Condorcet Winners and the Paradox of Voting: Probability Calculations for Weak Preference Orders

That individual preferences may he aggregated into a meaningful collective decision using the Condorcet criterion of majority choice is one of the central tenets of democracy. But that individual preferences may not yield majority winners is one of the classic findings of the social choice literature. Given this problem, social choice theorists have attempted to estimate the probability of Condorcet winners, given certain empirical or theoretical conditions. We shall estimate the probabilities of Condorcet winners and intransitive aggregate orders for various numbers of individuals with strong or weak preference orders across various numbers of alternatives. We find, using computer simulation, a stark contrast between these estimates assuming strong individual preferences and the estimates allowing for individuals' indifference between pairs of alternatives. In contrast to earlier work, which depends on the strong-preference assumption, we suggest that the problem is most acute for small committee decision making and least acute for mass elections with few alternatives.