Stability and Welfare in (Dichotomous) Hedonic Diversity Games

Abstract

In a hedonic diversity game (HDG) there are two types of agents (red and blue agents) that need to form disjoint coalitions, i.e., subgroups of agents. Each agent’s preferences over the coalitions depend on the relative number of agents of the same type in her coalition. In the special case of a dichotomous hedonic diversity game (DHDG) each agent distinguishes between approved and disapproved fractions only. We aim at outcomes that are stable against agents’ deviations, and at outcomes that maximize social welfare. In particular, we show that the strict core of a DHDG may be empty even in instances with only three agents, while each HDG with two agents has a non-empty strict core. We also provide several computational complexity results for DHDGs with respect to the number of fractions approved per agent. For instance, we prove that deciding whether a DHDG has a non-empty strict core is NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {NP}$$\\end{document}-complete even when each agent approves of at most three fractions. In addition, we show that deciding whether a DHDG admits a Nash stable outcome is NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {NP}$$\\end{document}-complete even in restricted settings with only two approved fractions per agent—therewith, improving a result in the literature. For the task of maximizing social welfare, we apply approval scores and Borda scores from voting theory. For DHDGs and approval scores, we draw the sharp separation line between polynomially solvable and NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {NP}$$\\end{document}-complete cases with respect to the fixed number of approved fractions per agent. We complement these findings with an NP\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extsf {NP}$$\\end{document}-completeness result for HDGs under Borda scores.