The Arrow Impossibility Theorem

Voting for class captains

Imagine you’re back in elementary school and there’s an upcoming election for classroom captain. This year, there are three nominees: Julius, Kim, and Napoleon—each pledging more computers, longer recesses, and less detentions to your fellow classmates. Your teacher asks you to implement a voting procedure that allows the classroom to elect their preferred candidate for captainship. Assuming that you’re not a sixth-grade-demagogue who wants power for yourself, what sort of voting mechanism would you recommend?

Plurality and majority rule

Perhaps, you’ll run the show using the plurality rule, where the winner is the candidate who receives the largest share of votes. Alternatively, you might consider a majority rule, where voters submit a ranking of their preferences, and the winner is the candidate who is most preferred relative to each alternative. Now, the sixth-grader in you realizes that while the distinction between majority and plurality rule is subtle, there are situations in which they lead to different outcomes.

Julius wins office

Consider, for instance, a poll of class preferences for each of the three candidates (Table 1). Here, 40 percent of the classroom prefers Julius over Kim, and Kim over Napoleon. 35 percent of the classroom, however, want Kim over Napoleon, and Napoleon over Julius. The remaining 25 percent want Napoleon over Kim, and Kim over Julius.

Table 1: Class captain elections

Rank	40%	35%	25%
1	J	K	N
2	K	N	K
3	N	J	J

Adapted from Maskin, Eric. & Sen, Amartya. (2014). The Arrow Impossibility Theorem.

Under plurality rule, Julius would win the election as 40 percent of first preference votes favours him the most. Kim, by contrast, would win under majority rule. This is because 75 percent of students prefer Kim over Napoleon (40% + 35%), while 60 percent prefer Kim over Julius (25% + 35%). (Poor Napoleon never had a chance.)

Indeed, the result should frighten any sixth-grader. Where class captains can shape the course of school history, voting design and outcomes can be world altering. What’s more, we’ve only discussed two voting rules when countless more exist. Can we say anything meaningful about the correct or preferred voting procedure?

Nobel laureate Kenneth Arrow thought about this problem deeply (not as a sixth-grader, but as an economist). For starters, he suggests that society should outline the properties or axioms of a voting rule that we deem desirable. Only then can we search for a voting rule that meets our needs.

Decisiveness, consensus, and non-dictatorship

Arrow himself proposes four axioms that he believes are reasonable. Firstly, it helps if the voting rule is decisive. The procedure should produce a clear winner without ambiguity. Secondly, it should satisfy the Pareto Principle or consensus principle. That is, if every voter prefers candidate X to candidate Y, then candidate Y should not win. Thirdly, it goes without saying that voting rules should be non-dictatorial. No individual or special interest group should have ultimate say on the outcome.

Independence of irrelevant alternatives

Finally, we have what economists call independence of irrelevant alternatives. This axiom seeks to prevent “spoilers” in voting rules. Let us say, for example, that Julius was to win an election. The voting rule satisfies independence if the removal of a losing or “irrelevant” candidate does not “spoil” the original outcome. Here, we can draw a parallel to a footrace. It’d be strange, for instance, for a gold medalist sprinter to fall to second place if the bronze medalist did not participate.

Spoiling the party

Of course, history is no stranger to breakdowns of independence. The economist Eric Maskin points in particular to the U.S. presidential race between George Bush and Al Gore in 2000. If you recall, a lot of hubbub was made about the result in Florida (which uses a plurality voting system). Indeed, Bush had beaten Gore by less than six hundred votes in The Alligator State—which decided the election overall.

Some analysts note, however, that had Green Party candidate Ralph Nader not run, pro-Nader Floridians (~100,000 voters) might have voted instead for Gore. Now, Nader accounted for less than two percent of Florida’s votes. He was “irrelevant” in the sense that he had zero winning chances. But the lawyer-turned-environmentalist “ended up determining the outcome of the election” by running, Maskin writes.

As the results showed, plurality rule does not satisfy the axiom of independence. It is susceptible to spoilage. Majority rule, on the other hand, exhibits independence. After all, if candidate X is preferred by the majority to every other candidate, then dropping an ‘irrelevant’ candidate should not matter.

The Condorcet Paradox

Unfortunately, the majority rule runs into another hurdle. It does not satisfy the decisiveness criterion, and may not always generate a definitive winner. The problem is best illustrated by the Condorcet Paradox, where it is possible for voters to express a collective preference for A over B, B over C, and C over A. Consider, for instance, the poll of classmate preferences for each of the three class captain candidates below (Table 2).

Table 2: The Condorcet Paradox

Rank	35%	33%	32%
1	J	K	N
2	K	N	J
3	N	J	K

Adapted from Maskin, Eric. & Sen, Amartya. (2014). The Arrow Impossibility Theorem.

The data says that 35 percent of voters prefer Julius over Kim over Napoleon; 33 percent prefer Kim over Napoleon over Julius; while 32 percent prefer Napoleon over Julius over Kim.

Under these results, the plurality rule elects Julius to school captain as he receives the largest share of first-preference votes (35 percent). The majority rule, however, cannot find a winner. While 67 percent of voters prefer Julius over Kim (35 + 32), 68 percent prefer Kim over Napoleon (35 + 33), and 65 percent prefer Napoleon over Julius (33 + 32). The result is circular, intransitive, and problematic—a bad case of Condorcet paradox.

The deficiencies and paradoxes troubled Arrow. So he searched for other voting rules that might satisfy these conditions. Arrow began to suspect, however, that such a rule does not exist. This in turn led him to his famous insight. While the mathematical proof is involved, Arrow’s Impossibility Theorem tells us that for sets with three or more candidates or social states, we cannot devise a voting rule that satisfies the four conditions of decisiveness, consensus, independence, and nondictatorship simultaneously. “These axioms are collectively contradictory”, writes Maskin.

Does the theorem condemn democratic societies to collective dysfunction and paradox? Well, not quite. While an ideal procedure is out of reach, Maskin warns against overstating the negative. For starters, extreme scenarios like the Condorcet Paradox may not occur frequently. And there are voting rules that work reasonably well for the most part. While we won’t go into the details here, the “domination theorem” suggests that “majority rule dominates every other voter rule”, writes Maskin. “Whenever another voting rule works well, majority rule must work well too, and there will be cases where the majority rule works well and the other voting rule does not.” [1]

Indecisive deliberation and group dynamics

While Maskin is right to encourage us, I cannot help but wonder where these axioms might fail elsewhere in life. Whether we’re talking about our family’s next holiday destination or a Fortune 500 company’s strategy, collective choice and decision-making involves implicit and explicit voting.

Many companies, for example, operate under a quasi-dictatorship that violate the principle of consensus. In a toxic work environment, employees may simply acquiesce to the whims and directives of a hair-raising manager. By contrast, other bureaucracies are so deliberative that they grow cumbersome and indecisive. They may not see or hear beyond the noise of endless meetings as the paradox of Condorcet feeds their paralysis.

Similarly, behavioral economics shows how the inclusion of irrelevant alternatives can skew our decision-making. Because we humans are not rational computers, arbitrary items can affect the reference points and anchors we use to form our preferences and choices. Of course, we also know that social dynamics play into how people think and act in groups. Ego, fear, herding, incentives, and other motivators influence how people vote and decide. Procedures that neglect human nature are vulnerable to its effects.

Indeed, families, companies, and economies make countless decisions everyday without a clear understanding of the mechanisms, choices, and contexts that structure our behaviour and outcomes. But as the Arrow Impossibility Theorem and Condorcet Paradox reveal, voting and decision rules matter, whether intentional or not.

Footnotes

[1] For more on the “domination theorem”, see: (1) Maskin, Eric. & Sen, Amartya. (2014). The Arrow Impossibility Theorem; And (2) Maskin, Eric. & Dasgupta, Partha. (2008). On the Robustness of Majority Rule.

Sources and further reading

Maskin, Eric. & Sen, Amartya. (2014). The Arrow Impossibility Theorem.
Maskin, Eric. & Dasgupta, Partha. (2008). On the Robustness of Majority Rule.
Ariely, Dan. (2008). Predictably Irrational: The Hidden Forces That Shape Our Decisions.