Altruism in Coalition Formation Games

Nguyen et al. [1] introduced altruistic hedonic games in which agents' utilities depend not only on their own preferences but also on those of their friends in the same coalition. We propose to extend their model to coalition formation games in general, considering also the friends in other coalitions. Comparing our model to altruistic hedonic games, we argue that excluding some friends from the altruistic behavior of an agent is a major disadvantage that comes with the restriction to hedonic games. After introducing our model and showing some desirable properties, we additionally study some common stability notions and provide a computational analysis of the associated verification and existence problems.


Introduction
We consider coalition formation games where agents have to form coalitions based on their preferences.Among other compact representations of hedonic coalition formation games, Dimitrov et al. [2] in particular proposed the friends-and-enemies encoding with friend-oriented preferences which involves a network of friends: a (simple) undirected graph whose vertices are the players and where two players are connected by an edge exactly if they are friends of each other.Players not connected by an edge consider each other as enemies.Under friend-oriented preferences, player i prefers a coalition C to a coalition D if C contains more of i's friends than D, or C and D have the same number of i's friends but C contains fewer enemies of i's than D. This is a special case of the additive encoding [3].For more background on these two compact representations, see Section 2 and the book chapter by Aziz and Savani [4].
Based on friend-oriented preferences, Nguyen et al. [1] introduced altruistic hedonic games where agents gain utility not only from their own satisfaction but also from their friends' satisfaction.However, Nguyen et al. [1] specifically considered hedonic games only, which require that an agent's utility only depends on her own coalition.In their interpretation of altruism, the utility of an agent is composed of the agent's own valuation of her coalition and the valuation of all this agent's friends in this coalition.While Nguyen et al. [1] used the average when aggregating some agents' valuations, Wiechers and Rothe [5] proposed a variant of altruistic hedonic games where some agents' valuations are aggregated by taking the minimum.
Inspired by the idea of altruism, we extend the model of altruism in hedonic games to coalition formation games in general.That is, we propose a model where agents behave altruistically to all their friends, not only to the friends in the same coalition.Not restricting to hedonic games, we aim to capture a more natural notion of altruism where none of an agent's friends is excluded from her altruistic behavior.
In order to model altruism globally, we release the restriction to hedonic games and introduce altruistic coalition formation games where agents behave altruistically to all their friends, independently of their current coalition.

Related Work
Coalition formation games, as considered here, are closely related to the subclass of hedonic games which has been broadly studied in the literature, addressing the issue of compactly representing preferences, conducting axiomatic analyses, dealing with different notions of stability, and investigating the computational complexity of the associated problems (see, e.g., the book chapter by Aziz and Savani [4]).
Closest related to our work are the altruistic hedonic games by Nguyen et al. [1] (see also the related minimization-based variant by Wiechers and Rothe [5]), which we modify to obtain our more general models of altruism.Based on the model due to Nguyen et al. [1], Schlueter and Goldsmith [6] defined super altruistic hedonic games where friends have a different impact on an agent based on their distances in the underlying network of friends.More recently, Bullinger and Kober [7] introduced loyalty in cardinal hedonic games where agents are loyal to all agents in their so-called loyalty set.In their model, the utilities of the agents in the loyalty set are aggregated by taking the minimum.They then study the loyal variants of common classes of cardinal hedonic games such as additively separable and friend-oriented hedonic games. 1ltruism has also been studied for noncooperative games.Most prominently, Ashlagi et al. [8] introduced social context games where a social context is applied to a strategic game and the costs in the resulting game depend on the original costs and a graph of neighborhood.Their so-called MinMax collaborations (where players seek to minimize the maximal cost of their own and their neighbors) are related to our minimizationbased equal-treatment model.Still, the model of Ashlagi et al. [8] differs from ours in that they consider noncooperative games.Other work considering noncooperative games with social networks is due to Bilò et al. [9] who study social context games for other underlying strategic games than Ashlagi et al. [8], Hoefer et al. [10] who study considerate equilibria in strategic games, and Anagnostopoulos et al. [11] who study altruism and spite in strategic games.Further work studying altruism in noncooperative games without social networks is due to Hoefer and Skopalik [12], Chen et al. [13], Apt and Schäfer [14], and Rahn and Schäfer [15].

Our Contribution
Conceptually, we extend the models of altruism proposed by Nguyen et al. [1] and Wiechers and Rothe [5] from hedonic games to general coalition formation games.We argue how this captures a more global notion of altruism and show that our models fulfill some desirable properties that are violated by the previous models.We then study the common stability concepts in this model and analyze the associated verification and existence problems in terms of their computational complexity.
This work extends a preliminary version that appeared in the proceedings of the 29th International Joint Conference on Artificial Intelligence (IJCAI'20) [16].Parts of this work were also presented at the 16th and 17th International Symposium on Artificial Intelligence and Mathematics (ISAIM'20 and ISAIM '22) and at the 8th International Workshop on Computational Social Choice (COMSOC'21), each with nonarchival proceedings.

The Model
In coalition formation games, players divide into groups based on their preferences.Before introducing altruism, we now give some foundations of such games.

Coalition Formation Games
Let N = {1, . . ., n} be a set of agents (or players).Each subset of N is called a coalition.A coalition structure Γ is a partition of N, and we denote the set of all possible coalition structures for N by C N .For a player i ∈ N and a coalition structure Γ ∈ C N , Γ(i) denotes the unique coalition in Γ containing i. Now, a coalition formation game (CFG) is a pair (N, ), where N = {1, . . ., n} is a set of agents, = ( 1 , . . ., n ) is a profile of preferences, and every preference i ∈ C N × C N is a complete weak order over all possible coalition structures.Given two coalition structures Γ, ∆ ∈ C N , we say that i weakly prefers Γ to ∆ if Γ i ∆.When Γ i ∆ but not ∆ i Γ, we say that i prefers Γ to ∆ (denoted by Γ ≻ i ∆), and we say that i is indifferent between Γ and ∆ (denoted by Note that hedonic games are a special case of coalition formation games where the agents' preference relations only depend on the coalitions containing themselves.In a hedonic game (N, ), agent i ∈ N is indifferent between any two coalition structures Γ and ∆ as long as her coalition is the same, i.e., Γ(i) = ∆(i) =⇒ Γ ∼ i ∆.Therefore, the preference order of any agent i ∈ N in a hedonic game (N, ) is usually represented by a complete weak order over the set of coalitions containing i.

The "Friends and Enemies" Encoding
Since |C N |, the number of all possible coalition structures, is extremely large in the number of agents, 2 it is not reasonable to ask every agent for her complete preference over C N .Instead, we are looking for a way to compactly represent the agents' preferences.In the literature, many such representations have been proposed for hedonic games, such as the additive encoding [19,3,20], the singleton encoding due to Cechlárová and Romero-Medina [21] and further studied by Cechlárová and Hajduková [22], the friends-and-enemies encoding due to Dimitrov et al. [2], and FEN-hedonic games due to Kerkmann et al. [23] and also used by Rothe et al. [24].Here, we use the friends-and-enemies encoding due to Dimitrov et al. [2].We focus on their friend-oriented model and will later adapt it to our altruistic model.
In the friend-oriented model, the preferences of the agents in N are given by a network of friends, i.e., a (simple) undirected graph G = (N, A) whose vertices are the players and where two players i, j ∈ N are connected by an edge {i, j} ∈ A exactly if they are each other's friends.Agents not connected by an edge consider each other as enemies.For an agent i ∈ N, we denote the set of i's friends by F i = { j ∈ N | {i, j} ∈ A} and the set of i's enemies by E i = N \ (F i ∪ {i}).Under friend-oriented preferences as defined by Dimitrov et al. [2], between any two coalitions players prefer the coalition with more friends, and if there are equally many friends in both coalitions, they prefer the coalition with fewer enemies: This can also be represented additively.Assigning a value of n to each friend and a value of , and v i (C) > 0 if and only if there is at least one friend of i's in C. For a given coalition structure Γ ∈ C N , we also write v i (Γ) for player i's value of Γ(i).
Furthermore, we denote the sum of the values of i's friends by sum

Three Degrees of Altruism
When we now define altruistic coalition formation games based on the friend-oriented preference model, we consider the same three degrees of altruism that Nguyen et al. [1] introduced for altruistic hedonic games. 2 The number of possible partitions of a set with n elements equals the n-th Bell number [17,18], defined as  However, we adapt them to our model, extending the agents' altruism to all their friends, not only to their friends in the same coalition.
• Selfish First (SF): Agents first rank coalition structures based on their own valuations.Only in the case of a tie between two coalition structures, their friends' valuations are considered as well.
• Equal Treatment (EQ): Agents treat themselves and their friends the same.That means that an agent i ∈ N and all of i's friends have the same impact on i's utility for a coalition structure.
• Altruistic Treatment (AL): Agents first rank coalition structures based on their friends' valuations.They only consider their own valuations in the case of a tie.
We further distinguish between a sum-based and a min-based aggregation of some agents' valuations.Formally, for an agent i ∈ N and a coalition structure In the case of F i = / 0, we define the minimum of the empty set to be zero.For any coalition structures Γ, ∆ ∈ C N , agent i's sum-based SF preference is then defined by Γ sumSF ) are defined analogously, using the respective utility functions.The factor M, which is used for the SF and AL models, ensures that an agent's utility is first determined by the agent's own valuation in the SF model and first determined by the friends' valuations in the AL model.Similarly as Nguyen et al. [1] prove the corresponding properties in hedonic games, we can show that for An altruistic coalition formation game (ACFG) is a coalition formation game where the agents' preferences were obtained by a network of friends via one of these cases of altruism.Hence, we distinguish between sum-based SF, sum-based EQ, sum-based AL, min-based SF, min-based EQ, and min-based AL ACFGs.For any ACFG, the players' utilities can obviously be computed in polynomial time.

Monotonicity and Other Properties in ACFGs
Nguyen et al. [1] focus on altruism in hedonic games where an agent's utility only depends on her own coalition.As we have already seen in Example 1, there are some aspects of altruistic behavior that cannot be realized by hedonic games.The following example shows that our model crucially differs from the models due to Nguyen et al. [1] and Wiechers and Rothe [5].
One can observe that agent 1 and all her friends assign a greater value to ∆ than to Γ. Consequently, also the aggregations of the friends' values (sum F  1 , sum F+ 1 , min F 1 , min F+ 1 ) are greater for ∆.Hence, 1 prefers ∆ to Γ under all our sum-based and min-based altruistic preferences.The hedonic models due to Nguyen et al. [1] and Wiechers and Rothe [5], however, are blind to the fact that agent 1 and all her friends are better off in ∆ than in Γ.Under their altruistic hedonic preferences, player 1 compares the two coalition structures Γ and ∆ only based on her own coalitions Γ(1) = {1, 2} and ∆(1) = {1, 5, . . ., 10}.She then only considers her friends that are in the same coalition, i.e., player 2 for Γ and players 5 and 6 for ∆.This leads to 1 preferring Γ(1) to ∆(1) under altruistic hedonic EQ and AL preferences.In particular, the average (and minimum) valuation of 1's friends in Γ(1) is 10 while the average (and minimum) valuation of 1's friends in ∆(1) is 5. Also considering 1's own value for EQ, the average (and minimum) in Γ(1) is 10 while the average (respectively, minimum) value in ∆( 1) is 8.6 (respectively, 5).

Some Basic Properties
As we have seen in Example 2, altruistic hedonic games [1,5] allow for players that prefer coalition structures that make themselves and all their friends worse off.To avoid this kind of unreasonable behavior, we focus on general coalition formation games.In fact, all our altruistic coalition-formation preferences fulfill unanimity: For an ACFG (N, ) and a player i ∈ N, we say that i is unanimous if, for any two coalition structures This property crucially distinguishes our preference models from the corresponding altruistic hedonic preferences, which are not unanimous under EQ or AL preferences, as Example 2 shows.Note that Nguyen et al. [1] define a restricted version of unanimity in altruistic hedonic games by considering only the agents' own coalitions.Other desirable properties that were studied by Nguyen et al. [1] for altruistic hedonic preferences can be generalized to coalition formation games.We show that these desirable properties also hold for our models.First, we collect some basic observations: Observation 3. Consider any ACFG (N, ) with an underlying network of friends G.
1.All preferences i , i ∈ N, are reflexive and transitive.

For any player i ∈ N and any two coalition structures Γ, ∆ ∈ C N , it can be decided in polynomial time
(in the number of agents) whether Γ i ∆.

The preferences i , i ∈ N, only depend on the structure of G.
Note that the third statement of Observation 3 implies that the properties that Nguyen et al. [1] call anonymity and symmetry are both satisfied in ACFGs.Another desirable property they consider is called sovereignty of players and inspired by the axiom of "citizens' sovereignty" from social choice theory: 3 Given a set of agents N, a coalition structure Γ ∈ C N , and an agent i ∈ N, we say that sovereignty of players is satisfied if there is a network of friends G on N such that Γ is i's most preferred coalition structure in any ACFG induced by G.

Proposition 4. ACFGs satisfy sovereignty of players under all sum-based and min-based SF, EQ, and AL altruistic preferences.
Proof.Sovereignty of players in ACFGs can be shown with an analogous construction as in the proof of Nguyen et al. [1,Theorem 5]: For a given set of players N, player i ∈ N, and coalition structure Γ ∈ C N , we construct a network of friends where all players in Γ(i) are friends of each other while there are no other friendship relations.Then Γ is i's (nonunique) most preferred coalition structure under all sum-based and min-based SF, EQ, and AL altruistic preferences.

Monotonicity
The next property describes the monotonicity of preferences and further distinguishes our models from altruistic hedonic games.In fact, Nguyen et al. [1] define two types of monotonicity, which we here adapt to our setting.
Definition 5. Consider any ACFG (N, ), agents i, j ∈ N with j ∈ E i , and coalition structures Γ, ∆ ∈ C N .Let further ′ i be the preference relation resulting from i when j turns from being i's enemy to being i's friend (all else remaining equal).We say that i is Theorem 6.Let (N, ) be an ACFG.

If (N, ) is min-based, its preferences satisfy type-II-but not type-I-monotonicity.
Proof.Let (N, ) be an ACFG with an underlying network of friends G = (N, H).Consider i ∈ N, Γ, ∆ ∈ C N , and j ∈ E i and denote with G ′ = (N, H ∪ {{i, j}}) the network of friends resulting from G when j turns from being i's enemy to being i's friend (all else being equal).Let (N, ′ ) be the ACFG induced by G ′ .For any agent a ∈ N and coalition structure a , and a's friends and enemies in (N, ′ ) by F ′ a and E ′ a , respectively.That is, we have and ′ i might differ from v i , v j , and i , while the friends, enemies, and values of all other players stay the same, i.e., Type-I-monotonicity under sum-based preferences.
Type-II-monotonicity under sum-based and min-based preferences.
To see that minEQ and minAL are not type-I-monotonic, consider the game G 2 with the network of friends in Figure 3c.Consider the coalition structures Γ = {{1, 2, 3, 4}, {5}} and ∆ = {{1, 2, 3, 4, 5}} and players i = 1 and j = 2 with 2 ∈ Γ(1)∩∆(1), and ∆. Now, making 2 a friend of 1's leads to the game G ′ 2 with the network of friends in Figure 3d.For this game, we have min F+′ Note that the hedonic models of altruism [1,5] violate both type-I-and type-II-monotonicity for EQ and AL.Hence, it is quite remarkable that all three degrees of our extended sum-based model of altruism satisfy both types of monotonicity.

Stability in ACFGs
The main question in coalition formation games is which coalition structures might form.There are several stability concepts that are well-studied for hedonic games, each indicating whether a given coalition structure would be accepted by the agents or if there are other coalition structures that are more likely to form.Although we consider more general coalition formation games, we can easily adapt these definitions to our framework.
Let (N, ) be an ACFG with preferences = ( 1 , . . ., n ) obtained from a network of friends via one of the three degrees of altruism and with either sum-based or min-based aggregation of the agents' valuations.We use the following notation.For a coalition structure Γ ∈ C N , a player i ∈ N, and a coalition C ∈ Γ ∪ { / 0}, Γ i→C denotes the coalition structure that arises from Γ when moving i to C, i.e., In addition, we use Γ C→ / 0 , with C ⊆ N, to denote the coalition structure that arises from Γ when all players in C leave their respective coalition and form a new one, i.e., Finally, for any two coalition structures Γ, ∆ ∈ C N , let # Γ≻∆ = |{i ∈ N | Γ ≻ i ∆}| be the number of players that prefer Γ to ∆.Now, we are ready to define the common stability notions.Definition 7. A coalition structure Γ is said to be • Nash stable if no player prefers moving to another coalition: • individually rational if no player would prefer being alone: • individually stable if no player prefers moving to another coalition and could deviate to it without harming any player in that coalition: • contractually individually stable if no player prefers another coalition and could deviate to it without harming any player in the new or the old coalition: • totally individually stable if no player prefers another coalition and could deviate to it without harming any other player: • core stable if no nonempty coalition blocks Γ: • strictly core stable if no coalition weakly blocks Γ: • popular if for every other coalition structure ∆, at least as many players prefer Γ to ∆ as there are players who prefer ∆ to Γ: • strictly popular if for every other coalition structure ∆, more players prefer Γ to ∆ than there are players who prefer ∆ to Γ: • perfect if no player prefers any coalition structure to Γ: Note that "totally individual stability" is a new notion which we introduce here.It strengthens the notion of contractually individual stability and makes sense in the context of coalition formation games because players' preferences may also be influenced by coalitions they are not part of.We now study the associated verification and existence problems in terms of their computational complexity.We assume the reader to be familiar with the complexity classes P (deterministic polynomial time), NP (nondeterministic polynomial time) and coNP (the class of complements of NP sets).For more background on computational complexity, we refer to, e.g., the textbooks by Garey and Johnson [25] and Rothe [26].Given a stability concept α, we define: • α -VERIFICATION: Given an ACFG (N, ) and a coalition structure Γ ∈ C N , does Γ satisfy α?
• α -EXISTENCE: Given an ACFG (N, ), does there exist a coalition structure Γ ∈ C N that satisfies α? Table 2 summarizes the results for these problems under sum-based and min-based SF preferences.We will also give results for EQ and AL in this section.In Table 2, however, we only mark if the results for EQ and AL match those for SF.

Individual Rationality
Verifying individual rationality is easy: We just need to iterate over all agents and compare two coalition structures in each iteration.Since players' utilities can be computed in polynomial time, individual rationality can be verified in time polynomial in the number of agents.The existence problem is trivial, since Γ = {{1}, . .., {n}} is always individually rational.Furthermore, we give the following characterization.
Theorem 8. Given an ACFG (N, ), a coalition structure Γ ∈ C N is individually rational 2. under min-based EQ preferences if and only if for all players i ∈ N, Γ(i) contains a friend of i's or i is alone or there is a friend of i's whose valuation of Γ is less than or equal to i's valuation of Γ, formally: Proof. 1.To show the implication from left to right, if Γ is individually rational, we assume for the sake of contradiction that Γ(i) ∩ F i = / 0 and Γ(i) = {i} for some player i ∈ N. First, we observe that for all j ∈ F i we have v j (Γ) = v j (Γ i→ / 0 ), as their respective coalition is not affected by i's move.It directly follows that, for all considered models of altruism, player i's utilities for Γ and Γ i→ / 0 only depend on her own valuation, which is greater for Γ i→ / 0 than for Γ (since there are enemies in Γ(i) but not in Γ i→ / 0 (i)).Hence, i prefers Γ i→ / 0 to Γ, so Γ is not individually rational.This is a contradiction.
The implication from right to left is obvious for all considered models of altruism.
2. From left to right, we have that Γ is individually rational and, for the sake of contradiction, we assume that there is a player i ∈ N with Γ(i) ∩ F i = / 0 and Γ(i) = {i} and for all j ∈ F i we have v j (Γ) > v i (Γ).Since i is the least satisfied player in F i ∪ {i}, we have . This is a contradiction to Γ being individually rational.
From right to left, we have to consider two cases.First, if Γ(i) ∩ F i = / 0 or Γ(i) = {i} for some i ∈ N, we obviously have Γ minEQ i Γ i→ / 0 .Second, if Γ(i) ∩ F i = / 0 and Γ(i) = {i}, we know that there is at least one j ∈ F i with v j (Γ) ≤ v i (Γ) < 0. Let j ′ denote a least satisfied friend of i's in Γ (pick one randomly if there are more than one).Since Γ(i) ∩ F i = / 0, it holds that Γ( j) = Γ i→ / 0 ( j) for all j ∈ F i .Consequently, j ′ is i's least satisfied friend in both coalition structures and we have 0 , so Γ is individually rational.

Nash Stability
Since there are at most |N| coalitions in a coalition structure Γ ∈ C N , we can verify Nash stability in polynomial time: We just iterate over all agents i ∈ N and all the (at most |N| + 1) coalitions C ∈ Γ ∪ { / 0} and check whether Γ i Γ i→C .Since we can check a player's altruistic preferences over any two coalition structures in polynomial time and since we have at most a quadratic number of iterations (|N| • (|N| + 1)), Nash stability verification is in P for any ACFG.Nash stability existence is trivially in P for any ACFG; indeed, the same example that Nguyen et al. [1] gave for altruistic hedonic games works here as well.Specifically, for C = {i ∈ N | F i = / 0} = {c 1 , . . ., c k } the coalition structure {{c 1 }, . . ., {c k }, N \ C} is Nash stable.

Individual Stability
For individual stability, contractually individual stability, and totally individual stability, existence is also trivially in P. Nash stability implies all these three concepts, hence, the Nash stable coalition structure given above is also (contractually; totally) individually stable.
Verification is also in P for these stability concepts.Similarly to Nash stability, we can iterate over all players and all coalitions and check the respective conditions in polynomial time.

Core Stability and Strict Core Stability
We now turn to core stability and state some results for sum-based and min-based SF ACFGs.We first show that (strict) core stability existence is trivial for SF ACFGs.Proof.For the sake of contradiction, assume that Γ were not strictly core stable, i.e., that there is a coalition D = / 0 that weakly blocks Γ.Consider some player i ∈ D. Since i weakly prefers deviating from Γ(i) to D, there have to be at least as many friends of i's in D as in Γ(i).Since Γ(i) contains all of i's friends, D also has to contain all friends of i's.Then all these friends of i's also have all their friends in D for the same reason, and so on.Consequently, D contains all players from the connected component Γ(i), i.e., Γ(i) ⊆ D.
Since D weakly blocks Γ, D cannot be equal to Γ(i) and thus needs to contain some ℓ / ∈ Γ(i).Yet, this is a contradiction, as ℓ is an enemy of i's and i would prefer Γ to Γ D→ / 0 if D contains the same number of friends as Γ(i) but more enemies than Γ(i).
However, the coalition structure from Theorem 9 is not necessarily core stable under EQ and AL preferences.
Example 10.Let N = {1, . . ., 10} and consider the network of friends G shown in Figure 4. Consider the coalition structure consisting of the connected component of G (i.e., of only the grand coalition: Γ = {N}) and the coalition C = {8, 9, 10}.C blocks Γ under sum-based and min-based EQ and AL preferences.To see this, consider how players 7, 8, 9, and 10 value Γ and Γ C→ / 0 : We then obtain Turning to (strict) core stability verification, we can show that this problem is hard under SF preferences, and we suspect that this hardness also extends to EQ and AL.
Theorem 11.Strict core stability verification and core stability verification are in coNP for any ACFG.For (sum-based and min-based) SF ACFGs, core stability verification is even coNP-complete.
Proof.To see that strict core stability verification and core stability verification are in coNP, consider any coalition structure Γ ∈ C N in an ACFG (N, ).Γ is not (strictly) core stable if there is a coalition C ⊆ N that (weakly) blocks Γ.Hence, we nondeterministically guess a coalition C ⊆ N and check whether C (weakly) blocks Γ.This can be done in polynomial time since the preferences of the agents in C for the coalition structures Γ and Γ C→ / 0 can be verified in polynomial time for all our altruistic models.To show coNP-hardness of core stability verification under min-based SF ACFGs, we use RX3C, which is a restricted variant of EXACT COVER BY 3-SETS and known to be NP-complete [25,27].We provide a polynomial-time many-one reduction from RX3C to the complement of our verification problem.Let (B, S ) be an instance of RX3C, consisting of a set B = {1, . . ., 3k} and a collection S = {S 1 , . . ., S 3k } of 3-element subsets of B, where each element of B occurs in exactly three sets in S .The question is whether there exists an exact cover for B in S , i.e., a subset S ′ ⊆ S with |S ′ | = k and S∈S ′ S = B. We assume that k > 4.
From (B, S ) we now construct the following ACFG.The set of players is N = {β b |b ∈ B}∪{ζ S , α S,1 , α S,2 , α S,3 , δ S,1 , . . . ,δ S,4k−3 | S ∈ S } and we define the sets Figure 5 shows the network of friends, where a dashed rectangle around a group of players means that all these players are friends of each other: • All players in Beta are friends of each other.
• For every S ∈ S , ζ S is friend with every β b with b ∈ S and with α S,1 , α S,2 , and α S,3 .
• For every S ∈ S , all players in {δ S,1 , . . ., δ S,4k−3 } are friends of each other.Furthermore, consider the coalition structure Γ = {Beta, Q S 1 , . . ., Q S 3k }.We will now show that S contains an exact cover for B if and only if Γ is not core stable under the min-based SF model.
Only if: Assume that there is an exact cover S ′ ⊆ S for B.
Γ for all i ∈ C, because (a) every β b ∈ Beta has 3k friends in C but only 3k − 1 friends in Beta and (b) every ζ S with S ∈ S ′ has 3 friends and 4k − 4 enemies in C but 3 friends and 4k − 3 enemies in Q S .
If: Assume that Γ is not core stable and let C ⊆ N be a coalition that blocks Γ.Then Γ C→ / 0 ≻ minSF i Γ for all i ∈ C. First, observe that every i ∈ C needs to have at least as many friends in C as in Γ(i).So, if any α S, j or δ S, j is in C, it follows quite directly that Q S ⊆ C.However, since Q S is a coalition in Γ and since every other player (from N \ Q S ) is an enemy of all δ -players, any coalition C with Q S ⊆ C cannot be a blocking coalition for Γ.This contradiction implies that no α S, j or δ S, j is in C.
We now have C ⊆ Beta ∪ Zeta.Since any β b ∈ C has 3k − 1 friends and no enemies in Γ(β b ) and prefers Γ C→ / 0 to Γ, one of the following holds: (a) β b has at least 3k friends in C or (b) β b has 3k − 1 friends and no enemies in C and β b 's friends assign a higher value to Γ C→ / 0 than to Γ.For a contradiction, assume that (b) holds for some β b ∈ C. First, observe that there are exactly 3k players in C (namely, β b and β b 's 3k − 1 friends).We now distinguish two cases: Case 1: All the 3k − 1 friends of β b 's are β -players.Then C consists of all β -players, i.e., C = Beta.This is a contradiction, as Beta is already a coalition in Γ.Then each ζ S ∈ C has 4k − 3 enemies in C. Since ζ S prefers Γ C→ / 0 to Γ, this implies that ζ S has exactly three friends and 4k − 3 enemies in C and the minimal value assigned to Γ C→ / 0 by ζ S 's friends is higher than the minimal value assigned to Γ by ζ S 's friends.In both coalition structures, the minimal value is given by ζ S 's α-friends.However, since these α-players lose ζ S as a friend when ζ S deviates to C, the minimal value assigned to Γ is higher than for Γ C→ / 0 .This is a contradiction.Hence, there are exactly k ζ -players in C. Finally, since every of the 3k β b ∈ C has one of the k ζ S ∈ C as a friend, it holds that {S | ζ S ∈ C} is an exact cover for B. This completes the coNP-hardness proof for min-based SF ACFGs.
For sum-based SF ACFGs, coNP-hardness of core stability verification can be shown by a similar construction.Again, given an instance (B, S ) of RX3C, with B = {1, . . ., 3k}, S = {S 1 , . . ., S 3k }, and k > 8, we construct the following ACFG.The set of players is The network of friends is given in Figure 6, where a dashed rectangle around a group of players means that all these players are friends of each other: • All players in Beta are friends of each other.
• For every S ∈ S , all players in Q S are friends of each other.
• For every S ∈ S , ζ S is friend with α S,1 , α S,2 , and α S,3 and with every β b with b ∈ S. Similar arguments as above show that the coalition structure Γ = {Beta} ∪ {{ζ S } ∪ Q S | S ∈ S } is not core stable under sum-based SF preferences if and only if S contains an exact cover for B.

Popularity and Strict Popularity
Now we take a look at popularity and strict popularity.For all considered models of altruism, there are games for which no (strictly) popular coalition structure exists.
Example 12. Let N = {1, . .., 10} and consider the network of friends shown in Figure 7. Then there is no strictly popular and no popular coalition structure for any of the sum-based or min-based degrees of altruism.Since perfectness implies popularity, there is also no perfect coalition structure for this ACFG.Recall from Footnote 2 that there are 115, 975 possible coalition structures for this game with ten players, which we all tested for this example by brute force.
We now show that, under sum-based and min-based SF preferences, it is hard to verify if a given coalition structure is popular or strictly popular, and it is also hard to decide whether there exists a strictly popular coalition structure for a given SF ACFG.
Theorem 13.Popularity verification and strict popularity verification are in coNP for any ACFG.For (sum-based and min-based) SF ACFGs, popularity verification and strict popularity verification are coNPcomplete and strict popularity existence is coNP-hard.
Proof.First, we observe that the verification problems are in coNP: To verify that a given coalition structure Γ is not (strictly) popular, we can nondeterministically guess a coalition structure ∆, compare both coalition structures in polynomial time, and accept exactly if ∆ is more popular than (or at least as popular as) Γ.
To show coNP-hardness of strict popularity verification for min-based SF ACFGs, we again employ a polynomial-time many-one reduction from RX3C.Let (B, S ) be an instance of RX3C, consisting of a set B = {1, . . ., 3k} and a collection S = {S 1 , • • • , S 3k } of 3-element subsets of B. Recall that every element of B occurs in exactly three sets in S and the question is whether there is an exact cover S ′ ⊆ S of B.
We now construct a network of friends based on this instance.The set of players is given by N = {α 1 , . . . ,α 2k } ∪ {β b | b ∈ B} ∪ {ζ S , η S,1 , η S,2 | S ∈ S }, so in total we have n = 14k players.For convenience, we define Alpha = {α 1 , . . ., α 2k }, Beta = {β b | b ∈ B}, and Q S = {ζ S , η S,1 , η S,2 | S ∈ S } for S ∈ S .The network of friends is shown in Figure 8, where a dashed square around a group of players means that all these players are friends of each other: All players in Alpha ∪ Beta are friends of each other; for every S ∈ S , all players in Q S are friends of each other; and ζ S is a friend of every β b with b ∈ S.
We consider the coalition structure Γ = {Alpha∪Beta} ∪{Q S |S ∈ S } and will now show that S contains an exact cover for B if and only if Γ is not strictly popular under min-based SF preferences.
Only if: Assuming that there is an exact cover S ′ ⊂ S for B, we define the coalition structure ∆ = {Alpha ∪ Beta ∪ S∈S ′ Q S } ∪ {Q S | S ∈ S \ S ′ }.We will now show that ∆ is as popular as Γ under minbased SF preferences.
Alpha ∪ Beta First, all 2k α-players prefer Γ to ∆, since they only add enemies to their coalition in ∆.Second, the 3k β -players prefer ∆ to Γ, as each β -player gains a ζ -friend and then has 5k friends instead of 5k − 1. Next, we consider the Q S -groups for S ∈ S ′ , i.e., the groups that were added to Alpha ∪ Beta in ∆.We observe that every ζ S -player in these Q S -groups prefers ∆ to Γ, since ζ S gains three additional β -friends.For the η-players, on the other hand, the new coalition only contains more enemies, so the η-players prefer Γ to ∆.Since we have |S ′ | = k, this means k ζ -players prefer ∆ to Γ, and 2k η-players prefer Γ to ∆.Finally, we consider the remaining Q S -groups with S ∈ S \ S ′ .Here, the coalition containing these players is the same in Γ and ∆.Hence, for each player p ∈ Q S , we have v p (Γ) = v p (∆).Thus the players have to ask their friends for their valuations.For ζ S ∈ Q S with S ∈ S \ S ′ , the minimum value of her friends is in both structures given by an η-friend, since η S,1 and η S,2 value Γ and ∆ both with n • 2, while the β -friends of ζ S assign values n • (5k − 1) to Γ and n • 5k − (3k − 1) to ∆.So we have u minSF (∆) and, therefore, 2k ζ -players that are indifferent.The η-players in Q S , S ∈ S \ S ′ , are also indifferent, as all their friends value Γ and ∆ the same.In total, # ∆≻Γ = |Beta ∪ {ζ S | S ∈ S ′ }| = 4k = |Alpha ∪ {η S,1 , η S,2 | S ∈ S ′ }| = # Γ≻∆ and, therefore, ∆ is exactly as popular as Γ, so Γ is not strictly popular.
If: Assuming that Γ is not strictly popular, there is some coalition structure ∆ ∈ C N with ∆ = Γ such that ∆ is at least as popular as Γ under min-based SF preferences.We will now show that this implies the existence of an exact cover for B in S .
First of all, we observe that all α-players' most preferred coalition is Alpha ∪ Beta, as it contains all their friends and no enemies.Thus we have For the sake of contradiction, we assume that Alpha ∪ Beta ∈ ∆.As ∆ = Γ, the players in the Q S -groups have to be partitioned differently.However, that would not increase any player's valuation since every player in Q S can only lose friends and gain enemies.That means that no β -player prefers ∆ to Γ, as they are in the same coalition as in Γ and their friends are not more satisfied.We also have at least three players of a Q Sgroup that are no longer in the same coalition, so they prefer Γ to ∆.This is a contradiction, as we assumed that ∆ is at least as popular as Γ.Thus we have Alpha ∪ Beta / ∈ ∆.Now consider the η-players.For every S ∈ S , we know that Q S is the best valued coalition for η S,1 and η S,2 .So again, η S,1 and η S,2 prefer Γ to ∆ if and only if Q S / ∈ ∆, and they are indifferent otherwise.Define k ′ = |{S ∈ S | Q S / ∈ ∆}|.So 2k ′ is the number of η-players that prefer Γ to ∆, and the remaining 6k − 2k ′ η-players are indifferent between Γ and ∆.We first collect some observations: 1.All 2k α-players prefer Γ to ∆.
3. 3k − k ′ ζ -players are in the same coalition in both coalition structures, so their utilities depend on their friends' valuations.In Γ, the minimum value of their friends is given by an η-player.Since this ηplayer is also in the same coalition in ∆ and thus assigns the same value, it is not possible that the minimum value of the friends is higher in ∆ than in Γ.So 3k − k ′ ζ -players are indifferent or prefer Γ to ∆.
Second, let us assume k ′ < k: Since every ζ -player has three β -friends and there are k ′ ζ -players that are not in their respective Q S coalition in ∆, there are at most 3k ′ β -players that gain a ζ -friend in ∆.The 3k − 3k ′ other β -players have at most 5k − 1 friends in ∆, namely all other αand β -players.But as Alpha ∪ Beta / ∈ ∆, they would also gain at least one enemy, so we have 3k − 3k ′ β -players that prefer Γ.That means we have # and therefore, # Γ≻∆ > # ∆≻Γ , which again is a contradiction.Thus we conclude that k ′ ≥ k and, in total, k ′ = k.
Consequently, we know that 4k players prefer Γ to ∆, namely all α-players and the 2k η-players that are not in Q S anymore.Subtracting all the indifferent players, we observe that all other players have to prefer ∆ to Γ in order to ensure # Γ≻∆ ≤ # ∆≻Γ .These other players are the 3k β -players and the k ζ -players that are not in Q S anymore.Finally, that is only possible if every β -player gains a ζ -friend in ∆.Hence each one of those k ζ -players has to be friends with three different β -players.Therefore, the set {S ∈ S | Q S / ∈ ∆} is an exact cover for B.
To show coNP-hardness of strict popularity verification for sum-based SF ACFGs, we use a similar construction.For an instance (B, S ) of RX3C with B = {1, . . ., 3k} and S = {S 1 , . . ., S 3k }, where each element of B occurs in exactly three sets in S , we construct the following ACFG.The set of players is given by N = {α 1 , . . . ,α 5k } ∪ {β b | b ∈ B} ∪ {ζ S , η S | S ∈ S }.Let Al pha = {α 1 , . . . ,α 5k }, Beta = {β b | b ∈ B}, and Q S = {ζ S , η S } for each S ∈ S .The network of friends is given in Figure 9, where a dashed rectangle around a group of players means that all these players are friends of each other: All players in Al pha ∪ Beta are friends of each other and, for every S ∈ S , ζ S is friends with η S and every β b with b ∈ S.
Consider the coalition structure Γ = {Al pha ∪ Beta, Q S 1 , . . ., Q S 3k }.We show that S contains an exact cover for B if and only if Γ is not strictly popular.
If: Assuming that Γ is not strictly popular, i.e., that there is a coalition structure ∆ ∈ C N , ∆ = Γ, with # Γ≻∆ ≤ # ∆≻Γ , it can be shown similarly as before that the set {S ∈ S | Q S / ∈ ∆} is an exact cover for B.
The results for strict popularity existence and popularity verification can be shown by slightly modifying the above reductions.
To show that strict popularity existence is coNP-hard for min-based and sum-based SF ACFGs, we consider the same two reductions as before but the coalition structures Γ are not given as a part of the problem instances.Then, there is an exact cover for B if and only if there is no strictly popular coalition structure.In particular, if there is an exact cover for B, Γ and ∆ as defined in the proofs above are in a tie and every other coalition structure is beaten by Γ.And if there is no exact cover for B then Γ beats every other coalition structure and thus is strictly popular.
Popularity verification for min-based and sum-based SF ACFGs can be shown to be coNP-complete by using the same constructions as for strict popularity verification (see Figure 8 and 9) but reducing the numbers of α-players to 2k − 1 and 5k − 1, respectively.Then there is an exact cover for B if and only if Γ, as defined above, is not popular.

Perfectness
Turning now to perfectness, we start with the SF model.Proof.From left to right, assume that the coalition structure Γ ∈ C N is perfect.It then holds for all agents i ∈ N and all coalition structures ∆ ∈ C N , ∆ = Γ, that i weakly prefers Γ to ∆.It follows that v i (Γ) ≥ v i (∆) for all ∆ ∈ C N , ∆ = Γ, and i ∈ N. Hence, every agent i ∈ N has the maximal valuation v i (Γ) = n • |F i | and is together with all of her friends and none of her enemies.This implies that each coalition in Γ is a connected component and a clique.
The implication from right to left is obvious.
Since it is easy to check this characterization, perfect coalition structures can be verified in polynomial time for sum-based and min-based SF ACFGs.It follows directly from Theorem 14 that the corresponding existence problem is also in P.

Corollary 15. For any sum-based or min-based SF ACFG (N, ) with an underlying network of friends G, there exists a perfect coalition structure if and only if all connected components of G are cliques.
We further get the following upper bounds.Proposition 16.For any ACFG, perfectness verification is in coNP.
Proof.Consider any ACFG (N, ).A coalition structure Γ ∈ C N is not perfect if and only if there is an agent i ∈ N and a coalition structure ∆ ∈ C N such that ∆ ≻ i Γ.Hence, we can nondeterministically guess an agent i ∈ N and a coalition structure ∆ ∈ C N and verify in polynomial time whether ∆ ≻ i Γ.
Furthermore, we initiate the characterization of perfectness in ACFGs.The diameter of a connected graph component is the greatest distance between any two of its vertices.For sum-based EQ ACFGs, we get the following implication.
Proposition 17.For any sum-based EQ ACFG with an underlying network of friends G, it holds that if a coalition structure Γ is perfect for it, then Γ consists of the connected components of G and all these components have a diameter of at most two.
Proof.We first show that, in a perfect coalition structure, all agents have to be together with all their friends.For the sake of contradiction, assume that Γ is perfect but there are i, j ∈ N with j ∈ F i and j / ∈ Γ(i).We distinguish two cases.
Case 1: All f ∈ F i ∩ Γ(i) have a friend in Γ( j).Consider the coalition structure ∆ that results from the union of Γ(i) and Γ( j), i.e., ∆ = Γ \ {Γ(i), Γ( j)} ∪ {Γ(i) ∪ Γ( j)}.It holds that i and all friends of i's either gain an additional friend in ∆ or their coalition stays the same: First, i keeps all friends from Γ(i) and gets j as an additional friend.Hence, i has at least one friend more in ∆ than in Γ and we have v i (∆) > v i (Γ).Second, all friends f ∈ F i ∩ Γ(i) have a friend in Γ( j) and therefore also gain at least one additional friend from the union of the two coalitions.Hence, v f (∆) > v f (Γ) for all f ∈ F i ∩ Γ(i).Third, all friends f ∈ F i ∩ Γ( j) have i as friend.Hence, they also gain one friend from the union.Thus v f (∆) > v f (Γ) for all f ∈ F i ∩ Γ( j).Finally, all f ∈ F i who are not in Γ(i) or Γ( j) value Γ and ∆ the same because their coalition is the same in both coalition structures.Hence, v f (∆) = v f (Γ) for all f ∈ F i with f / ∈ Γ( j) and f / ∈ Γ(i).Summing up, we have u sumEQ i (∆) > u sumEQ i (Γ), so i prefers ∆ to Γ, which is a contradiction to Γ being perfect.
Case 2: There is an f ∈ F i ∩ Γ(i) who has no friends in Γ( j).Consider the coalition structure ∆ that results from j moving to Γ(i), i.e., ∆ = Γ j→Γ(i) .Let k ∈ F i ∩ Γ(i) be one of the agents who have no friends in Therefore, k prefers ∆ to Γ, which again is a contradiction to Γ being perfect.Next, assume that Γ is perfect but there is a coalition C in Γ that has a diameter greater than two.Then there are agents i, j ∈ C with a distance greater than two.Thus j is an enemy of i's and an enemy of all of i's friends.It follows that i prefers coalition structure Γ j→ / 0 to Γ, which is a contradiction to Γ being perfect.Summing up, in a perfect coalition structure Γ for a sum-based EQ ACFG every agent is together with all her friends and every coalition in Γ has a diameter of at most two.Together this implies that Γ consists of the connected components of G and all these components have a diameter of at most two.
From Propositions 16 and 17, we get the following corollary.
However, Proposition 17 is not an equivalence.The converse does not hold, as the following example shows.Hence, Γ is not perfect.

Conclusions and Open Problems
We have proposed to extend the models of altruistic hedonic games due to Nguyen et al. [1] and Wiechers and Rothe [5] to coalition formation games in general.We have compared our more general models to altruism in hedonic games and have motivated our work by removing some crucial disadvantages that come with the restriction to hedonic games.In particular, we have shown that all degrees of our general altruistic preferences are unanimous while this is not the case for all altruistic hedonic preferences.Furthermore, all our sum-based degrees of altruism fulfill two types of monotonicity that are violated by the corresponding hedonic equaland altruistic-treatment preferences.
We have furthermore studied the common stability notions and have initiated a computational analysis of the associated verification and existence problems (see Table 2 for an overview of our results).We also gave characterizations for some of the stability notions, using graph-theoretical properties of the underlying network of friends.For future work, we propose to complete this analysis, close all gaps between complexitytheoretic upper and lower bounds, and get a full characterization for all stability notions.

Figure 1 :
Figure 1: Network of friends for Example 1 For example, for n = 10 agents, we have B 10 = 115,975 possible coalition structures.

Figure 2 :
Figure 2: Network of friends for Example 2

2 Figure 3 :
Figure 3: Networks of friends in the proof of Theorem 6

Figure 4 :
Figure 4: Networks of friends for Example 10

Theorem 9 .
Let (N, SF ) be a (sum-based or min-based) SF ACFG with the underlying network of friends G. Let further C 1 , . . .,C k be the vertex sets of the connected components of G. Then Γ = {C 1 , . . .,C k } is strictly core stable (and thus core stable).

Figure 5 :
Figure 5: Network of friends in the proof of Theorem 11 that is used to show coNP-hardness of core stability verification in min-based SF ACFGs.A dashed rectangle around a group of players indicates that all these players are friends of each other.

Case 2 :
There are some ζ -players in C that are β b 's friends.Since β b has three ζ -friends in total and no enemies in C, there are between one and three ζ -players in C. Hence, there are between 3k − 3 and 3k − 1 β -players in C. Then one of the β -players has no ζ -friend in C. (The at most three ζ -players are friends with at most nine β -players, but 3k − 3 > 9 for k > 4.) Consequently, this β -player has only the other (at most 3k − 2) β -players as friends in C and does not prefer Γ C→ / 0 to Γ.This is a contradiction.Hence, option (a) holds for each β b ∈ C. In total, each β b has exactly three ζ -friends and 3k − 1 β -friends.Thus at least 3k − 3 of β b 's friends in C are β -players and at least one of β b 's friends in C is a ζ -player.Also counting β b herself, there are at least 3k − 2 β -players in C. Since all of these 3k − 2 β -players have at least one ζ -friend in C, there are at least k ζ -players in C. (Note that k − 1 ζ -players are friends with at most 3k − 3 β -players.)Consider some ζ S ∈ C. Since ζ S has three friends and 4k − 3 enemies in Q S , at most three friends in C, and prefers Γ C→ / 0 to Γ, ζ S has exactly three friends and at most 4k − 3 enemies in C. Hence, C contains at most 4k − 3 + 3 + 1 = 4k + 1 players.So far we know that there are at least 3k − 2 β -players in C. If C contains exactly 3k − 2 (or 3k − 1) β -players then each of this players has only 3k − 3 (or 3k − 2) β -friends in C and additionally needs at least three (or two) ζ -friends in C. Hence, we have at least (3k − 2) • 3 = 9k − 6 (or 6k − 2) edges between the β -and ζ -players in C. Then there are at least 3k − 2 (or 2k) ζ -players in C. Thus there are at least (3k − 2) + (3k − 2) = 6k − 4 (or 5k − 1) players in C which is a contradiction because there are at most 4k + 1 players in C. Hence, there are exactly 3k β -players in C. Summing up, there are exactly 3k β -players, at least k ζ -players, and at most 4k + 1 players in C. Hence, there are k or k + 1 ζ -players in C. For the sake of contradiction, assume that there are k + 1 ζ -players in C.

Figure 6 :Figure 7 :
Figure 6: Network of friends in the proof of Theorem 11 that is used to show coNP-hardness of core stability verification in sum-based SF ACFGs.A dashed rectangle around a group of players indicates that all these players are friends of each other.

Figure 8 :
Figure 8: Network of friends in the proof of Theorem 13 that is used to show coNP-hardness of strict popularity verification in min-based SF ACFGs.A dashed rectangle around a group of players indicates that all these players are friends of each other.

Figure 9 :
Figure 9: Network of friends in the proof of Theorem 13 that is used to show coNP-hardness of strict popularity verification in sum-based SF ACFGs.A dashed rectangle around a group of players indicates that all these players are friends of each other.

Theorem 14 .
For any sum-based or min-based SF ACFG (N, ) with an underlying network of friends G, a coalition structure Γ ∈ C N is perfect if and only if it consists of the connected components of G and all of them are cliques.

Figure 10 :∑
Figure 10: Network of friends for Example 19

Table 1 :
Values for the game in Example 2 with the network of friends in Figure2v 1 v 2 v 5 v 6 sum F

Table 2 :
Complexity results in sum-based and min-based SF ACFGs is in coNP for sum-based EQ ACFGs 1. under sum-based SF, sum-based EQ, sum-based AL, min-based SF, or min-based AL preferences if and only if it holds for all players i ∈ N that Γ(i) contains a friend of i's or i is alone, formally: 3