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 bottom 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 responsive 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 natural 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: ∃ (Th.3) (ASHG) A hedonic game is additively CS+EF: (Prop.3) separable if for each agent i ∈ N there exists a utility function BR PO+JEF: ∃ (Th.5) — vi : N 7→ R such that vi(i) = 0 and for any two coalitions S, T ∈ N , S < T ⇔ P v (j) ≥ P v (j). IS+EF: NP-c (Prop.4, Th.4) i i j∈S i j∈T i SBR — CS/PO+WJEF: @ (Prop.4/5) An ASHG is symmetric if any two agents, i, j ∈ 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. ∃ 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) \{j}) ∪ 1.2 Related Work {i} 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 ∈π(j)\{j}, (π(j)\{j})∪{i} j0 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 ∈ π(j)\{j}, et al. [2016] introduced the price of Pareto optimality. (π(j) \{j}) ∪ {i}

68 Proceedings of the Twenty-Eighth International Joint Conference on Artiﬁcial Intelligence (IJCAI-19)

Perfect 5 6 3 5 7 3 SSNS 1 2 1 1

4 4 6 SNS SCS EF 3

NS SIS PO WJEF Figure 2: A symmetric ASHG with no IS and JEF partition.

IS CS 3 Justified Envy in Symmetric ASHGs In symmetric ASHGs, it is known that Nash (and thus in- [ IR JEF dividually) stable partitions always exist Bogomolnaia and Jackson, 2002]. However, Aziz et al. [2013b] show that an Figure 1: Implications between stability and envy-freeness notions. individually stable and envy-free partition may not exist, and that deciding the existence of such partition is NP-complete. We strengthen both results with Proposition 1 and Theorem 1, showing that they even hold for justified envy-freeness. that X ∪ {i} i π(i). • Strong Nash stable (SNS): partition π0 and set of agents H Proposition 1. In symmetric ASHGs, an individually stable H 0 0 and justified envy-free partition may not exist. such that 1) π −→ π and 2) for all i ∈ H, π (i) i π(i). • Strict strong Nash stable (SSNS): partition π0 and set of The proof is based on the following counterexample. H 0 0 Example . agents H such that 1) π −→ π , 2) for all i ∈ H, π (i)

69 Proceedings of the Twenty-Eighth International Joint Conference on Artiﬁcial Intelligence (IJCAI-19)

In addition, the reduction holds as it is for any conjunction of {IS, NS} and {JEF, WJEF}. 1 7 y1 Finally, Pareto optimal partitions always exist by defini- 5 tion. However, we show that Pareto optimal and justified 1 1 envy-free partitions may not exist in symmetric ASHGs. 5 2 7 s1 y1 5 Proposition 2. In symmetric ASHGs, a Pareto optimal and 1 justified envy-free partition may not exist. 3 7 y1 We omit the argument of the proof based on Example 3.

s2 Example 3. Consider the symmetric ASHG with nine agents {1,..., 9} and values v1(2) = 8, v2(3) = 7, v2(4) = 4, y4 1 v3(4) = 2, v3(5) = 6, v4(5) = 3, v4(6) = 4, v5(6) = 2, v5(7) = v6(7) = 1, and v3(8) = v5(8) = 1. All other values s 5 are equal to −10. Figure 3:3 Symmetric ASHG obtainedy1 by reduction of an X3C instance with s1 = {1, 2, 3}, s2 = {2, 3, 4}, . . . , with focus on s1. 4 Top Responsiveness and Envy-freeness 6 y1 Top responsiveness is a condition on agent’s preference, intro- Consider (Y,S) an instance of X3C. From (Y,S), we con- duced in Alcalde and Revilla [2004]. Intuitively, it requires struct a symmetric ASHG as follows, illustrated by Figure 3: that an agent’s preference over two coalitions is responsive i • For each i ∈ Y , we create six agents (yj)j=1,...,6 that form to her best subsets (choice set) in each coalition. Given agent an instance of Example 2. i∈N and coalition X ∈Ni, let Ch(i, X) denote the choice set 0 0 • For each set s = {i, j, k} ∈ S, we create agent s such of i in X, i.e., Ch(i, X) = {Y ∈Xi : ∀ Y ∈ Xi,Y

70 Proceedings of the Twenty-Eighth International Joint Conference on Artiﬁcial Intelligence (IJCAI-19)

Algorithm 1 Extended Top Covering Algorithm Proof of Theorem 2. Consider a top responsive hedonic game Input: Top responsive hedonic game (N, <) (N, <) and let π denote the outcome of Algorithm 1. Output: π, a partition of N [IS] Toward a contradiction, assume ﬁrst that agent i ∈ πk k0 k0 k 1: R1 := N, π := ∅ individually deviates toward π , i.e., π ∪ {i} i π and k0 k0 k0 2: for k = 1 to |N| do for all j ∈ π , π ∪ {i}

71 Proceedings of the Twenty-Eighth International Joint Conference on Artiﬁcial Intelligence (IJCAI-19)

Finally, when a top responsive hedonic game satisfies mu- Proof. In Example 5, partitions {{1, 2, 3}} and {{1, 3}, {2}} tuality, Aziz and Brandl [2012] show that the outcome of the are not individually rational for agents 1 and 3. Also, partition Top Covering Algorithm is strict strong Nash stable. We fur- {{1}, {2}, {3}} is not core or individually stable, since {1, 2} ther show that its outcome is envy-free. is a deviation. For partitions {{1, 2}, {3}} and {{1}, {2, 3}}, Theorem 3. Top responsiveness and mutuality guarantee the agent 3 has weakly justified envy toward agent 1 in the for- existence of strict strong Nash stable and envy-free partitions. mer, and 1 has weakly justified envy toward 3 in the latter. Preferences are strong bottom responsive and mutual. Proof. Consider a top responsive and mutual hedonic game (N, <) and let π be the outcome of the Top Cover Algorithm. Proposition 4 directly holds for the conjunction of {WJEF, By Theorem 3 and Lemma 1 in Aziz and Brandl [2012] re- EF} and {NS, SIS, SNS}. An interesting question is then the spectively, π is strict strong Nash stable, and for all i ∈ N, complexity of deciding the existence of such partitions. ch(i, N) ⊆ π(i). Toward a contradiction, assume that agent i Theorem 4. Given a strong bottom responsive hedonic game, has envy toward agent j in π, i.e., (π(j) \{j}) ∪ {i} i π(i). deciding whether an individually stable and envy-free parti- First assume π(i) = {i}. It implies (π(j) \{j}) 6= ∅, and tion exists is NP-complete, even with mutual preferences. then {i} ( (π(j) \{j}) ∪ {i}. Since ch(i, N) ⊆ π(i), it holds that ch(i, N) = {i}. Hence, by definition of ch(i, N), Proof sketch. We show NP-hardness by reduction from X3C. {i} = ch(i, {i}) = ch(i, (π(j) \{j}) ∪ {i}). With {i} ( Consider (Y,S) an instance of X3C with Y = {1,..., 3n} (π(j) \{j}) ∪ {i}, top responsiveness implies {i} = π(i) i and S = {s1, . . . , sm}. We construct a symmetric and strong (π(j) \{j}) ∪ {i}, which is contradiction. bottom responsive ASHG, as follows: i i Thus π(i) 6= {i}. Since π is individually rational, it • For each i ∈ Y , we create two agents {y1, y2} and a i 0 i 0 implies ch(i, N) 6= {i}, and thus ch(i, N) * (π(j) \ “clique” of seven agents K7 where for all l, l ∈ K7, vl(l ) = i i i {j}) ∪ {i}. By definition of ch(i, N), for all S ∈ Ni 1. We set for all l ∈ K7, vl(y1) = vl(y2) = 1. and S 6= ch(i, N), ch(i, N) i S. Hence, for all S ⊆ • For each set s = {i, j, k} ∈ S, we create a “clique” of s 0 s 0 (π(j) \{j}) ∪ {i}, ch(i, N) i S, and thus ch(i, N) i five agents K5 where for all l, l ∈ K5 , vl(l ) = 1. We set s i j k ch(i, (π(j) \{j}) ∪ {i}). Moreover, since ch(i, N) ⊆ π(i), for all l ∈ K5 , vl(y1) = vl(y1) = vl(y1 ) = 1, and also j we obtain ch(i, π(i)) = ch(i, N), and thus ch(i, π(i)) i k i vyi (y1) = vyj (y1 ) = vyk (y1) = 1. ch(i, (π(j) \{j}) ∪ {i}). Hence, top responsiveness implies 1 1 1 • All other values are equal to −|N| = −(3n · 9 + m · 5). π(i) (π(j) \{j}) ∪ {i}, which is a contradiction. i The constructed ASHG is both strong bottom responsive 5 Bottom Responsiveness and Envy-freeness and mutual (indeed, it is a symmetric enemy aversion hedonic game [Dimitrov et al., 2006]). Notice that, in individually sta- In opposition to top responsiveness, bottom responsiveness i s ble partitions, agents in each “clique” (K7)i∈Y and (K5 )s∈S [Suzuki and Sung, 2010] is based on the notion of avoid set. are in the same coalition. The main argument is then similar Given agent i ∈ N and coalition X ∈ Ni, the avoid set of i in to the proof of Theorem 1 and we omit it. 0 0 X is Av(i, X) = {Y ∈ Xi : ∀ Y ∈ Xi,Y

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