The Smoothed Possibility of Social Choice

We develop a framework to leverage the elegant "worst average-case" idea in smoothed complexity analysis to social choice, motivated by modern applications of social choice powered by AI and ML. Using our framework, we characterize the smoothed likelihood of some fundamental paradoxes and impossibility theorems as the number of agents increases. For Condrocet's paradox, we prove that the smoothed likelihood of the paradox either vanishes at an exponential rate, or does not vanish at all. For the folklore impossibility on the non-existence of voting rules that satisfy anonymity and neutrality, we characterize the rate for the impossibility to vanish, to be either polynomially fast or exponentially fast. We also propose a novel easy-to-compute tie-breaking mechanism that optimally preserves anonymity and neutrality for even number of alternatives in natural settings. Our results illustrate the smoothed possibility of social choice---even though the paradox and the impossibility theorem hold in the worst case, they may not be a big concern in practice in certain natural settings.