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Arrow’s Impossibility Theorem Exposes a Big Problem with Democracy

Summary:
Listen to the Audio Mises Wire version of this article. There is arguably nothing more sacrosanct to today’s elites than “democracy”—by which they mean “a political outcome we endorse.” And yet ironically, one of the most surprising and powerful results in social choice theory, namely Kenneth Arrow’s so-called impossibility theorem, shows that even in principle ...

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Listen to the Audio Mises Wire version of this article.

There is arguably nothing more sacrosanct to today’s elites than “democracy”—by which they mean “a political outcome we endorse.” And yet ironically, one of the most surprising and powerful results in social choice theory, namely Kenneth Arrow’s so-called impossibility theorem, shows that even in principle there is no coherent way to aggregate individual preferences into a collective will.

In a sense, Arrow did to democracy what Kurt Gödel did to the attempts to place mathematics on an axiomatic foundation. Yet while everyone from philosophers to cognitive scientists to computer programmers cites Gödel—even when they don’t really understand what he demonstrated—hardly anyone discusses Arrow when it comes to politics. My simple and cynical explanation is that his result is so devastating that it’s hard to say anything at all in its wake. (Note that free market economists also might suffer from this problem, if we speak too glibly about the “optimality” of a market outcome.)

Why Majority Rule Doesn’t Work

Before explaining Arrow’s shocking result, let me set the table with a demonstration of why simple majority rule isn’t a viable rule for making group decisions. Suppose Alice, Bob, and Charlie have the following subjective rankings of three candidates:

Specifically, if we ask “Does ‘society’ think Trump is better than Biden?,” the answer is yes, because Bob and Charlie think Trump is better than Biden—they outvote Alice on that narrow question. Using majority rule, we can also conclude that “society” thinks Biden is better than Jorgensen, because Alice and Charlie outvote Bob. So since “society” thinks Trump beats Biden and Biden beats Jorgensen, for the group to be rational we would also expect “society” to think Trump beats Jorgensen. And yet, as the table indicates, on that narrow question the voters would say the opposite: Alice and Bob would vote for Jo Jo over the Donald. Now with only three voters and three possible candidates, it should be relatively simple to determine what “the group” thinks about the best candidate, right? Yet if we happen to have the political preferences shown in the table above, then simple majority rule leads to intransitivity in the social ranking.

And so we see, even with a very simple example, that there is a fundamental problem with using simple majority rule as the mechanism for aggregating individual preference rankings into a single ‘social’ ranking. To repeat, there is no guarantee that the resulting ‘social’ rankings will obey transitivity.

Besides being worrisome conceptually, intransitive rankings also suffer from the practical problem that the overall winner is dependent on the order of pairwise contests. In our example above, if the group first pitted Biden versus Trump and then had the winner face Jo Jo, then Jo Jo would win. But if instead the elites wanted Biden, they would have the voters first decide between Trump and Jo Jo, then have that winner go head to head against Sleepy Joe.

Because a robust social choice rule shouldn’t be vulnerable to such manipulation, political thinkers have known since Condorcet that majority rule isn’t the answer.

Arrow’s Approach

I don’t know if this backstory is apocryphal, but I was taught that Kenneth Arrow set out in grad school in the late 1940s/early 1950s to get rigorous in social choice theory. Economists and other formal social scientists had known there were many types of undesirable social choice procedures (or rules). Arrow, so the story goes, originally wasn’t trying to find the best one, but instead he was merely trying to weed out the obviously bad rules in order to focus attention on the pool of surviving candidate procedures.

Arrow’s framework was a generalization of our table above. Specifically, Arrow assumed there were a finite number of citizens who each had subjective rankings of the possible “states of the world,” and he further assumed that each citizen’s preferences were complete (meaning the citizen had a definite opinion on any pairwise comparison, including the possibility of being indifferent between two outcomes) and transitive.

Taking this list of citizens’ complete and transitive preference rankings, Arrow wanted a procedure that would generate a complete and transitive “social” preference ranking of the various possible “states of the world.” In order to rule out what seemed self-evidently undesirable procedures, Arrow insisted that the eligible procedures also obey the following principles:

  • Nondictatorship: there shouldn’t exist one person in society such that, no matter what everyone else says, the procedure always makes the “social” ranking identical to the one person’s preferences. To be clear, it’s fine if in any particular example of everybody’s rankings the rule just so happens to make the “social” ranking the same as Jim’s ranking. But if, no matter what Jim and everybody else preferred, the rule always made “society” agree with Jim’s personal views, then he would be a dictator in Arrow’s sense.
  • Weak Pareto optimality: if every citizen thinks outcome A is preferable to outcome B, then it better be the case that the procedure spits out the result that “society” prefers A to B.
  • Independence of irrelevant alternatives (IIA): this is the least intuitive of the axioms, but when you understand it, it also sounds reasonable. This criterion says that the “social” ranking of outcome A versus B should only depend on how the citizens compare A to B.

To get a sense of what Arrow is after with IIA, suppose a child is ordering ice cream at a restaurant. The waiter says, “We have vanilla or chocolate.” The child chooses vanilla. Then the waiter comes back a minute later and explains, “Sorry, I just realized we still have some strawberry ice cream as well. Would you like to change your order?” The child responds, “Yes! I’ll order chocolate instead.”

I hope the reader can see why our hypothetical child would here be exhibiting unusual choices. This is in the spirit of what the IIA criterion is prohibiting.

Arrow’s Shocking Result

To continue the story, apparently Arrow set about to prune away the possible procedures that violated any of the above criteria and ended up with…the empty set. In other words, Arrow realized that there did not exist a procedure for generating “social” preference rankings that obeyed his list of seemingly innocuous requirements.

What’s really fun is that any interested reader can see an actual proof of Arrow’s result that doesn’t rely on prior mathematical knowledge. See, for example, this version that Amartya Sen formulated. I heartily encourage the curious to give it a shot. You’ll see that it’s not based on a trick; Arrow really did demonstrate something with devastating relevance to the notion of political sovereignty.

Conclusion

As the media elites urge Americans to vote and celebrate the wonders of “democracy,” both at home and foisted by firepower around the world, always keep in mind the elephant in the room: Kenneth Arrow showed in 1951 that the entire project of social choice theory rested on quicksand.

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