Fair and Square: Issues of Fairness and Computation in Partition Problems

This thesis deals with fair division of indivisible goods and with hedonic games. Both are vibrant fields in the multiagent systems and artificial intelligence community. Fair division of indivisible goods is concerned with the distribution of goods under possibly conflicting interests of agents, whereas hedonic games model issues of coalition formation. For fair division of indivisible goods, we introduce scoringbased allocation rules and correspondences. Agents submit ordinal preferences over single goods to the social planner, who then uses a fixed scoring vector to generate additive representations of these ordinal preferences. Next, this representation is leveraged for social welfare optimization, focusing on utilitarian and egalitarian social welfare, and on the leximin preorder. We study these rules and correspondences with respect to axiomatic properties and computational complexity. Then we extend the scoring-based model to analyze strategy-proofness using set extensions from social choice theory. For utilitarian social welfare, we characterize strategy-proofness using the Gärdenfors extension in terms of the structure of the scoring vector. For the case when agents submit cardinal preferences, we study a fairness hierarchy proposed by Bouveret and Lemaître [2016] with respect to k-additive utility functions and answer a question raised by the same authors about the relationship between efficiency and a strong fairness notion. In the same work we study inequality-reducing rank-weighted utilitarianism and show that the approximability of rank dictator functions is intimately connected with the approximability of egalitarian social welfare. Then we look at the connection between both approaches to fairness for the max-min share and rank-weighted utilitarianism with inequality-reducing weight vectors in an exploratory manner. We introduce local fairness notions for hedonic games. These are fairness notions that depend on an agent’s individual preferences only and are satisfied if the utility of every agent is above a certain threshold. Inspired by fairness notions from fair division, we study max-min, grand-coalition, and min-max fairness in the framework of local fairness. We show that every Nash-stable coalition structure is min-max fair, and that the three fairness notions form a hierarchy, even under general preferences. For additively separable hedonic games, the hierarchy collapses. Computing individual thresholds is possible in polynomial time for grand-coalition and max-min fairness but is coNP-complete for min-max fairness. The existence problems are NP-hard in all three cases. The last contribution of this thesis is a new representation of preferences for hedonic games. In contrast to prior work that assumes that preferences are independent among agents, we introduce social preferences in the context of friend-oriented hedonic games. We distinguish three degrees of altruism. Depending on the degree, an agent either uses the classical preferences first and the average opinion of this agent’s friends as a tie-breaker, or vice versa. Between these two extremes we consider a degree where an agent’s preference is given the same weight as the friends’ preferences. For these preference types, we show that each is different from classical representations. Using properties from social choice, we check that these preferences behave naturally. Moreover, in all three cases Nash-stable coalition structures always exist and we initiate the study of the computational complexity for other stability notions.