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Stable and Envy-Free Partitions in Hedonic Games Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence (IJCAI-19) Stable and Envy-free Partitions in Hedonic Games Nathanael¨ Barrot1;2 and Makoto Yokoo2;1 1Riken, AIP Center 2Kyushu University [email protected], [email protected] Abstract In hedonic games, a partition is said to be envy-free if no agent prefers another agent’s coalition. Similarly to resource In this paper, we study coalition formation in he- allocation, envy-freeness is always trivially achieved by the donic games through the fairness criterion of envy- grand coalition and the partition of singletons. Since these freeness. Since the grand coalition is always envy- two partitions are not always relevant, requiring only envy- free, we focus on the conjunction of envy-freeness freeness may not be restrictive enough. Hence, it makes sense with stability notions. We first show that, in sym- to impose additional requirements, like stability or feasibility, metric and additively separable hedonic games, an in conjunction with envy-freeness. However, only a couple individually stable and justified envy-free partition of works have done so in hedonic games, Aziz et al. [2013b] may not exist and deciding its existence is NP- and Ueda [2018], following two different directions: The first complete. Then, we prove that the top responsive- paper considers envy-freeness in conjunction with stability ness property guarantees the existence of a Pareto and efficiency requirements. The second one restricts the set optimal, individually stable, and envy-free parti- of feasible partitions and formulates justified envy-freeness, a tion, but it is not sufficient for the conjunction of stronger fairness notion motivated by matching theory. core stability and envy-freeness. Finally, under bot- tom responsiveness, we show that deciding the ex- Following the first direction, we investigate the conjunc- istence of an individually stable and envy-free par- tion of envy-freeness with stability or efficiency notions in tition is NP-complete, but a Pareto optimal and jus- three subclass of hedonic games where some level of stability tified envy-free partition always exists. is guaranteed: additively separable hedonic games (ASHG) which form a natural subclass, and top or bottom respon- sive hedonic games which encompass well-studied hedonic 1 Introduction games with realistic interpretations, e.g., friend-enemy ori- Coalition formation plays a major role in our social, eco- ented hedonic games [Dimitrov et al., 2006] or B-hedonic nomic, or politic life. In examples as diverse as academic re- games [Cechlarov´ a´ and Romero-Medina, 2001]. search or labor unions, individuals (agents) perform activities in groups (coalitions) rather than on their own. In coalition 1.1 Our Contribution formation with hedonic preferences, or hedonic games, each agent’s preference over the set of coalitions only concerns the First, we propose a natural weakening of justified envy- coalitions that she joins. The outcome of such game is a set freeness and explore the implications between stability and of disjoint coalitions of all agents (partition). A natural ques- envy-freeness notions (Figure 1). In symmetric ASHGs, we tion is which coalitions are expected to form based on agents’ strengthen results from Aziz et al. [2013b], by proving that preferences. The main criterion to analyze this question is an individually stable and justified envy-free partition may stability, for which many notions have been discussed in the not exist and deciding its existence is NP-complete. We also literature (see handbook chapter Aziz and Savani [2016]). show that a Pareto optimal and justified envy-free partition Introduced by Foley [1967], envy-freeness is a notion of may not exist. In top responsive hedonic games, we pro- fairness which has a broad range of applicability. This no- pose the Extended Top Covering Algorithm which returns a tion has received considerable attention in resource allocation Pareto optimal, individually stable, and envy-free partition. [Chevaleyre et al., 2006]. Since envy-freeness can be triv- In bottom responsive hedonic games, we show that deciding ially achieved (by not allocating any resources), the central the existence of an individually stable and envy-free partition issue in resource allocation is the study of envy-free outcomes is NP-complete, but a Pareto optimal and justified envy-free that additionally satisfy efficiency or feasibility requirements. partition is guaranteed. Table 1 summarizes our results. Contrary to other well-studied fairness notions, e.g. maxmin Note that the partition of singletons is always envy-free and or proportionality, envy-freeness does not require to compare individually rational. Hence, our paper settles the existence inter-personnal preferences, and thus, it can be meaningfully of partitions satisfying any conjunction of stability and envy- expressed in ordinal settings like hedonic games. freeness notions from Figure 1, in the three class at hand. 67 Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence (IJCAI-19) General Symmetric/Mutual Additively separable hedonic games (ASHG) form a natu- ral class of hedonic game where each agent has a value for IS+JEV: NP-c (Prop.1, Th.1) ASHG — @ any other agent and the utility that an agent derives from a PO+JEV: (Prop.2) coalition is the sum of the values she has for its members. PO+IS+EF: P (Th.2) Definition 1 . (N; <) TR @ SSNS+EF: 9 (Th.3) (ASHG) A hedonic game is additively CS+EF: (Prop.3) separable if for each agent i 2 N there exists a utility function BR PO+JEF: 9 (Th.5) — vi : N 7! R such that vi(i) = 0 and for any two coalitions S; T 2 N , S < T , P v (j) ≥ P v (j): IS+EF: NP-c (Prop.4, Th.4) i i j2S i j2T i SBR — CS/PO+WJEF: @ (Prop.4/5) An ASHG is symmetric if any two agents, i; j 2 N, asso- ciate the same value to each other, i.e., vi(j) = vj(i). Table 1: Summary of the results. TR, BR, and SBR refer to top, Envy-freeness is a notion of fairness in which no agent has bottom, and strong bottom responsiveness. 9 and @ means that the envy toward another agent. Informally, a partition is envy-free desired partition always exists and may not exist. P and NP-c stand if no agent prefers another agent’s coalition over his own. for polynomial computation and NP-complete existence problem. Definition 2 (Envy-freeness (EF)). In partition π, agent i has envy toward an agent j with π(i) 6= π(j), if (π(j) n fjg) [ 1.2 Related Work fig i π(i). Partition π is envy-free if no agent has envy. In hedonic games, initiated by Banerjee et al. [2001] and Bo- Justified envy-freeness is a stronger notion of fairness, for- gomolnaia and Jackson [2002], the central question is to iden- mulated in hedonic games by Ueda [2018]. We propose a tify the conditions on preferences that guarantee stability. An natural weakening of justified envy. Agent i has (weakly) important contribution is Aziz and Brandl [2012] which clar- justified envy toward agent j if i has envy toward j and each ified the inclusions between well-studied stability concepts. agent in j’s coalition is (weakly) in favor of replacing j by i. In fractional hedonic games, introduced by Aziz et al. [2014], Definition 3 (Justified envy-freeness (JEF)). In partition π, Brandl et al. [2015] showed that individually stable outcomes i has justified envy toward j with π(i) 6= π(j), if i has envy 0 may not exist, Bilo` et al. [2018] extensively studied Nash sta- toward j and for all j 2π(j)nfjg, (π(j)nfjg)[figj0 π(j). bility, and Monaco et al. [2018] explored Nash and core sta- Partition π is justified envy-free if no agent has justified envy. bility in a slightly modified setting. Concerning efficiency Definition 4 (Weakly justified envy-freeness (WJEF)). In [ ] notions, Aziz et al. 2013a proposed an algorihm computing partition π, agent i has weakly justified envy toward j with Pareto optimal and individually rational partitions, and Elkind π(i) 6= π(j), if i has envy toward j and for all j0 2 π(j)nfjg, et al. [2016] introduced the price of Pareto optimality. (π(j) n fjg) [ fig <j0 π(j). Partition π is weakly justified Among many complexity results, Ballester [2004] showed envy-free if no agent has weakly justified envy. that deciding the existence of individually stable partitions is NP-complete, and Peters and Elkind [2015] developed a The following example illustrates these fairness notions. framework for proving NP-hardness of existence problems. Example 1. Consider a symmetric ASHG with four agents Envy-freeness has been extensively studied as a fairness f1; 2; 3; 4g and values v1(2) = v2(3) = v3(1) = 2, v2(4) = criterion in resource allocation. In ordinal settings, Brams et v3(4) = x, and v1(4) = −7. In this game, consider partition al. [2003] analyzed conflicts between Pareto optimality and ff1; 2; 3g; f4gg. If x = 1 then agent 4 has envy toward agent envy-freeness. When the preferences are incomplete, Bou- 1, but not weakly justified envy. If x = 2 then 4 has weakly veret et al. [2010] proposed possible and necessary envy- justified envy toward 1, but not justified envy. Finally, if x = freeness. Justified envy-freeness is motivated in two-sided 3 then 4 has justified envy toward 1. matching theory, particularly in the school choice setting, By definition, envy-freeness implies weakly justified envy- where Abdulkadiroglu˘ and Sonmez¨ [2003] argued that it freeness, which implies justified envy-freeness. Example 1 could lead to legal actions, and Fragiadakis et al.
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