Selçuk J. Appl. Math. Selçuk Journal of
Vol. 6. No.2. pp. 53-58, 2005 Applied Mathematics
Fuzzy Ballot Paper and Its Election System Naim Ça¼gman1 and Hac¬Akta¸s
1Department of Mathematics, Faculty of Science and Art, Gaziosmanpa¸sa University,
60100 Tokat,Turkey; e-mail:ncagm an@ yaho o.com
Received : October 21, 2005
Summary. In this work, we will design a fuzzy ballot paper (FBP) in order to obtain a more suitable election system based on the fuzzy sets. This new system gives the voter a chance to vote for less supported candidates. The voting procedures and use of the equipments of the FBP are easy for voters to use and understand. FBP reduces problems with mismarked ballots, disputed results and recounts.
Key words: Ballot paper, election systems, fuzzy sets, linguistic variables, linguistic terms.
1. Introduction
The ballot is one of the most important tools of an election and its design can aid or inhibit clarity in the elections. A poor design leads to confusion and potentially chaos when large numbers of voters go for voting. In this work, we will introduce a new ballot paper and its election system based on the fuzzy logic. Fuzzy logic refers to using of fuzzy sets in the representation and manipulation of vague information for the purpose of making decisions. The concepts of fuzzy sets used in this work can be found in [8], [10] and [16].
The concept of linguistic variables is one of the most important tools in appli-cations of the fuzzy set theory. The value of a linguistic variable is not numbers but words or sentences in a natural or arti…cial language. The linguistic terms are subjective categories for the linguistic variable. Fuzzy linguistic terms do not hold very exact meaning and may be understood di¤erently by di¤erent people. Each linguistic term is associated with a fuzzy set, each of which has a de…ned membership function. Formally, in a universe U , a fuzzy set eA was de…ned by Zadeh [15] as
(1) A = f(x;e Ae(x)) : x 2 U; Ae(x) 2 [0; 1]g
where the function Ae(x) is called a membership function that gives the degree of membership of x. This indicates the degree to which x belongs in set eA. It is quite natural that, before any election, the candidates are introduced by their teams with some fuzzy propositions such as “it is better than others", “it is the best one", etc. Such propositions have in…nity possible truth values, such as "true", "false", "very true", "very false", "approximately true", "possibly true", etc. So a voter has many options to evaluate the truthfulness or falsehood of these candidates’ sentences. Therefore, the truth value of each proposition is expressed by a fuzzy set designed to interpret the meaning of the linguistic truth values. The criteria of linguistic truth values depend on the person who judges them and we cannot construct a strict basis for them. Since the truth or falsity of a fuzzy proposition is a matter of degree, it is not always possible for the voter to assert that it is either true or false. Therefore, it is necessary to express degree of being true.
In [6], a special ballot paper was designed to obtain a voting method where a vote is cut into k-many parts, and the ballot paper can allow k-many options (parts) to mark for the candidates. The voter can distribute their prefers without any restriction for many candidates by making k-many marks. This ballot paper is not completely fuzzy because there are only k-many marks for all candidates, for instance, when a voter use k-many marks for only one candidate, there will be not left any mark for other candidates.
In this work, we will redesign the balled paper, called fuzzy ballot paper (FBP), in order to obtain a more suitable election system. In FBP, the voter can use as may marks as she/he wants to express the degree of truth of the candidate proposition to re‡ect their prefers. The most election usually force yes-or-no answer, there may be information lost in doing so. The FBP solve this problem and will be more adequate in realizing democratic ideals. We then use fuzzy sets theory to count the FBP.
Formal study of voting methods date to the French Revolution and the works of Borda [4] and Condorcet [7] raised important ideas related to voting systems. The literature is now quite vast. The researchers who made contributions to mathematical ideas that involve elections include Black [2], Arrow [1], Brams [5], Nurmi [12], Saari [13] are well known.
The paper is organized as follows. In the next section, the FBP is designed. In section 3, counting of the FBP is introduced by using fuzzy sets. In section 4, an illustrative example of the method is described to demonstrate the process of the method. In the …nal section, some concluding comments are presented and there is a comparison of the FBP with the well known classical voting systems. For the classical voting systems, we can refer to [5], [11], [12], [13], and [14].
2. Fuzzy Ballot Paper
In this section, we can design the FBP is a special ballot paper of an election for n-many candidates, A = fx1; x2; :::; xng, as in Figure 1.
1 2 k x1 x2 .. . ... ... . .. ... xn Figure 1. An unmarked FBP
where in the left side of the each row, xi, i = 1; 2; :::; n, stands for candidates, and in the right side of the each row, the k-many circles which will be marked by the voters, asN, to re‡ect their prefers easily.
In an election, there is a fuzzy predicate p(x) for each candidate x 2 A such that
p(x) = \x is the best"
where the word “best" is a fuzzy linguistic term. The truth value of this proposi-tion is a fuzzy set de…ned, hence by its membership funcproposi-tion, which are designed to interpret the meaning of the linguistic truth values. There are in…nity possible truth values, such as "true", "false", "very true", "very false", "approximately true", "possible true", etc. So a voter has many options to evaluate the truth-fulness or falsehood of these candidates’sentence. The truth or falsity of a fuzzy proposition is a matter of degree. Here, we can categorize truth value of the linguistic variable for a …nite number, assume that k-many. Therefore one of the possible truth values of the proposition for candidate x 2 A, will be indicated by voter with following possible marked circles
Number of
marked circles Marked circles Meaning of marked circles
0 p(x) is false, 1 N p(x) is less true, 2 N N p(x) is fairly true, .. . ... ... k N N N N p(x) is absolutely true.
Table 1. Meaning of possible marked circles for a candidatex
For example, for a candidate x if a voter things that “p(x) is fairly true", then the voter has to mark two circles in the own ballot-paper asN N .
Note that it is not important being together successively for marked circles in a row of a FBP. For example, the meaning ofN N N N , N N , , N N, etc. are same because each of them has two marked circles.
3. Counting
Each voter marks as many circle as they wish for all of the candidates on their FBP in order of preference.
Let V be the crisp set of all voters, and let A be the crisp set of all candidates in an election. Then each voter’s FBP can be represented by a fuzzy set as follows (2) Aev= f(x; Aev(x)) : x 2 A; Aev(x) =
vx kg
for all v 2 V . Where v(x) indicates number of marked circle for candidate x by voter v in the FBP while voter v voting, and k is maximum number of circles (marked or unmarked) in a row of the FBP.
In an election, after each voter’s FBP transfers to a fuzzy set, we can de…ne a fuzzy election set by using arithmetics mean as follows
(3) A = f(x;e Ae(x)) : x 2 A; Ae(x) = 1 jV j X v2V e Av(x)g
where jV j is cardinality of V . In fuzzy election set eA, if one element (candidate) x has bigger membership degree Ae(x) than every other candidate, that candidate is declared the winner.
Now, we give a numerical demonstration for the process of the system. 4. An Illustrative Example
Let C = fc1; c2; c3; c4g be a candidate set, V = fv1; v2; v3; v4; v5; v6; v7g be a voter set, and k = 5 that means the set of fuzzy linguistic variable has 5 elements which are , N, N N, N N N, N N N N, N N N N N stand for "P is not true", "P is less true", "P is true", "P is very true", "P is very very true", "P is absolutely true" respectively. Suppose that the voter v1 cast vote as following FBP 1 2 3 4 5 c1 N N N N c2 N c3 c4 N N N N N Figure 2. A ballot-paper ofv1
According to (2), this fuzzy ballot-paper gives the following fuzzy set as e
Cv1 = f(c1; 4=5); (c2; 1=5); (c3; 0); (c4; 1)g
Suppose that each voter vi, i 2 I = f2; 3; 4; 5; 6; 7g, give their opinions by the following FBP sets e Cv2 = f(c1; 0:8); (c2; 0:2); (c3; 0:2); (c4; 0:8)g e Cv3 = f(c1; 1:0); (c2; 0:0); (c3; 0:0); (c4; 0:0)g e Cv4 = f(c1; 1:0); (c2; 1:0); (c3; 1:0); (c4; 1:0)g e Cv5 = f(c1; 0:2); (c2; 0:2); (c3; 0:2); (c4; 0:8)g e Cv6 = f(c1; 0:6); (c2; 0:2); (c3; 0:4); (c4; 0:6)g e Cv7 = f(c1; 0:6); (c2; 0:2); (c3; 0:6); (c4; 0:6)g
Then, according to (3) we can get the fuzzy election set as e
C = f(c1; 0:71); (c2; 0:28); (c3; 0:34); (c4; 0:68)g where the winner is c1.
5. Conclusion
This election system is a kind of range voting. In the range voting, for each candidate, each voter expresses the utility of that candidate’s election to them in the form of a number. In "pure" range voting, each voter may give any candidate any real number. For example, each voter might give a real number between -1 and 1, or an integer between 1 and 10. The scores for each candidate are summed, and the candidates with the highest sums are declared the winners. In the FBP, we use marks which is easy to use for every kind of voters. There is some similarity with the Borda count. In the Borda count, a number k is selected as in the FBP; this number can be smaller than or equal to the number of candidates, it is not necessary in the FBP. Each voter lists their top k choices, in order of preference. In the FBP, it is not necessary listing candidates as in the Borda count. In the FBP, each voter may vote for as many candidate as they wish, for example, a voter can do same number of marks for more then one candidate.
In the trivial case of k=1, this is mathematically identical to plurality voting. This is equivalent to saying that each voter may "approve" or "disapprove" each option by marking or not marking for them. In this case, it’s also equivalent to voting +1 or 0 in a range voting system.
Some features of using the FBP include: the voting procedures or equipments are easier for voters to use and understand. The FBP lists all the candidates and allows mark to the voter by hand, and it is also suitable for electronic voting.
This reduces problems with mismarked ballots, disputed results and recounts. It provides less incentive for negative campaigning than many other systems. The FBP gives the voter a chance to vote for less supported candidates. References
1. Arrow K.J. (1951): Social Choice and Individual Values, New York.
2. Black D. (1958): The Theory of Committees and Elections, Cambridge, Cambridge University Press.
3. Blin J.M., and Whinston (1973): Fuzzy set and social choice, J. of Cybernetics, 3, pp.28-36.
4. Borda J-C. (l78l): Mémoire sur les elections au scrutin, Histoire de l’Academie Royale des Sciences, Paris.
5. Brams S.J.(1994): Voting Systems, in Handbook of Game Theory, Volume 2, ed. R. Aumann and S. Hart, Elsevier Science, New York.
6. Ça¼gman N. (2003): A new election system via fuzzy logic (in Turkish), I. National Symposium of Logic, Mathematics and Philosophy (Assos-Çanakkale, 26-28 September 2000) Kultur University Publication, no:41, pp. 35-42.
7. Condorcet M.(l785): Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix, Paris.
8. Dubois D. and Prade H. (1980): Fuzzy Set and Systems: Theory and Applications, Academic Press, New York.
9. Dummett M. (l984): Voting Procedures, University Press, Oxford.
10. Klir G.J. and Folger T.A. (1988): Fuzzy Sets, Uncertainty and Information, Prentice-Hall.
11. Malkevitch J.(2002): Voting and elections, American Mathematical Society, March. 12. Nurmi H.(l987): Comparing Voting Systems, Dordrecht, Holland: D. Reidel. 13. Saari D.G.(2001): Decisions and Elections, Cambridge U. Press, New York. 14. Stra¢ n P.(1980): Topics in the Theory of Voting, Birkhauser, Boston. 15. Zadeh L.A.(1965): Fuzzy Sets, Inform. and Control8, pp. 338-353.
16. Zimmermann H.-J.(1993): Fuzzy Set Theory and Its Applications, Kluwer Acad-emic Publishers.
