Kenneth Arrow's Democracy Impossibility Theorem
Discussion
Ben Steil:
"Kurt Gödel—the first central figure in Hofstadter’s bestseller—suggests that a fully self-regulating rules-based political order is unattainable. Gödel’s famed “incompleteness theorem,” published in 1931, used an ingenious mathematical argument to demonstrate that any rules-based system elaborate enough to govern and interpret its own operations will eventually confront questions its rules cannot answer.
In a political context, such a system must have rules not just for governing society, but for governing, interpreting, and modifying the rules themselves. Once a political order becomes sufficiently recursive, however, situations inevitably arise in which the rules themselves become contested.
For example, if constitutions must be created through pre-constitutional means, who decides when they are legitimate? Who determines whether an emergency justifies suspending rights? When may judges overrule elected majorities? What if institutions deadlock? Who decides whether a state is acting in legitimate self-defense? Does a treaty obligation supersede national sovereignty?
Though Hofstadter does not mention him, the Nobel laureate economist Kenneth Arrow’s famed “impossibility theorem,” published in 1950, is closely related in structure and implication to Gödel’s theorem. Arrow used mathematics to answer a deceptively simple but profoundly important question about democracy: Is it possible to design a voting system that converts individual preferences into a logically coherent collective decision, while simultaneously satisfying basic standards of democratic fairness?
The answer, perhaps surprisingly, is no. Though countless political theorists had assumed democracy to be a valid mechanism for discovering the public will, Arrow showed that the aggregation problem is insoluble. Democracy cannot reliably transform private preferences into a coherent collective rationality. It tends, instead, to produce incoherence or unfairness.
Gödel’s and Arrow’s findings cast logical doubt on Francis Fukuyama’s famous proposition that liberal democracy represented “the end of history”—the telos of human political evolution. The “liberal” part of liberal democracy relates to the rule of law, which Gödel’s theorem showed to be insufficient for a fully stable political order. The “democracy” part relates to elections, which Arrow’s theorem showed to be insufficient for identifying a coherent collective will."