| Abstract | Teorija društvenog odabira interdisciplinarno je područje na presjeku informacijskih, eko- nomskih i matematičkih znanosti. U radu proučavamo modele definirane na profilima strogih linearnih preferencija (poredaka) nekog broja kandidata (ili opcija). U okviru tako definirane teorije, proučavamo načine modeliranja pojma „kompromisa“ kroz de- finiciju mjere odmaka od kompromisa odabira pojedinog kandidata na dano mjesto u strogom linearnom poretku. Kompromis se kao cilj realizira kroz minimizaciju tako defi- nirane mjere. Prateći formalnu metodologiju matematičkog dokaza, dokazujemo da tako definirana mjera odmaka od kompromisa djelomično potvrđuje neformalna očekivanja od pojma kompromisa u okviru funkcija društvenog izbora: Borda metoda društvenog izbora uvijek bira kao pobjednika kandidata koji ima manju mjeru odmaka od kom- promisa od većinske metode – u slučaju izbora između tri kandidata. U slučaju četiri ili više kandidata, pokazujemo kako postoje profili društvenih preferencija u kojima ve- ćinska metoda može izabrati kandidata s manjom mjerom odmaka od kompromisa. Na minimizaciji mjere odmaka od kompromisa temeljimo definiciju novih funkcija društvenog izbora. SdM metoda definirana je kroz minimizaciju odmaka od kompromisa oko izbora pobjedničkog kandidata. Formalno dokazujemo kako SdM metoda ispunjava Youngovu karakterizaciju, te pripada klasi pozicijskih bodovnih funkcija društvenog izbora. GdM metoda je definirana kao pohlepna metoda koja redom minimizira odmake od kompro- misa oko izbora kandidata na sva mjesta u linearnom poretku. Pokazujemo kako GdM posjeduje neželjena svojstva poput ne ispunjavanja Paretovog aksioma. Konačno, defi- niramo TdM metodu koja minimizira zbroj odmaka od kompromisa oko izbora na sva mjesta u linearnom poretku, na skupu svih permutacija (mogućih poredaka) kandidata. Dokazujemo kako TdM metoda zadovoljava Paretov aksiom, te da u posebnom slučaju (za tri kandidata) ispunjava Miharinu karakterizaciju, te je ekvivalentna Bordinoj metodi |
| Abstract (english) | Social choice theory is an interdisciplinary theory that connects information sciences, mathematics and economics. In this paper, we study models defined on profiles of strict linear preferences of a number of candidates (or options). Within the framework of the theory defined in this way, we study the ways of modeling the concept of ”compromise” through the definition of a measure of divergence from the compromise of selecting an individual candidate to a given position in a strict linear ordering. The main idea is that for some candidate, that greater distance between a position in a preference, and a certain position in strict linear ordering (or resulting preference) should have more than linear contribution to the measure of divergence. Therefore, we introduce the measure of divergence as an sum of distances (between position in the given preference and selected position in linear ordering) raised to the power of d > 1. In such model, value of d > 1 represents the level of compromise that society finds acceptable. Since the core phenomenon of the compromise is the concept of vagueness, selection of the value for d allows us to analyze the compromise as a version of a Sorites paradox. Through selection of d society has to decide where to ”draw the line” on the subject of compromise winner. Compromise as a version of the Sorites paradox is analyzed in Section 2.2. This model also enables a never-top-ranked candidate to have the least measure of divergence from the compromise about first position, which is consistent with the notion of Compromise axiom (see [14]). Definition of the d-measure of divergence from compromise allows us to compare some standard social choice function. In Chapeter 2, three standard social choice functions are compared with regard to the d-measure of divergence from the winning position: Borda count, plurality count and Condorcet method. Plurality count is usually considered as ”the least compromise” function since it selects winners using just number of winning preferences for a candidate. On the other hand, Borda count is often considered as a social choice function which fairly represents compromise, since it utilizes the information about all positions of candidates in all preferences. Our main result in Section 2.5. justifies in part such colloquial expectations. We proved that winner of the Borda count (if different from the winner of Plurality count) always has lesser d-measure of divergence from the compromise about first position than the Plurality count winner, in the three-candidate scenario. Proof of this claim is combinatorial. To reduce number of profiles which should be analyzed, we used the property both Borda and Plurality count are satisfying: both of those SCF are invariant to the removal of Condorcet triplets (just as a d-measure of divergence itself), that is, removal of maximal symmetrical sub profiles. However, we also showed that in case of four or more candidates, there are profiles of preferences on which Plurality count winner has lesser d-measure of divergence from the compromise about first position than Borda winner. In Chapter 2 we also compared winners of Borda count and Condorcet method, with regard to d-measure of compromise. Since Borda count does not always elects as a winner a candidate with the least d-measure of divergence from compromise about first position, there is a possibility that some other SCF elects such candidate on a given profile. Since Condorcet method is not invariant to the removal of maximal symmetrical sub profiles, this allows the construction of profiles on which Condorcet method elects a winner with smaller d-measure of divergence than Borda count. This result is shown in Section 2.6. In Chapter 3 we analyzed possibilites for construction of new social choice function, based around minimization of the d-measure of divergence from the compromise. In first approach we took, main goal eas to minimize d-measure of divergence from the compromise about first position. This approach resulted with SdM (Simple d-Measure) social choice (welfare) function that ranks candidates with regard to their d-measure of divergence from compromise about winner, i.e., it elects winning candidate with the least d-measure of divergence from compromise about winner. In Section 3.1 we analyzed properties of SdM. We proved that such social choice fun- ction satisfies number of desirable axioms of the social choice theory, namely anonymity, neutrality, monotonicity and Pareto efficiency. Furthermore, we proved that SdM sa- tisfies axioms of reinforcement and continuity, thus fulfilling Young characterization of point scoring social choice functions. In Section 3.2 we wish to utilize complete information produced by the d-measure of divergence, not just information of the d-measure of divergence from the compromise about winner. Therefore we define a greedy approach to minimization of d-measure of divergence, where candidate is electer to some position in the linear ordering if s/he mini- mizes d-measure of divergence from the compromise about that position. Although sounds reasonable, we prove that greedy approach to minimization has unwanted properties, such as lack of Pareto efficiency which is usually considered as one of the most important axi- oms of the social choice theory. In same section we propose an improvement of the greedy method, though so-called ”cumulative greedy” approach to the minimization. We do not pursue this line of reasoning. Finaly, in Section 3.3 we define social choice (welfare) function TdM (Total d-Measure) which utilizes the complete information about d-measures of divergence. This method compares the sum of d-measures for all candidates over the set of all possible orderings (permutations). In this section we prove that TdM also satisfies number of axioms of the social choice theory: anonymity, neutrality, continuity and Pareto efficiency. We also prove that in three candidates scenario, for d = 2, TdM satisfies monotonicity and intensity of independence of irrelevant alternatives (IIIA). Those properties allows us to use Mihara characterization of the Borda count, proving that TdM is equivalent to Borda count for d = 2 at three candidates scenario. |
