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

Abstract

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.

Condorcet efficiency: A preference for indifference

“…Exceptions to this have dealt with the probability that there is a Condorcet winner when indifference is allowed [Van Deemen (1999), Lepelley and Martin (1998), Fishburn and Gehrlein (1980), Jones, Radcliff, Taber and Timpone (1995)]. The purpose of the current study is to consider the impact that voter indifference on candidates will have on the Condorcet efficiency of voting rules.…”

Trump, Condorcet and Borda: Voting paradoxes in the 2016 Republican presidential primaries

“…Gehrlein and Fishburn 1976;Radcliff 1994;Jones et al 1995). However, the outcome was also due to the process: Had the primary process involved either head-to-head contests or one or more run-offs between the contenders, it is quite plausible that Trump would not have won the nomination and therefore not become president.…”

Computing the Perfect Model: Why Do Economists Shun Simulation?

“…The likelihood of the occurrence of the Condorcet paradox (or rather cyclic preferences) has been studied via both analytical and simulation approaches. Although there are some exceptions, most of the papers in which the main contribution is based on simulation are published in political science journals (e.g., Klahr 1966;Jones et al 1995), but those based on analytical models are published either in economics journals (e.g., DeMeyer and Plott 1970) or journals devoted to formal methodologies (e.g., Van Deemen 1999). Furthermore, some scholars who started studying this topic by means of simulations (Fishburn and Gehrlein 1976) subsequently adopted an analytical framework (e.g., Gehrlein 1983Gehrlein , 2002).…”