Optimal Partitions in Additively Separable Hedonic Games

We conduct a computational analysis of fair and optimal partitions in additively separable hedonic games. We show that, for strict preferences, a Pareto optimal partition can be found in polynomial time while verifying whether a given partition is Pareto optimal is coNP-complete, even when preferences are symmetric and strict. Moreover, computing a partition with maximum egalitarian or utilitarian social welfare or one which is both Pareto optimal and individually rational is NP-hard. We also prove that checking whether there exists a partition which is both Pareto optimal and envy-free is $\Sigma_{2}^{p}$-complete. Even though an envy-free partition and a Nash stable partition are both guaranteed to exist for symmetric preferences, checking whether there exists a partition which is both envy-free and Nash stable is NP-complete.Comment: 11 pages; A preliminary version of this work was invited for presentation in the session `Cooperative Games and Combinatorial Optimization' at the 24th European Conference on Operational Research (EURO 2010) in Lisbo

Boolean Hedonic Games

We study hedonic games with dichotomous preferences. Hedonic games are cooperative games in which players desire to form coalitions, but only care about the makeup of the coalitions of which they are members; they are indifferent about the makeup of other coalitions. The assumption of dichotomous preferences means that, additionally, each player's preference relation partitions the set of coalitions of which that player is a member into just two equivalence classes: satisfactory and unsatisfactory. A player is indifferent between satisfactory coalitions, and is indifferent between unsatisfactory coalitions, but strictly prefers any satisfactory coalition over any unsatisfactory coalition. We develop a succinct representation for such games, in which each player's preference relation is represented by a propositional formula. We show how solution concepts for hedonic games with dichotomous preferences are characterised by propositional formulas.Comment: This paper was orally presented at the Eleventh Conference on Logic and the Foundations of Game and Decision Theory (LOFT 2014) in Bergen, Norway, July 27-30, 201

Mechanism Design for Team Formation

Team formation is a core problem in AI. Remarkably, little prior work has addressed the problem of mechanism design for team formation, accounting for the need to elicit agents' preferences over potential teammates. Coalition formation in the related hedonic games has received much attention, but only from the perspective of coalition stability, with little emphasis on the mechanism design objectives of true preference elicitation, social welfare, and equity. We present the first formal mechanism design framework for team formation, building on recent combinatorial matching market design literature. We exhibit four mechanisms for this problem, two novel, two simple extensions of known mechanisms from other domains. Two of these (one new, one known) have desirable theoretical properties. However, we use extensive experiments to show our second novel mechanism, despite having no theoretical guarantees, empirically achieves good incentive compatibility, welfare, and fairness.Comment: 12 page

Role Based Hedonic Games

In the hedonic coalition formation game model Roles Based Hedonic Games (RBHG), agents view teams as compositions of available roles. An agent\u27s utility for a partition is based upon which role she fulfills within the coalition and which additional roles are being fulfilled within the coalition. I consider optimization and stability problems for settings with variable power on the part of the central authority and on the part of the agents. I prove several of these problems to be NP-complete or coNP-complete. I introduce heuristic methods for approximating solutions for a variety of these hard problems. I validate heuristics on real-world data scraped from League of Legends games

Formation of Segregated and Integrated Groups

A model of group formation is presented where the number of groups is fixed and a person can only join a group if the group’s members approve the person’s joining. Agents have either local status preferences (each agent wants to be the highest status agent in his group) or global status preferences (each agent wants to join the highest status group that she can join). For both preference types, conditions are provided which guarantee the existence of a segregated stable partition where similar people are grouped together and conditions are provided which guarantee the existence of an integrated stable partition where dissimilar people are grouped together. Additionally, in a dynamic framework we show that if a new empty group is added to a segregated stable partition, then integration may occur.Group Formation, Stable Partition, Segregation, Integration

Status-Seeking in Hedonic Games with Heterogeneous Players

We study hedonic games with heterogeneous player types that reflect her nationality, ethnic background, or skill type. Agents' preferences are dictated by status-seeking where status can be either local or global. The two dimensions of status define the two components of a generalized constant elasticity of substitution utility function. In this setting, we characterize the core as a function of the utility's parameter values and show that in all cases the corresponding cores are non-empty. We further discuss the core stable outcomes in terms of their segregating versus integrating properties.Coalitions, Core, Stability, Status-seeking

Coalition Formation with Local Public Goods and Network Effect

Many local public goods are provided by coalitions and some of them have network effects. Namely, people prefer to consume a public good in a coalition with more members. This paper adopts the Drèze and Greenberg (1980) type utility function where players have preferences over goods as well as coalition members. In a game with anonymous and separable network effect, the core is nonempty when coalition feasible sets are monotonic and players' preferences over public goods have connected support. All core allocations consist of connected coalitions and they are Tiebout equilibria as well. We also examine the no-exodus equilibrium for games whose feasible sets are not monotonic.Coalition formation, core, network effect, local public goods

Hedonic Seat Arrangement Problems

In this paper, we study a variant of hedonic games, called \textsc{Seat Arrangement}. The model is defined by a bijection from agents with preferences to vertices in a graph. The utility of an agent depends on the neighbors assigned in the graph. More precisely, it is the sum over all neighbors of the preferences that the agent has towards the agent assigned to the neighbor. We first consider the price of stability and fairness for different classes of preferences. In particular, we show that there is an instance such that the price of fairness ({\sf PoF}) is unbounded in general. Moreover, we show an upper bound $\tilde{d}(G)$ and an almost tight lower bound $\tilde{d}(G)-1/4$ of {\sf PoF}, where $\tilde{d}(G)$ is the average degree of an input graph. Then we investigate the computational complexity of problems to find certain ``good'' seat arrangements, say \textsc{Maximum Welfare Arrangement}, \textsc{Maximin Utility Arrangement}, \textsc{Stable Arrangement}, and \textsc{Envy-free Arrangement}. We give dichotomies of computational complexity of four \textsc{Seat Arrangement} problems from the perspective of the maximum order of connected components in an input graph. For the parameterized complexity, \textsc{Maximum Welfare Arrangement} can be solved in time $n^{O(\gamma)}$, while it cannot be solved in time $f(\gamma)^{o(\gamma)}$ under ETH, where $\gamma$ is the vertex cover number of an input graph. Moreover, we show that \textsc{Maximin Utility Arrangement} and \textsc{Envy-free Arrangement} are weakly NP-hard even on graphs of bounded vertex cover number. Finally, we prove that determining whether a stable arrangement can be obtained from a given arrangement by $k$ swaps is W[1]-hard when parameterized by $k+\gamma$, whereas it can be solved in time $n^{O(k)}$

Algorithmic aspects of fixed-size coalition formation

We study algorithmic aspects of models in which a set of agents is to be organised into coalitions of a fixed size. Such models can be viewed as a type of hedonic game, coalition formation game, or multidimensional matching problem. We mostly consider models in which coalitions have size three and are formalisms of Three-Dimensional Roommates (3DR). Models of 3DR are characterised by a combination of the system by which agents have preferences over coalitions, and the solution concept (e.g. stability). Since the computational problems associated with 3DR are typically hard, we consider approximate solutions and restricted cases, with the aim of characterising the boundary between tractable and intractable variants. Part of our contribution relates to two new models of 3DR, which involve two existing systems of preferences called [B- and W-preferences]. In each model, we consider the existence of matchings that are stable. We show that the related decision problems are NP-complete and devise approximation algorithms for corresponding optimisation problems. In a model of 3DR with additively separable preferences, we investigate stable matchings and envy-free matchings, for three successively weaker definitions of envy-freeness. We consider restrictions on the agents’ preferences including symmetric, binary, and ternary valuations. We identify dichotomies based on these restrictions and provide a comprehensive complexity classification. Interestingly, we identify a general trend that, for successively weaker solution concepts, existence and polynomial-time solvability hold under successively weaker preference restrictions. We also consider a variant of 3DR known as Three-Dimensional Stable Matching with Cyclic preferences (3-DSM-CYC), which has been of independent interest. It was recently shown that finding a stable matching is NP-hard, so we consider a related optimisation problem and present an approximation algorithm based on serial dictatorship. We also consider a situation in which the preferences of some agents are sufficiently similar to some master list, and show that the approximation ratio of this algorithm can be improved in relation to a specific similarity measure. Finally, we consider a problem in graph theory that generalises the notion of assigning agents to coalitions of a fixed size. Rather than organising a set of agents, the problem is to find a maximum-cardinality set of r-cliques in an undirected graph subject to that set being either vertex disjoint or edge disjoint, for a fixed integer r ≥ 3. This general problem is known as the Kr- packing problem. Here we study the restriction of this problem in which the graph has a fixed maximum degree ∆. It is known for r = 3 that the vertex-disjoint (edge-disjoint) variant is solvable in linear time if ∆ = 3 (∆ = 4) but APX-hard if ∆ ≥ 4 (∆ ≥ 5). We generalise these results to an arbitrary but fixed r ≥ 3, and provide a full complexity classification for both the vertex- and edge-disjoint variants in graphs of maximum degree ∆, for all r ≥ 3