When basing the decision about whom players share a task with on these players’ preferences, a well-established model are hedonic games (ban-kon-soen:j:core-simple-coalition-formation-game; bog-jac:j:stability-hedonic-coalition-structures). As a form of a coalition formation game, players aim to partition into a so-called coalition structure. In a hedonic game the individual preferences over such partitions only depend on coalitions, i.e. sets of players, they belong to. The key idea of this paper is to model a distance-based representation of hedonic games and study it from an axiomatic and computational point of view.
In general, the number of possible coalitions is exponential in the number of players. Hence, from an algorithmic point of view, it is relevant to find reasonable preference representations that are succinct but also expressive (cha-elk-woo:b:ccgt). In very large settings, especially, it is reasonable to assume that for each player the number of known players is small, e.g., bounded by a constant (pet:graphical-hg-bounded-tw:16; fic-kri-rey:c:hgpt:2019). Furthermore, we assume a subdivision of these known co-players into those appreciated (friends) and those disapproved of (enemies) (see, e.g., dim-etal:hg-fe:06). A player prefers to share a coalition with their friends rather than their enemies. Additionally, we follow the encoding proposed by lan-rey-rot-sch-sch:c:hgopt in which each player ranks their friends and enemies, respectively. Their model of how a player compares two coalitions is based on the polarized responsive extension principle which we are going to discuss in Section 5. This encoding allows the expression of a variety of opinions while at the same time players do not have to specify a detailed information, such as a numerical evaluation or knowledge of the whole player set.
The model at hand calculates the distance between a player’s preference and a coalition which expresses the player’s dissatisfaction with this coalition. While distance-based preferences play an important role in decision making, preference extensions in hedonic games have not, to the best of our knowledge, so far, been based on ordinal distances. We attend to the well-studied Kendall-tau distance which counts the minimal necessary swaps to transform one preference into another. In order to allow for indifferences in a player’s preference we consider the Hausdorff-Kendall-tau distance. To this end a comparable encoding of a preference and a coalition is necessary. We show that this model meets desired properties and compare it with other known hedonic game representations in terms of expressibility and computational complexity of stability problems.
1.1 Related Work
Hedonic games have been introduced by ban-kon-soen:j:core-simple-coalition-formation-game and bog-jac:j:stability-hedonic-coalition-structures and studied from an axiomatic and algorithmic point of view since (see, e.g. cha-elk-woo:b:ccgt; azi-sav:b:handbook-hg). Applications can be found in particular instances, such as stable roommate problems (stable-roommates; irv:j:stable-roommates) and group activity selection problems (DarGro18). Several convenient representations encode preferences over individual players which are then extended to preferences over coalitions. They include network approaches (dim-etal:hg-fe:06), numerical approaches (such as additively separable encodings bog-jac:j:stability-hedonic-coalition-structures) and ordinal approaches (such as the singleton encoding cec-rom:j:singleton-encoding). More recently, encodings have been considered that do not assume that each player knows every other player in the game, but that they only know a subset of players and consider the others as neutral (ota-etal:hg-fen:17; pet:graphical-hg-bounded-tw:16; lan-rey-rot-sch-sch:c:hgopt), sometimes referred to as FEN-hedonic games. Hedonic games with ordinal preferences and thresholds (lan-rey-rot-sch-sch:c:hgopt; ker-lan-rey-rot-sch-sch:j:hgopt) combine trichotomous preferences with ordinal preferences. In this paper we refer to this encoding. So far, the extensions to coalitions comprise a set of possible and necessary extensions as well as a numerical approach involving Borda scores (lan-rey-rot-sch-sch:c:hgopt). The additional requirement of a constant number of known players enables computational lower bounds to decrease and we cannot transfer proof techniques to our model immediately. pet-elk:c-coopmas:simple-causes-complexity-hg study causes of complexity for hedonic games. Unfortunately, the results are not applicable here. We do, however, fulfil the requirements of a graphical hedonic game (pet:graphical-hg-bounded-tw:16). Consequently, several problems regarding stability of a game are fixed-parameter tractable with respect to treewidth and degree of the underlying dependency graph. Note that social distance games (social-distance-games) and distance hedonic games (fla-kod-ols-var:c:distance-hedonic-games) are a different approach concerning distances of players in a graph.
Preference extensions from ordinal preferences over single items to a preference order over subsets of items (see, e.g. bar-bos-pat:b:ranking-sets-of-objects) find applications in various topics within the field of Computational Social Choice (bra-con-end-lan-pro:b:handbook-comsoc). Distance-based approaches in order to define these extensions are suggested in several contexts, but have not, to the best of our knowledge, yet been modelled for hedonic games. Closely related to our research is the field of committee elections (see, e.g., fal-sko-sli-tal:b:multiwinner-voting-trends-comsoc) in which a subset of candidates, the so-called committee, is elected based on voters’ preferences. In this context bra-kil-san:j:minimax-electing-committees proposed to use distances as closeness measures and established a minimax approach, i.e., the election winner minimizes the maximal distance between preferences and committee. As in our work, this method includes the necessity of representing the committee (as in our case the coalition) as well as the preference in the same form. Their approach, however, regards votes which solely express binary opinions (approval or disapproval) of candidates. Thereby, it suffices to make use of the Hamming distance. This approach was extended to other forms of preferences, including linear orders (bau-den:c:voter-dissatisfaction-committee-elections). Similarly, in judgment aggregation (see, e.g. end:b:handbook-ja), aiming at a collective judgment set out of individual judgments, distance-based procedures measure the Hamming distance of a complete and consistent judgment set to the individual judgments and return the one minimizing the sum of distances.
We refer to a number of axiomatic properties which are desirable for a hedonic game. These are in parts inspired by different fields of Computational Social Choice as described above and preference-based matching. Fairness properties such as anonymity and nonimposition stem from a voting context. Responsiveness as introduced by roth:j:responsive in the context of many-to-one matching markets refers to a guaranteed benefit from replacing a former match by a match with a preferred item. Basic properties for extensions to subsets of any size are defined by bar-bos-pat:b:ranking-sets-of-objects and bos-sch:j:minimal-paths-on-ordered-graphs, whose extension principle is defined for polarized preferences by lan-rey-rot-sch-sch:c:hgopt.
We define the model of hedonic games with distance-based preferences in Section 3: In order to extend a player’s opinion encoded by polarized ordinal preferences over a subset of known players (3.1) to a preference order over coalitions which constitute the game (3.4), we represent a coalition also as a preference order from the player’s perspective (3.2), and compute a distance between these two preference orders (3.3). We support this definition by an easily accessible characterization. In Section 4 we discuss why this particular coalition representation and distance outperforms other approaches. In Section 5 we analyse our model axiomatically: Desired properties are fulfilled, for instance, that a coalition cannot become less favourable when a preferred player enters the coalition. We observe this distance-based approach to be a reasonable and natural completion of existing compact representations which is at the same time flexible enough in terms of expressivity. We study computational aspects of individual-based stability in Section 6 and show that, e.g., Nash stability of a given outcome can be verified in linear time while it is NP-complete to decide whether there exists a Nash-stable outcome in a given game.
Our work is based on the game-theoretic study of hedonic games as well as on distance measures which are outlined in the following two subsections.
2.1 Hedonic Games
We consider a class of cooperative games, where players want to partition into subsets of players, so-called coalitions. A hedonic game (bog-jac:j:stability-hedonic-coalition-structures; ban-kon-soen:j:core-simple-coalition-formation-game) is such a coalition formation game with a set of players and a preference profile such that each player ’s preference only depends on the coalitions is contained in. The set of such coalitions is denoted by . A partition of the players into subsets is called a coalition structure and denotes the coalition in containing player .
A common goal of these games is to achieve stable coalition structures no player or group of players has an intention to deviate from. A coalition structure is called Nash stable, if there is no player who prefers another coalition with to . If a deviation is restricted to coalitions where is welcome ( for each ) and if there is no such deviation, is called individually stable. If additionally can only deviate if not bound to a contract in ’s former coalition ( for each ), is called contractually individually stable. Moreover, is called perfect if each is one of player ’s favourite coalitions ( for each ). For an analysis of stability, usually, two decision problems are distuinguished: the verification problem of whether a given coalition structure is stable, and the existence problem of whether a game allows a stable coalition structure.
In order to achieve succictness, often a player’s opinion is encoded as a preference order over the set of players which can be extended to a preference order over coalitions. We consider to be an ordinal preference order with possible indifferences, i.e., a linear order (reflexive, transitive), that is not necessarily antisymmetric. For two players , is weakly preferred to if ; is (strictly) preferred to if (i.e., and not ); and and are considered indifferent if (i.e., and ). Similarly, for , we have if and only if and not . By we denote the restriction of to the players within . Let the rank of a player be the position among equally liked players within a preference order: is among the top-ranked players (of rank ) if for each ; is of rank if there exist other players such that , but no players such that . Due to indifferences, several players can share the same rank, such that the maximal rank is possibly smaller than .
2.2 Distance Measures
We study the distance between two preference orders, i.e., linear orders over the set of players. To be precise, in our setting a player’s opinion is compared to a coalition, the former given in form of a preference order over a subset of players; the latter interpreted as a preference order from the player’s perspective. Before we go into detail in Section 3, we present background on how to compare preference orders in general. A distance measure on a space is a metric , i.e. for each it fulfils non-negativity (), identity of indiscernibles (), symmetry (), and the triangle inequality (). A directed distance is a quasimetric which does not fulfil symmetry, or a pseudoquasimetric, if it additionally does not fulfil identity of indiscernibles.
For strict preference orders, that is, antisymmetric linear orders, a well-known distance metric is
the Kendall-tau distance. Based on Kendall’s measure of rank correlation (ken:j:a-new-measure-of-rank-correlation), it
calculates the minimal number of inversions of adjacent players necessary to convert one
strict preference order into another, :
In our context, preference orders allow indifferences (i.e., are not necessarily antisymmetric).
Any such order can be interpreted as an equivalence class containing all strict preferences orders which are consistent with (i.e., if )
and each indifference is resolved by some permutation (either or , if and ).
In order to include indifferences, we consider a Hausdorff distance (hau:b:mengenlehre).
The Hausdoff–Kendall-tau distance is defined by
for two preference orders and interpreted as equivalence classes of strict preferences with maximal ranks and , respectively. Intuitively, for the first part, a worst-case resolution of is chosen for which the the minimal number of swaps between some resolving and is maximized. For our interpretation of a coalition depending on a player’s perspective, we only consider this first part; see Section 3 for details. We denote this directed distance by
where the latter is a characterization (see, e.g.,cri:b:metric-ranked-data) with the number of items that are ranked on position in and on position in . While in general is a pseudoquasimetric, the directed distance as defined in Section 3.3 allows a quasimetric. See Section 5 for details. This matches the Hausdorff-Kendall-tau metric via .
3 The Model
This section presents a compactly encoded model of hedonic games which yet gives credit to each player’s detailed opinion on a subset of candidates. The preference extension is based on calculating the Kendall-tau distances between a player’s preference and coalitions containing this player. Thus, a player compares two coalitions by ranking the coalition with minimal distance, i.e., the minimal dissatisfaction, highest. We create a hedonic game with distance-based preferences as follows: Given a player’s ordinal preferences over the known players, we define
(3.2) a preference-based encoding of a coalition from a player’s perspective and
(3.3) a distance between preference and coalition from a player’s perspective
which determines the extension to preferences over coalitions for each player.
3.1 Preference Encoding
We define the preference encoding similar to
hedonic games with ordinal preferences and
thresholds (lan-rey-rot-sch-sch:c:hgopt). For each player the set of other players is partitioned into accepted players (friends),
unaccepted players (enemies),
and neutral (e.g. unknown) players .
This can be represented by a directed graph, the underlying dependency graph with two edge labels and .
By we denote ’s neighbourhood
assume that for a constant .
Additionally, each player specifies a partial preference relation over and
a partial preference relation over such that a ballot contains information of
the form , where for any denotes that all players in are
This induces a preference relation : For players it holds that if and only if
and ; and ; and ; and ; or and .
In order to obtain our final preference encoding we
conduct two steps. Firstly, since the neutral players in
do not have any effect on the evaluation
of a coalition we abbreviate the ballot notation by
. Secondly, in order to facilitate a comparison with a coalition, we subdivide the ballot into a part regarding player ’s friends
and enemies, respectively,
This again induces preference orders, namely and in which for all ( for all ) complements (, respectively).
For instance, let know five other players, , , , , and , three of which likes (, , ) and two of which doesn’t like (, ), and let specify preferences with whom to cooperate by .
3.2 Coalition Encoding
In order to compare a player ’s ballot with a coalition , we interpret from
friends in and absent enemies are ranked according to
’s opinion, whereas missing good friends and present unfavourable enemies have a reversed impact.
By we denote player ’s reversed preference order over all players in
such that the separate representation is defined matching the form of two ballots
Note that this is again the abbreviated notion, omitting neutral players which are implicitly ranked equally to player without influencing player ’s preference order over coalitions. By this we allow a partial ballot to be compared to a partial coalition containing the identical subset of players.
Continuing Example 1, from ’s perspective, we can now encode coalitions contained in . For a coalition (and any combination with further neutral players), we obtain and . For , we have and . The coalition containing all neighbouring players, is encoded by and ; containing all friends, by and .
3.3 Distance Between Preference and Coalition
Given representations of both a player’s preferences and a coalition, we calculate their distance by summing up the separate distances regarding all friends and all enemies. Our model defines this distance between and by
for which and ( and ) induce, as defined, equivalence classes containing all antisymmetric linear orders over () dissolving occurring indifferences in the ballots.
Note that we consider the directed distance here, since we have a dependence between the coalition representation and the preferences of a player. Considering the worst case of possible swaps within indifferences in the preference encoding, would contradict this intuition. In fact, it always holds that . Furthermore, note that could be any other function in and , for instance the Euclidean distance. If not stated otherwise, we consider the 1-norm in this paper.
We can characterize this distance as follows.
It holds that
Figure 1 depicts
the characterization of (Equation (1), see also cri:b:metric-ranked-data)
on the left hand side.
Note that each row contains at most two non-zero entries, since all equally ranked players are either positioned in the same rank within or not within the coalition.
each column only has one non-zero entry, since each player in the same rank in the coalition (and outside the coalition) origins from a common rank in the preference.
In the bottom row, marks a fixed position in both orders.
We sum up the following parts.
For each shifting from rank in the preference to in the coalition, there is no non-zero entry with and .
Each shifting from rank in the preference to outside the coalition is multiplied with the entries in the rectangle below and left of and . These are non-zero for (a) each shifting from (i.e., is preferred to ) to any (both inside and outside the coalition) and (b) the -marker in the bottom row.
3.4 Preference Extension
Now, we make use of this distance notion in order to define our game’s preference extension. A player weakly prefers a coalition to a coalition if and only if is at most as far from preference order as . Formally,
Finally, for player in Examples 1, 2, and 3, we obtain the following preferences over the example calitions: is ’s favourite coalition with distance . Coalition is preferred to () which is also reasonable since contains more and better friends, while has an additional enemy. Coalition cannot be compared to that easily with more friends, but also more enemies. The distance, however, is the same for both and . Therefore, and .
Hence, with we obtain a hedonic game with distance-based preferences . In Section 5 we show that this model satisfies desired axiomatic properties.
4 The Model’s Background
In this section we argue the steps toward the above defined model to achieve our aim of finding a Kendall-tau-based preference extension. For a Hausdorff–Kendall-tau approach extending player ’s polarized ordinal preference encoding , we rule out the below described alternative representations. As a starting point we consider a representation of coalition where players within and outside of remain unranked, such as . This way the number of necessary swaps is minimal for player ’s best friend and maximal for their least preferred player. Strongly opposing the advantage of a coalition independent of the player’s preference is the following example which shows that adding a friend to a coalition can make the coalition worse.
Consider player ’s preference . Coalition is preferred to coalition .
Discarding equivalences between all candidates within and outside the coalition leads to the adjusted representation . Two different problems arise in this setting. Firstly, still, adding a friend to a coalition can increase player ’s dissatisfaction as Example 7 shows. Note that reversing the order of players outside the coalition does not change this. Secondly, this ordering contains friends and enemies alike, thus weighting friends higher than enemies, a constraint not going in line with the idea of the model.
Again consider . It holds that .
Resulting from this, we consider distances separately for friends and enemies. We subdivide the preference ballot in the same manner in order to compare ballots which contain the exact same set of players. Keeping the previous coalition representation intact, a subdivision would lead to the coalition ballots and . Example 8 shows that adding a friend again increases the distance.
For , prefers to .
As a last step this postulates that the order of those players who are not in the desired positions is reversed in the coalition, i.e., the order of all friends who are not within the coalition and of all enemies who are within the coalition is reversed. At this point we work with the representations presented in Section 3 for which we show desirable properties in Section 5
Apart from the representation of preferences and coalitions we make use of a directed distance measure. Next to the above discussed argument that the coalition is represented in relation to the preference, an axiomatic analysis underlines the importance of basing the model on a directed version of the Hausdorff-Kendall-tau distance. Example 9 shows, that again adding a friend to a coalition can make a coalition worse when considering the undirected distance.
Consider player ’s preference . Then, player prefers coalition to when calculating the distance via the (undirected) Hausdorff-Kendall-tau distance.
This leads us to both the representation of coalitions and preferences defined in Section 3 as well as to the use of a directed distance.
5 Axiomatic Analysis
In the following, we study properties of the directed distance between a player’s preference and a coalition as well as the preference extension. Since these properties are from one player ’s perspective, we omit the index in this section.
By definition of , naturally satisfies non-negativity and the triangle inequality.
In general, the directed distance does not satisfy the identity of indiscernibles.
For instance, it holds that .
Nevertheless, for each over as divided into and we obtain two unique preference orders and which represent a unique coalition (up to neutral players), such that . Indeed, is ’s favourite coalition (up to neutral players) and contains all of ’s friends and none of ’s enemies.
The distance between a player’s preference order and a coalition is if and only if and .
Moreover, the comparability of two coalitions is efficient since we assume that for each player the number of known players is bounded by a constant. This conforms to the definition of a graphical hedonic game (pet:graphical-hg-bounded-tw:16). In Proposition 4 we explicitly state the calculation of the underlying distances.
The distance between a player’s preference and a coalition can be computed in constant time.
Our model satisfies reflexivity ( for each ) and transitivity ( and implies for each ), since the comparison between the coalitions is based on the relation for natural numbers. As renaming the players has no influence on the preference over coalitions, anonymity is fulfilled by definition. A further property, adapted from the definitions of citizen’s sovereignty and nonimposition in the context of (committee) elections requests the possibility for each coalition to become a player ’s favourite coalition: Nonimposition holds if for a player and each there exists a preference order such that ends up as ’s most preferred coalition. Considering for some , and equal and , respectively. Hence, equals .
Hedonic games with distance-based preferences satisfy nonimposition.
5.1 Changes within the Coalition
Among the many options of how to extend a preference order over single players to an order over coalitions, there are some arguably reasonable basic rules for the comparison of two coalitions which only vary in the exchange or addition of one player (see, e.g. bar-bos-pat:b:ranking-sets-of-objects). For instance, if considers two players to be indifferent, coalitions which differ only in these players are ranked equally by .
Let , , and be players in a hedonic game with distance-based preferences with . Then, for all coalitions , it holds that and .
In the same way, the idea requires that adding a friend to a coalition never downgrades this coalition while adding an enemy never makes it more preferable.
From a player ’s perspective in a hedonic game with distance-based preferences, adding a friend to a coalition always improves this coalition while adding an enemy always makes it less favourable.
Formally, a player’s increasing satisfaction with a coalition when adding a friend to this coalition is shown by proving that for . Via the characterization in Equations 2 and 3 it suffices to observe that . Similarly, to show that for and it suffices that . A weak variant of both findings is also implied by the following general notion. On the basis of these properties and an extension principle for ranked sets of objects by bos-sch:j:minimal-paths-on-ordered-graphs, lan-rey-rot-sch-sch:c:hgopt; ker-lan-rey-rot-sch-sch:j:hgopt define an extension principle for ordinal preference ballots with two thresholds. The general idea is that more and better friends are preferred, while more and worse enemies are less preferred.
Definition 15 (polarized responsive extension principle (ker-lan-rey-rot-sch-sch:j:hgopt)).
Let be player ’s preference order over players . Moreover, let and be two coalitions in . The partial extension principle to preferences over coalitions is defined by
|there exist two injective functions|
We show that our model is compatible with this extension principle. That is, if two coalitions satisfy , it also holds that . Note that the reverse implication is not required here, since allows indecisions between coalitions.
The preferences of a player over coalitions in a hedonic game with distance-based preferences are compatible with .
Let and let and be the two injective functions as in Definition 5.1. We want to show that , i.e., .
Firstly, by Equation (3) for the players in it holds that
Since is injective, contains at least as many players as .
Furthermore, for each it holds that if some satisfies , then . Hence,
By (4) and summing up (5) over each , we obtain . Secondly, for the players in , Equation (2) can be used to argue that . Since , is implied. The sum over can be divided into the sum over all minus the sum over , such that a similar argument as for can be applied. All in all, it holds that . ∎
5.2 Changes within the Preference
If a player’s position is changed within a preference order, it is common to assume certain monotonicity properties (see, e.g. bra-con-end-lan-pro:b:handbook-comsoc). The following relaxation of monotonicity holds: If a player ’s position is improved in a preference ranking, a coalition not containing can only outperform a previously preferred coalition containing by the number of swaps during the improvement.
Let be the preference order of player over the set of players in a hedonic game with distance-based preferences, two players in , and two coalitions with and and . If player changes their preference to by shifting to a better position than , it holds that
Note that we do not obtain a difference of at most . This relates to a classical notion of monotonicity in which a player is improved within player ’s preference. At the same time it is crucial that this improvement is detached from any changes regarding other players. In our setting, however, player ’s improvement is immediately associated with a deterioration of at least one other player. Thus, that notion of monotonicity is not applicable.
5.3 Distinction from other Hedonic Games
The notions of and allow a certain degree of flexibility in the expressivity of hedonic games with distance-based preferences. In its current form, being the -norm of and neither specifies a tendency towards friend appreciation nor enemy aversion, but a combination of both. Either edge case can be expressed by multiplying or with an approriate weight.
Hedonic games with distance-based preferences do satisfy additive separability. In fact, equivalent preferences can be encoded by additive utilities and for players , , and . If is altered to, e.g., the -norm, this no longer holds. Then, we can express independent coalition relations such as for as in Example 1, but .
The model distinguishes from other known extensions of ordinal player preferences. For instance, there exists games for this model (e.g., the relation via ) which cannot be expressed by - or -preferences (cec-rom:j:singleton-encoding). However, there also exist -preferences () and -preferences ( and indifference between and ) which cannot be expressed here.
A frequent question for hedonic games is whether a coalition structure is stable in some sense. The best outcome would be a perfect coalition structure, where each player is in their favourite coalition. We obtain Proposition 18 for the verification and the existence problem for perfection. By Observation 10, we know that each player’s favourite coalition satisfies . Hence, in order to decide whether a given coalition structure is perfect, it has to be verified whether for each which by Observation 11 can be determined in constant time. For the existence problem, the proof is constructive: Similar to a breadth first search, players are added consecutively to coalitions until they either have a conflict or form a perfect coalition structure. Since for each player there are at most known players, the search runs in time.
It can be verified in time whether a given coalition structure in a hedonic game with distance-based preferences is perfect. Furthermore, it can be decided in time whether a given hedonic game with distance-based preferences allows a perfect coalition structure.
A more likely stable outcome is that of a coalition structure no player has an incentive to deviate from. Example 19 shows that a Nash-stable and an individually stable coalition structure do not always exist. However, it can be shown that a contractually individually stable coalition structure always exists. It can be seen that when a player wants to deviate and is welcome in the new coalition and not bound to the former coalition the sum of all players’ distances decreases which can reach a minimum.
Consider a game consisting of five players , , , , and with , , etc. continued rotationally symmetricly. Then for each coalition structure, it can be seen that there is always at least one player who rather wants to play alone or is welcome joining a friend. Thus, no outcome is individually stable and consequently also not Nash-stable.
The verification problem for these stability notions can be decided in polynomial time. For (contractual) individual stability, we need to distinguish between graphs which also have a bounded in-degree (players can only be known by a limited number of players and followers are encoded in the graph) and those which allow an unbounded in-degree (players can have many followers they do not know themselves).
Given a hedonic game with distance-based preferences and a coalition structure , it can be verified whether is
Nash-stable in time in ,
(contractually) individually stable in time in for bounded in-degree,
(contractually) individually stable in time in for unbounded in-degree.
Due to the degree bound , there are at most possible coalitions a player might want to deviate to, namely those containing a friend of ’s, for each , or the singleton coalition. For each player, we only need to find out whether prefers to play in one of those candidate coalitions to ’s current coalition. This can be determined in time independent from by Observation 11. If one player prefers one candidate coalition, is not Nash-stable.
For individual stability, it additionally needs to be verified whether the members of the new coalition welcome . There is only an impact on those players who know . For players with , is welcome (see Proposition 14); for those with , is not welcome. We need to check whether there are any of the latter players in a candidate coalition. If ’s followers are known and the in-degree is also bounded, this can easily be verified independently from . If the incoming edges are not encoded or the in-degree is unbounded, in the worst case we have to ask every player in the candidate coalition whether they don’t like . This requires an additional factor for the running time. The coalition structure is not individually stable, if one player prefers one candidate coalition which does not contain any players who consider an enemy.
Similarly, for contractual individual stability, it additionally needs to be verified whether is not bound to the current coalition. The same distiction for incoming edges needs to be made. If one player prefers one candidate coalition which does not contain any players who consider an enemy, and does not contain any players who consider a friend, is contractually individually stable. ∎
Since hedonic games with distance-based preferences are graphical hedonic games, it follows from a result by pet:graphical-hg-bounded-tw:16 that if additionally the treewidth of the underlying dependency graph is bounded by a constant, it can be decided in time linear in the number of players whether a Nash-stable coalition structure exists. In general, this problem is NP-complete. The NP-hardness reduction is inspired by another result by pet:graphical-hg-bounded-tw:16, adapted to our model and a degree bound of , but an unbounded treewidth. Note that the same proof holds for being the -norm instead of the -norm.
Given a hedonic game with distance-based preferences, it is NP-complete to decide whether a Nash-stable coalition structure exists.
The problem is contained in NP, since, for a chosen coalition structure, it can be verified whether it is Nash-stable in polynomial time in the number of players by Theorem 20.
For the lower bound, we consider a reduction from Exact Cover by Three Sets. We may assume that each element of an instance occurs in at most three sets . Given such an instance, we create a game with players (, for each , and , , , for each ) in linear time with the following preferences for , , , and , respectively (see also Figure 2).
For each there are two players and ,
for each there are four players and , and ,
’s preferences are ,
’s preferences are ,
’s preferences are , and
’s preferences are .
It holds that the original instance allows an exact cover if and only if there exists a Nash-stable coalition structure in the constructed game. If, on the one hand, an exact cover exists, the coalition structure is Nash-stable. In fact, no has an incentive to move, since they are indifferent between playing alone and playing with , but also with a corresponding . Each is as happy as possible, since they are not playing together with or any . Moreover, each cannot improve, since is indifferent between and . The players don’t want to move, as they are in one of their favourite coalitions.
If, on the other hand, a stable coalition structure exists, we can observe the following: For each , , all three , have to play in one coalition, otherwise there exists one that has an incentive to move to another player of that group. Each , , cannot play together with or , otherwise would move to the singleton coalition. Therefore, can only be in a coalition with at most three friends, either the group of players or corresponding players. If the latter are less than three players, moves to the former group. Thus, ’s coalition contains exactly three friends. If some does not play together with one corresponding , then would move to ’s coalition. Hence, there exists an exact cover, namely by those for which plays together with those with . ∎
7 Conclusion and Future Work
We have developed a model of hedonic games based on distances between players ordinal preferences and a suitable representation of coalitions. An axiomatic and computational study indicates that this model is a reasonable completion of compact hedonic game representations. For future work, it is interesting to study this model with respect to further stability concepts, such as the core. Furthermore, we are interested in a comparison to a different preference model, e.g., relaxing the constant bound on known players. Additionally, we can extend the model to other distance norms or adapt further ordinal distance measures. Moreover, the distance-based approach enables a comparison between games, in scenarios which require a distance between games.