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A mathematical solution for the probability of the paradox of voting
Author(s) -
Niemi Richard G.,
Weisberg Herbert F.
Publication year - 1968
Publication title -
behavioral science
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.371
H-Index - 45
eISSN - 1099-1743
pISSN - 0005-7940
DOI - 10.1002/bs.3830130406
Subject(s) - voting , mathematical economics , condorcet method , a priori and a posteriori , rank (graph theory) , interpretation (philosophy) , social choice theory , majority rule , order (exchange) , probability distribution , mathematics , computer science , econometrics , economics , statistics , artificial intelligence , epistemology , combinatorics , philosophy , finance , programming language , politics , law , political science
The paradox of voting occurs when individual rank orders of three or more alternatives lead to an intransitive social ordering. This means, for example, that with a majority decision rule for voting between pairs of alternatives, it is possible that no alternative will receive a majority vote over all of the other alternatives. The a priori probability of the paradox, based on certain probability assumptions, has been sought in order to judge how serious the paradox is for societal decision‐making. In this paper, the authors specify the model underlying these attempted calculations. In the process, new problems of interpretation are raised. Through the use of this model, a general solution for the probability of the paradox is derived, together with an approximation for computational convenience. Some numerical results are given to demonstrate the nontrivial probability of the paradox with a moderate number of alternatives and the assumption that all possible rank orders are equally likely.