Computing stable outcomes in hedonic games with voting-based deviations Martin Gairing Rahul Savani Department of Computer Science Department of Computer Science University of Liverpool University of Liverpool [email protected] [email protected] ABSTRACT 1. INTRODUCTION We study the computational complexity of finding stable Hedonic games were introduced in the economics litera- outcomes in hedonic games, which are a class of coalition ture as a model of coalition formation where each player formation games. We restrict our attention to a nontrivial cares only about those within the same coalition [12]. Such subclass of such games, which are guaranteed to possess sta- games can be used to model a variety of settings ranging ble outcomes, i.e., the set of symmetric additively-separable from multi-agent coordination to group formation in social hedonic games. These games are specified by an undirected networks. This paper studies the computational complex- edge-weighted graph: nodes are players, an outcome of the ity of finding stable outcomes in hedonic games. We con- game is a partition of the nodes into coalitions, and the sider and extend the stability requirements introduced in utility of a node is the sum of incident edge weights in the the work of Bogomolnaia and Jackson [6], which includes same coalition. We consider several stability requirements a detailed discussion of real-life situations in which purely defined in the literature. These are based on restricting hedonic models are reasonable. feasible player deviations, for example, by giving existing An outcome is called Nash-stable if no player prefers to be coalition members veto power. We extend these restrictions in a different coalition. Here a deviation depends only on by considering more general forms of preference aggregation the preferences of the deviating player. Less stringent stabil- for coalition members. In particular, we consider voting ity requirements restrict feasible deviations: a coalition may schemes to decide if coalition members will allow a player try to hold on to an attractive player or block the entry of to enter or leave their coalition. For all of the stability re- an unattractive player. In [6], deviations are restricted by quirements we consider, the existence of a stable outcome is allowing members of a coalition to “veto” the entry or exit guaranteed by a potential function argument, and local im- of a player. They introduce individual stability, where there provements will converge to a stable outcome. We provide is a veto for entering - a player can deviate to another coali- an almost complete characterization of these games in terms tion only if everyone in this coalition is happy to have her. of the tractability of computing such stable outcomes. Our They also introduce contractual individual stability, where, findings comprise positive results in the form of polynomial- in addition to a veto for entering, coalition members have a time algorithms, and negative (PLS-completeness) results. veto to prevent a player from leaving the coalition - a player The negative results extend to more general hedonic games. can deviate only if everyone in her coalition is happy for her to leave. The case where every member of a coalition has a veto on Keywords allowing players to enter and/or leave the coalition can be seen as an extreme form of voting. This motivates the study Hedonic games, coalition formation, voting, local search, of more general voting mechanisms for allowing players to PLS-completeness. enter and leave coalitions. In this paper, we consider general voting schemes, for example, where a player is allowed to join a coalition if the majority of existing members would like the Categories and Subject Descriptors player to join. We also consider other methods of preference I.2.11 [Distributed Artificial Intelligence]: Multiagent aggregation for coalition members. For example, a player is Systems; F.2.0 [Analysis of Algorithms and Problem allowed to join a coalition only if the aggregate utility (i.e., Complexity]: General the sum of utilities) existing members have for the entrant is non-negative. These preference aggregation methods are also considered in the context of preventing a player from General Terms leaving a coalition. We study the computational complexity Algorithms, Economics, Theory of finding stable outcomes under stability requirements with various restrictions on deviations. Cite as: Computing stable outcomes in hedonic games with voting- based deviations, M. Gairing and R. Savani, Proc. of 10th Int. Conf. on Autonomous Agents and Multiagent Systems (AAMAS The model. 2011), Tumer, Yolum, Sonenberg and Stone (eds.), May, 2–6, 2011, In this paper, we study hedonic games with symmetric Taipei, Taiwan, pp. 559-566. additively-separable utilities, which allow a succinct repre- Copyright c 2011, International Foundation for Autonomous Agents and sentation of the game as an undirected edge-weighted graph Multiagent Systems (www.ifaamas.org). All rights reserved. 559 G = (V, E, w). For clarity of our voting definitions, we any Tin 2/3. Since there are no deviations that are both ≤ assume w.l.o.g. that we = 0 for all e E. Every node veto-in and veto-out feasible, this is a contractual individ- i V represents a player.6 An outcome∈ is a partition p of ual stable outcome. The outcome a, b, d , c , e, f is an V ∈into coalitions. Denote by p(i) the coalition to which individual stable outcome, and {{a, b, d, e,} f{ }, {c is}} Nash i V belongs under p, and by E(p(i)) the set of edges stable. {{ } { }} ∈i, j E j p(i) . {{The}utility ∈ | of∈i V}under p is the sum of edges to others in ∈ Justification of the model. the same coalition, i.e., e E(p(i)) w(e). Each player wants With the goal of understanding how difficult it is for agents to maximize her utility, so a∈ player wants to deviate if there to find stable outcomes, we focus on a model in which they P exists a (possibly empty) coalition c where are guaranteed to exist. The computational complexity of a problem is measured in terms of the size of its input and w(e) < w( i, j ). { } therefore depends on the representation of the problem in- e E(p(i)) i,j E j c ∈ X {{ }∈X| ∈ } stance. For games, we desire that the size of the input is We consider different restrictions on player deviations. Those polynomial in the number of players, as this is the natu- restrict when players are allowed to join and/or leave coali- ral parameter with which to measure the size of the game. tions. A deviation of player i to coalition c is called We consider only such succinct representations, since other- wise we can find solutions using trivial algorithms (enumer- Nash feasible if player i wants to deviate to c. ation of strategy profiles) that are polynomial in the input • size. Our focus on additively-separable games is motivated vote-in feasible with threshold Tin if it is Nash feasible • by the hardness of even deciding the existence of stable out- and either at least a Tin fraction of i’s edges to c are comes and other solution concepts for more general (uni- positive or i has no edge to c. versal) succinct representations, such as hedonic nets [14]. A non-symmetric additively-separable game, which is rep- vote-out feasible with threshold Tout if it is Nash-feasible • resented by a edge-weighted directed graph, may not have and either at least a Tout fraction of i’s edges to p(i) are negative or i has no edges within p(i). a Nash-stable outcome [6,4], and deciding existence is NP- complete. We study a more restrictive model where stable sum-in feasible if it is Nash feasible and outcomes (for all of the stability requirements we consider) • are guaranteed to exist, noting that our hardness results ex- w( i, j ) 0. tend to all more general models where existence of stable { } ≥ i,j E j c {{ }∈X| ∈ } outcomes is either guaranteed or promised, i.e., instances are restricted to those possessing stable outcomes. sum-out feasible if it is Nash feasible and • In a symmetric additively-separable hedonic game, for each of the stability requirements we consider, a stable out- w(e) 0. ≤ come always exists by a simple potential function argument: e E(p(i)) ∈ X the potential function is the total happiness of an outcome, Outcomes where no corresponding feasible deviation is pos- i.e., the sum of players’ utilities. Unilateral player deviations sible are called Nash stable, vote-in stable, vote-out stable, improve the potential. So for all our considered stability re- sum-in stable, and sum-out stable, respectively. Outcomes quirements, local improvements will find a stable outcome, and all the problems we consider are in the complexity class which are vote-in (resp. vote-out) stable with Tin = 1 (resp. PLS (polynomial local search) [20], which we introduce next. Tout = 1) are also called veto-in (resp. veto-out) stable. Note that a veto-in stable outcome is an individual stable outcome i.e., any player can veto a player joining a coali- Local search and the complexity class PLS. tion; an outcome that is veto-in and veto-out stable is a Local search is one of few general and successful approaches contractual individual stable outcome. to difficult combinatorial optimisation problems. A local search algorithm tries to find an improved solution in the An example. neighborhood of the current solution. A solution is locally optimal if there is no better solution in its neighborhood. 5 −1 Johnson et al.