TY - JOUR

T1 - Bounds on the disparity and separation of tournament solutions

AU - Brandt, Felix

AU - Dau, Andre

AU - Seedig, Hans Georg

N1 - Publisher Copyright:
© 2015 Elsevier B.V.

PY - 2015/5/31

Y1 - 2015/5/31

N2 - A tournament solution is a function that maps a tournament, i.e., a directed graph representing an asymmetric and connex relation on a finite set of alternatives, to a non-empty subset of the alternatives. Tournament solutions play an important role in social choice theory, where the binary relation is typically defined via pairwise majority voting. If the number of alternatives is sufficiently small, different tournament solutions may return overlapping or even identical choice sets. For two given tournament solutions, we define the disparity index as the order of the smallest tournament for which the solutions differ and the separation index as the order of the smallest tournament for which the corresponding choice sets are disjoint. Isolated bounds on both values for selected tournament solutions are known from the literature. In this paper, we address these questions systematically using an exhaustive computer analysis. Among other results, we provide the first tournament in which the bipartisan set and the Banks set are not contained in each other.

AB - A tournament solution is a function that maps a tournament, i.e., a directed graph representing an asymmetric and connex relation on a finite set of alternatives, to a non-empty subset of the alternatives. Tournament solutions play an important role in social choice theory, where the binary relation is typically defined via pairwise majority voting. If the number of alternatives is sufficiently small, different tournament solutions may return overlapping or even identical choice sets. For two given tournament solutions, we define the disparity index as the order of the smallest tournament for which the solutions differ and the separation index as the order of the smallest tournament for which the corresponding choice sets are disjoint. Isolated bounds on both values for selected tournament solutions are known from the literature. In this paper, we address these questions systematically using an exhaustive computer analysis. Among other results, we provide the first tournament in which the bipartisan set and the Banks set are not contained in each other.

KW - Disjointness

KW - Disparity

KW - Minimal examples

KW - Tournament solutions

UR - http://www.scopus.com/inward/record.url?scp=84928204316&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2015.01.041

DO - 10.1016/j.dam.2015.01.041

M3 - Article

AN - SCOPUS:84928204316

SN - 0166-218X

VL - 187

SP - 41

EP - 49

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

ER -