Collective Indecision

Abstract

A desirable property of a collective choice procedure in human groups is that its outcome should be clear‐cut. The plurality and Condorcet procedures are investigated under a variety of conditions to determine their susceptibility to indecisiveness. Both plurality indecision and Condorcet indecision can attain a large likelihood of occurrence when group size is small. However, both likelihoods, but especially the latter's, prove highly volatile in the presence of minor variations in group size when group size is small. In general, the relative decisiveness of the two procedures fluctuates as a function of the pattern of the group's preference probabilities. An exception is groups of four and six members where plurality indecision is always less likely than Condorcet indecision. When it is required that the plurality winner receive more than 50 percent of the votes, the probability of Condorcet indecision is always less than or equal to the probability of plurality indecision, irrespective of the pattern of the group's preference probabilities.

Generating random weak orders and the probability of a Condorcet winner

“…Garman & Kamien [1968, p. 314] computed 1 -P(n, m) as multinomial probabilities for some small and odd n and some small m. The rational fractions they report are the complements of the corresponding ones in Table 2. The same holds for Gillett [1977] who computed 1 -P(n, 3) also for some even n. The estimates of P (n, m) that Jones et al [1995] obtained by simulations are confined to odd n and values of m up to 15. The estimates they report for n = 3, 5 deviate at most 0.002 from the corresponding ones in Table 2.…”

Simulation

“…Such formal analyses may prove very useful for understanding the nature of decision making, leading some to call for implementing Condorcet's system of voting in practice (Felsenthal & Machover, 1992). Further work has examined and extended this research to investigate the nature of many different electoral systems such as (Felsenthal & Maoz, 1992;Felsenthal, Maoz, & Rapoport, 1990;Gillett, 1977Gillett, , 1980Merrill, 1984;Merrill & Nagel, 1987;Niemi, 1984;Niemi & Frank, 1982. And although single-peaked individual preferences generally reduce the probability of not having a Condorcet winner, they actually increased the probability of having plurality choice distortions.…”

“…Obviously, one must question the uniform distribution of individual preferences. However, Gillett (1977) has shown that these results do not hold equally for even-sized electorates (and by extension possibly for any electorate that allows indifference) due to the greater possible variety in types of indecision. Arrow (1963) and Black (1948and Black ( , 1958, for example, both demonstrate that when all voters use a common dimension of evaluation, and preferences are single peaked, there will always be a Condorcet winner.…”

“…With few exceptions (Gillett, 1977;Kelly, 1974), investigations of Condorcet's paradox have focused on cases with odd numbers of voters. With an odd number of voters and strong preferences, ties at the top are impossible and the different definitions of a Condorcet winner are functionally equivalent.…”

The comparative likelihood of an equivocal outcome under the plurality, Condorcet, and Borda voting procedures

“…Formulae for the probability of plurality indecision and for the probability of Condorcet indecision are given in Gillett (1977). To this end, an expression for the probability of Borda indecision in case m = 3 is presented.…”