Boolean Hedonic Games

09/23/2015 ∙ by Haris Aziz, et al. ∙ 0

We study hedonic games with dichotomous preferences. 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. The assumption of dichotomous preferences means that, additionally, 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. We develop a succinct representation for such games, in which each player's preference relation is represented by a propositional formula. We show how solution concepts for hedonic games with dichotomous preferences are characterised by propositional formulas.

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1 Introduction

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.

2 Preliminaries

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

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.

Example 1

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.

Syntax

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.

Semantics

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 .

Axiomatisation

We have the following axiom schemes for mutually distinct players , and ,

  1. all propositional tautologies

  2. (transitivity)

as well as modus ponens as the only rule of the system:

  1. from and infer . (modus ponens)

The resulting logic we refer to as  and write if there is a derivation of  from , 3, and i, using 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:

In fact, formulas in  are strictly more expressive in the sense that they can represent any dichotomous preference relation over partitions rather than just preference relations over partitions as induced by a preference relation  for a player  over coalitions in . We find, however, that every Boolean hedonic game  represents a hedonic game with dichotomous preferences provided that every player’s goal  is equivalent to a formula in the language , the sublanguage of  in which only variables in occur. Intuitively, formulas in  only convey information about the coalitions player  is in or she is not in.

Proposition 1

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.

Substitution

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 

. 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.

Lemma 1

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,

  1. iff ,

  2. and iff .

  • We prove i; the proof for ii is by structural induction on  and relies on similar principles as i. As  and are fixed throughout the proof, for better readability, we write  for .

    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:

    1. there is some  with and , or

    2. 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.

Example 2

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.

Proposition 2

Let  be a partition,  a player, and  a formula of . Then,

iff for some ,
  • First assume . Then,  for some . Define . By Lemma 1ii, we then obtain .

    For the opposite direction, assume that for some . Define as the Boolean vector of length  such that for every ,

    Then, clearly, . By Lemma 1ii, it follows that . We may conclude that .    

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.

Lemma 2<