The asymptotic likelihood of agreement between plurality and condorcet outcomes

Abstract

This article deals 4 t h collective decision making by voting procedures in living systems at the levels of groups, organizations, societies, or supranational systems. General principles are outlined which permit the determination of the likelihood of agreement, and disagreement, between plurality and Condorcet outcomes when system size is large, for any number of alternatives in virtually any culture. A method is developed which ascertains, for any number of alternatives, the system in which the likelihood of plurality/Condorcet disagreement reaches a maximum. The hypothesis that an inverse relation exists between the probability of plurality distortion and the probability of the paradox of voting is disproved. Methodological issues arising from previous work are considered. In pairwise simple majority contests, A1 defeats Az by 13 votes to 7, Az defeats A3 by 12 votes to 8, and A3 defeats A1 by 11 votes to 9. Thus, there is intransitivity in the social ordering, in the form of a cyclical majority, and no Condorcet winner emerges. Consequently, no alternative can be said to represent the will of the majority. The cyclical majority problem, known also as the paradox of voting, has generated a substantial body of research concerning both the conditions for its occurrence (e.g., Black, 1958;Sen, 1970) and the likelihood of its occurrence (e.g., Gehrlein & Fishburn, 1976; Gillett, 1977Gillett, , 1978May, 1971).A method of collective choice which does not give rise to the paradox of voting is the plurality procedure. The winner under this scheme is the alternative which receives the largest number of fist preference votes.Thus, in the previous example, application of the plurality procedure results in the emergence of a clear-cut winner, A1, with 8 first preference votes. However, by concentrating on first preferences, the plurality system fails to take into account all the information contained in the voters' preference orderings. Consequently, it suffers from the disadvantage that it sometimes selects an alternative over which at least one other alternative is preferred by a simple majority of voters. That is, collective 23

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.…”

“…This includes research on preference &dquo;cultures,&dquo; in which preference patterns systematically cluster in society (Garman & Kamien, 1968;Gillett, 1980;May, 1971), and research on the impact of varying levels of unidimensionality or of multidimensional continuous and finite alternative distributions on the probability of Condorcet winners (Browne et al, 1991;Feld & Grofman, 1987;Fishburn, 1974;McKelvey, 1976;Niemi, 1969;Tullock, 1967;Tullock & Campbell, 1970). This includes research on preference &dquo;cultures,&dquo; in which preference patterns systematically cluster in society (Garman & Kamien, 1968;Gillett, 1980;May, 1971), and research on the impact of varying levels of unidimensionality or of multidimensional continuous and finite alternative distributions on the probability of Condorcet winners (Browne et al, 1991;Feld & Grofman, 1987;Fishburn, 1974;McKelvey, 1976;Niemi, 1969;Tullock, 1967;Tullock & Campbell, 1970).…”

Condorcet's paradox

“…A number of studies have addressed this issue, including Garman and Kamien (1968), Niemi and Weisberg (1968), DeMeyer and Plott (1970), Weisberg and Niemi (1973) and Gillett (1980a). dimensions to pro.…”

Dimensions of election procedures: Analyses and comparisons

“…Other comparisons of social choice functions include the relative likelihoods that different non-Condorcet procedures will choose a majority candidate (Fishburn [41] ; Chamberlain and Cohen [20] ; Gillett [55] ; Fishburn and Gehrlein [49]), the probability that different functions yield the same choice set (Fishburn [38]; Gehrlein and Fishburn [52]), and the robustness of different Borda-type scoring systems to preserve the winner when a loser is removed (Gehrlein and Fishburn [53]). …”