Condorcet's Paradox and the Median Voter Theorem for Randomized Social Choice

Condorcet's paradox is one of the most prominent results in social choice theory. It says that there may not exist any alternative that a net majority prefers over every other alternative. When outcomes need not be deterministic alternatives, we show that a similar paradox still exists even if preferences are dichotomous. Thus relaxing the requirement of discrete alternatives does not help in circumventing Condorcet's paradox. On the other hand, we show that a fractional/randomized version of Black's Median Voter Theorem still holds for single-peaked preferences.