Abstract— Voting plays an important role in many decision making problems. Various conventional preferential voting methods where the voters rank candidates in order of preference are reviewed for syntactic patterns and categorized. Several other new voting methods are devised from the conventional procedural patterns and metrics as well. Explicit formulae for over fifty different voting methods are presented and the hierarchical clustering technique is adopted to reveal semantic similarities among them. A nomenclature for voting methods is suggested to reveal their syntactic patterns. All preferential voting methods perform significantly different from the simplest plurality method. Index Terms—decision, hierarchical clustering, preference, voting, nomenclature I. INTRODUCTION ONSENSUS of a group plays an important role in decision making such as elections [1-3] and combining multiple classifiers [4]. It is essential in most democratic societies and has received great attention in artificial intelligence and computer science communities as well [5]. Consider an ordered set of four candidates, C = {„A‟, „B‟, „C‟, „D‟} and they received the corresponding votes, V = {11, 6, 7, 6}. The notations in Table I shall be used throughout the rest of this article. The most widely used and simplest voting method is called the „plurality‟, i.e. the winner is one who has the most votes as defined in (1). The majority voting method in (2) is the same as the plurality but rejects if the winner does not receive more than half votes.   x C x V V plurality   max arg ) ( (1)             otherwise 2 max if max arg ) ( void m V V V majority x C x (2) TABLE I Basic Notations notation meaning and/or example C an ordered set, e.g., {„A‟, „B‟, „C‟, „D‟} and C2 = „B‟. V the corresponding votes, e.g., {11, 6, 7, 6} and V3 = 7. c the number of candidates, c = |C| = 4 in the example. m the total number of voters, e.g., 30 in the example. n the number of unique preference order ballots. 5 in Table II. p(i, j) the candidate in the ith ballot and jth choice, p(2, 1) = „A‟ r(i, x) the choice rank for the candidate x in the ith ballot, r(4,„A‟) = 3 Manuscript received July 23, 2012; revised August 10, 2012. S.-H. Cha is with Computer Science department, Pace University, New York, NY 10038 USA (e-mail: scha@pace.edu). Y.J. An is with the Division of Engineering Technologies and Computer Sciences, Essex County College, Newark, NJ 07102 USA, (e-mail: yan@essex.edu). Starting September 2012. TABLE II Sample Preference Ballot Table Choice\ votes 7 11 1 6 5 1 C A D B D 2 D D B C C 3 B C C A A 4 A B A D B (a) stacked pies (b) multiple bars Fig. 1. Distribution of preference votes. Due to flaws in the simple top choice voting system, the preferential voting system in which the voter ranks candidates in order of preference has been proposed [1-3,5-7]. Voters are asked to rank the candidates where omissions and ties are not allowed and quantities are not important but only the strict order matters as exemplified in Table II. Fig 1 shows the distribution of preference voting in stacked pies and multiple bars. The outer most shell is the first choice and the inner shells are the next choices and so on. The winner of Table I case differs depending on voting methods used and there is a zoo of diverse methods. Various voting methods are used in diverse social groups and different regions. There are so many variation and alternatives and thus a comprehensive study is necessary because even names for certain voting methods are fluid and promulgated differently. Also, a nomenclature for voting methods is necessary as even today another new voting method is invented. The rest of the paper is organized as follows. In section 2, various conventional preference voting methods are given to reveal their syntactic similarities. In section 3, conventional methods are generalized and other new methods are devised from the existing methods‟ patterns. In order to provide a better perspective on similarity among different methods, section 3 presents the hierarchical cluster tree of over fifty different preference voting methods. Finally, section 4 concludes this work. II. CONVENTIONAL VOTING METHODS In this section, different syntactic patterns of conventional voting methods are examined and expressed in as generic forms as possible. Perhaps, the most common paradigm for many methods is finding the argmax of certain measurable score values for each candidate as given in (3). Taxonomy and Nomenclature of Preferential Voting Methods Sung-Hyuk Cha and Yoo Jung An C Proceedings of the World Congress on Engineering and Computer Science 2012 Vol I WCECS 2012, October 24-26, 2012, San Francisco, USA ISBN: 978-988-19251-6-9 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) WCECS 2012