Arrow’s Impossibility Theorem Exposes a Big Problem with Democracy
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 3 voters and 3 possible candidates, it should be relatively simple to determine what “the g
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