. As a group, the collective judgment of the Senate was to prefer no bill to the bill with the provision and the bill with the provision to the bill without. Therefore, for a logical outcome, it would seem no bill would be preferred to the bill without the Davis-Bacon provision. The actual outcome was just the opposite—the bill without the Davis-Bacon provision passed, not even requiring a roll call.
Such situations make it possible for voters to deliberately vote against their preference on a given issue in order to tilt the outcome in their favor. And this in fact happens. A classic example is an amendment by conservative Republicans attached to the proposed 17th Amendment to the Constitution, which deals with the direct election of Senators. The proposed amendment by these Republicans attracted liberal Republicans and alienated enough southern Democrats who would otherwise have supported passage, thus delaying passage in the Senate for nearly 10 years.
Critics from a wide political spectrum have challenged the voting system for a multitude of sins, ranging from slick ad campaigns to the outright stealing of votes. But the paradox of voting shows that, amid all the other problems of collective decision making, the voting process itself permits logical inconsistencies. Schemes such as maximizing social welfare through voting can often lead to irrational, intransitive outcomes, even when all affected persons are given an equal opportunity to vote. If each member of the group is rational and consistent in his individual preference patterns, the collective voting method still has the possibility of irrational and inconsistent results.
Don Reading is an associate professor of economics at Idaho State University. His work has been published in various academic journals.
This article originally appeared in print under the headline "The Paradox of Voting".