Coalition formation is everywhere in human activities. Companies group their employers into project teams. Countries form coalitions to promote international trade among them. Individuals interact with each other and form groups in order to achieve objectives they cannot seek for on their own.
Hedonic coalition formation games (for short, hedonic games), introduced by Bogomolnaia and Jackson (2002) and Banerjee et al. (2001), provide an elegant framework to formulate coalition formation. In these games, each player has preferences over the coalitions to which she or he belongs, and desirable outcomes often correspond to stable partitions. The basic intuition behind stable partitioning is that group structures need to be robust under certain changes within the system; that is, outcomes must be immune to players’ coalitional or individual deviations to other coalitions. Stability can prevent internal conflicts among members of a coalition, or it can help us nurture the relationship among members of a project team.
In many real-world scenarios, however, groups may encounter unexpected changes and challenges, imposed from the outside of the system. For instance, a certain country can go bankrupt and be enforced to leave a political alliance. In this respect, a group structure that satisfies a standard stability requirement can become immediately unstable due to unexpected circumstances. A case in point is a political coalition of three countries with one intermediate country connecting two other countries who are enemies to each other: if the intermediate player happens to disappear from the coalition, one cannot maintain the stability of the whole system.
In this paper, we propose a novel criterion that redefines stability from robustness aspect. We define an outcome to be robust with respect to a certain stability requirement if removing any set of at most players still preserves . Besides the preceding example of a political alliance, there are several applications of hedonic games, such as project team formation (Okimoto et al., 2015), research team formation (Alcalde and Revilla, 2004), and group activity selection (Darmann et al., 2017), in which unexpected players’ non-participation may severely affect stability of the system. To the best of our knowledge, however, no attempt has been ever made to connect two important considerations, robustness and stability. Our goal is to make the first step filling this gap.
Our contribution We focus on friend-oriented games, introduced by Dimitrov et al. (2006), where players’ preferences are succinctly encoded via the binary friendship relations. While it is known that such games always guarantee the existence of stable outcomes, we observe that a simple example of one player connecting two enemies shows impossibility in maintaining most of the stability properties, such as core stability, Nash stability, individual stability, and contractual individual stability. Not surprisingly, this negative result holds even under a very small change of the system, i.e., only a single player can disappear.
Given these non-existence results, we investigate the computational complexity of deciding the existence of a robust outcome in a symmetric friend-oriented game. Specifically, we show that we can efficiently decide the existence of an outcome that is robust with respect to Nash stability, irrespective of the number of players leaving the game. We then prove that any symmetric friend-oriented game admits a polynomial time algorithm that finds a robust outcome with respect to contractual individual stability in case of removing a single player. To this end, we obtain a non-trivial characterization of games whose corresponding robust outcomes are non-empty. Moreover, we complement this result by showing that the problem becomes NP-hard when . We also show that the positive results do not extend to an intermediate stability property, individual stability: we prove that the associated problem for individual stability is NP-hard even when only a single player is allowed to leave.
Finally, we consider the question of whether a minimum stability requirement, individual rationality, can be maintained while ensuring that an outcome of a game itself satisfies stronger stability desiderata. It turns out that when players have symmetric additively separable preferences, an individually stable partition which is robust with respect to individual rationality always exist. Our complexity results are summarized in Table 1.
|NS-robustness||poly time (Cor. 4.3)|
|IS-robustness||NP-complete () (Th. 5.8)|
|CIS-robustness||poly time () (Th. 5.6)|
|NP-complete () (Th. 5.7)|
|IS & IR-robustness||exists and polytime (Th. 6.2)|
Related work Several papers considered robustness against agent failures in the context of cooperative games. Bachrach et al. (2011)
proposed the reliability extension of cooperative games where each agent has an independent failure probability. This probabilistic model has been also applied to subclasses of cooperative games, such as totally-balanced games(Bachrach et al., 2012), weighted voting games (Bachrach and Shah, 2013), and cooperative max-games (Bachrach et al., 2014). Okimoto et al. (2015) introduced the concept of -robustness for team formation problems; under their definition, each team still needs to accomplish their task even after agents fail.
Crudely, two different approaches deal with cooperative games with uncertainty. One is the Bayesian approach assuming a known prior over agent capabilities (Chalkiadakis and Boutilier, 2004; Chalkiadakis et al., 2007; Myerson, 2007). Another approach, initiated by Balcan et al. (2015), is to apply the PAC (probably approximately correct) learning model to cooperative games. Specifically, Balcan et al. (2015) studied the learnability of probable stable payoffs, given random samples of coalitions, which has been extended to hedonic games
Our work is also related to the rich body of the literature on the study of hedonic games. Bogomolnaia and Jackson (2002) and Banerjee et al. (2001) were the first to model hedonic coalition formation games in which players’ preferences solely depend on the members of each coalition. Bogomolnaia and Jackson (2002) considered various possibilities of players’ deviations, which gives rise to different concepts of stability outcomes. Several important subclasses of hedonic games have been later proposed, including additively separable hedonic games (ASHGs) (Bogomolnaia and Jackson, 2002), friend and enemy oriented games (Dimitrov et al., 2006), fractional hedonic games (Aziz et al., 2014; Aziz et al., 2017), to name a few.
Several works explored the relation between stability and the networks capturing agents’ preferences, in which the nodes of a graph represent players and edges correspond to the degree of preference. Bilò et al. (2014) analyzed the ratio between the social welfare of a Nash stable outcome and social optimum in fractional hedonic games, for different topologies such as bipartite graphs and trees. In a more general setting, Peters (2016) introduced graphical hedonic games and obtained a number of computational complexity results of stability outcomes for the case when the graph describing agents’ preferences has a bounded treewidth. Igarashi and Elkind (2016) used a different approach and considered hedonic games where players are located on a graph and coalitions are only allowed to form if they are connected in this graph; they proved both existence and complexity results of some stability concepts on acyclic graphs.
Our definition of robustness is arguably the most stringent requirement one could aim for, as it requires an outcome to be immune to any possibility of deterministic agent failure. However, the definition resembles some graph connectivity concepts, such as the -vertex-connectivity, capturing the robustness of a given network (see, e.g., Schrijver (2003)). We also note that the notion of robustness in a stable matching is fundamentally different from ours. For instance, Kojima (2011) considers instability that results from agents’ manipulation; a mechanism is considered to be robustly stable if it is strategy-proof and immune to a blocking pair before and after an agent misrepresents her preference. This does not take into account the possibility of making the system unstable due to agent failures.
For a natural number , we write . A hedonic game is defined as a pair where is a finite set of players and each is a preference over the subsets of (also referred to as coalitions); specifically for every , we let denote the collection of all coalitions containing ; each describes a complete and transitive preference over the sets in . Let denote the strict preference derived from , i.e., if , but . For and , we say that player strictly prefers a coalition to another coalition if ; player weakly prefers to if . We call a coalition individually rational if every player weakly prefers to .
A preference profile is said to be additively separable if there exists a weight function such that for each and each we have if and only if (Bogomolnaia and Jackson, 2002); we will assume that for each . An additively separable preference is said to be symmetric if the weight function is symmetric, i.e., for all . We use the notation to denote an additively separable game with weight function . For additively separable games, each player can consider every other player to be either a friend, a neutral player, or an enemy; specifically, for each pair of distinct players , we say that is a friend of if , and is an enemy of if .
Dimitrov et al. (2006) introduced a subclass of additively separable preferences, which they called friend-oriented preferences. Under friend-oriented preferences, each player has strong favour towards her friends: for each with . For a symmetric additively separable game , let denote the friendship graph where the set of vertices is given by the set of players and two players are adjacent if and only if they are friends; each coalition is said to have minimum degree if each player in has at least other friends in .
An outcome of a hedonic game is a partition of players into disjoint coalitions. Given a partition of and a player , let denote the unique coalition in that contains . Much of the existing literature is concerned with outcomes that satisfy certain stability requirements. A minimum stability property we require is individual rationality. A partition of is said to be individually rational (IR) if each player prefers their coalition to staying alone, i.e., all coalitions in are individually rational. If we extend this to a group deviation, we obtain the definition of the core. Specifically, a coalition strongly blocks a partition of if every player strictly prefers to her own coalition . A partition of is said to be core stable (CR) if no coalition strongly blocks . We also consider deviations based on individual movements. Specifically, consider a player and a pair of coalitions , . A player accepts a deviation of to if weakly prefers to ; a player accepts a deviation of to if weakly prefers to . A deviation of from to is an NS-deviation if strictly prefers to , an IS-deviation if it is an NS-deviation and all players in accept it, and a CIS-deviation if it is an IS-deviation and all players in accept it. A partition is called Nash stable (NS) (respectively, individually stable (IS) and contractually individually stable (CIS)) if no player has an NS-deviation (respectively, an IS-deviation and a CIS-deviation) from to another coalition or to .
Trivially, Nash stability implies individual stability, which also implies contractually individual stability. Usually, core stability does not imply the stability based on individual deviations. However, we note that for a symmetric friend-oriented game, core stability implies individual stability.
Lemma 2.1 ().
For a symmetric friend-oriented game , if a partition is core stable, then is individually stable.
Suppose towards a contradiction that is core stable but does not satisfy individual stability. Then, there is a player who has an IS-deviation to some coalition . This means that strictly prefers to , and and every player in are friends to each other. Thus, the players strongly block , a contradiction. ∎
We also note that contractual individual stability does not normally imply individual rationality as a player’s deviation under the stability concept needs to be approved by the members of her coalition. With symmetric friend-oriented preferences, nevertheless, the implication holds.
Lemma 2.2 ().
For a symmetric friend-oriented game , if a partition is contractually individually stable, then is individually rational.
Suppose towards a contradiction that is contractually individually stable but does not satisfy individual rationality. Then, there is a player who strictly prefers to his own coalition . This means that contains at least two players and every other player is an enemy of . Thus, has a CIS-deviation to the emptyset, a contradiction. ∎
3. Agent failure in hedonic games
Earlier we defined the robustness informally: A sudden deletion of players upon an outcome should preserve the property it has achieved before. We are now in a position to make the definition more formal. For each and , we denote by the preference relation restricted to .
Definition 3.1 ().
Given and a natural number , a partition is said to be -robust under deletion of at most players if satisfies the property , and for any with , the partition still satisfies the property in the subgame . When is clear from the context, we will simply call such partition -robust.
By definition, if an outcome is -robust under deletion of players, then it is -robust under deletion of any players. Also, fixing parameter , the relations between the above robustness concepts are the same as those among the corresponding stability concepts. Namely, we have the following containment relation among the classes of outcomes: , and . Also, by Lemma 2.2, CR-robustness implies IS-robustness and CIS-robustness implies IR-robustness for a symmetric friend-oriented game.
It is known that a stable outcome of a symmetric friend-oriented game is guaranteed to exist and can be found in polynomial time: a partition that divides the players into the connected components satisfies the preceding stability requirements. However, the example below illustrates that even when players have symmetric friend-oriented preferences, an -robust partition under deletion of a single player may not exist for any .
Example 3.2 ().
Consider a symmetric friend-oriented game with three players , , and . The friendship graph forms a star with the center being (Figure 1).
Suppose towards a contradiction that is CIS-robust under deletion of a single player.
First, suppose . Then, without , the coalition is not individually rational, a contradiction.
Second, suppose or . Then, if the player who belongs to the same coalition as disappears, player would have a CIS-deviation to the other coalition, a contradiction.
Third, if , it would not satisfy contractually individual stability, a contradiction.
Finally, if , then would not be individually rational, a contradiction.
We have exhausted all possible cases and obtained a contradiction. Hence the game admits no CIS-robust partition. This means that the game does not have an -robust outcome for any .∎
We saw that a symmetric friend-oriented game may not admit an NS-robust outcome. In this section, we show that deciding the existence of an NS-robust outcome remains easy for a symmetric friend-oriented game. We warm up by observing that in order to preserve individual rationality, each coalition must be a clique or have minimum degree at least .
Lemma 4.1 ().
For any symmetric friend-oriented game, , and any IR-robust partition , each is either a clique or has minimum degree at least .
Let be an IR-robust partition. Suppose towards a contradiction that there is a coalition such that does not form a clique and there is a player who has at most friends in . This means that by IR-robustness, has size at most ; otherwise, removing all ’s friends in would violate individual rationality for . Now since is not a clique, there is a player who has an enemy in . Observe that has at most friends in , since has size at most and at least one of the players is an enemy of . Hence if all the friends of in disappear, this would cause the deviation of to staying alone, contradicting IR-robustness. ∎
Observe that if there is a coalition of size at most and some player has a friend in other coalitions, the player would have an NS-deviation to the other coalition after removal of players. Hence, any NS-robust outcome cannot contain such coalition, which leads to the following characterization of the classes of friend-oriented games whose NS-robust outcomes are non-empty.
Theorem 4.2 ().
The following conditions are equivalent for any symmetric friend-oriented game and any natural number :
There exists an NS-robust partition.
Each connected component of is either a clique or has minimum degree at least .
Suppose towards a contradiction that there exists an NS-robust outcome but for some connected component of , is not a clique and there is a player with at most friends. Let . By Lemma 4.1, is a clique of size at most . Now since is not a clique, is non-empty. By connectivity of , there is a player having a neighbor . But this implies that when all the players in except for disappear, player would have an incentive to deviate to the coalition of , a contradiction.
Conversely suppose that each connected component of is a clique, or has minimum degree at least . Let be a partition that divides the players into the connected components of . Clearly, no player has an incentive to deviate to another coalition at . Also, removing at most players does not affect Nash stability. Indeed, after removal of at most players, each player has at least one friend in his coalition of the resulting partition and has no friend in the other coalitions; or forms a singleton and has no friend in the other coalitions. Hence, is NS-robust. ∎
Corollary 4.3 ().
For a symmetric friend-oriented game , deciding the existence of NS-robust outcomes can be done in polynomial time.
The condition of Theorem 4.2 can be easily verified by checking the size and minimum degree of each connected component, which can be done in by depth-first search, where is the number of edges in . ∎
5. CIS-robustness and IS-robustness
We now turn our attention to a weaker stability concept, contractually individual stability. Usually, such stability requirement is not difficult to achieve: Gairing and Savani (2010) observed that a CIS partition is guaranteed to exist for any symmetric additively separable game and can be efficiently computed. As we have seen before, the presence of a star with two leaves complicates the existence of CIS-robust outcomes. In what follows, we will show that by decomposing the friendship graph appropriately, one can determine the existence of CIS-robust outcomes under deletion of a single player. We start by showing that a leaf player and its unique neighbor playing a role of pseudo-center form a pair in a CIS-robust outcome. For a graph and a subset , we denote by the subgraph of induced by . We say that a vertex is a pseudo-center in a graph if at most one neighbor of is a non-leaf vertex.
Lemma 5.1 ().
For a symmetric friend-oriented game and , let be an arbitrary CIS-robust partition. If is the unique friend of , and is a pseudo-center in , then .
By Lemma 4.1, ’s coalition is either the singleton or the pair . Assume towards a contradiction that . If , then has a CIS-deviation to ’s coalition, a contradiction. If , then removing the player in would cause the CIS-deviation of to ’s coalition, a contradiction. If , then this means that has at least two friends in by Lemma 4.1. However, this means that at least one of the players has only one friend but has at least one enemy in , contradicting Lemma 4.1. In either case, we obtain a contradiction. ∎
The above lemma can recursively apply to all such pairs in the following way: as long as there is an edge satisfying the property in Lemma 5.1, we need to put the players into a pair and examine whether such an edge still exists in the remaining instance. This allows us to partially determine the structure of a CIS-robust outcome. Figure 2 illustrates the sequence of pairs of players that need to be formed in a robust outcome. We now formalize the above idea as follows. For a friendship graph , a sequence of edges for is called an outer elimination sequence if the following two hold:
is the unique friend of in , and is a pseudo-center in ; or
is the unique friend of in , and is a friend of some player in .
Here for each . An outer elimination sequence is said to be maximal if it cannot be made any longer, i.e., there is no outer elimination sequence .
Lemma 5.2 ().
For a symmetric friend-oriented game and , let be an arbitrary CIS-robust partition. If there is an outer elimination sequence for , then we have for each .
We prove the statement by induction on . When , the claim holds due to Lemma 5.1. Suppose that the claim holds for and we prove it for . Now by the induction hypothesis, for each , and thus players and form a coalition within . Now, we have either or by Lemma 4.1. Assume towards a contradiction that . First suppose that is a pseudo-center in . Again, if , then contains at least two friends of by Lemma 4.1 where at least one of the players has only one friend in , contradicting Lemma 4.1. Thus, either stays alone at or forms a coalition with his another friend in ; however, in the former case, would have a CIS-deviation to ; and in the latter case, deleting the other friend of would cause the CIS-deviation of to , a contradiction. Second, suppose that is a friend of some player . Then, by removing with , player would have a CIS-deviation to , a contradiction. We thus conclude that . ∎
Given a symmetric friend oriented game , we say that a pair of players is an elimination pair if it appears in some outer elimination sequence. We denote by the set of players who belong to some elimination pair, by the set of players who have exactly one friend in , by the set of players who have at least two friends in , and by the set of remaining players, i.e., . Before we proceed, we observe the following.
Lemma 5.3 ().
For a symmetric friend oriented game , each player in has no friend in .
Suppose that there is a player who is a friend of some player in . Let be the unique friend of in . Then, the pair of players and is an elimination pair satisfying the condition E2, a contradiction. ∎
Lemma 5.4 ().
For a symmetric friend oriented game , let . Then, has no friend in . Further if there is a CIS-robust outcome, has no friend in .
Consider . Assume towards a contradiction that has some friend in . If has a friend , then together with is an elimination pair and must be included in , a contradiction. If has exactly one friend in , then this means that , a contradiction. If has at least two friends in , then this means that , a contradiction. Hence has no friend in .
Figure 3 illustrates the partition of the player set into , and . Now, a CIS-robust outcome must include all the elimination pairs, and hence if such outcome exists, there is at most one maximal outer elimination sequence. We can thus completely characterise the class of symmetric friend-oriented games that admit a CIS-robust outcome under deletion of a single player.
Theorem 5.5 ().
For a symmetric friend-oriented game and , a CIS-robust outcome exists if and only if the following holds:
the set of elimination pairs that appear in each maximal elimination sequence is the same; and
there are no elimination pairs and where is a friend of both and ; and
for each player and each player , and are enemies to each other; and
if there is a player who is a friend of every player in , then every player is a friend of exactly one player in and an enemy of at least one player in each elimination pair.
Suppose that there is a CIS-robust outcome . To show i, take any maximal elimination sequence and . If there are two elimination pairs and with and , this would imply that by Lemma 5.2, a contradiction. If the two sequences are disjoint, then one can create a longer elimination sequence by adding edge to the last position of the other sequence, contradicting maximality. To see ii, if there are two pairs and where is a friend of both and , then has to include both pairs, which however implies that would have a CIS-deviation to the coalition after the removal of player , a contradiction. The statement iii holds due to Lemma 5.4. To see iv, assume that some player is a friend of every player in . Consider any player . If , then by deleting the other friend with , would have a CIS-deviation to , a contradiction. Thus we have . Further, each player in has no friend and hence stays alone at by Lemma 4.1. This means , and thus . However, if has no friend in or multiple friends in , player in who belongs to ’s coalition has no friend in or has at most one friend and at least one enemy in , contradicting Lemma 4.1. Hence has exactly one friend in ; by Lemma 4.1 and by the fact that , we have . If there is an elimination pair where both of them are adjacent to , then would have a CIS-deviation to the coalition after removal of . Hence, is an enemy of at least one player in each elimination pair.
Conversely, suppose that all the properties iiv hold. Let where is a maximal outer elimination sequence. We note that is empty if there is no elimination pair. We define the partition as follows: First, for each , we set . Second, for each player , we set . Finally, we partition the players in and as follows.
If there is a player who is a friend of every player in , we put each and the unique friend of in into a pair.
Otherwise, all players in form a coalition and put each player in into a singleton.
Since each player belongs to a clique or has at least two friends in his coalition, can be easily seen to be IR-robust under deletion of a single player. Now take any player . We will show that has no CIS-deviation even after removal of a single player. Now consider the following cases.
: By Lemma 5.4 and iii, is an isolated vertex of the friendship graph and hence has no incentive to deviate to the other coalitions, even after deletion of a single player.
: By definition, has at least two friends in his coalition, or forms a pair with his friend in . If contains at least two friends of , has no CIS-deviation even after removal of an arbitrary single player, as there is at least one friend of in who would be worse off by the deviation of . Suppose forms a pair with his friend in . In order for to have a CIS-deviation to other coalitions, must disappear; but then, every other coalition still contains an enemy of , thereby implying that has no CIS-deviation to any other coalition even a single player disappears.
: Note that is adjacent to only one player in . Thus, has an incentive to deviate to another coalition only if . However, has at least two friends in and hence contains at least one enemy of even after removal of an arbitrary single player. Thus, has no CIS-deviation even after removal of a single player.
: First, by Lemma 5.3 and iii, has no incentive to join a singleton included in or . Second, player has no CIS-deviation to another elimination pair even after deletion of any single player, since by ii, at least one player in his coalition or the deviating pair would be worse off by the deviation of . Finally, the property iv further ensures that player has no CIS-deviation to a coalition containing a player . Indeed, player has a CIS-deviation to such coalition only when the other player of ’s coalition disappears; but then ’s coalition contains at least one enemy of . Thus, has no CIS-deviation even after removal of a single player.
The proof is complete. ∎
Building on the above characterization, it is easy to see that we can decide in polynomial time whether a symmetric friend-oriented game admits a CIS-robut outcome under deletion of a single player. The proof employs a simple procedure, which iteratively expands an outer elimination sequence and eventually decompose the player set into , , , and .
Theorem 5.6 ().
For a symmetric friend-oriented game and , deciding the existence of a CIS-robust outcome can be done in polynomial time.
Consider the following algorithm which takes as input a friendship graph and returns a CIS-robust outcome if it exists. The algorithm recursively finds an elimination pair and divides the players into the three sets , , , and . By Theorem 5.5, the algorithm returns a CIS-robust partition if it exists. Indeed, if there is a CIS-robust outcome, the algorithm correctly computes ; and we have shown in the proof of Theorem 5.5, the output of Algorithm 1 is CIS-robust under deletion of a single player. Conversely, if there is no CIS-robust outcome, is not CIS-robust and thus Algorithm 1 fails. Clearly, Algorithm 1 runs in polynomial time.
The above result turns out to be tight in several aspects. We first show that for , finding a CIS-robust outcome of a symmetric friend-oriented game is NP-hard.
Theorem 5.7 ().
For a symmetric friend-oriented game , it is NP-complete to decide the existence of a CIS-robust outcome even for .
CIS-robustness can be verified in polynomial time: for each set of size at most two, one can check in polynomial time whether is contractually individually stable. So our problem is in NP. To show hardness, we give a reduction from Exact-3-Cover (X3C). Recall that an instance of X3C is given by a set of elements and a family of three-element subsets of ; it is a ‘yes’-instance if and only if there is an exact cover with and .
Construction: Given an instance of X3C, we construct an instance of a friend-oriented game as follows. For each , we create a vertex player . For each vertex , we create a vertex gadget , which enforces the corresponding vertex player to have at least two friends in his robust coalition. Specifically, consists of vertex player , two friends and of , and one enemy of . All the three players , , and are friends to each other, and are enemies of all the vertex players except for , and the player is an enemy of all the vertex players. Figure 4(a) illustrates . For each , we create a set gadget which consists of its vertex players , and cliques for . Specifically, and are a friend of for each ; form a clique; and the pairs of and , and , and and are friends to each other. See Figure 4(b) for an illustration. Unless specified otherwise, players are enemies to each other. Finally, we set .
Correctness: Suppose that there is an exact cover . Then, we define as follows. For each set and each , we set ; the remaining players of the set gadget forms a coalition, i.e., . For each , the non-vertex players in the set gadget form a coalition, i.e., . For each , we set . Since is an exact cover, it can be easily verified that is a partition of the player set. Since each coalition is either a clique or a coalition with minimum degree at least three, is IR-robust under deletion of at most two players. To show that satisfies CIS-robustness, we note that in order for player to have a CIS-deviation to some coalition , all her friends in and all her enemies in must disappear. Thus, no player has a CIS-deviation to the coalitions of form as every player not in has at least three enemies in . Also, no player has a CIS-deviation to the coalitions of form or as each player has at least two friends in his coalition and one enemy in .
To show the opposite direction, let be a CIS-robust partition. Consider a player for ; we will show that . First, assume towards a contradiction that . Recall that by Lemma 4.1, each player must be in a clique or in a coalition with minimum degree . As there is no coalition containing with minimum degree three, forms a clique of size at most three and we have either or . When , player forms a singleton or a pair with ; however, either case admits the CIS-deviation of to when player disappears, a contradiction. When , forms a singleton and hence player would have a CIS-deviation to after player disappears, a contradiction. Thus, and hence . If , then there is a player who forms a singleton and has a CIS-deviation to the other coalition of with . Thus, for each , .
This means that each vertex player needs to have at least two friends in his coalition as otherwise removing his enemy and one friend of ’s coalition would cause the CIS-deviation of to the remaining players . Now the only IR-robust way to do this is to select triples of form where and , and put them into a coalition.
We will next show that if for some , then it must be the case that and . Suppose towards a contradiction that for some but for some with . We note that needs to have at least two friends in his coalition; otherwise he would deviate to the coalition after and one friend of in his coalition disappear. Thus, by Lemma 4.1, is in a clique of size three and it must be the case that . Then by Lemma 4.1, we have the following cases:
and : This would cause the CIS-deviation of to .
and : This would cause the CIS-deviation of to after deletion of .
: This would cause the CIS-deviation of to after removing two friends of .
In either case, we obtain a contradiction, and thus . Similarly, we have . Now let
Clearly, each player appears in some set in Also, there is no pair of distinct sets that includes the same vertex player , as otherwise, this would mean that , a contradiction. We conclude that is an exact cover. ∎
A similar proof of Theorem 5.7 shows that finding an IS-robust outcome is NP-hard even if only a single player is allowed to disappear.
Theorem 5.8 ().
For a symmetric friend-oriented game , deciding the existence of an IS-robust outcome is NP-complete even for .
Clearly, our problem is in NP. Again, we give a reduction from Exact-3-Cover (X3C).
Construction: Given an instance of X3C, we construct an instance of a friend-oriented game as follows. For each , we create a vertex player . For each vertex , we create the same vertex gadget as in the proof of Theorem 5.7 with the additional dummy players and , each of which is adjacent to and , respectively. See Figure 5(a) for an illustration. For each , we create a set gadget which consists of its vertex players , and cliques for . Specifically, and are a friend of for each , and form a clique. See Figure 5(b) for an illustration. Unless specified otherwise, players are enemies to each other. Finally, we set .
Correctness: Suppose that there is an exact cover . Again, we define as follows:
for each , we set , , , and ; and
for each , we set ; and
for each , we set , and .
Since is an exact cover, it can be easily verified that is a partition of the player set. We will show that satisfies IS-robustness. No player has an IS-deviation to the coalitions for , since each player outside the coalition has at least three enemies, and there is at least one player who would not accept such a deviation even after removal of a single player. Now fix . The only players who would have an incentive to deviate to the coalition are the vertex player or the dummy players and . However, since has two friends in his coalition , and since there are two friends and one enemy of in , would not have an IS-deviation even after removal of an arbitrary player. Also, no dummy player has an IS-deviation to even after removal of a single player, as at least one enemy of that player remains in the coalition. Further, the only player who has an incentive to deviate to a coalition