Hedonic games are cooperative games in which players desire to form coalitions, but only care about the makeup of the coalitions of which they are members; they are indifferent about the makeup of other coalitions (Drèze and Greenberg, 1980; Chalkiadakis et al., 2011). Because the specification of a hedonic game requires the expression of each player’s ranking over all sets of players including him, in general, such a specification requires exponential space – and, when used by a centralised mechanism, exponential elicitation time. Such an exponential blow-up severely limits the practical applicability of hedonic games, and for this reason researchers have investigated compactly represented hedonic games. One approach to this problem has been to consider possible restrictions on the possible preferences that players have. For example, one may assume that each player specifies only a ranking over single players, and that her preferences over coalitions are defined according to the identity of the best (respectively, worst) element of the coalition (Cechlárová and Hajduková, 2004; Cechlárová, 2008). One may also assume that each player’s preferences depend only on the number of players in her coalition (Bogomolnaia and Jackson, 2002). These representations come with a domain restriction, i.e., a loss of expressivity: Elkind and Wooldridge (2009) consider a fully expressive representation for hedonic games, based on weighted logical formulas. In the worst case, the representation of Elkind and Wooldridge requires space exponential in the number of players, but in many cases the space requirement is much smaller.
In this paper, we consider another natural restriction on player preferences. We consider hedonic games with dichotomous preferences. The assumption of dichotomous preferences means that each player’s preference relation partitions the set of coalitions of which that player is a member into just two equivalence classes: satisfactory and unsatisfactory. A player is indifferent between satisfactory coalitions, and is indifferent between unsatisfactory coalitions, but strictly prefers any satisfactory coalition over any unsatisfactory coalition.
While to the best of our knowledge dichotomous preferences have not been previously studied in the context of hedonic games, they have of course been studied in other economic settings, such as by Bogomolnaia et al. (2005), Bogomolnaia and Moulin (2004), and Bouveret and Lang (2008) in the context of fair division, by Harrenstein et al. (2001) in the context of Boolean games, by Konieczny and Pino-Pérez (2002) in the context of belief merging, by Bogomolnaia and Moulin (2004) in the context of matching, and by Brams and Fishburn (2007) (and many others) in the context of approval voting.
When the space of all possible alternatives has a combinatorial structure, propositional formulas are a very natural representation of dichotomous preferences. In such a representation, variables correspond to goods (in fair division), outcome variables (Boolean games), state variables (belief merging), or players (coalition formation). In the latter case, which we will be concerned with in the present paper, each player can express her preferences over coalitions containing her by using propositional atoms of the form (), meaning that is in the same coalition as . Thus, for example, player 1 can express by the formula that he wants to be in a coalition with player or with player , but not with player . Our primary aim in this paper is to present such a propositional framework for specifying hedonic games and computing various solution concepts. We will first define a propositional logic using atoms of the form , together with domain axioms expressing that the output of the game should be a partition of the set of players. Then we consider a range of solution concepts, and show that they can be characterised by some specific classes of (sometimes polysize) formulas, and solved using propositional satisfiability solvers. The result is a simple, natural, and compact representation scheme for expressing preferences, and a machinery based on satisfiability for computing partitions satisfying some specific stability criteria such as Nash stability or core stability.
In this section, we recall some definitions relating to coalitions, coalition structures (or partitions), and hedonic games. See, e.g., Chalkiadakis et al. (2011) for an in-depth discussion of these and related concepts.
Coalitions and Partitions
We consider a setting in which there is a set of players with typical elements . Players can form coalitions, which we will denote by . A coalition is simply a subset of the players . One may usefully think of the players as getting together to form teams that will work together. A coalition structure is an exhaustive partition of the players into disjoint coalitions, i.e., and for all such that . For technical convenience, we slightly deviate from standard conventions and require that every coalition structure contains the empty set . We commonly refer to coalition structures simply as partitions. In examples, we also write, e.g., rather than the more cumbersome . For each player in , we let denote the set of coalitions over that contain . If is a partition, then refers to the coalition in that player is a member of.
The notion of players leaving their own coalition and joining another lies at the basis of many of the solution concepts that we will come to consider. We introduce some notation to represent such situations. For a group of players (not necessarily a coalition in ), by we refer to the partition and we write for . Moreover, for a coalition in partition , we use to refer to the partition that results if the players in leave their respective coalitions in and join coalition . We also allow to form a coalition of its own, in which case we write . Formally, we have, for ,
If is a singleton we also write and instead of and , respectively. Thus, e.g., and .
Finally, define as the partition where and exchange their places, i.e.:
Thus, for partition , we have and . Furthermore, and . Also, , , and .
Hedonic games are the class of coalition formation games in which each player is only interested in the coalition he is a member of, and is indifferent as to how the players outside his own coalition are grouped. Hedonic games were originally introduced by Drèze and Greenberg (1980) and further developed by, e.g., Bogomolnaia and Jackson (2002). Also see Hajduková (2006) for a survey from a more computational point of view. Formally, a hedonic game is a tuple , where represents ’s transitive, reflexive, and complete preferences over the set of all coalitions containing . Thus, intuitively signifies that player considers coalition at least as desirable as coalition , where and are coalitions in . By and we denote the strict and the indifferent part of , respectively. The preferences of a player are said to be dichotomous whenever can be partitioned into two disjoint sets and such that strictly prefers all coalitions in to those in and is indifferent otherwise, i.e., if and only if and . A coalition in is acceptable to if (weakly) prefers to coalition , where he is on his own, i.e., if . By contrast, we say that a coalition is satisfactory or desirable for if . Satisfactory partitions are thus generally acceptable to all players. The implication in the other direction, however, does not hold.
We lift preferences on coalitions to preferences on partitions in a natural way: player prefers partition to partition whenever prefers coalition to coalition . We also extend the concepts of acceptability and desirability of coalitions to partitions.
Consider the following Boolean game with four players, , , , and , whose (dichotomous) preferences are as follows. (Indifferences are indicated by commas.)
Thus, player wants to be in a coalition of at least three and player wishes to be in a coalition of exactly three. Moreover, player wants to be in the same coalition as player or as . He does not want to be in a coalition with player . Finally, player does not want to be with players and together. There is exactly one partition that is satisfactory for all four players, namely . For players , , and , all coalitions are acceptable. For player , however, and are unacceptable.
Solution Concepts for Hedonic Games
A solution concept associates with every hedonic game a (possibly empty) set of partitions of . Here we review some of the most common solution concepts for hedonic games.
Individual rationality captures the idea that every player prefers the coalition he is in to being on his own, i.e., that coalitions are acceptable to its members. Thus, formally, is individually rational if, for all players in ,
This condition is obviously equivalent to .
For dichotomous hedonic games, a partition is said to be social welfare optimal if it maximises the number of players who are in a satisfactory coalition, that is, if maximises . In a similar way, a partition is Pareto optimal if it maximises the set of players being in a satisfactory coalition with respect to set-inclusion, that is, if there is no partition with
In the extreme case in which every player is in a most preferred coalition, is said to be perfect (cf., Aziz et al., 2013). A perfect partition satisfies any other of our stability concepts.
A partition is Nash stable if no player would like to unilaterally abandon the coalition he is in and join any other existing coalition or stay on his own, that is, if, for all and all ,
Observe that this condition is equivalent to .
Core stability concepts consider group deviations instead of individual ones. A group of players, possibly from different coalitions, is said to block a partition if they would all benefit by joining together in a separate coalition. Formally, blocks (or is blocking) partition if, for all ,
Thus, blocks if and only if for all . A group weakly blocks (or is weakly blocking) if holds for all and holds for some . Then, is core stable if no group is blocking it and is strict core stable if no group is weakly blocking it.
Partition is envy-free if no player is envious of another player, that is, if no player would prefer to change places with another player . Formally, partition is envy-free if, for all players and ,
If we also say that player envies player .
Example 1 (continued)
In our example, in partition each player is in a most preferred coalition. As such is perfect as well as social welfare optimal and satisfies all solution concepts mentioned above. Moreover, all partitions except and individually rational.
Now, consider partition . Here, player does not want to abandon her coalition and join another as she prefers none of the following partitions to : , , , and . As, however, and , partition is not Nash stable.
Also observe that for the group is strongly blocking, as and for all . Thus, is not core stable. By contrast, is core stable as only player and are not satisfied and both of them will only be if they can form a blocking coalition of exactly three. However, is still weakly blocking, and as such is not strict core stable.
For envy-freeness, consider partition . Then, player envies player , as and . By contrast, player does not envy player : we have but not .
3 A Logic for Coalition Structures
In this section, we develop a logic for representing coalition structures. We will then use this logic as a compact specification language for dichotomous preference relations in hedonic games.
Given a set of players, we define a propositional language built from the usual connectives and with for every (unordered) pair of distinct players a propositional variable . The set of propositional variables we denote by . Observe that . For notational convenience we will write for . Thus, and refer to the same symbol. The language is interpreted on coalition structures on and the informal meaning of is “ and are in the same coalition”. Formally, the formulas of the language , with typical element is given by the following grammar
where and . By we denote the size of .
For a given coalition of players, we write for the propositional variables in which some appears, i.e.,
Note that for distinct players and we have . The propositional language over we denote by . We write and for and , respectively. The remaining classical connectives , , , , and are defined in the usual way. Moreover, for formulas of formulas, we have and abbreviate and , respectively. We also make use of the following useful notational shorthand:
Thus, conveys that are in the same coalition and each of them in another coalition than . Thus, where , abbreviates and signifies that player is in a coalition of two players.
We interpret the formulas of on partitions as follows.
For , we have if for all implies . If , we write and say that is valid.
Notice that partitions play a dual role in our framework: both their initial role as coalition structures, and the role of models in our logic. This dual role is key to using formulas of our propositional language as a specification language for preference relations. Thus, e.g., partition satisfies the following formulas of : , , , , and .
We have the following axiom schemes for mutually distinct players , , and ,
all propositional tautologies
as well as modus ponens as the only rule of the system:
from and infer . (modus ponens)
Theorem 1 (Completeness)
Let . Then,
|if and only if .|
Soundness is straightforward. For completeness a standard Lindenbaum construction can be used. To this end, assume . Then, is consistent and can as such be extended to a maximal consistent theory . Define a relation such that for all ,
if and only if .
The axiom schemes 3 and i ensure that is a well-defined equivalence relation. Let be the equivalence class under to which player belongs. Then define the partition . By a straightforward structural induction, it can then be shown that for all ,
if and only if .
It follows that and . Hence, .
Alternatively, one can reason with coalition structures in standard propositional logic, by writing the transitivity axiom directly as a propositional logic formula. Let
Then, for any propositional formulas and of ,
that is, checking whether a formula implies another formula in is equivalent to saying that together with the transitivity constraint implies . This means that reasoning tasks in can be done with a classical propositional theorem prover. In what follows we say that two formulas and are -equivalent whenever their equivalence can be proven in , i.e., .
4 Boolean Hedonic Games
The denotation of a formula of our propositional language is a set of coalition structures, and we can naturally interpret these as being the desirable or satisfactory coalition structures for a particular player. Thus, instead of writing a hedonic game with dichotomous preferences as a structure , in which we explicitly enumerate preference relations , we can instead write , where is a formula of our propositional language that acts as a specification of the preference relation . Intuitively, represents player ’s ‘goal’ and player is satisfied if his goal is achieved and unsatisfied if he is not. We refer to a structure as a Boolean hedonic game. Thus, a Boolean hedonic game represents the (standard) hedonic game with for each ,
|if and only if implies .|
Observe that, defined thus, the preferences of each player in a hedonic Boolean game are dichotomous.
It should be clear that every dichotomous preference relation can be
specified by a propositional formula , and hence our
propositional language forms a fully expressive representation
scheme for Boolean hedonic games.111Let be a player with dichotomous preferences and let be the set of coalitions most preferred by , i.e.,
if and only if for all coalitions and containing .
Then, is represented by following formula of in disjunctive normal form:
If a Boolean hedonic game represents a hedonic game with dichotomous preferences, then for every player there is a formula that is -equivalent to . Moreover, if for every player there is a formula that is -equivalent to , then represents a hedonic game with dichotomous preferences.
For a player and a formula in , a straightforward inductive argument shows that
if and only if for all with .
Then, the result follows as a corollary.
Often, the use of propositional formulas gives a ‘concise’ representation of the preference relation , although of course in the worst case the shortest formula representing may be of size exponential in the number of players. In what follows, we will write , understanding that we are referring to the game corresponding to this specification.
Example 1 (continued)
The hedonic game with dichotomous preferences in Example 1 is represented by the Boolean hedonic game with and the players’ goals given by:
For each player we then have that if and only if .
5 Substitution and Deviation
We establish a formal link between substitution in formulas of our language and the possibility of players deviating from their respective coalition in a given partition and joining other coalitions.
We first introduce some formal notation and terminology with respect to substitution of formulas for variables in our logic.
For a propositional variable in and and formulas of , we denote by the uniform substitution of variable by in . If is a sequence of distinct variables in and a sequence of formulas,
denotes the simultaneous substitution of each by ().
Thus, e.g., .
A special case, which recurs frequently in what follows, is if every is a Boolean, i.e., if .
Sequences where we will also refer to as Boolean vectors of length
Boolean vectors of length. Thus, e.g., is a Boolean vector of length and .
Characterising individual deviations
Some of the stability concepts for Boolean hedonic games we consider in this paper, e.g., Nash stability, are based on which coalitions an individual player can join given a partition . Recall that these coalitions are given by . Of course, not all groups of agents are included in . For instance, let partition be given by . Then, player can join coalition but cannot form a coalition with players and by unilaterally deviating from . We find that the set can be characterised in our logic. This furthermore yields a logical characterisation of when a player can unilaterally break loose from his coalition, join another one and thereby guarantee that a given formula will be satisfied. A particularly interesting case is if implies the respective player’s goal. We thus gain expressive power with respect to whether a player can beneficially deviate from a given partition, a crucial concept.
Let be a partition, a player, a group of players in . Let furthermore be a Boolean vector of length and an enumeration of such that . Then,
and iff .
For the “only if”-direction, assume that as well as . Observe that Accordingly, there are some (mutually distinct) , , and such that . It suffices to consider the following three cases.
Case cannot occur as we would have , , , and is a theorem of the system.
If , then It follows that , , and . Observe that in this case . Hence, . Also notice that and, thus, and . Accordingly, but . As and having assumed , a contradiction follows:
If , we have Thus, , , and . Observe that . Hence, . Moreover, , from which follows that and . Accordingly, both . With , we obtain that , a contradiction.
For the “if”-direction, assume and . Because of the latter, there is some . Accordingly, . As , and thus in particular , there are two possibilities:
there is some with and , or
there is some with and .
If , we have as well as . As , it holds that but . If , however, we have and . As , it holds that but . In either case it follows that .
The following example illustrates Lemma 1.
Consider the partition . Then, . Let be a fixed enumeration of . Also let and be Boolean vectors (of length ). Then,
(This may be established, somewhat tediously, by painstakingly checking all conjuncts of the form of .) Now, observe that and that . On the other hand, observe that . It is easily established, however, that does not satisfy and, hence, neither . Finally, observe that and that is not in .
We now introduce the following abbreviation, where is assumed to be a fixed enumeration of .
Thus, can be understood as the operation of forgetting everything about player (in the sense of Lin and Reiter (1994)) while taking the transitivity constraint into account. Intuitively, signifies that given partition player can deviate to some coalition such that that is satisfied.
Let be a partition, a player, and a formula of . Then,
|iff for some ,|
It is important to note, however, that the number of Boolean vectors of length is exponential in . Accordingly, abbreviates a formula whose size is exponential in the size of .
Characterising group deviations
Besides a single player deviating from its coalition and joining another, multiple players (from possibly different coalitions) could also deviate together and form a coalition of their own. This concept lies at the basis of, e.g., the core stability concept. We establish a formal connection between substitution and group deviations.
Let be a group of players. Observe that and let be a fixed enumeration of . By the -separating Boolean vector (given ) we define as the unique Boolean vector of length such that for all and all ,
Intuitively, represents the choice of group to form a coalition of their own. Whenever is clear from the context we omit the subscript in and . The following characterisation now holds.