Kenneth Arrow on the Irrationality of Our Voting System

As the political vitriol heats up in anticipation of this year’s elections and presidential candidates attack the legitimacy of the nominating process, we ought to look at a more systemic, often overlooked, issue of legitimacy — the legitimacy of the democratic process. The question of whether our voting system adequately represents individual voter preferences is something we rarely consider, but it’s a pertinent question. And it’s a question that lies at the core of modern social choice theory, which concerns itself with how individual preferences are combined to reach a collective social decision.

In discussing the problems and limitations of our voting system, it is instructive to look back at Nobel laureate Kenneth Arrow’s research. Arrow’s book Social Choice and Individual Values, written in 1951, is generally acknowledged to be the foundational cornerstone of modern social choice theory. It contains Arrow’s Impossibility Theorem, which asserts that it is impossible for a voting system to aggregate individual voter preferences into a rational group preference, whilst simultaneously fulfilling a set of four minimal conditions:

1. Universal Admissibility
   Each individual in the group can adopt any set of rational preferences.
2. Transitivity
   If every individual in the group prefers A to B, then the group preference reflects a preference for A over B.
3. Independence from Irrelevant Alternatives
   If every individual in the group prefers A to B, then any change in other irrelevant preferences (e.g. preferences regarding another choice C) should not result in a change in the group preference between A and B.
4. Dictatorship
   There is no particular individual whose own preferences dictate the group preference, independent of the other individuals in the group.

When each is considered individually, the four conditions seem to be perfectly reasonable demands to have of a voting system. But when these four seemingly straightforward conditions are combined, every voting system is in violation of at least one or more of these conditions, or in other words, of “exhibiting irrationality.”

In a colloquium hosted by Stanford’s Center for Ethics in Society on Wednesday, March 9 to mark the 65th anniversary of the publication of Arrow’s work, panelist Nobel Laureate Amartya Sen pointed out that Arrow’s theorem exposes the limitations of our current plurality voting system (an electoral process in which the candidate who receives more votes than any other candidate is elected). The problem with the plurality system is that it encourages “tactical voting”. Instead of voting for the best candidate, people vote for the candidate they think are most likely to defeat the candidate they most dislike. As a result, the plurality system is less disposed to take into account the nature of the candidate, or other important aspects like stances on income inequality and welfare issues. The theorem suggests that ordinal preferences — that is, placing preferences in order of ranking without asking how much one is preferred over the other — are insufficient for making social choices.

“Take Trump as an example,” said Sen, who was a first year undergraduate in Calcutta when Arrow’s book came out in the 1950s. “Mr. Trump won many votes in numerous states, but not the majority. He would’ve been defeated by the other candidates had it been a pairwise contest.” His example attests to a commonly voiced concern with the plurality system — it can violate the condition of independence from irrelevant alternatives. Consider the recent North Carolina Republican primary. Trump won the primary with about 40% of the vote, Cruz had 37% of the vote, and Kasich had 13% of the vote. Suppose that for the sake of argument, all Kasich voters would have voted for Cruz had Kasich dropped out of the race. Then by 50% to 40%, Cruz would have won the North Carolina Republican primary, which roughly speaking violates the independence from irrelevant alternatives condition. As panelist John Ferejohn, Samuel Tilden Professor of Law at NYU, adds, the implications of Arrow’s theorem are breathtaking for political scientists, especially concerning voting practices and rules.

The existence of these “defects” in social choices raises the question of how much authority an elected leader should have. If every system has the capability of exhibiting irrationality, then to what extent does social preference represent individual values? “Can a government call you anti-national for not agreeing with them?” Sen asks, clearly referring to his own relationship with Indian Prime Minister’s Narendra Modi’s government (Sen disapproves of Modi’s government). The answer is a resounding no. Therefore this raises the issue of how social choice exercises should be conducted. When asked whether one condition or another should be left out of the criteria for voting systems (in essence creating an “imposed rationality” by leaving one of the conditions out of our definition of rationality), the speakers all asserted that it wasn’t a solution. Rather, John Rawl’s idea of a reflective equilibrium is a necessary part of a social choice exercise; in making judgments and decisions, we have to work back and forth between considered judgments, making constant revisions in order to achieve coherence amongst them. That is arguably the only acceptable way we can make any sort of aggregate decision — that is, to give consideration to different viewpoints and judgments, and to keep an open mind.

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Ruru Hoong, a freshman studying economics, is a staff writer at Stanford Political Journal.