Stability and Welfare in (Dichotomous) Hedonic Diversity Games

In a hedonic diversity game (HDG) there are two types of agents (red and blue agents) that need to form disjoint coalitions, i.e., subgroups of agents. Each agent’s preferences over the coalitions depend on the relative number of agents of the same type in her coalition. In the special case of a dichotomous hedonic diversity game (DHDG) each agent distinguishes between approved and disapproved fractions only. We aim at outcomes that are stable against agents’ deviations, and at outcomes that maximize social welfare. In particular, we show that the strict core of a DHDG may be empty even in instances with only three agents, while each HDG with two agents has a non-empty strict core. We also provide several computational complexity results for DHDGs with respect to the number of fractions approved per agent. For instance, we prove that deciding whether a DHDG has a non-empty strict core is NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf {NP}$$\end{document}-complete even when each agent approves of at most three fractions. In addition, we show that deciding whether a DHDG admits a Nash stable outcome is NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf {NP}$$\end{document}-complete even in restricted settings with only two approved fractions per agent—therewith, improving a result in the literature. For the task of maximizing social welfare, we apply approval scores and Borda scores from voting theory. For DHDGs and approval scores, we draw the sharp separation line between polynomially solvable and NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf {NP}$$\end{document}-complete cases with respect to the fixed number of approved fractions per agent. We complement these findings with an NP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsf {NP}$$\end{document}-completeness result for HDGs under Borda scores.


Introduction
In a hedonic diversity game (HDG) there are two types of agents (red and blue agents) that need to form disjoint coalitions, i.e., subgroups of agents.Each agent's preferences over the coalitions depend only on the relative number of agents of the same type in her coalition.An important subclass of hedonic games are so-called Bakers and Millers games (Aziz et al. [3] and Bredereck et al. [13]), where each red agent (baker) wishes to be in a coalition in which the fraction of blue agents (millers) is as large as possible, and each blue agent (miller) wishes to be in a coalition in which the fraction of red agents (bakers) is as large as possible.For another example of a HDG consider the situation in which two academic departments (say R and B) merge into one: while some of the members of R (red agents) are eager to closely collaborate with their new colleagues, i.e., the members of B (blue agents), others might be reluctant to do so and prefer to work on their next project together with other members of R, who they are used to collaborate with; hence, some red agents may prefer a high fraction of blue agents in their coalition, while others prefer a low fraction.Analogously, some of the blue agents prefer a high relative number of red agents in their coalition to a low one, whereas others prefer it the other way round.Similar applications of HDGs appear in student group formation (e.g., among local and exchange students) and international collaboration (see Bredereck et al. [13]).
We are hence concerned with a coalition formation problem, where the aim would be a reasonable outcome, i.e., partition of agents into disjoint coalitions.In this respect we take into account two kinds of solution concepts.On the one hand, we consider concepts that deal with stability against agents' deviations: here, we focus on the concepts of Nash stability, a famous concept of stability against individual deviations, and strict core stability, which is concerned with stability against group deviations.On the other hand, by adapting scores from voting theory we aim at outcomes that maximize the induced social welfare (i.e., sum of scores).In this work we lay particular focus on the setting of a dichotomous hedonic diversity game (DHDG), the special case of a HDG in which each agent, in a binary way, states her preferences by distinguishing only between "good" fractions, i.e., fractions she approves of, and "bad" fractions, i.e., fractions she disapproves of.
In this paper, we show that deciding whether a DHDG admits a Nash stable outcome is NP-complete, even when each agent approves of only two fractions (see also Table 1 Overview of computational complexity results for different stability notions; the results provided in this paper are in bold.In the table, "in P" indicates that a respective outcome always exists and can be found in polynomial time."NP-c" means that the corresponding decision problem is NP-complete Is there a stable outcome?HDG DHDG Nash stable NP-c [13] NP-c for s approvals per agent, for any s ≥ 2 individually stable in P [9] i n P [9] core stable NP-c [13] i n P [8] strictly core stable NP-c NP-c for s approvals per agent, for any s ≥ 3 Table 1).Therewith, we improve upon a result in the literature.In addition, we prove that for instances with two agents the strict core of a HDG-and thus of a DHDG-is always non-empty, while for any number n ≥ 3 there is an instance of a DHDG with n agents in which the strict core is empty, even when each agent approves of only one fraction.Next, we show that the corresponding decision problem whether a DHDG admits a non-empty strict core is NP-complete, even when each agent approves of three fractions only.Adapting scores from voting theory, we then aim at outcomes that maximize social welfare (see also Table 2).In that respect, for approval scores we draw the sharp separation line between polynomially solvable and NP-complete cases with respect to the fixed number of approved fractions per agent: when each agent approves of exactly one fraction, by a non-trivial reduction to the two-constraint knapsack problem we show that an outcome that maximizes social welfare can be found in polynomial time; as soon as the agents may approve of more fractions, the corresponding decision problem becomes NP-complete-more precisely, we prove that NP-completeness holds for any fixed number s ≥ 2 of fractions approved per agent.Finally, we show that maximizing social welfare under Borda scores is computationally difficult.

Related literature
Hedonic diversity games.The literature on hedonic diversity games is a very recent one.Introduced by Bredereck et al. [13], the main solution concepts considered in HDGs deal with stability against individual or group deviations such as Nash stability, individual stability, and core stability.For the special case of single-peaked preferences, Bredereck et al. [13] show that a Nash stable outcome may fail to exist, whereas an individually stable outcome always exists and can be found in polynomial time.They also show that the core might be empty even when each agent's preferences are single-peaked, and prove that in general, deciding whether a HDG admits a non-empty core is an NP-complete problem.In a follow-up paper, Boehmer and Elkind [9] focus on stability against individual deviations by considering Nash stability and individual stability.They generalize a previous result by Bredereck et al. [13] by showing that an individually stable outcome is guaranteed to exist in any HDG, and that such an outcome can be found in polynomial time.On the negative side, Boehmer and Elkind [9] show that it is NP-complete do decide whether a DHDG-and thus a HDG-admits a Nash stable outcome, even when each agent approves of at most 4 fractions.In this work, we sharpen that result and take a step towards a computational complexity dichotomy with respect to the fixed number of approved fractions 123 per agent: we show that the decision problem whether a DHDG admits a Nash stable outcome is NP-complete even in a restricted setting in which each agent approves of at most 2 fractions.More generally, we prove that for any fixed number s ≥ 2 the corresponding decision problem is NP-complete even when each agent approves of exactly (or at most) s fractions.Turning to stability against group deviations, it is known that-in contrast to HDGs-every DHDG has a non-empty core (this observation follows from a more general result for dichotomous hedonic games (see Aziz et al. [2] and also Peters [24])); Boehmer [8] then provides a polynomial time algorithm to find such an outcome.In this paper, we prove that in contrast to the above finding for the core, the strict core of a DHDG might be empty, even in instances with only three agents each of which approving of only one fraction.From a computational complexity perspective, we show that the corresponding decision problem of whether a DHDG admits a non-empty strict core is NP-complete even in a restricted setting with exactly (or at most) s approved fractions per agent, for any fixed number s ≥ 3.
A hedonic game is a coalition formation game, in which each agent has preferences over the members of her coalition (for a survey, see, e.g., Aziz and Savani [1]).In anonymous hedonic games, the agents are only concerned with the size of their possible coalitions (and not with the identity of the members in their coalition).For different stability notions such as Nash, individual, and (strict) core stability, stable coalition formation in (anonymous) hedonic games has been well-studied from a computational complexity viewpoint, for instance by Bogomolnaia and Jackson [10], Ballester [4], Olsen [23], and Peters [24].In fractional hedonic games, each agent links a certain value with each other agent; an agent's value of her coalition is then given by the average value of the other agents' values in the coalition.For fractional hedonic games, the computational complexity of finding stable outcomes or deciding whether stable outcomes exist has also been studied in several papers including the ones by Bilò et al. [6], Brandl et al. [12], and Aziz et al. [3].We point out that a Bakers and Millers game can be understood as a special case of a fractional hedonic game (see also Bredereck et al. [13]); in a Bakers and Millers game, a finest partition in the strict core-which is guaranteed to be non-empty-can be determined in linear time (Aziz et al. [3]).Note that in contrast to the settings of an anonymous hedonic game and a fractional hedonic game, in the problem of a HDG considered in our paper we are concerned with two types of agents who have preferences over the possible fractions of agents of their own type.
Finally, we remark that positional scores from voting theory (see Brams and Fishburn [11] for a survey), originally designed to determine the winner(s) of an election, have been applied to several other settings in order to evaluate outcomes.Most prominently, approval and Borda scores have been used, for instance, in combinatorial optimization problems such as the traveling salesperson problem (Klamler and Pfer-schy [22]), in fair division problems (see, e.g., Baumeister et al. [5], Darmann and Schauer [16], and Kilgour and Vetschera [21]), and in the group activity selection problem (Darmann [14]).
The structure of this paper is as follows.In Section 2 we formally introduce the model of a (dichotomous) hedonic diversity game and the solution concepts considered.In Sections 3 and 4 we consider dichotomous hedonic diversity games: in Section 3 we present our results for Nash stability and strict core stability, and in Section 4 we focus on outcomes maximizing social welfare measured by means of the total number of approvals.In Section 5, we consider the problem of maximizing social welfare under the use of Borda scores in hedonic diversity games with strict preferences.A preliminary version of this paper appeared as [15].

Preliminaries
A hedonic diversity game G = (R, B, ( i ) i∈R∪B ) consists of two disjoint sets R, B of agents1 -the agents in R are called red agents, the agents in B are called blue agents-and we set N = R ∪ B. Each agent i ∈ N specifies a weak order i (with indifference part ∼ i and strict preference part i ) over the set of all fractions of red agents in some subset of N containing agent i.Hence, for a red agent we have ; observe that the cardinality of is the same for a red and a blue agent.A subset C ⊆ N is called coalition, and C i denotes the set of all coalitions containing agent i ∈ N .We interpret i as the preferences of agent i over all possible fractions of red agents in some coalition containing her.For coalition C, we denote the fraction of red agents in C by θ R (C).A coalition is mixed if it contains both blue and red agents, otherwise it is pure.A purely red (blue) coalition consists of red (blue) agents only.An outcome π is a partition of R∪ B into disjoint coalitions.For outcome π , let π(i) denote the coalition containing agent i; conversely, we write In a dichotomous hedonic diversity game (DHDG) G = (R, B, (A i ) i∈R∪B ), each agent i specifies a set A i of approved fractions in ; agent i is indifferent between all fractions in A i (i.e., for θ, θ ∈ A i we have θ ∼ i θ ), strictly prefers any θ ∈ A i to any θ / ∈ A i , and is indifferent between all fractions not contained in A i .

Solution Concepts
We will consider two kinds of solution concepts: On the one hand, we take into account the game-theoretic notions of stability against individual and group deviations, where we focus on Nash stability and strict core stability; on the other hand, we apply approval scores and Borda scores from voting theory to our setting in order to determine outcomes that maximize (utilitarian) social welfare, i.e, the total sum of scores.

Stability Notions
Nash stable outcomes require that no agent can make herself better off by forming a singleton coalition or by deviating towards some other coalition.Formally, an outcome π of a hedonic diversity game is Nash stable, if there is no agent i with S ∪{i} i π(i) for some S ∈ π ∪ {∅}.In a DHDG, an outcome π is hence Nash stable if there is no Strictly core stable outcomes require that there is no group of agents S such that, by forming a deviating coalition, at least one member of S is better off while no member of S changes for the worse.This can be formalized as follows.A coalition S ⊆ N weakly blocks an outcome π of N if for every agent i ∈ S we have S i π(i), and for some i ∈ S we have S i π(i).An outcome π is said to be strictly core stable (or in the strict core) if there is no weakly blocking coalition for π .

Social Welfare
The score of outcome π for agent i, sc π (i), is a non-negative integer assigned to π(i).Under given scores, the social welfare of outcome π , SW (π ), is the sum of the scores over all agents: SW (π ) = i∈N sc π (i).We consider the following two kinds of scores.
In a DHDG, the approval score of outcome π for agent i is 1 if θ R (π(i)) ∈ A i and 0 otherwise.In a DHDG, using approval scores the social welfare SW (π ) (or total approval score) of outcome π is hence the number of agents i ∈ N for which θ R (π(i)) ∈ A i holds.
Given a hedonic diversity game G = (R, B, ( i ) i∈R∪B ) with strict preferences i over the set , the Borda score of outcome π for agent i is given by sc  1+1 = 1 2 which r 2 approves of whereas she does not approve of 2 3 , i.e., the fraction of coalition {r 2 , r 3 , b 3 } she is assigned to under π .On the other hand, π is strictly core stable.The only agents that have an incentive to deviate are the agents r 2 and r 3 .Observe that r 2 (and r 3 respectively) is the only agent approving of 1 2 (resp.), which would require a mixed coalition.However, each blue agent would be worse off in a coalition of fraction 1 2 (resp. 3 7 ).Next, consider the outcome μ given by the coalitions {r in a coalition with fraction she approves of and hence has no incentive to deviate; agent r 3 does not approve of 2 3 , 1, or 1  2 , so she does not have an incentive to deviate either; likewise, r 1 (and b 3 respectively) does not approve of 1  2 or2 3 (resp. 1 4 , 1 2 or 0) and hence does not have an incentive to deviate.On the other hand, μ is not strictly core stable because the coalition S = {r 1 , b 1 , b 2 } weakly blocks μ since by forming the deviating coalition S agent r 1 is better off while b 1 and b 2 are not worse off when compared with the coalition assigned under μ.
Finally, observe that the approval score of outcome μ is 4, while the approval score of π is 5.In particular, π is an outcome that maximizes the approval score.

DHDGs: Nash Stability and the Strict Core
In this section, we consider Nash stability and strict core stability in restricted settings with a fixed number of approvals per agent.In that context, we state that even in instances with a small number of approvals per agent such outcomes may fail to exist, and then turn to the decision problems whether a DHDG admits a Nash stable outcome and a strictly core stable outcome respectively.

Nash Stability in DHDGs
Nash stable outcomes may fail to exist in a DHDG even in small instances with only two agents as shown in [9].It is straightforward to generalize their example to any number of agents greater than or equal to two; for the sake of completeness, however, we state this in the proposition below.
Proposition 1 For each n ≥ 2, there is an instance of a dichotomous hedonic diversity game G = (R, B, (A i ) i∈R∪B ) with |N | = n where each agent approves of exactly one fraction that does not admit a Nash stable outcome.
, where the red agent approves of fraction 1 only and each blue agent approves of fraction 1  2 . 2 Any outcome that contains a mixed coalition C is not Nash stable, since the red agent r 1 , who must be a member of C, prefers forming the singleton coalition containing only herself to being engaged in C.However, any outcome with only pure coalitions is not Nash stable, because each of the blue agents would prefer to join the unique purely red coalition {r 1 }.
Our first computational complexity result states that deciding whether a DHDG admits a Nash stable outcome is computationally intractable, even when restricted to instances with at most two approvals per agent and one type of agents approving of mixed coalitions only.

Theorem 1 The problem of deciding whether a dichotomous hedonic diversity game G = (R, B, (A i ) i∈R∪B ) admits a Nash stable outcome is NP-complete, even when (i) each agent approves of at most two fractions and (ii) none of the red agents approves of a purely red coalition.
Proof We provide a reduction from Exact Cover by 3-Sets (X3C).An instance of X3C is a pair (X , Y), where X = {1, . . ., 3q} and Y = {Y 1 , . . ., Y p } is a collection of 3-element subsets (3-sets) of X ; it is a "yes"-instance iff X can be covered by exactly q sets from Y. We assume that every element of X appears in exactly three sets in Y; X3C is known to be NP-complete even under this restriction [18].Observe that the restriction implies p = 3q, which allows us to omit q.Given such a restricted instance of X3C, we construct an instance G = (R, B, (A i ) i∈R∪B ) of a dichotomous hedonic diversity game as follows.We set the three sets of Y that contain x k .We identify x k ∈ X with the six agents b k,t and r k,t , and the set Y k t with the fraction 4+3k t 7+3k t .The agents' approved fractions are as follows.For each k, • each red agent ri, j approves of 1 i+1 exclusively.Observe that • each fraction θ = 4+3k i 7+3k i -induced by the 3-set Y k i -is approved of by exactly three blue agents, since for each element x k ∈ Y k i exactly one blue agent approves of θ ; • each fraction θ = 5+3k i 8+3k i is approved of by exactly six red agents, because for each element x k ∈ Y k i exactly two red agents approve of θ .
We show that (X , Y) admits an exact cover by 3-sets from Y iff G has a Nash stable outcome.
Assume (X , Y) admits an exact cover, say Z ⊂ Y, by 3-sets from Y. We derive the following partition of the agents in G: • For each set Y k ∈ Z form a coalition made up of the three blue agents approving of 4+3k 7+3k together with the six red agents approving of 5+3k 8+3k plus exactly (4 + 3k) − 6 arbitrarily chosen red agents ri, j .The remaining 2 p blue agents form the purely blue coalition S. The remaining red agents form singleton coalitions each.
Observe that each blue agent approves of its coalition's fraction, hence no such agent has an incentive to deviate.No red agent approves of 1 or 1  2 p+1 , hence no red agent wants to deviate towards a purely red coalition or towards S. In addition, for any choice of k, ∈ N we have (4+3k)+1 (7+3k)+1 = 1 because otherwise = 8+3k 5+3k = 1 + 3 5+3k in contradiction with , k ∈ N. Therefore, no red agent ri, j has an incentive to deviate towards a mixed coalition.Finally, for any coalition of fraction 4+3k 7+3k , by construction the coalition contains all the six agents r k,t approving of 5+3k 8+3k .Therefore, none of the agents r k,t has an incentive to deviate towards a mixed coalition either.Thus, the partition is Nash stable.
On the other hand, let π be a Nash stable outcome.Let C be a mixed coalition in π .Coalition C must contain exactly the three blue agents approving of its fraction: C cannot contain a blue agent not approving of its fraction since she would otherwise wish to form a singleton coalition instead.Also, each mixed coalition requires at least three blue agents, and in case C contains more than three blue agents at least one of them wishes to form a singleton coalition instead because for each fraction θ = 4+3k 7+3k there are exactly three blue agents approving of θ .Now, we show that π cannot contain a purely blue coalition of size s = 2 p, s ≥ 1. Assume the opposite and let S be such a purely blue coalition of size s = 2 p, s ≥ 1.
Observe that there are exactly (3 p + 1) red agents rs, j approving of 1 s+1 for any choice of s = 2 p.Each agent rs, j hence prefers S ∪ {r s, j } over its current coalition-and hence has an incentive to deviate-unless she is already in a coalition of fraction 1 s+1 .However, it is impossible that each such agent rs, j is in a coalition of fraction 1 s+1 since this would require (3 p + 1) • s > 3 p blue agents.Thus, π cannot contain a purely blue coalition of size s = 2 p.
Hence, there must be at least p blue agents which are engaged in some mixed coalitions.Since each blue agent needs to approve of the fraction of the mixed coalition C she is part of, C must be of fraction θ = 4+3k 7+3k for some k.Also, recall that fraction θ = 4+3k 7+3k is induced by set Coalition C is thus made up of 1. exactly the three blue agents approving of θ , i.e., three agents b u,h u , b v,h v , b w,h w for some choices of h u , h v , h w ∈ {1, 2, 3}, and 2. exactly 4 + 3k red agents including the six red agents who approve of 5+3k 8+3ksay r u,t u , r u,t u r v,t v , r v,t v r w,t w , r w,t w for some choices of t u , tu , t v , tv , t w , tw -since otherwise π is not Nash stable.
Note that any two mixed coalitions must have different fractions since each mixed coalition must have exactly three blue agents, all of which approving of its fraction, and by construction for each fraction there are exactly three such agents.
In addition, observe that due to Point 2. above, for each u at most one of b u,1 , b u,2 , b u,3 can be contained in a mixed coalition: the fact that r u,t u and r u,t u are contained in mixed coalition C with θ = 4+3k 7+3k implies that r u,t u with tu / ∈ {t u , tu } who approves of 5+3 8+3 , = k,cannot be contained in a mixed coalition D = C with fraction 4+3 7+3 because (i) one of {r u,t u , r u,t u } approves of 5+3 8+3 , and (ii) D would need to contain all the six red agents approving of 5+3 8+3 .Since at least p blue agents need to be engaged in some mixed coalition it follows that for each u exactly one of b u,1 , b u,2 , b u,3 is contained in some mixed coalition.Due to 1. that coalition C has to contain also the two other blue agents approving of its fraction.As a consequence, the collection Z of sets Y k for which π contains a coalition of size 4+3k 7+3k forms an exact cover by 3-sets in instance (X , Y).
As we show below, a corresponding hardness result holds for instances in which each agent approves of exactly s fractions, for any choice of s ≥ 2.
Proposition 2 For any fixed s ≥ 2, s ∈ N, the problem of deciding whether a dichotomous hedonic diversity game G = (R, B, (A i ) i∈R∪B ) admits a Nash stable outcome is NP-complete, even when (i) each agent approves of exactly s fractions and (ii) none of the red agents approves of a purely red coalition.
Proof We assume |R|, |B| > s since s is fixed.We adapt the proof of Theorem 1 by letting • each agent b k,t also approve of h h+|B| for 3 ≤ h ≤ s, • each agent r k,t also approve of h h+|B| for 4 The "only if"-part follows analogously to the proof of Theorem 1.For the "if" part, let π be Nash stable.Recall that each blue agent approves of fraction 0. Hence Nash stability implies that each blue agent approves of her coalition's fraction in π , because otherwise she would deviate towards forming a singleton coalition instead.Hence, each mixed coalition C in π must be either of fraction 4+3k 7+3k for some k or of fraction h h+|B| for some 3 ≤ h ≤ s.Observe that the latter is impossible however: by s < |B| = 3 p at least one of the red agents r k,t must be in a purely red coalition because all the blue agents are in C-hence, such an agent r k,t wants to join C. Recalling that 4+3k 7+3k = 1 , for k, ∈ N, the proof then follows analogously to the proof of Theorem 1.

Strict Core Stability in DHDGs
Analogously to the case of Nash stability, in a DHDG also strict core stable outcomes may fail to exist.However, in instances with only two agents the strict core is guaranteed to be non-empty in any HDG and therefore also in any DHDG (Proposition 3).On the other hand, we show that as soon as a third agent (or more agents) emerges, the strict core may be empty (Proposition 4).In addition, we prove that-from a computational viewpoint-it is difficult to decide whether a given instance has a non-empty strict core, even when the number of approved fractions per agent is at most three and one type of agents approves of mixed coalitions only; we conclude the section by showing that hardness also holds for exactly s approved fractions, for any fixed number s ≥ 3.

Proposition 3 Each instance of a hedonic diversity game G = (R, B, ( i ) i∈R∪B ) with two agents has a non-empty strict core.
Proof Let r and b denote the red and blue agent respectively.For agent r , the set of possible fractions is = { 1 2 , 1}, and for agent b the set of possible fractions is = {0, 1  2 }.If both agents have 1 2 top-ranked, then the outcome made up of the grand coalition {r , b} is strictly core stable.Otherwise, the outcome made up of the singleton coalitions {r }, { b} is strictly core stable, because at least one of the agents strictly prefers her singleton coalition over the grand coalition.
Remark.Observe that the above proposition implies that for instances with two agents also the core of a HDG is always non-empty.

Proposition 4
For each n ≥ 3, there is an instance of a dichotomous hedonic diversity game G = (R, B, (A i ) i∈R∪B ) with |N | = n where each agent approves of exactly one fraction such that the strict core is empty.
, where the agents r 1 , r 2 , b 1 approve of fraction 1  2 , and the remaining agents approve of fraction 0. Consider an outcome π .Each of the blue agents approving of fraction 0 must be in a purely blue coalition, otherwise π is not strictly core stable because a respective agent would prefer to form a singleton coalition instead.In addition, each of r 1 , r 2 , b 1 must be in a coalition of fraction 1  2 : for the sake of contradiction, assume the opposite, and let w.l.o.g.r 1 be in a coalition which has some other fraction; then, S = {r 1 , b 1 } (which has fraction 1  2 ) weakly blocks π , because r 1 is made better off and b 1 is not made worse off.However, each of r 1 , r 2 , b 1 being in a coalition of fraction 1  2 requires at least one blue agent different from b 1 , which is ruled out by the fact that each blue agent different from b 1 must be in a purely blue coalition.Therewith, π is not strictly core stable.
We now turn to the decision problem whether a DHDG has a non-empty strict core and prove that this problem is computationally hard even in a restricted setting with at most three approved fractions per agent.

Theorem 2 The problem of deciding whether a dichotomous hedonic diversity game G = (R, B, (A i ) i∈R∪B ) admits a strictly core stable outcome is NP-complete, even when (i) each agent approves of at most three fractions and (ii) none of the blue agents approves of a purely blue coalition.
Proof We provide a reduction from the restricted NP-complete version of Exact Cover by 3-Sets (X3C) used in the proof of Theorem 1.Given such an instance (X , Y) of X3C, where X = {1, . . ., 3q} and Y = {Y 1 , . . ., Y p } is a collection of 3element subsets of X such that every element of X appears in exactly three sets in Y, we construct an instance G = (R, B, (A i ) i∈R∪B ) of a dichotomous hedonic diversity game.Recall that we have p = 3q.We set the three sets of Y that contain x k .We identify x k ∈ X with the agents b k and r k , and we associate set Y i ∈ Y with the fraction 1+3i 4+3i .The agents' approvals are as follows: • for each k, agent b k 's and agent r k 's set of approved fractions is { 1+3k t 4+3k t | 1 ≤ t ≤ 3}, and • for each k and j, agent rk, j 's set of approved fractions is {1, 1+3k 4+3k }.Observe that by construction each fraction 1+3i 4+3i , 1 ≤ i ≤ p, is approved of by exactly three blue agents.We now show that (X , Y) admits an exact cover by 3-sets from Y iff G admits a non-empty strict core.
Assume that in instance (X , Y) there is an exact cover Z by 3-sets.We construct partition π of N as follows.For each set and let each rk, j with Y k / ∈ Z form a singleton coalition.Each of the agents in a singleton coalition approves of its fraction.Observe that C i contains exactly three blue agents and (3 + 3i − 2) red agents.The fraction of coalition C i is hence 1+3i 4+3i which, due to x k ∈ Y i , is approved of by all of its agents.Note that by the fact that Z is an exact cover each of the agents r k , b k is in exactly one mixed coalition.Therefore, each agent is engaged in some coalition and approves of its fraction.Thus, partition π is strictly core stable.
On the other hand, assume that there is a strictly core stable outcome π .For the sake of contradiction, assume that at least one blue agent b k is in a coalition with a fraction she disapproves of.Let Y i ∈ Y denote one of the three sets that contain element x k .As above, form the coalition 4+3i which is approved of by all members of C i .Since b k ∈ C i holds we can conclude that C i weakly blocks π , in contradiction with the assumption that π is strictly core stable.Therewith, each blue agent must be in a coalition with a fraction θ she approves of.Note that by construction (for each blue agent, the denominator of each approved fraction exceeds the numerator by three), this requires that all the three agents approving of θ must be in the same coalition.Thus, the set 4+3k } forms an exact cover by 3-sets in (X , Y).
Proposition 5 For any fixed s ≥ 3, s ∈ N, the problem of deciding whether a dichotomous hedonic diversity game G = (R, B, (A i ) i∈R∪B ) admits a strictly core stable outcome is NP-complete, even when (i) each agent approves of exactly s fractions and (ii) none of the blue agents approves of a purely blue coalition.
Proof We assume |R|, |B| > s since s is fixed.We adapt the proof of Theorem 2 as follows: • each agent b k and each agent r k also approves of |R| |R|+4 , |R| |R|+5 , . . ., |R| |R|+s , • each agent rk, j also approves of 1  1+|B| and of 1 1+(|B|−t) for 4 ≤ t ≤ s.The "only if"-part follows analogously to the proof of Theorem 2. For the "if"-part, analogously to the proof of Theorem 2 it follows that in a strictly core stable outcome, each blue agent must be in a coalition with a fraction θ she approves of.Observe that an outcome that has a coalition of fraction |R| |R|+t for some t ≥ 4 is not strictly core stable, since the agents rk, j would deviate towards a purely red coalition.Hence θ must correspond to some 1+3k 4+3k for some 1 ≤ k ≤ p. Therewith, the proof follows analogously to the proof of Theorem 2.

Maximizing Social Welfare: a Dichotomy for DHDGs
Apart from stability notions, from a social choice perspective an outcome that maximizes social welfare is of interest.In this section, we consider DHDGs and use approval scores to measure the social welfare induced by an outcome.We first show that an outcome that maximizes social welfare, i.e., total approval score, can be found in polynomial time when each agent approves of exactly one fraction.However, we then prove that the corresponding decision problem turns NP-complete already as soon as agents may approve of two fractions.Therewith we draw the sharp separation line between polynomially solvable and NP-complete cases with respect to the fixed number of approved fractions per agent.
We introduce some additional notation.For set N ⊆ N of agents and fraction θ , let R θ (N ) and B θ (N ) denote the set of red and blue agents in N approving of θ respectively.Let #r (N ) and #b(N ) denote the number of red and blue agents in set N respectively.Theorem 3 In a dichotomous hedonic diversity game G = (R, B, (A i ) i∈R∪B ) with approval scores in which each agent approves of exactly one fraction, an outcome that maximizes social welfare can be found in polynomial time.
Proof We will reduce a dichotomous hedonic diversity game with a single approval per agent to a two-constraint knapsack problem 3 .An instance of the two-constraint knapsack problem consists of a set J of items, where each item j ∈ J is associated with a profit p j , a weight w j and a volume v j ; the goal is to select a subset J * ⊆ J of items of maximum total profit p * = j∈J * p j such that the total weight does not exceed a given weight bound W and the total volume does not exceed a given volume bound V (i.e., j∈J * w j ≤ W and j∈J * v j ≤ V ).By dynamic programming, the maximum profit in an instance of the two-constraint knapsack problem can be determined in O(nW V ) time, determining both the optimal profit and the profit maximizing set of items can be done in O(n 2 W V ) time (see Ch. 9.3.2 of [20]).
Given a dichotomous hedonic diversity game G = (R, B, (A i ) i∈R∪B ) with exactly one approval per agent, we construct an instance I of the two-constraint knapsack problem.W.l.o.g., we assume that the fractions in G cannot be reduced anymore, i.e., the numerator and denominator of each fraction θ in G are coprime.
For each fraction θ = r θ r θ +b θ (with 0 = 0 1 and 1 = 1 1+0 ) approved of by at least one agent we first partition the set of agents approving of θ into sets S θ (i) and introduce the items for instance I on basis of these sets.
In order to construct the sets, the idea is that while there are red or blue agents approving of θ , add them to S θ (1) as long as it contains less than r θ red agents (b θ blue agents); then continue with S θ (2) , etc.We proceed as follows: and B θ (N ) exclusively, such that -each such set contains at most r θ red agents and b θ blue agents, the set S θ ( 1) is non-empty, and -for red agent r ∈ R θ (N ) we have r ∈ S θ (i+1) iff S θ (i) contains r θ agents (for blue agent b of B θ (N ) we have b ∈ S θ (i+1) iff S θ (i) contains b θ agents).
We say that set S θ (i) is full, if it contains exactly r θ red agents and b θ blue agents.
Observe that each agent of N is contained in exactly one of the sets S θ (i) (hence we have at most n such sets), and each agent in set S θ (i) approves of θ .blue agents; however, the weight and volume of the corresponding item θ (i) are r θ and b θ , respectively.Hence, for each non-full set S θ (i) ∈ S * the weight contribution of θ (i) exceeds the number of red agents in S θ (i) by r θ − |R ∩ S θ (i) | and the volume contribution of θ (i) exceeds the number of blue agents in S θ (i) by b θ − |B ∩ S θ (i) |.Together with the choice of W = |R| (and V = |B| respectively) it follows that there must be at least red agents and at least blue agents in N that are not contained in some set of S * .Therefore, for all sets S θ (i) ∈ S * which are not full we are able to construct a coalition C θ (i) of fraction θ by "filling up" S θ (i) with such red and blue agents-i.e., create C θ (i) by adding to S θ (i) red and blue agents of N \ S θ (i) ∈S * S θ (i) until it contains exactly r θ red and b θ blue agents.Now let π be the outcome made up of the coalitions C θ (i) for S θ (i) ∈ S * plus coalition D containing all remaining agents.Observe that for each C θ (i) ∈ π at least |S θ (i) | agents in C θ (i) approve of its fraction θ .In addition, recall that by definition p θ (i) = |S θ (i) |.Thus, for outcome π we have In our running example, let J * = { 2 Let S be the set of coalitions C ∈ π in which all agents approve of its fraction θ R (C), and let S be the set of coalitions C for which at least one agent disapproves of θ R (C ).Let N be the set of agents engaged in some coalition C ∈ S , and let N a ⊆ N be the set of agents of N who approve of its coalition fraction, and N d ⊆ N be the set of agents of N who disapprove of its coalition fraction.From π we construct a new partition π by regrouping agents of N as follows: • for all θ approved of by an agent in N a , build the sets S θ (i) from agents in N a analogously to building the sets S θ (i) in the construction of instance I; • Any outcome μ containing a coalition D of fraction h h+|B|−1 for some 1 ≤ h ≤ s − 1 contains h red agents and all but one of the blue agents.Each of the red agents in D may approve of its fraction, but no blue agent does.Besides, each of the agents not in D is either in a purely red coalition or in a coalition E containing exactly one blue agent.There is no agent approving of a pure coalition.As μ cannot contain a coalition of fraction |R| |R|+1 due to the fact that at least one red agent is engaged in D, there is no agent approving of the fraction of coalition E either.Hence μ yields a social welfare of at most s − 1 < 3 p.With these observations, the "only if"-part follows analogously to the proof of Theorem 4.

Maximizing Social Welfare in HDGs Under Borda Scores
We now leave the setting of DHDGs and consider the case in which each agent's preferences are given by means of a strict order over the possible coalition fractions.It turns out that in such a scenario under the use of Borda scores maximizing social welfare is computationally hard.
Theorem 5 Given integer , the problem of deciding whether a hedonic diversity game G = (R, B, ( i ) i∈R∪B ), with strict order i over for i ∈ N , under Borda scores admits an outcome with SW ≥ is NP-complete.

Proof
We reduce from the NP-complete variant of Exact Cover by 3-Sets (X3C) restricted to instances (X , Y) with X = {1, . . ., 3q} and Y = {Y 1 , . . ., Y p } such that every element of X appears in exactly three sets in Y. Let I be such a restricted instance of X3C (recall that p = 3q holds).From I we derive instance G = (R, B, ( i ) i∈R∪B ) of a hedonic diversity game as described below.The set of agents is made up of the sets R = {r i | i ∈ {1, . . ., p 5 }} and B = {b k | 1 ≤ k ≤ p}.Again, for x k ∈ X we denote the three sets containing

and we associate fraction j+3
j+6 with set Y j ∈ Y.The agents' rankings--up to the respective position where fraction 1  |R|+|B| is ranked-are given in Table 3.Let T = | | − 1, i.e., T is the maximum possible Borda score for a single agent.
We claim that I is a "yes"-instance of X3C iff G admits an outcome π with total Borda score SW (π ) ≥ = (T − 2) p + (T − p) p 5 .
"⇒": Let Z be an exact cover by 3-sets in instance I. Consider partition π which • for each Y j ∈ Z forms a coalition C j made up of the three blue agents who have j+3 j+6 among their top 3 ranked fractions together with ( j + 3) arbitrarily chosen red agents, • and assigns the remaining agents (who, by the fact that Z is an exact cover by 3-sets, must all be red agents) to the single coalition D.
By the fact that Z is an exact cover by 3-sets each blue agent is in a coalition with a fraction she ranks first, second, or third.Each red agent is in a coalition of fraction 1 or j+3 j+6 for some 1 ≤ j ≤ p.Thus, we have SW (π ) ≥ (T − 2) p + (T − p) p 5 = .
T − 2 "⇐": Let π be an outcome with SW (π ) ≥ .Note that any outcome in which all blue agents are engaged in the same coalition yields a total Borda score of at most (T − 3) p + (T − p − 1) p 5 < .Hence, any outcome meeting the desired bound splits the set of blue agents into at least two coalitions.
Assume there is a blue agent b k who is not in a coalition with fraction ranked among her top 3 fractions.Since the blue agents are not in a single coalition, the maximum possible Borda score for that agent is sc π (i) = T − 2 − |R| − 1 = T − 3 − p 5 .For the remaining p − 1 blue agents the maximum Borda score is T .Next, observe that any coalition's fraction among the first p ranked fractions of agents r i corresponds to k+3 k+6 for some 1 ≤ k ≤ p, and hence the number of blue agents required in such a coalition is a multiple of 3. Since there are only p blue agents in total, there are less than p the largest possible total Borda score contributed by all red agents is bounded by As a consequence, each blue agent's coalition must have a fraction ranked among her top 3 fractions.Since each such fraction is among the top 3 fractions of exactly three blue agents, each respective coalition requires exactly 3 blue agents (because such a coalition requires a multiple of 3 agents).Therewith, the collection of sets Z = {Y j | ∃C ∈ π : θ R (C) = j+3 j+6 } forms an exact cover by 3-sets in I.

Conclusion
A hedonic diversity game can be understood both as a game-theoretic problem and a social choice problem; accordingly, we have considered two kinds of solution concepts: stability notions that originate from game theory on the one hand, and social welfare which stems from social choice theory on the other hand.With respect to the latter, we have taken into account the two most prominent types of scores from voting theory, namely approval scores and Borda scores.Besides several computational complexity results, we have also shown that-in contrast to the core, which is known to be always non-empty (see [2,24])-the strict core of a dichotomous hedonic diversity game may be empty even in instances with a small number of agents and only one approved fraction per agent.Some interesting questions, however, remain open.For instance, what is the computational complexity of deciding whether a dichotomous hedonic diversity game admits a Nash stable outcome in the case of exactly one approved fraction per agent?How hard is it to decide whether a dichotomous hedonic diversity game admits a strictly core stable outcome when each agent approves of exactly one fraction or at most two fractions?
An interesting direction for future research would also be to study the computational complexity of the considered problems when, instead of the number of approved fractions, the number of disapproved fractions per agent is fixed.E.g., can we derive a similar dichotomy for the task of maximizing social welfare when each agent approves of all but a fixed number of fractions?
More generally, a possible future research direction could be to study which (additional) plausible domain restrictions allow for positive results related to computing stable outcomes or outcomes that maximize social welfare.

Example 1
In the following instance of a DHDG with R = {r 1 , r 2 , r 3 } and B = {b 1 , b 2 , b 3 , b 4 }, each agent approves of exactly one fraction as listed below.

Table 2
Overview of results for maximizing social welfare under approval and Borda scores; the results provided in this paper are in bold.Here, for the problem that is "in P", a respective outcome can be found in polynomial time."NP-c" means that the corresponding decision problem is NP-complete, while "pref." and "p.a." stand for "preferences" and "per agent" respectively DHDG: approval scores in P for 1 approval p. a.; NP-c for s approvals p. a., for any s ≥ 2 HDG with strict pref.: Borda scores NP-c

Table 3
Rankings of agents b k and r i up to fraction