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Testing Stability Properties in Graphical Hedonic Games

In hedonic games, players form coalitions based on individual preferences over the group of players they belong to. Several concepts to describe the stability of coalition structures in a game have been proposed and analyzed. However, prior research focuses on algorithms with time complexity that is at least linear in the input size. In the light of very large games that arise from, e.g., social networks and advertising, we initiate the study of sublinear time property testing algorithms for existence and verification problems under several notions of coalition stability in a model of hedonic games represented by graphs with bounded degree. In graph property testing, one shall decide whether a given input has a property (e.g., a game admits a stable coalition structure) or is far from it, i.e., one has to modify at least an ϵ-fraction of the input (e.g., the game's preferences) to make it have the property. In particular, we consider verification of perfection, individual rationality, Nash stability, (contractual) individual stability, and core stability. Furthermore, we show that while there is always a Nash-stable coalition (which also implies individually stable coalitions), the existence of a perfect coalition can be tested. All our testers have one-sided error and time complexity that is independent of the input size.

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1 Introduction

Hedonic games are a form of coalition formation games, in which players form teams in a decentralized manner based on individual preferences over coalitions, i.e., subsets of players. The solution to such a game is a coalition structure, i.e., a partition of the set of players. The main idea of hedonic games is that the players’ evaluation of a coalition structure only depends on their own coalitions and not on how other players work together Drèze and Greenberg (1980). These games have been formalized by Banerjee et al. (2001) and Bogomolnaia and Jackson (2002), independently. In order to evaluate the quality of a coalition structure, several solution concepts have been considered. These include, e.g., Nash stability, which states that no individual player wants to deviate from the coalition structure, and core stability, which requires that no group of players wants to peel off and form a coalition of their own.

Algorithmically, a key issue is to find suitable representations of hedonic games: Since the number of possible coalitions for a player is exponential in the number of players, there is a trade-off between compactness and expressivity of the preference profile. In the areas of Cooperative Games in Multiagent Systems (see, e.g., (Chalkiadakis et al., 2011)) and Computational Social Choice (see, e.g., (Aziz and Savani, 2016)), a number of representations and stability concepts are analysed with respect to the computational complexity of deciding whether there exists a stable solution, verifying whether a given solution is stable, and finding a stable solution. Even for restricted representations such as additively separable games, these questions are often intractable. For instance, it is often -complete to decide whether a given game allows a Nash-stable coalition structure, see e.g., Peters (2016). The existence of core-stability is often even -complete to decide, see e.g., Woeginger (2013) and Ota et al. (2017). This strikes even harder when the considered game instances are very large because they arise from, e.g., social networks or the assignment of advertisements to available slots on web pages such that adjacent ads do not interfere. Here, it might already be impractical to read the whole input once because the data does not fit into memory or the access is slow or restricted.

In this paper, we study sublinear algorithms for hedonic games. We aim to decide in sublinear time whether a game has a stable coalition structure or is far from this with respect to the number of required changes of preferences such that it admits one, as well as whether a game is stable under a given coalition structure or is far from being stable under . When a coalition can be stabilized by only few compromises on the preferences, it may be acceptable to sustain the situation or (if possible) make the changes. When, however, too many modifications are required to obtain any stable situation, the current situation is too far off the goal.

Graph representations provide a compact means to encode structural connections between players. A formal study of graphical hedonic games is provided by Peters (2016). A popular variant is to encode a game as a network where players correspond to vertices and edges illustrate friendship relations. Players that are no friends are often referred to as enemies. Preferences are extended either by prioritising appreciation of friends or aversion to enemies Dimitrov et al. (2006). However, if the game is very large, many players may not be involved in any relationship. In this scenario, it is natural to consider a more general model. For each player, the set of other players is divided into three subsets: friends, enemies and neutral players Ota et al. (2017), which is what we call the FEN-encoding. Furthermore, we bound the number of friends and enemies per player by a constant (for example, if the players are humans, this phenomenon is known as Dunbar’s number Dunbar (1992), which describes the maximum number of stable relationships a single person can maintain). Under restrictions such as bounded degree and bounded treewidth, some stability questions become solvable in linear time Peters (2016). Nevertheless, this still incurs the evaluation of the whole game in order to verify whether a coalition structure is stable. Given the local views of individual vertices within hedonic games, it would be preferable and much more practical to ask only a sample of players for their individual preferences and deduce global properties.

The area of property testing provides a framework to relax such decision problems in favour of sublinear complexity (see, e.g., (Goldreich, 2017)

for an overview). A property tester is a randomized algorithm that decides, with error probability at most

, whether the input satisfies some property  or is far from satisfying by probing only a small part of it. In the setting of graph properties, a graph  with bounded vertex degree is -far from satisfying some property  (e.g., bipartiteness) if one has to modify at least edges to make have property . If the property tester always accepts graphs in , it has one-sided error; otherwise, it has two-sided error. The input graph  may be probed by the algorithm through an oracle that provides access to the entries of the adjacency lists of , and the computational complexity of the property tester is measured in terms of queries it asks.

In comparison to classic decision problems, property testing problems allow for algorithms with sublinear complexity. For example, a randomized decision algorithm for graph connectivity needs to read the whole input to achieve constant error probability, which implies a linear lower bound on the complexity. In contrast, a property tester for connectivity has only constant complexity Goldreich and Ron (2002). This difference arises because the property tester does not need to read the whole input, and, in fact, sublinear complexity renders this impossible. Therefore, the input model plays an important role in property testing. While there is a characterization for constant query testable properties in dense graphs (graphs with edges) Alon et al. (2009), less is known for graphs with bounded degree and general graphs.

1.1 Our Contribution

We study property testing of stability problems in FEN-hedonic games, where each player has a bounded number of symmetric relationships to friends and enemies as represented by labelled edges of an undirected graph, and preferences are extended to coalitions by any utility function linear in the number of friends and enemies in a coalition. The setting of hedonic games enhances graphs by rich semantics, which stands in contrast to purely combinatorial and geometric properties previously studied in graph property testing. We model the semantics of hedonic games as an additional layer on top of the combinatorial graph structure and analyze existence and verification problems for various stability concepts. In particular, we study common individual-based stability concepts such as perfection, individual rationality, Nash stability, and (contractual) individual stability as well as core stability.

While individually rational, Nash-stable, individually stable, and contractually individually stable coalitions always exist, there are games which do not allow a perfect coalition structure.

Theorem 1.

Given a FEN-hedonic game  with bounded degree , it can be tested whether admits a perfect coalition structure with bounded coalition size  with one-sided error and query complexity .

While the existence problem as to whether a game allows a stable outcome is a property of edge-labelled graphs, the verification problem of whether a game satisfies stability according to a fixed coalition structure requires additional modelling: We assume that next to oracle access to the adjacency lists of the underlying bounded-degree graph of a game , we have additional access to an oracle to , i.e., a partition of the vertex set.

We show the testability of verification problems independent of any bound on the coalition size.

Theorem 2.

Given a FEN-hedonic game  with bounded degree and a coalition structure , it can be tested whether is stable under  with respect to perfection, individual rationality, Nash stability, individual stability and contractual individual stability with one-sided error and query complexity . For core stability we obtain a tester with query complexity , where is the maximum coalition size.

Note that while we consider and to be of constant size, independent of the input size , our statements remain valid if, for instance, .

1.2 Related Work

Hedonic games were formally defined by Banerjee et al. (2001) and Bogomolnaia and Jackson (2002). A well-known application of a restricted variant with size-two coalitions is the stable-roommates problem Cechlárová (2002) for the allocation of student houses. Mostly, hedonic games have been analysed from a computational complexity point of view with respect to a trade-off between expressivity, succinct representation and tractability of stability decision problems. The complexity of general hedonic games has first been studied by Ballester (2004). The worst-case complexity of stability problems for various representations and different stability concepts has been studied extensively: Popular representations include additively separable hedonic games Bogomolnaia and Jackson (2002); Aziz et al. (2013); Woeginger (2013), singleton encodings Cechlárová and Romero-Medina (2001), hedonic coalition nets Elkind and Wooldridge (2009), and dichotomous preferences Aziz et al. (2016); see also Aziz and Savani (2016) and Chalkiadakis et al. (2011) for an overview. Peters and Elkind (2015) analyse causes of and conditions for hardness. The existence of Nash stability and other individual stability concepts is often (if not guaranteed) -complete to decide. For core stability, this is often even harder, namely -complete Woeginger (2013); Ota et al. (2017). Dimitrov et al. (2006) define restricted hedonic games based on a network of friends and enemies. A more general version including neutral players is defined by Ota et al. (2017). Games with neutral players and partial individual evaluations are studied by Lang et al. (2015) and Peters (2016). Peters, in particular, considers a constant bound on the number individual preferences and studies graphical hedonic games with bounded treewidth. With this restriction, it can be decided in linear time whether, for instance, a Nash stability coalition structure exists. A graphical model restricting the formation of feasible coalitions is studied by Igarashi and Elkind (2016).

Goldreich and Ron (2002) showed that classic graph problems like connectivity, being Eulerian and cycle freeness are testable with constant query complexity. On the other hand, it is known that testing bipartiteness Goldreich and Ron (1999) and expansion Czumaj and Sohler (2010); Kale and Seshadhri (2011); Nachmias and Shapira (2010) have upper and lower bounds of roughly . Turning to more general results, Benjamini et al. (2010) proved that every minor-closed property is constant query testable, and Newman and Sohler (2013) extended this result to hold for every hyperfinite property. Property testing of annotated (or labelled) graphs has been studied for geometric graphs mainly. For example, testing whether a graph that is embedded into the plane is a Euclidean minimum spanning tree has been studied by Ben-Zwi et al. (2007) and Czumaj and Sohler (2008). Ben-Zwi et al. show that any non-adaptive tester has to make queries, and that any adaptive tester has query complexity . Czumaj and Sohler provide a one-sided error tester with query complexity . Hellweg et al. (2010) develop a tester for Euclidean -spanners.

Learning hedonic games was studied by Sliwinski and Zick (2017). While property testing focuses on testing whether a game admits a stable outcome or whether an outcome is stable with sublinear complexity, PAC learning constructs a good hypothesis and PAC stabilization uses this hypothesis to learn a stable outcome (if possible) using a superlinear number of samples (with possibly linear size). As far as we know, no sublinear algorithms have been developed for hedonic games, yet.

2 Preliminaries

In this paper, we consider undirected graphs with vertex degrees bounded by a constant . For a graph at hand, we write . Without loss of generality, we assume that .

2.1 Hedonic Games

A hedonic game  consists of a set of players  and a preference profile , where is player ’s preference relation over . A subset of players is called coalition. An output of a hedonic game is a coalition structure, i.e., a partition  of the player set. Let be the coalition containing . We say that a player  weakly prefers a coalition  to a coalition , if . Player  prefers to , denoted by , if , but ; is indifferent between and , denoted by , if and .

Since the set of coalitions a player is contained in, has an exponential size in the number of players, a central question in the study of hedonic games is to define representations that are adequately compact and at the same time as expressive as possible.

One common representation is that of a graph network , where the players are vertices in the graph. In the encoding as defined by Ota et al. (2017), for each player , there exists a set of friends, set of enemies, . The remaining players are considered as neutral . We call this representation FEN-encoding. It can be represented by a labelled graph  with , where if and only if , and if and only if . This conforms to the definition of a graphical hedonic game Peters (2016) such that a player ’s preference of a coalition over a coalition only depends on ’s neighbourhood :

(1)

Here, we extend the players’ relations to preferences in the following manner. A value function is specified such that each player  assigns a positive value  to each and a negative value  to each . The corresponding utility function , , is defined additively by . For instance, under friends appreciation we have and , and under enemies aversion this corresponds to and . The preference extension is obtained by .

Definition 3.

We call a hedonic game represented by an FEN-encoding with a preference profile extended via a utility function linear in the number of friends and enemies, FEN-hedonic game.

Note that responsiveness is always satisfied by the considered preference extensions, i.e., and , for each , and each and , . Since we consider undirected graphs, we obtain symmetric preferences, i.e., if and only if and if and only if .

Furthermore, we make the following assumptions. We consider graphs of bounded degree represented by an adjacency list; in particular, it can be decided in time independent of the number  of players whether , and independent of the coalition size and . Moreover, it is often useful to restrict the coalition size, e.g., when players are people that have to communicate or when a coalition represents all ads displayed on a single web page. Therefore, we also consider a bounded coalition size of .

By we denote the set of graphs with vertices that represent such a game. The set of coalition structures partitioning players is denoted by . For a stability concept, questions of interest are:

  • Verification: Given a game and a coalition structure, is it stable?

  • Existence: Is a given game stable, i.e., does there exist a stable coalition structure?

  • Search: Find a stable coalition structure for a given game.

In the following, let be a graph that represents a FEN-hedonic game and let be a coalition structure solving this game. There are several solutions concepts motivated from different perspectives on the game.

On the one hand, is called

perfect

if each player  weakly prefers to every coalition, i.e., for each , .

This property reflects an ideal situation, but is rather rarely fulfilled. On the other hand, is called

individually rational

if for each player , is acceptable, i.e., .

Individual rationality is guaranteed by .

Other stability notions are based, for example, on the lack of deviations of a single player to another (possibly empty) existing coalition. Let denote the set of ’s favourite coalitions of size at most , i.e., those coalitions that weakly prefers over all other coalitions of size at most . A coalition structure is called

Nash-stable

if no player wants to move to another existing or empty coalition, i.e., for each player  and each coalition  with , it holds that ;

individually stable

if no player can move to another preferred coalition without making a player in the new coalition worse off, i.e., for each player  and for each coalition  with , it holds that or there exists a player such that ;

contractually individually stable

if no player can move to another preferred coalition without making a player in the new coalition or in the old coalition worse off, i.e., for each player  and for each coalition  with , it holds that , or there exists a player such that , or there exists a player such that .

Note that Nash stability implies individual stability, which, in turn, implies contractual individual stability. A further popular stability concept is based on group deviation. A coalition structure is called

core stable

if no coalition blocks , i.e., for each non-empty coalition , there exists a player  such that .

2.2 Graph Property Testing

Let be a graph with vertex degrees bounded by  and let be a graph property, i.e., a set of graphs (e.g., let be all graphs that admit a perfect coalition structure). We say that is -far from a property  if more than edges of have to be modified in order to convert it into a graph that satisfies the property , otherwise is -close to . A property tester has access to by querying a function , where denotes the neighbour of if has at least  neighbours. Otherwise, .

Definition 4 (One-sided testers).

A one-sided error -tester for a property of bounded degree graphs with query complexity is a randomized algorithm that makes queries to for a graph . The algorithm accepts if has the property . If is -far from , then rejects with probability at least .

A graph property  is edge monotone if for every , . In other words, every subgraph of is also in .

3 Property Testing of Stability Concepts

To test stability concepts, we generalize the standard edit distance of graph property testing as follows. Since we consider graphs that represent FEN-hedonic games, we have to account for the two types of edges: friends and enemies. Therefore, an edge modification is one insertion of an element to or one removal of an element from , respectively, while maintaining . In particular, turning a friend edge into an enemy edge is counted as two edge modifications (removing it from and inserting it into ). The intuition of these semantics is that edge modifications measure the number of compromises that are needed to reach a stable situation. If a partition is too far from being stable, too many compromises are necessary, and the partition should be discarded. Everything in-between is not an ideal situation, but only a few compromises may be affordable.

Now, the existence of a stable outcome in a game is modelled as a graph property as follows.

Definition 5 (stability existence property).

The set of stable graphs with respect to some stability concept (e.g., Nash stability) is the set of all graphs that admit a stable coalition structure.

For some stability concepts, the existence of a stable outcome is guaranteed. Nevertheless, the question of whether a given partition  satisfies the stability property can still be hard to decide. The worst case time that is needed to verify stability of for all stability concepts mentioned above is at least linear in the number of players. We can, however, tackle the following problem in sublinear time: Given a graph  and a partition of vertices , is a stable outcome for the game represented by , or is -far from being a stable instance for ?

Definition 6 (-stability verification property).

Let , and let  be a partition of . The set of -stable graphs with respect to some stability concept (e.g., Nash stability) is the set of -vertex graphs  such that  is a stable coalition structure of .

Note that, unlike the existence of a stable coalition structure, a stability property is not closed under isomorphism as long as is not permuted additionally. Therefore, extending the basic model of graph property testing to reflect the semantics of hedonic games is the foundation of our main contribution. Access to is provided by a set oracle that supports two queries. A find query returns, given a vertex , the key of the set that contains . A member query returns, given a key  and an index , the -th element of the set represented by , or if no such element exists.

One beneficial feature of bounded degree graphs is the bounded size of neighbourhoods, i.e., the number of graphs that have constant distance to a given vertex. In hedonic games, this is mirrored by the maximum coalition size, and therefore, we take the coalition size as a parameter into our analysis.

4 Property Testing in the FEN-Model

In this section we study property testers for stability verification problems, resulting in Theorem 2, as well as stability existence problems, resulting in Theorem 1, for various individual-based stability notions within the previously defined model of FEN-hedonic games.

4.1 Testing Verification Problems

In the following we aim to prove Theorem 2, the testability of verification problems with query complexity dependent only on the degree bound, but independent of the graph size or any coalition bound, which is restated in Theorem 12 below. In our constructions we relate to the players’ favourite coalitions and make use of the following lemma that states that we can easily modify a player’s local surroundings to turn the current coalition into a favourite coalition. In other words, only a constant number of compromises suffice to optimise one player’s current situation.

Lemma 7.

For every graph  with bounded by , each coalition structure  of , it holds that for each , queries and edge modifications are sufficient to turn into one of ’s favourite coalitions in the FEN-hedonic game represented by .

Proof.

If for player , it already holds that , no modification is required. Otherwise we can proceed as follows: Accessing the (at most ) members of requires at most oracle queries. Moreover, we can ask one oracle query each in order to find out, whether a player  is contained in . For each , delete the edge from ; for each : remove the edge from . This requires at most edge modifications. Note that this is independent of any bound  of the coalition size. The obtained coalition now only contains friends of ’s and does not have any friends outside of . Hence, for all considered preference extensions, no coalition is preferred to the current coalition . ∎

Many stability concepts are of the form such that stability holds if and only if no player  satisfies a certain condition . If there exists a player  that satisfies this condition , we call a witness for non-stability.

Next, we observe that due to responsiveness and symmetry in all considered preference extensions, an edge modification that benefits one player, can never be a disadvantage for other players.

Observation 8.

Let be two players in the FEN-hedonic game represented by a graph . Furthermore, let be ’s original preference relation, and the preference relation of after a modification of edge , i.e., in the new game represented by with the same preference extension. The following statements hold:

  1. If is deleted from , for each , , it holds that and ,

  2. If is deleted from , for each , , it holds that and ,

  3. If is deleted from , and for two coalitions with , it holds that .

  4. If is deleted from , and for two coalitions with , it holds that .

The considered stability concepts are defined via properties that need to be avoided for each player. Let denote such a player property assigning each player  either value  ( is a witness against the property) or value  ( is not a witness). In the following proofs we require certain conditions to hold for and show that all considered concepts share these conditions, which enables us to devise a unified testing scheme.

Definition 9.

Let be a stability concept for which there exists a property , , such that for any game represented by , and a coalition structure  it holds that for each , if and only if is stable in with respect to . We say that is feasible in our setting if for each and each and the following conditions are met for each .

  1. [label=()]

  2. .

  3. The value can be determined with a constant number of queries to the oracles for and (i.e., dependent on , but independent of ).

  4. If and an edge , is removed from , resulting in a new game , it holds that remains valid.

  5. If and an edge , is removed from , resulting in a new game , it holds that remains valid.

We note that if is feasible, then is edge monotone.

Lemma 10.

Let be a graph that represents a FEN-hedonic game and a coalition structure of . Let, furthermore, be a stability concept, for which there exists a feasible player property . If there are at most witnesses, edge modifications are sufficient to make the game stable with respect to .

Proof.

By Lemma 7, for each witness , edge modifications are enough to turn into a favourite coalition, thus, is no longer satisfied. For each player  that is not a witness, already holds which does not change due to Conditions 3 and 4 of Definition 9. If there are at most witnesses, edge modifications are sufficient such that no player satisfies , thus, stability with respect to holds. ∎

With the help of this lemma we are now ready to prove that Algorithm 1 provides a property tester for the verification problem of each stability concept with a feasible player property.

1:access to and is provided by an oracle, is the corresponding boolean stability function
2:function VerificationTester(, , , )
3:     
4:     sample players iid from
5:     for each sampled player  do
6:         if  then
7:              return reject               
8:     return accept
Algorithm 1
Theorem 11.

Let be a stability concept for which there exists a feasible player property . It holds that Algorithm 1 is a one-sided error property tester for -stability verification with respect to .

Proof.

If holds, there is no witness for non-stability, i.e., for each sampled vertex holds. Therefore, the tester decides in Line 6 that holds with probability .

If is -far from being stable with respect to , at least edge modifications are required. Thus, by Lemma 10, there are at least witnesses. Hence, the probability that a sampled player is a witness is at least .

Then, the algorithm correctly rejects if at least one witness is sampled, i.e., the condition in Line 5 is true. The probability of this event is minus the probability that for each sampled player the condition in Line 5 is false, i.e., . The latter probability is at most

Thus, the probability that the tester correctly rejects is at least .

Since is feasible, can be determined in constant query time. Hence, the tester requires constant query time dependent on the applied function . ∎

Now it remains to show that each considered stability property has such a feasible player property . By Theorem 11, we obtain a verification tester by Algorithm 1 with a query complexity depending on .

Theorem 12.

For the FEN-hedonic game model, the -stability verification property can be tested with respect to

  1. perfection and individual rationality with query complexity in ,

  2. Nash stability, individual and contractual individual stability with query complexity in .

  3. core stability with query complexity , where is the maximum coalition size.

Proof.

For each considered stability concept we show that there exists a feasible player property such that Theorem 11 can be applied. In each case we determine the query complexity of the tester.

perfect:

The corresponding player property is

This is the case if and only if does not contain all of ’s friends and none of ’s enemies, which can be verified in constant query time by asking whether is in the same coalition as for each . Therefore, condition 2 is met with queries per sampled player. In total, the query complexity is in . Condition 1 holds by definition of perfection. Conditions 3 and 4 can be implied immediately by Observation 8, since the relation remains valid in each relevant case.

individually rational:

Here, the player property is

Condition 1 holds, since implies that . Since the decision whether only depends on , queries are sufficient, i.e., a total query complexity in . Again, Conditions 3 and 4 can be implied by Observation 8.

Nash-stable:

A witness  against Nash stability satisfies

Condition 1 holds, since implies in particular for such that , .

(a)

If , wants to deviate to a coalition  with . Due to the linearity of the preferences, this can only be (with ) or a coalition in that contains at least one friend. There are at most coalitions in that contain a friend, namely , . Hence, at most comparisons of coalitions are sufficient, which can be done with at most  neighbour and  find queries by Equation (1).

(b)

If , but , the analysis is analogous to (a).

(c)

If and , is already one of ’s favourite coalitions, hence .

(d)

If and , wants to deviate to the single player coalition , hence .

It can be decided with neighbour queries which of the four cases holds for . The at most  coalition comparisons require at most additional find queries. Therefore, can be decided with queries, satisfying Condition 2. The total query complexity of Algorithm 1 is .

Again, Conditions 3 and 4 can be implied by Observation 8.

individually stable:

A witness  against individual stability satisfies

Hence, if is not a witness for a Nash deviation, it cannot be a witness here, either. Therefore, Condition 1 holds. If wants to deviate, this is due to one of the cases (a), (b), or (d) above. In the latter case, again, is not acceptable, and is always welcome in . In cases (a) and (b), we have to consider at most candidate coalitions, can deviate to. For each neutral player , it holds that . Thus, we only have to ask ’s neighbours for permission to enter the new coalition, which are in total at most . In fact, due to the symmetry of preferences, friends always welcome , and enemies always don’t.

We obtain if and only if there are no enemies in , which we can decide with at most queries. Thus, we can employ the same queries as for Nash stability in order to determine , which satisfies Condition 2. The total query complexity of Algorithm 1 is .

Observe that if is not a witness due to some and a player  with , is ’s enemy due to Observation 8. That means, if an edge is deleted from , the witness condition is unchanged. Thus, Condition 3 is met. If an edge is deleted from and , Condition 4 can only be false if . In that case holds, a contradiction to . If is not a witness, because there does not exist any preferred coalition to move to, the arguments for Nash stability can be applied.

contractually individually stable:

A witness  against contractual individual stability satisfies which holds if and only if there exists some such that the two conditions for individual stability ( and ) and an additional condition () hold. Condition 1 holds analogously to individual stability.

Observe that for neutral players  it holds that and for enemies  it holds that . Again, if wants to deviate to a coalition, cases (a), (b) and (d) remain. In case (a) has friends in that contractually depends on. Here . In cases (b) and (d) there are no friends in , which means there is no contractual dependence. Then, is a witness against contractual individual stability if and only if it is a witness against individual stability. Thus, in case (d) and in case (b) that same queries as above can be applied. Thus, we need at most queries in order to determine . The total query complexity of Algorithm 1 is .

Conditions 3 and 4 hold with analogous arguments as above.

core-stable:

A player  is a witness against core-stability if takes part in a blocking coalition, i.e., such that each prefers to . Observe that if there exists a coalition blocking , its connected components also block . Hence, we can assume that is connected. It either holds that is not acceptable for , and or is a neighbouring coalition to .

Condition 1 holds, since cannot strictly prefer a blocking coalition to a favourite coalition.

Now, in order to detect a connected blocking coalition  that contains , we have to query the oracle as follows: We verify whether is acceptable for  by considering the intersection of ’s neighbours with . If it is not acceptable, holds, and thus, . If is acceptable, can only be any of ’s neighbouring coalitions, that are connected and of size . There are at most

possible such coalitions. If for each of these coalitions , we ask the at most contained players  whether they prefer to , this is sufficient to verify whether one of the coalitions blocks and, thus, whether is a witness. Therefore, Condition 2 holds.

Now, let , i.e., for each neighbouring coalition  to , there exists some such that . Condition 3 holds by the following argument. Assume, an edge is removed from , . Then, it holds that: If and , remains valid. If , implies that by Observation 8. If and , implies that by Observation 8. Similarly, Condition 4 is obtained by Observation 8. ∎

4.2 Testing Existence Problems

Now we prove Theorem 1. In general, there always exists an individually rational coalition structure. Bogomolnaia and Jackson (2002) show that in symmetric additively separable hedonic games there always exists a Nash stable coalition structure. Note that in our model the argument that if a player deviates, the social welfare increases, remains valid, even for bounded coalition size.

Observation 13.

Each symmetric FEN-hedonic game (all considered preference extensions) allows a Nash-stable, and consequently individually stable and contractually individually stable coalition structure.

There does not necessarily exist a perfect coalition structure. For example, there does not exist any perfect coalition structure for , where , .

Theorem 14.

There is a tester with constant query complexity for the existence of a perfect coalition structure in the FEN-hedonic game model with a constant coalition size bound.

Proof.

Let , and observe that is a favourite coalition of if and only if and . It follows that there exists a perfect coalition structure if and only if there does not exist any edge in between vertices of the same connected component of , where is the set of endpoints in , i.e., . This suggests the following algorithm: first, sample a set of vertices at random. For each , we run a BFS that follows only edges in . If one of these BFSs sees more than vertices or it discovers two endpoints of the same edge , the tester rejects. Otherwise it accepts the graph.

By the above observation, every path in that contains only edges from must be in the same coalition in a perfect coalition structure. The algorithm rejects only when it finds a path such that for every coalition structure such that some coalition contains , also contains , which is a witness against the existence of a perfect coalition structure.

If is -far from having a perfect coalition structure, then at least edges in have to be removed in order to make have a perfect coalition structure because having a perfect coalition structure is an edge-monotone property. Let be a minimal set of edges that have to be removed. Since

contains a vertex that is incident to an edge with probability at least . As argued above, then the tester finds and rejects. ∎

5 Open Questions

A natural question that is related to finding stable partitions is the following: Given a graph and a partition , is the partition  far from being stable in (instead of the graph being far from -stable)? This can be generalized further: Property testing is a special case of local computation algorithms (LCA), where one shall provide oracle access to a solution, given oracle access to the input. In property testing, the solution is a single bit (accept or reject). While it is beyond the scope of sublinear algorithms to actually compute a stable partition, one may seek to develop an LCA that gives oracle access to it.

Generalizing the results we obtained, one may seek to obtain sublinear algorithms for games with unbounded coalition size. Here, the main difficulty is to obtain insights into the local structure of very large, say, linear sized coalitions.

Following a slightly different line of thought, one may consider other graphs models like the dense model, where (almost) all players relate to each other and one may ask how two players relate, or the general graph model, where vertices have arbitrary degrees.

As mentioned in the introduction, there exist also plenty of other stability concepts like Pareto-optimality and popularity that can operate on the same preference extension, which may be interesting to analyse in order to obtain a deeper understanding of locality mechanics in FEN-hedonic games. Here, the main difficulty is to circumvent the usually high computational complexity of the exact decision problems.

Finally, one may study other models of hedonic games, in particular with ordinal preferences (e.g., rankings over known edges Lang et al. (2015)). This requires further modelling of the oracle access and considered distance measures.

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