Farsighted Stability in Hedonic Games - Semantic Scholar
Farsighted Stability in Hedonic Games E¤rosyni Diamantoudi ¤ and Licun Xue¤ November 2000
Abstract We investigate how rational individuals partition themselves into di¤erent coalitions in “hedonic games” [see Banerjee, Konishi and Sönmez (1998) and Bogomolnaia and Jackson (2000)],where individuals’ preferences depend solely on the composition of the coalition they belong to. We show that the four solution concepts studied in the literature (core, Nash stability, individual stability and contractual individual stability) exhibit myopia on the part of the players. We amend these notions by endowing players with foresight in that they look many steps ahead and consider only credible outcomes. We show the existence and study the properties of the new solutions, as well as their relation to the previous notions.Journal of Economic Literature Classi…cation Number: C71. Keywords: Hedonic games, Coalition structures, Foresight.
1
Introduction
The signi…cance of coalition formation is manifested through its presence in many facets of our economic, political, and social life as well as by the numerous studies attempting to model it. The plethora of e¤orts highlights the complexities embedded in the problem in question. In order to circumvent some of these complexities, the problem of coalition formation is often limited to speci…c contexts with more structured preferences, or payo¤ distributions ¤
We would like to thank the seminar audience at the University of Copenhagen for useful comments. Both authors are at the Department of Economics, University of Aarhus, Building 350, DK-8000 Aarhus C., Denmark. Email addresses are [email protected] and [email protected] respectively.
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or even constrained coalition formation options. Under such restrictions, it is often possible to study more e¤ectively various solution concepts and their properties. In this spirit, we study coalition formation in hedonic games, that is, in situations where individuals’ preferences depend solely on the composition of the coalition they belong to. The hedonic aspect in players’ preferences was originally introduced by Dréze and Greenberg (1980) in a context concerning local public goods, where agents’ preferences depend on their consumption of the public good as well as the coalition they belonged to. However, the hedonic aspect transcends the local public goods context and spans over a much greater area of economics and sociology. The formation of societies, communities, social clubs and groups are just a few cases where the hedonic aspect is the central element steering their structure. Recently, several works concentrate on the hedonic notion and its implications in a more general framework. Notably, Banerjee, Konishi and Sönmez (1998) and Bogomolnaia and Jackson (2000) introduced the afore-mentioned pure hedonic games. Various existing as well as new solution concepts are applied to hedonic games and their rendition is studied. The Core, identi…es coalition structures that are immune to any bene…cial coalitional deviation, addressing, situations where coalitional deviations are feasible and costless. However, in a signi…cant number of situations coalitional deviations are not viable, either because of institutional (legal or physical) constraints or simply because agents are not able to coordinate their actions, especially in cases where the entire set of agents is rather large. In such instances solutions concepts that consider only individual deviations are warranted. Nash Stability identi…es coalition structures where no player wishes to migrate to another coalition in the structure. Such an analysis is relevant in situations where no permission is required to join a new coalition, a simple example being that of moving from one city to another. Individual Stability examines not only the preferences of the migrating player but also the preferences of the coalition this player plans to join, a coalition structure therefore, is not individually stable if some player wishes to migrate to another coalition where he is not hurtful. Any situation where an individual is hired by a business 2
entity (a coalition), and thus is allowed to join them, serves as an example where individual stability is a more appropriate notion. Lastly, even stronger is the notion of Contractual Individual Stability that requires the permission of the original coalition to allow a player to migrate, in the sense that they are not hurt by the departure of this member. In the spirit of the previous example, consider the case where the newly hired employee has to …rst break his contract with his previous employer. Unfortunately, most of the aforementioned notions do not always provide an answer even to simple problems as we will illustrate later in this section. Therefore, restrictions on players’ preferences need to be made to guarantee the existence of di¤erent solutions. In particular, Banerjee, Konishi and Sönmez (1998) study the Core in hedonic games. They de…ne the top coalition property, a property that imposes a degree of commonality on players’ preferences, while it guarantees the non-emptiness of the Core. The motivation for such a property is served from two perspectives: (i) the multiplicity of economic applications, some of which are formally presented in the paper, where the property naturally holds, and (ii) the possibility of an empty Core, illustrated through examples, even when strong assumptions such as additive separability, anonymity or single peakedness are imposed. Bogomolnaia and Jackson (2000) study primarily individually stable coalition structures in hedonic games and they propose the condition of ordered characteristics – a condition that builds on single peaked preferences- and consistency to guarantee the existence of individually stable coalition structures and partially their e¢ciency, while it provides an algorithm to identify weakly Pareto e¢cient and individually stable coalition structures. Analogously, the motivation for the condition stems from examples satisfying already strong assumptions where individually stable coalition structures do not exist. This project attempts to build on such endeavors, extending the existing analysis by identifying and rectifying some problems associated with the solution concepts used. While the most analyzed problem concerning the Core is its “frequent” emptiness, the notion has also being criticized for the easiness with which it discards feasible outcomes -which often results to emptiness. In particular, an outcome is excluded from the Core because a 3
set of agents can induce an other outcome that bene…ts all of its members. The likelihood of this outcome surviving a further deviation by some other set of agents is not examined. Especially, when such a further replacement may render the original deviating coalition much worse o¤ than it was in the very beginning, the original deviation may be deterred. Alternatively, the Core has been criticized for including outcomes that can be dominated if a group of agents does not behave myopically. To illustrate this point consider a group of agents that can induce some outcome which does not necessarily improve upon the original one. It is possible however, that once this new outcome becomes the status quo, further deviations may occur which will render the original deviating coalition better than it was in the very beginning. Allowing therefore, a potential deviating coalition to foresee many steps ahead, it may proceed with a deviation which ultimately bene…ts its members, that otherwise may have been halted. Indeed, such an argument applies to Example 5 where a core element is ruled out. However, if players’ preferences are strict, then core outcomes exhibit a surprising robustness and are immune to foresight, as is formally shown in following sections. Put di¤erently, for a core outcome to be excluded from the farsighted solution we propose, it must be the case that some players are indi¤erent between some coalitions. In environments were moves can be followed by counter moves and so on, it is natural to allow any group of agents that considers deviating to “speculate” on the ultimate result of its action. Obviously, an “ultimate” result must be immune to further deviations; that is, it cannot itself be dominated. For an outcome to be ultimate, it must be the case that no group of agents has an incentive to deviate, again anticipating the ultimate outcomes of its deviation. Thus, an ultimate result must be in the solution set of the game. Such a reasoning is captured by the notion of abstract stable set …rst introduced by von Neumann & Morgenstern (1944). The notion is characterized by internal and external stability. Internal stability guarantees that the solution set is free from inner contradictions, that is, any two outcomes in the solution set cannot dominate each other and external stability guarantees that every outcome excluded from the solution is accounted for, 4
that is, it is dominated by some outcome inside the solution. Although the original abstract stable set accommodates the issue of credibility regarding a dominating outcome it does not address foresight. Harsanyi (1974) introduced the concept of indirect dominance to capture foresight. An outcome indirectly dominates another, if there exists a sequence of outcomes starting from the dominated outcome and leading to the dominating one, and at each stage of the sequence the group of players required to enact the inducement prefers the …nal outcome to its status quo. In criticizing the Core for its myopia and lack of consistency, we suggested that agents should look ahead and speculate on the ultimate result of their initial actions. Baring in mind that di¤erent sequences of actions may take place, it is natural to question how the players are to treat situations where the ultimate result may not be unique; moreover, how they are to behave when some ultimate outcomes may make them better o¤ while others make them worse o¤. The answer to such a question depends solely on the behavioral characteristic of the players making the decision. Two extreme approaches are that of optimism (implicit in the notion of abstract stable set), where a coalition would proceed with the deviation if at least one of the ultimate outcomes makes its members better o¤, whereas a conservative coalition would not proceed with the deviation unless every possible ultimate outcome makes its members better o¤. Amalgamations of stability and indirect dominance, are studied by Chwe (1994) and Xue (1998) where both behavioral assumptions are investigated along with the existence of the solution concepts. The criticism developed thus far concerning the Core applies also to the stability notions that allow for only individual deviations. In the same manner a deviating coalition considers all plausible outcomes that may arise from its initial deviation, an individual migrating from one coalition to another can raise the same concerns, and question the possibility of others joining later on, or existing members departing. Moreover, the welcoming coalition (in the case of individual stability) may, for example, wonder whether the admission of a new undesirable member may bring later on far more desirable migrants. Lastly, the remaining coalition (in the case of contractual 5
individual stability) may permit a favorite member to depart believing that it will, later on, be replaced by someone even more desirable, and so on. To amend existing solution concepts, we introduce notions that allow players to foresee arbitrarily many steps ahead, and consider only the plausible (as opposed to any feasible) outcomes that are likely to result from their actions. Although the next sections de…ne formally and motivate more precisely all proposed solution concepts, we illustrate through the well known problem of roommates how (most) solutions remain silent. Example 1 (The roommate problem) Three individuals have to agree in sharing a room that can only accommodate two of them. The third individual has to …nd a room of his own, and of course none of them wishes that all three squeeze in one room. Unfortunately, their preferences are rather contradicting: the …rst prefers to share the room with the second, than the third, while the second prefers to share the room with the third rather than the …rst, and lastly the third prefers to share the room with the …rst rather than the second. Their preferences are summarized as follows: f1; 2g Â
1 f1; 3g
Â1 f1g Â1 f1; 2; 3g
f2; 3g Â
2 f1; 2g
Â2 f2g Â2 f1; 2; 3g
f1; 3g Â
3 f2; 3g
Â3 f3g Â3 f1; 2; 3g
It is easy to see that the Core of the game is empty. For example consider the partition of ff1; 2g; f3gg; it is blocked by f2; 3g since 2 prefers to be with 3 than with 1; and 3 prefers to be with 2 than alone. The same argument blocks any partition involving a pair and a singleton. Unfortunately, the Core is not the only solution encountering problems. No coalition structure is Nash stable either. Consider the same partition of ff1; 2g; f3gg ; which is blocked via 2 that wishes to join 3. In fact, since 3 welcomes 2 this coalition structure is not individually stable either. Similar arguments render all partitions Nash and individually unstable. The only solution that provides an answer to this problem is contractual individual stability. Although the grand coalition and the singletons are excluded from the solution, the partitions involving a pair and a singleton are 6
contractually individually stable. Consider the partition f1; 2gf3g, although 2 would like to join 3 and 3 welcomes 2, 1 will not permit 2 to depart from their coalition. In fact, contractual individual stability o¤ers a very appealing analysis to the problem if 1 and 2 had indeed singed a contract together for their room. In the event, though, were no commitments have taken place the notion is not applicable, and no answer is given to the problem. In contrast, all four modi…cations of the solution concepts proposed in this paper suggest the same solution, that is, they support only all the coalition structures involving a pair and a singleton. The fundamental argument embedded in all the notions is as follows: consider again the partition f1; 2gf3g; player 2 will not join 3 fearing that 3 may, later on, join 1 and thus leave 2 alone. It is the conservative behavior imposed on the players, along with their foresight that prevents them from dissolving a “mediocre” partnership in pursuance of the perfect one. In Section 2 we introduce formally the model and the solution concepts. We also provide examples where the old notions are not silent to illustrate the distinction and re…nement of the new ones. Existence results concerning the solution concepts are presented in Section 3. In the same section we investigate the relation of the new notions to existing ones when di¤erent restrictions are imposed on players’ preferences. Lastly, in Section 4 we discuss conjectures and open questions of interest to the topic.
2
De…nitions
We …rst introduce the basic notations and the existing stability notions. Then we proceed to introduce stability notions under foresight.
2.1
Preliminaries
² Let N be a …nite set of players. ² A coalition S is a non-empty subset of N, and S(i) denotes that i 2 S: ² Let S denote the collection of all coalitions and for i 2 N; let S(i) denote the collection of coalitions that contain i, that is, S(i) = fS ½ 7
N j i 2 Sg: Hedonic game: A hedonic game G is a pair (N; fºi gi2N ), where N is the …nite set of players and for all i 2 N, ºi is a complete, re‡exive, and transitive binary relation on S(i); representing i’s preferences over coalitions that contain i: We use Âi to denote asymmetric part of º i (i.e., strict preferences) and » i the indi¤erence relation. ² A coalition structure, P , is a partition of N, that is, P = fS1 ; S2 ; :::; Sk g, Sk T Sj = ;: j =1 S j = N and for all i 6= j; S i ² Let P be the collection of all coalition structures and PS = fP j S 2 P; S 2 Sg be the collection of all coalition structures that contain coalition S: Similarly, PS denotes that S 2 P . ² For P 2 P and i 2 N, let SP (i) be the coalition S 2 P such that i 2 S: The four main stability notions in the literature are as follow. Core stability: A coalition structure P 2 P is core stable or in the core of G if there does not exist “a blocking coalition” S ½ N such that S  i SP (i) for all i 2 S: Non-cooperative stability notions: Nash stability: A coalition structure P 2 P is Nash stable if there do not exist i 2 N and S 2 P [ f;g such that S [ fig Âi SP (i): Individual stability: A coalition structure P 2 P is individually stable if there do not exist i 2 N and S 2 P [ f;g such that S [ fig Âi SP (i) and S [ fig º j S for all j 2 S: Contractural individual stability: A coalition structure P 2 P is contractually individually stable if there do not exist i 2 N and S 2 P [f;g such that (1) S [ fig  i SP (i) and S [ fig º j S for all j 2 S; (2) SP (i) n fig º j SP (i) for all j 2 SP (i) n fig: 8
2.2
Farsighted coalitional stability
According to core stability, if P is under consideration, any coalition S ½ N can form and object to P . When S forms and before other players regroup, the resulting coalition structure P 0 = fSg [ fT n S j T 2 P and T n S 6= ;g S
and in this case, we write P ! P 0 . We can extend players’ preferences over coalitions to coalitional preferences over coalition structures. Coalitional preferences: A coalition S ½ N strictly prefers P 0 to P; denoted P 0 Â S P; where P; P 0 2 P , if SP 0 (i) Âi SP (i) for all i 2 S; similarly, a coalition S ½ N prefers P 0 to P; denoted P 0 &S P; where P; P 0 2 P , if SP 0 (i) &i SP (i) for all i 2 S. Given a coalition structure P 2 P , if some coalition S ½ N has an incentive to form, thereby inducing P 0, we say P 0 dominates P: (Direct) Dominance: For P; P 0 2 P, P 0 (directly) dominates P , or P 0 > S P , if P ! P 0 and P 0 Â S P . Thus, core can be alternatively de…ned as follows: Core: Core of G is the set of coalition structures that are not dominated with respect to >, i.e., core of G is the (abstract) core of abstract system1 (P ; >): Formally, Core(P ; >) = fP 2 P j @P 0 2 P such that P 0 > P g : Myopia is re‡ected in the fact that a blocking coalition does not consider the possibility that the rest of the players may regroup. That is, given P 0 , another coalition can form to induce a new coalition structure and so on. If players are farsighted, they should consider the ultimate outcomes 1 An abstract system is a set and a binary relation on this set. See von Neumann and Morgenstern (1944).
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of their actions. Thus, a coalition may choose to “deviate” to coalition structure, which does not necessarily make its members better o¤, as long as their deviation leads to …nal coalition structures that bene…t all its members; similarly, a coalition may choose not to deviate to a coalition structure it prefers if their deviation eventually leads to coalition structures that make its members worse o¤. The following “indirect dominance” due to Harsanyi (1974) and Chwe (1994) captures foresight. Indirect dominance: For P; P 0 2 P, P 0 indirectly dominates P , or P 0 À P; if there exists a sequence of coalition structures P 1; P 2; : : : ; P k ; with P 1 = P and P k = P 0, and a sequence of coalitions S1 ; S2; : : : ; S k¡1 Sj such that P j ! P j+1 and P 0 Â Sj P j for all j = 1; 2; : : : ; k ¡ 1: We can now examine di¤erent solutions to the abstract system (P; À): The (abstract) core of (P; À) is one of these. The (abstract) core of (P ; À): Core(P; À) = fP 2 P j @P 0 2 P s.t. P 0 À P g : It is easy to see that Core(P; À) ½ Core(P; >): However, core does not consider the credibility of the dominating alternative. The following von Neumann-Morgenstern (vN-M) (abstract) stable set amends this shortcoming of the core. vN-M stable set of (P ; À): R ½ P is vN-M internally stable if there do not exist P; P 0 2 R such that P 0 À P ; R is vN-M externally stable if for all P 2 P n R; there exists P 0 2 R such that P 0 À P . R is vN-M stable is it is both internally and externally stable. As shown in Greenberg (1990), vN-M stable set implicitly assumes “optimistic behavior”. To see this, we …rst introduce the following notation. Likely outcomes given P : For Q ½ P and P 2 P , let QjP;À = fP 0 2 Q j P 0 = P or P 0 À P g : If Q is a “solution set”, then QjP;À is the set of likely outcomes. 10
Now we can rewrite the de…nition of vN-M stable set: vN-M stable set rede…ned: R is vN-M internally stable if Q 2 R implies S that there do not exist P 2 P and S ½ N such that Q ! P and P 0 ÂS Q for some P 0 2 RjP;À: R is vN-M externally stable if Q 2 P nR S implies that there exist P 2 P and S ½ N such that Q ! P and P 0 ÂS Q for some P 0 2 Rj P;À: R is vN-M stable if it is both internally and externally stable. However, the vN-M stable set for (P ; À) entails over-optimism on the part of a departing coalition as illustrated in Xue (1998). The following notion of conservative stable set amends this problem. Conservative stable set: Q is conservatively internally stable if Q 2 Q S implies that there do not exist P 2 P and S ½ N such that Q ! P; QjP;À 6= ;; and P 0 Â S Q for all P 0 2 QjP;À: Q is conservatively externally stable if Q 2 P n Q implies that there exist P 2 P and S S ½ N such that Q ! P; Qj P;À 6= ;; and P 0 ÂS Q for all P 0 2 QjP;À: Q is conservatively stable if it is both conservatively internally and externally stable. The conservative stable set is closely related to Greenberg’s (1990) “conservative stable standard of behavior” (CSSB) and Chwe’s (1994) “consistent set”. The following example illustrates the myopia embedded in the core and how it is recti…ed through the conservative stable set. Example 2 (Conciliatory partnerships) Consider a game with 4 players and the following preference ordering: f1; 4g Â
1 f1; 2; 3g
 1 f1; 2g Â1 f1g Â1 ¢ ¢ ¢
f1; 2; 3g Â
2 f1; 2g
Â2 f2; 3g Â2 f2g  2 ¢ ¢ ¢
f1; 2; 3g Â
3 f2; 3g
Â3 f3; 4g Â3 f3g  3 ¢ ¢ ¢
f3; 4g Â
4 f1; 4g
Â4 f4g Â4 ¢ ¢ ¢
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The rest of relevant coalitions are ranked strictly below the above coalitions. The core contains only the following partition: ff1; 4g; f2; 3gg: Note that partition ff1; 2g; f3; 4gg is not in the core since coalition f1; 2; 3g can strictly improve upon it. However, if players 2 and 3 are farsighted they will realize that once partition ff1; 2; 3g; f4gg is the status quo, coalition f1; 4g will form and bring about ff1; 4g; f2; 3gg which is not as good for 2 and 3 as the original one. Therefore, ff1; 2g; f3; 4gg should not be ruled out by f1; 2; 3g if 2 and 3 are farsighted. However, the reader may have already observed that although ff1; 2g; f3; 4gg may not be disrupted by coalition f1; 2; 3g; the following sequence of events may take place: ff1; 2g; f3; 4gg
f1;3g
f2;3g
¡! ff2g; f1; 3g; f4gg ¡! ff1g; f2; 3g; f4gg f1;4g
¡! ff1; 4g; f2; 3gg: Indeed partition ff1; 4g; f2; 3gg indirectly dominates ff1; 2g; f3; 4gg. But the reverse holds as well: ff1; 4g; f2; 3gg
f2;4g
f1;2g
¡! ff1g; f2; 4g; f3gg ¡! ff1; 2g; f3g; f4gg f3;4g
¡! ff1; 2g; f3; 4gg: In fact, both partitions ff1; 4g; f2; 3gg and ff1; 2g; f3; 4gg belong to the maximal conservative stable set:
2.3
Notions of farsighted non-cooperative stability
Nash stability, individual stability, and contractual individual stability consider only one-step “deviations” initiated by individuals. We can introduce a farsighted stability notion as an alternative for each of these three concepts. 2.3.1
Farsighted Nash stability
To de…ne farsighted Nash stability, we …rst formalized that if i 2 N leaves a coalition in P to join another coalition also in P , the resulting coalition 12
structure, P 0; is de…ned as follows: (1) SP (i) n fig 2 P 0 if SP (i) n fig 6= ;; (2) 9 T 2 P [ f;g such that T [ fig 2 P 0 ; (3) S 2 P 0 for all S 2 P such fig that S 6= SP (i) and S 6= T: We shall write P * P 0 in this case. Now, we can adapt the notions of farsighted stability in Section 2.1, observing that only fig individuals can move according to *. The following example illustrates the myopia embedded in Nash stability. Example 3 (An undesired guest ) 2 Consider a game with 3 players and the following preference orderings: f1; 2g  f1; 2g  f1; 2; 3g Â
1f1g 2f2g
 1 f1; 2; 3g  1 f1; 3g  2 f1; 2; 3g  2 f2; 3g
3f2; 3g
 3 f1; 3g  3 f3g
No partition is this game is Nash stable. For example, ff1; 2g; f3gg3 is not Nash stable because 3 prefers to joint f1; 2g than staying alone. However, ff1; 2; 3gg is not Nash stable either. In fact, players 1 and 2 will eventually bring back ff1; 2g; f3gg: If players are farsighted, then ff1; 2g; f3gg is stable. 2.3.2
Farsighted individual and contractual stability
Recall, in the previous subsection, player i 2 N can change P 2 P to another fig P 0 2 P such that P * P 0. However, if player i need the permission of SP 0 (i) n fig as in the case of individual stability, then i alone cannot change fig fig P to P 0 if P * P 0: We write P * P 0 to denote that i needs the permission SP 0 (i)
fig
of SP 0 (i) n fig to change P to P 0. Thus, P is individually stable if P *
SP 0 (i)
P 0 ; P 0 Âfig P; and, P 0 &SP 0 (i) P: If players are farsighted, they should consider that P 0 may be replaced by another coalition structure and it is the “…nal” coalition structure that matters. For this we can de…ne the following indirect dominance relation.
Indirect individual dominance For P; P 0 2 P, P 0 indirectly individually dominates P , or P 0 m P; if there exists a sequence of coalition structures P 1; P 2; : : : ; P k ; with P 1 = P and P k = P 0 , and a sequence of 2 3
See Bogomolnaia and Jackson (2000). This partition is, however, core stable.
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individuals i 1; i2 ; : : : ; ik¡1 such that P j
fij g
*
SP j +1 (i)
P 0 &SP j +1 (i) P j for all j = 1; 2; : : : ; k ¡ 1:
P j+1 and P 0 Âij P j and
Individually conservative stable set (under foresight): Q is conservatively internally stable if Q 2 Q implies that there do not exist P 2 P fig and i 2 N such that Q * P; Qj P;m 6= ;; P 0 Â i Q; and P 0 &SP (i) Q SP (i)
for all P 0 2 Qj P;m : Q is conservatively externally stable if Q 2 P n Q fig
implies that there exist P 2 P and i 2 N such that Q * P; QjP;m 6= SP (i)
;; P 0 Âi Q; and P 0 & SP (i) Q for all P 0 2 Qj P;m Q is conservatively stable if it is both conservatively internally and externally stable. Note that for Q ½ P and P 2 P, Qj P;m = fP 0 2 Q j P 0 = P or P 0 m P g : We can also de…ne a similar notion of contractual stable set under foresight by modifying the above de…nitions to capture the fact that a deviating player also needs the permission of the coalition from which he departs. The following example illustrates the implication of foresight. Example 4 (Switching partners) Consider a game with 4 players. The following is part of the preference ordering of the players. The rest of the relevant coalitions are ranked strictly below these coalitions. f1; 4g Â
1f1; 2g
f2; 3g Â
2f2; 3; 4g
f1; 4g Â
4f3; 4g
f2; 3g Â
3f3; 4g
 2 f1; 2g
ff1; 2g; f3; 4gg is individually stable and contractually individually stable: For example, if 2 leaves his current partner 1, then 1 will be worse o¤; moreover, if 1 joins the coalition of 3 and 4, both 3 and 4 will be worse o¤. If players are farsighted, however, they will anticipate that once 2 leaves 1 and joints f3; 4g, 4 will subsequently leave 2 and 3 to join 4. That is, f2g
f4g
f:g
f:g
ff1; 2g ; f3; 4gg * ff1g ; f2; 3; 4gg * ff1; 4g ; f2; 3gg : 14
Thus, not only players 2 and 4 have incentive to initiate their moves but also other relevant coalitions have incentive to approve their moves. Note that the …nal coalition structure ff1; 4g ; f2; 3gg Pareto dominates ff1; 2g ; f3; 4gg.
3
Results
In this section, we …rst establish the existence of the notions we propose and then proceed to analyze their properties.
3.1
Existence
The following theorem shows that the conservative stable set always exists; this is in contrast to the fact that the core may be empty and the vN-M stable set may fail to exist. Theorem 1 There exists a conservative stable set and a maximal one with respect to set inclusion. Proof. Let Q 0 = P and for k = 0; 1; : : : de…ne recursively, ( ) S k k k S ½ N s.t. Q ! P; Q jP;À 6= ;; : Qk+1 = Q 2 Q j @P 2 Q and and P 0 ÂS Q 8P 0 2 Qk jP;À ¤
¤
¤
We shall show that there exits k¤ such that Qk +1 = Qk and Qk is conservatively stable. First we claim that for all k and P 2 P, there exist P 0 2 Qk such that Qk jP;À ¾ Qk jP 0 ;À 3 P 0: Obviously, this is true for k = 0: Assume the claim is true for k. Let P 1 2 Qk n Qk+1; then there exist S Q 2 Qk and S ½ N such that P 1 ! Q; Qk jQ;À 6= ;; and P 0 Â S P 1 for all P 0 2 Qkj Q;À: Thus, P 0 À P 1 for all P 0 2 Qk j Q;À: Given that À is irre‡exive, Qk jP 1 ;À ) Qk j Q;À. Since the claim is true for k, then there exists P 2 2 Qk such that Qkj Q;À ¾ Qk+1j P 2 ;À 3 P 2. Thus, we Qk j P 1 ;À ) Qk j P 2 ;À 3 P 2: Obviously, Qk+1 jP 1;À ¾ Qk+1(P 2): If P 2 2 Qk+1jP 2 ;À; we are done. Otherwise, there is P 3 2 Qk such that Qk j P 2 ;À ) Qk j P 3 ;À 3 P 3: Since Qk is …nite, there exist ` such that Qk j P 1 ;À ¾ Qk j P 2 ;À ¾ ¢ ¢ ¢ ¾ Qk jP ` ;À 3 P `. Moreover, ¤ ¤ it follows that there exits k¤ such that Qk +1 = Qk : 15
¤
¤
¤
Qk +1 = Qk implies that Qk is conservatively internally stable. To ¤ show conservative external stability, suppose P 2 P n Qk : Then there exists k < k ¤ such that P 2 Qk nQk+1: Thus, there exist Q 2 Qk and S ½ N S such that P ! Q; Qk j Q;À 6= ;;and P 0 Â S P for all P 0 2 Qk jQ;À: Since ¤ ¤ ¤ Qk jQ;À ¾ Qk jQ;À 6= ;; we have P 0 ÂS P for all P 0 2 Qk j Q;À: Thus, Qk is conservatively external stable. ¤ To show that Qk is the largest conservative stable set, assume to the contrary that there exists another conservative stable set Q0 such that there exists P 2 Q0 n Q. There exists k such that Qk ¾ Q0 : Since Q0 is stable, Q` ¾ Q0 for all ` ¸ k. Contradiction. Existence of other farsighted notions can be established in the same fashion.
3.2
Farsighted Coalitional Stability and the Core
As mentioned in the introduction, the Core of hedonic games with strict preferences exhibits a surprising robustness. Although it may be expected that core partitions would be excluded from the conservative stable set on the grounds that coalitions may form (without any immediate improvement) looking ahead to future (improving) restructuring, it is actually not the case. In fact, the following result formalized by theorem 2, shows that core partitions are always included in the maximal conservative stable set, when preferences are strict. Concluding, therefore, that core partitions survive farsighted players. However, this in not the case when preferences are not strict as illustrated in Example 8. The critical element that collapses the intuitive “expectations” expressed above is the conservative aspect embedded in the players’ behavior. Indeed, it may be the case that such a sequence of events exists that improves an initial acting coalition, departing from a core partition, some steps after its action. However, along such a sequence of events many other sequences may originate, at least one them leading back to the starting core partition, which is not improving, discouraging therefore, due to the conservative behavior the initial acting coalition. Exactly this feature of core partitions is captured by
16
lemma 3. Theorem 2 Consider a game G with strict preferences and let Q¤ be the maximal conservative stable set. Then Core(P ; >) ½ Q¤: ¹ = fP g such that P 2 Core(P ; >): We will show that Proof. Consider Q ¹ is conservatively stable. Q Internal stability is trivial. External stability can also be established ¹ is with the direct use of Lemma 3, which asserts that any partition P 0 2 =Q indirectly dominated by P: Finally, according to Theorem 1 the maximal conservative stable set Q¤ ¹ and therefore, P 2 Q¤: ¾Q Lemma 3 Consider a game G with strict preferences and assume that the Core(P ; >) 6= ;. If partition P ¤ 2 Core(P; >); then P ¤ indirectly dominates any partition P 2 P nfP ¤g: Proof. We will show that P ¤ À P by constructing a sequence of moves, departing from P and ending at P ¤, where every acting coalition prefers P ¤ to its status quo. The sequence consists of two main parts: in the …rst part P is decomposed into singletons and in the second part P ¤ is constructed from the singletons. We start with the …rst part. For convenience let SP denote an arbitrary coalition in P: Since P ¤ 2 Core(P; >); @ S 2 P such that P ÂS P ¤: That is, for any SP there exists i 2 SP such that P ¤ Â i P: Let i initiate the sequence by breaking away from SP . That is, T1 = fig induces partition P1 such that T1 2 P1 and (SP nT1 ) 2 P1 : Again, since P ¤ 2 Core(P ; >); @ S 2 P1 such that P1 Â S P ¤: Any SP1 contains j such that P ¤ Â j P1 : Let T2 = fjg induce partition P2 such that T1 ; T2; (SP1 nT2 ) 2 P2: We can continue in this manner until we reach partition Pt such that every jSPt j = 1. Obviously, since P ¤ 2 Core(P; >); for every SPt = fig 2 Pt we have P ¤ Âfig Pt : Next we proceed with second part of the sequence which involves constructing P ¤ : 17
Let P ¤ = fS1; S2; :::; Sk g: Then, let the next step of the sequence be the formation of S1 ; since all of its members would rather be together (which is the case under P ¤ ) than alone (which is the case under Pt ): Denote the new partition by Pt1 . Let the next step of the sequence be the formation of S2 : Again all of S2 ’s members would rather be together (which is the case under P ¤ ) than alone, which is the case under Pt 1 : Denote the new partition by Pt2 : Continue in this manner, until P ¤ is formed. Note that along the way, when Sj forms from Pt(j ¡1) ; all of its members would rather be together (under P ¤) than alone (under Pt(j ¡1) ). In summarizing: 1st part : 2nd part :
T
T
T
T
T P !1 P1 !2 P2 !3 ::: ! Pt
S
S
S
Sk¡1
S
Pt !1 Pt1 !2 P t2 !3 ::: ! Pt(k¡1) !k P ¤:
While P ¤ ÂTi+1 Pi for i = 0; :::; T where P = P0 and P ¤ Â Sj+1 Ptj for j = 0; :::; k ¡ 1; where Pt0 = Pt . Thus, P ¤ À P: The following example asserts that theorem 2 holds only in the case of strict preferences, by illustrating a case (involving indi¤erences) where core partitions are actually excluded from the maximal conservative stable set. Example 5 (Core Re…nement ) Consider a game with 4 players and the following preference ordering: f1; 2g Â
1 f1; 3g
Â1 f1; 4g Â1 f1g  1 ¢ ¢ ¢
f2; 4g Â
2 f1; 2g
»2 f2; 3g Â2 f2g  2 ¢ ¢ ¢
f1; 3g Â
3 f3; 4g
»3 f2; 3g »3 f3g  3 ¢ ¢ ¢
f3; 4g Â
4 f2; 4g
Â4 f1; 4g Â4 f4g  4 ¢ ¢ ¢
The rest of relevant coalitions are ranked strictly below the above coalitions. The core contains the two partitions ff1; 2g; f3; 4gg and ff1; 3g; f2; 4gg:
18
Given ff1; 2g; f3; 4gg, however, players 2 and 3 will formed a coalition, expecting the following chain of events ff1; 2g; f3; 4gg
f2;3g
f1;3g
¡! ff1g; f2; 3g; f4gg ¡! ff1; 3g; f2g; f4gg f2;4g
¡! ff1; 3g; f2; 4gg:
More formally, the unique stable set contains only ff1; 3g; f2; 4g Remark 1 It is easy to see that if the jCore(P ; >)j > 1 then Core(P; À) is always empty, since any one core outcome indirectly dominates all the others. The reverse however, is not true. As is demonstrated by example 2, jCore(P ; >)j = 1, yet Core(P; À) = ; since the unique core outcome is indirectly dominated by another (non-core) outcome and vice versa, ruling each other out. Banerjee, Konishi and Sönmez (1998) …rst introduce the top-coalition property as a relaxation of the common ranking property which is due to Farell and Scotchmer (1988). It requires that there exists a coalition that is preferred the most by its members compared to any other coalition each member could possibly join. Moreover, the condition holds for any subset of players. Top-coalition property Given a non-empty set of players V ½ N, a coalition S ½ V is a top-coalition of V if for every i 2 S and any T (i) ½ V we have S ºi T (i): A hedonic game G satis…es the top-coalition property if for any non-empty set of players V ½ N, there exists a top-coalition of V . The authors show that under the top-coalition property and strict preferences the Core always contains a unique partition, related to top-coalitions, that we formally de…ne as a top-coalition partition: Top-coalition partition Given a game G that satis…es the top-coalition property, a partition P ¤ 2 P is a top-coalition partition if P ¤ is such that P ¤ = fS1; S2; :::; Sj ; :::Sk g where S1 is a top-coalition of N, S2 is a top-coalition of NnS1; S3 is a top-coalition of N nfS1 [ S2g,..., Sk is a top-coalition of Nnf[j): And since there is only one element in Q¤; even if there are more sequences stemming from Q 0; their terminal coalition structure is bound to be P ¤. However, we …nd the construction of the following sequence of coalition structures leading from any arbitrary coalition structure Q0 to P ¤ more intuitive due to the direct use of the top coalitions. In particular, the sequence proposed here omits the 1st part -the dissolution of the starting partition into singletons- of the sequence proposed in lemma 3 and starts immediately from the 2nd part, that is, the formation of the terminal sequence. Step 1: Consider an arbitrary partition Q0 6= P ¤: If S1 2 Q0 proceed to step 2; otherwise let the …rst step of the sequence be the formation of S1 since, by de…nition, Q0 ÁS1 P ¤ no matter what Q 0 is as long as S1 2 = Q0: Let the new coalition structure be denoted by Q1 ; and note that S1 2 Q1: 4 The
authors also de…ne a weaker notion of the top-coalition property, appropriately named weak-top-coalition property and they show that the Core is always non-empty under the weak-top-coalition property, yet they do not o¤er any characterization of the solution apart from the fact that it contains at least all the weak-top-coalition partitions. It is easy to construct examples where the Core along with the conservative stable set may contain partitions that are not weak-top-coalition.
20
Step 2: If S2 2 Q1 proceed to step 3, otherwise let the second step of the sequence be the formation of S2 , since by de…nition Q1 ÁS2 P ¤: More speci…cally, we know that all the members of S2 prefer to be in S2 than in any other coalition involving members of NnS1, which is exactly their situation under Q1; since S1 has already formed: Let the new coalition structure be denoted by Q2; and note that fS1 ; S2 g 2 Q2 : .. .
Step k: If Sk 2 Qk¡1 , then Qk¡1 is already P ¤; otherwise let the last step of the sequence be the formation of Sk , since be de…nition Qk¡1 ÁSk P ¤: Note that if Sk is not formed then it must partitioned in smaller coalitions, whose members would rather merge since Sk is a top-coalition of Nnf[j
Abstract We investigate how rational individuals partition themselves into di¤erent coalitions in “hedonic games” [see Banerjee, Konishi and Sönmez (1998) and Bogomolnaia and Jackson (2000)],where individuals’ preferences depend solely on the composition of the coalition they belong to. We show that the four solution concepts studied in the literature (core, Nash stability, individual stability and contractual individual stability) exhibit myopia on the part of the players. We amend these notions by endowing players with foresight in that they look many steps ahead and consider only credible outcomes. We show the existence and study the properties of the new solutions, as well as their relation to the previous notions.Journal of Economic Literature Classi…cation Number: C71. Keywords: Hedonic games, Coalition structures, Foresight.
1
Introduction
The signi…cance of coalition formation is manifested through its presence in many facets of our economic, political, and social life as well as by the numerous studies attempting to model it. The plethora of e¤orts highlights the complexities embedded in the problem in question. In order to circumvent some of these complexities, the problem of coalition formation is often limited to speci…c contexts with more structured preferences, or payo¤ distributions ¤
We would like to thank the seminar audience at the University of Copenhagen for useful comments. Both authors are at the Department of Economics, University of Aarhus, Building 350, DK-8000 Aarhus C., Denmark. Email addresses are [email protected] and [email protected] respectively.
1
or even constrained coalition formation options. Under such restrictions, it is often possible to study more e¤ectively various solution concepts and their properties. In this spirit, we study coalition formation in hedonic games, that is, in situations where individuals’ preferences depend solely on the composition of the coalition they belong to. The hedonic aspect in players’ preferences was originally introduced by Dréze and Greenberg (1980) in a context concerning local public goods, where agents’ preferences depend on their consumption of the public good as well as the coalition they belonged to. However, the hedonic aspect transcends the local public goods context and spans over a much greater area of economics and sociology. The formation of societies, communities, social clubs and groups are just a few cases where the hedonic aspect is the central element steering their structure. Recently, several works concentrate on the hedonic notion and its implications in a more general framework. Notably, Banerjee, Konishi and Sönmez (1998) and Bogomolnaia and Jackson (2000) introduced the afore-mentioned pure hedonic games. Various existing as well as new solution concepts are applied to hedonic games and their rendition is studied. The Core, identi…es coalition structures that are immune to any bene…cial coalitional deviation, addressing, situations where coalitional deviations are feasible and costless. However, in a signi…cant number of situations coalitional deviations are not viable, either because of institutional (legal or physical) constraints or simply because agents are not able to coordinate their actions, especially in cases where the entire set of agents is rather large. In such instances solutions concepts that consider only individual deviations are warranted. Nash Stability identi…es coalition structures where no player wishes to migrate to another coalition in the structure. Such an analysis is relevant in situations where no permission is required to join a new coalition, a simple example being that of moving from one city to another. Individual Stability examines not only the preferences of the migrating player but also the preferences of the coalition this player plans to join, a coalition structure therefore, is not individually stable if some player wishes to migrate to another coalition where he is not hurtful. Any situation where an individual is hired by a business 2
entity (a coalition), and thus is allowed to join them, serves as an example where individual stability is a more appropriate notion. Lastly, even stronger is the notion of Contractual Individual Stability that requires the permission of the original coalition to allow a player to migrate, in the sense that they are not hurt by the departure of this member. In the spirit of the previous example, consider the case where the newly hired employee has to …rst break his contract with his previous employer. Unfortunately, most of the aforementioned notions do not always provide an answer even to simple problems as we will illustrate later in this section. Therefore, restrictions on players’ preferences need to be made to guarantee the existence of di¤erent solutions. In particular, Banerjee, Konishi and Sönmez (1998) study the Core in hedonic games. They de…ne the top coalition property, a property that imposes a degree of commonality on players’ preferences, while it guarantees the non-emptiness of the Core. The motivation for such a property is served from two perspectives: (i) the multiplicity of economic applications, some of which are formally presented in the paper, where the property naturally holds, and (ii) the possibility of an empty Core, illustrated through examples, even when strong assumptions such as additive separability, anonymity or single peakedness are imposed. Bogomolnaia and Jackson (2000) study primarily individually stable coalition structures in hedonic games and they propose the condition of ordered characteristics – a condition that builds on single peaked preferences- and consistency to guarantee the existence of individually stable coalition structures and partially their e¢ciency, while it provides an algorithm to identify weakly Pareto e¢cient and individually stable coalition structures. Analogously, the motivation for the condition stems from examples satisfying already strong assumptions where individually stable coalition structures do not exist. This project attempts to build on such endeavors, extending the existing analysis by identifying and rectifying some problems associated with the solution concepts used. While the most analyzed problem concerning the Core is its “frequent” emptiness, the notion has also being criticized for the easiness with which it discards feasible outcomes -which often results to emptiness. In particular, an outcome is excluded from the Core because a 3
set of agents can induce an other outcome that bene…ts all of its members. The likelihood of this outcome surviving a further deviation by some other set of agents is not examined. Especially, when such a further replacement may render the original deviating coalition much worse o¤ than it was in the very beginning, the original deviation may be deterred. Alternatively, the Core has been criticized for including outcomes that can be dominated if a group of agents does not behave myopically. To illustrate this point consider a group of agents that can induce some outcome which does not necessarily improve upon the original one. It is possible however, that once this new outcome becomes the status quo, further deviations may occur which will render the original deviating coalition better than it was in the very beginning. Allowing therefore, a potential deviating coalition to foresee many steps ahead, it may proceed with a deviation which ultimately bene…ts its members, that otherwise may have been halted. Indeed, such an argument applies to Example 5 where a core element is ruled out. However, if players’ preferences are strict, then core outcomes exhibit a surprising robustness and are immune to foresight, as is formally shown in following sections. Put di¤erently, for a core outcome to be excluded from the farsighted solution we propose, it must be the case that some players are indi¤erent between some coalitions. In environments were moves can be followed by counter moves and so on, it is natural to allow any group of agents that considers deviating to “speculate” on the ultimate result of its action. Obviously, an “ultimate” result must be immune to further deviations; that is, it cannot itself be dominated. For an outcome to be ultimate, it must be the case that no group of agents has an incentive to deviate, again anticipating the ultimate outcomes of its deviation. Thus, an ultimate result must be in the solution set of the game. Such a reasoning is captured by the notion of abstract stable set …rst introduced by von Neumann & Morgenstern (1944). The notion is characterized by internal and external stability. Internal stability guarantees that the solution set is free from inner contradictions, that is, any two outcomes in the solution set cannot dominate each other and external stability guarantees that every outcome excluded from the solution is accounted for, 4
that is, it is dominated by some outcome inside the solution. Although the original abstract stable set accommodates the issue of credibility regarding a dominating outcome it does not address foresight. Harsanyi (1974) introduced the concept of indirect dominance to capture foresight. An outcome indirectly dominates another, if there exists a sequence of outcomes starting from the dominated outcome and leading to the dominating one, and at each stage of the sequence the group of players required to enact the inducement prefers the …nal outcome to its status quo. In criticizing the Core for its myopia and lack of consistency, we suggested that agents should look ahead and speculate on the ultimate result of their initial actions. Baring in mind that di¤erent sequences of actions may take place, it is natural to question how the players are to treat situations where the ultimate result may not be unique; moreover, how they are to behave when some ultimate outcomes may make them better o¤ while others make them worse o¤. The answer to such a question depends solely on the behavioral characteristic of the players making the decision. Two extreme approaches are that of optimism (implicit in the notion of abstract stable set), where a coalition would proceed with the deviation if at least one of the ultimate outcomes makes its members better o¤, whereas a conservative coalition would not proceed with the deviation unless every possible ultimate outcome makes its members better o¤. Amalgamations of stability and indirect dominance, are studied by Chwe (1994) and Xue (1998) where both behavioral assumptions are investigated along with the existence of the solution concepts. The criticism developed thus far concerning the Core applies also to the stability notions that allow for only individual deviations. In the same manner a deviating coalition considers all plausible outcomes that may arise from its initial deviation, an individual migrating from one coalition to another can raise the same concerns, and question the possibility of others joining later on, or existing members departing. Moreover, the welcoming coalition (in the case of individual stability) may, for example, wonder whether the admission of a new undesirable member may bring later on far more desirable migrants. Lastly, the remaining coalition (in the case of contractual 5
individual stability) may permit a favorite member to depart believing that it will, later on, be replaced by someone even more desirable, and so on. To amend existing solution concepts, we introduce notions that allow players to foresee arbitrarily many steps ahead, and consider only the plausible (as opposed to any feasible) outcomes that are likely to result from their actions. Although the next sections de…ne formally and motivate more precisely all proposed solution concepts, we illustrate through the well known problem of roommates how (most) solutions remain silent. Example 1 (The roommate problem) Three individuals have to agree in sharing a room that can only accommodate two of them. The third individual has to …nd a room of his own, and of course none of them wishes that all three squeeze in one room. Unfortunately, their preferences are rather contradicting: the …rst prefers to share the room with the second, than the third, while the second prefers to share the room with the third rather than the …rst, and lastly the third prefers to share the room with the …rst rather than the second. Their preferences are summarized as follows: f1; 2g Â
1 f1; 3g
Â1 f1g Â1 f1; 2; 3g
f2; 3g Â
2 f1; 2g
Â2 f2g Â2 f1; 2; 3g
f1; 3g Â
3 f2; 3g
Â3 f3g Â3 f1; 2; 3g
It is easy to see that the Core of the game is empty. For example consider the partition of ff1; 2g; f3gg; it is blocked by f2; 3g since 2 prefers to be with 3 than with 1; and 3 prefers to be with 2 than alone. The same argument blocks any partition involving a pair and a singleton. Unfortunately, the Core is not the only solution encountering problems. No coalition structure is Nash stable either. Consider the same partition of ff1; 2g; f3gg ; which is blocked via 2 that wishes to join 3. In fact, since 3 welcomes 2 this coalition structure is not individually stable either. Similar arguments render all partitions Nash and individually unstable. The only solution that provides an answer to this problem is contractual individual stability. Although the grand coalition and the singletons are excluded from the solution, the partitions involving a pair and a singleton are 6
contractually individually stable. Consider the partition f1; 2gf3g, although 2 would like to join 3 and 3 welcomes 2, 1 will not permit 2 to depart from their coalition. In fact, contractual individual stability o¤ers a very appealing analysis to the problem if 1 and 2 had indeed singed a contract together for their room. In the event, though, were no commitments have taken place the notion is not applicable, and no answer is given to the problem. In contrast, all four modi…cations of the solution concepts proposed in this paper suggest the same solution, that is, they support only all the coalition structures involving a pair and a singleton. The fundamental argument embedded in all the notions is as follows: consider again the partition f1; 2gf3g; player 2 will not join 3 fearing that 3 may, later on, join 1 and thus leave 2 alone. It is the conservative behavior imposed on the players, along with their foresight that prevents them from dissolving a “mediocre” partnership in pursuance of the perfect one. In Section 2 we introduce formally the model and the solution concepts. We also provide examples where the old notions are not silent to illustrate the distinction and re…nement of the new ones. Existence results concerning the solution concepts are presented in Section 3. In the same section we investigate the relation of the new notions to existing ones when di¤erent restrictions are imposed on players’ preferences. Lastly, in Section 4 we discuss conjectures and open questions of interest to the topic.
2
De…nitions
We …rst introduce the basic notations and the existing stability notions. Then we proceed to introduce stability notions under foresight.
2.1
Preliminaries
² Let N be a …nite set of players. ² A coalition S is a non-empty subset of N, and S(i) denotes that i 2 S: ² Let S denote the collection of all coalitions and for i 2 N; let S(i) denote the collection of coalitions that contain i, that is, S(i) = fS ½ 7
N j i 2 Sg: Hedonic game: A hedonic game G is a pair (N; fºi gi2N ), where N is the …nite set of players and for all i 2 N, ºi is a complete, re‡exive, and transitive binary relation on S(i); representing i’s preferences over coalitions that contain i: We use Âi to denote asymmetric part of º i (i.e., strict preferences) and » i the indi¤erence relation. ² A coalition structure, P , is a partition of N, that is, P = fS1 ; S2 ; :::; Sk g, Sk T Sj = ;: j =1 S j = N and for all i 6= j; S i ² Let P be the collection of all coalition structures and PS = fP j S 2 P; S 2 Sg be the collection of all coalition structures that contain coalition S: Similarly, PS denotes that S 2 P . ² For P 2 P and i 2 N, let SP (i) be the coalition S 2 P such that i 2 S: The four main stability notions in the literature are as follow. Core stability: A coalition structure P 2 P is core stable or in the core of G if there does not exist “a blocking coalition” S ½ N such that S  i SP (i) for all i 2 S: Non-cooperative stability notions: Nash stability: A coalition structure P 2 P is Nash stable if there do not exist i 2 N and S 2 P [ f;g such that S [ fig Âi SP (i): Individual stability: A coalition structure P 2 P is individually stable if there do not exist i 2 N and S 2 P [ f;g such that S [ fig Âi SP (i) and S [ fig º j S for all j 2 S: Contractural individual stability: A coalition structure P 2 P is contractually individually stable if there do not exist i 2 N and S 2 P [f;g such that (1) S [ fig  i SP (i) and S [ fig º j S for all j 2 S; (2) SP (i) n fig º j SP (i) for all j 2 SP (i) n fig: 8
2.2
Farsighted coalitional stability
According to core stability, if P is under consideration, any coalition S ½ N can form and object to P . When S forms and before other players regroup, the resulting coalition structure P 0 = fSg [ fT n S j T 2 P and T n S 6= ;g S
and in this case, we write P ! P 0 . We can extend players’ preferences over coalitions to coalitional preferences over coalition structures. Coalitional preferences: A coalition S ½ N strictly prefers P 0 to P; denoted P 0 Â S P; where P; P 0 2 P , if SP 0 (i) Âi SP (i) for all i 2 S; similarly, a coalition S ½ N prefers P 0 to P; denoted P 0 &S P; where P; P 0 2 P , if SP 0 (i) &i SP (i) for all i 2 S. Given a coalition structure P 2 P , if some coalition S ½ N has an incentive to form, thereby inducing P 0, we say P 0 dominates P: (Direct) Dominance: For P; P 0 2 P, P 0 (directly) dominates P , or P 0 > S P , if P ! P 0 and P 0 Â S P . Thus, core can be alternatively de…ned as follows: Core: Core of G is the set of coalition structures that are not dominated with respect to >, i.e., core of G is the (abstract) core of abstract system1 (P ; >): Formally, Core(P ; >) = fP 2 P j @P 0 2 P such that P 0 > P g : Myopia is re‡ected in the fact that a blocking coalition does not consider the possibility that the rest of the players may regroup. That is, given P 0 , another coalition can form to induce a new coalition structure and so on. If players are farsighted, they should consider the ultimate outcomes 1 An abstract system is a set and a binary relation on this set. See von Neumann and Morgenstern (1944).
9
of their actions. Thus, a coalition may choose to “deviate” to coalition structure, which does not necessarily make its members better o¤, as long as their deviation leads to …nal coalition structures that bene…t all its members; similarly, a coalition may choose not to deviate to a coalition structure it prefers if their deviation eventually leads to coalition structures that make its members worse o¤. The following “indirect dominance” due to Harsanyi (1974) and Chwe (1994) captures foresight. Indirect dominance: For P; P 0 2 P, P 0 indirectly dominates P , or P 0 À P; if there exists a sequence of coalition structures P 1; P 2; : : : ; P k ; with P 1 = P and P k = P 0, and a sequence of coalitions S1 ; S2; : : : ; S k¡1 Sj such that P j ! P j+1 and P 0 Â Sj P j for all j = 1; 2; : : : ; k ¡ 1: We can now examine di¤erent solutions to the abstract system (P; À): The (abstract) core of (P; À) is one of these. The (abstract) core of (P ; À): Core(P; À) = fP 2 P j @P 0 2 P s.t. P 0 À P g : It is easy to see that Core(P; À) ½ Core(P; >): However, core does not consider the credibility of the dominating alternative. The following von Neumann-Morgenstern (vN-M) (abstract) stable set amends this shortcoming of the core. vN-M stable set of (P ; À): R ½ P is vN-M internally stable if there do not exist P; P 0 2 R such that P 0 À P ; R is vN-M externally stable if for all P 2 P n R; there exists P 0 2 R such that P 0 À P . R is vN-M stable is it is both internally and externally stable. As shown in Greenberg (1990), vN-M stable set implicitly assumes “optimistic behavior”. To see this, we …rst introduce the following notation. Likely outcomes given P : For Q ½ P and P 2 P , let QjP;À = fP 0 2 Q j P 0 = P or P 0 À P g : If Q is a “solution set”, then QjP;À is the set of likely outcomes. 10
Now we can rewrite the de…nition of vN-M stable set: vN-M stable set rede…ned: R is vN-M internally stable if Q 2 R implies S that there do not exist P 2 P and S ½ N such that Q ! P and P 0 ÂS Q for some P 0 2 RjP;À: R is vN-M externally stable if Q 2 P nR S implies that there exist P 2 P and S ½ N such that Q ! P and P 0 ÂS Q for some P 0 2 Rj P;À: R is vN-M stable if it is both internally and externally stable. However, the vN-M stable set for (P ; À) entails over-optimism on the part of a departing coalition as illustrated in Xue (1998). The following notion of conservative stable set amends this problem. Conservative stable set: Q is conservatively internally stable if Q 2 Q S implies that there do not exist P 2 P and S ½ N such that Q ! P; QjP;À 6= ;; and P 0 Â S Q for all P 0 2 QjP;À: Q is conservatively externally stable if Q 2 P n Q implies that there exist P 2 P and S S ½ N such that Q ! P; Qj P;À 6= ;; and P 0 ÂS Q for all P 0 2 QjP;À: Q is conservatively stable if it is both conservatively internally and externally stable. The conservative stable set is closely related to Greenberg’s (1990) “conservative stable standard of behavior” (CSSB) and Chwe’s (1994) “consistent set”. The following example illustrates the myopia embedded in the core and how it is recti…ed through the conservative stable set. Example 2 (Conciliatory partnerships) Consider a game with 4 players and the following preference ordering: f1; 4g Â
1 f1; 2; 3g
 1 f1; 2g Â1 f1g Â1 ¢ ¢ ¢
f1; 2; 3g Â
2 f1; 2g
Â2 f2; 3g Â2 f2g  2 ¢ ¢ ¢
f1; 2; 3g Â
3 f2; 3g
Â3 f3; 4g Â3 f3g  3 ¢ ¢ ¢
f3; 4g Â
4 f1; 4g
Â4 f4g Â4 ¢ ¢ ¢
11
The rest of relevant coalitions are ranked strictly below the above coalitions. The core contains only the following partition: ff1; 4g; f2; 3gg: Note that partition ff1; 2g; f3; 4gg is not in the core since coalition f1; 2; 3g can strictly improve upon it. However, if players 2 and 3 are farsighted they will realize that once partition ff1; 2; 3g; f4gg is the status quo, coalition f1; 4g will form and bring about ff1; 4g; f2; 3gg which is not as good for 2 and 3 as the original one. Therefore, ff1; 2g; f3; 4gg should not be ruled out by f1; 2; 3g if 2 and 3 are farsighted. However, the reader may have already observed that although ff1; 2g; f3; 4gg may not be disrupted by coalition f1; 2; 3g; the following sequence of events may take place: ff1; 2g; f3; 4gg
f1;3g
f2;3g
¡! ff2g; f1; 3g; f4gg ¡! ff1g; f2; 3g; f4gg f1;4g
¡! ff1; 4g; f2; 3gg: Indeed partition ff1; 4g; f2; 3gg indirectly dominates ff1; 2g; f3; 4gg. But the reverse holds as well: ff1; 4g; f2; 3gg
f2;4g
f1;2g
¡! ff1g; f2; 4g; f3gg ¡! ff1; 2g; f3g; f4gg f3;4g
¡! ff1; 2g; f3; 4gg: In fact, both partitions ff1; 4g; f2; 3gg and ff1; 2g; f3; 4gg belong to the maximal conservative stable set:
2.3
Notions of farsighted non-cooperative stability
Nash stability, individual stability, and contractual individual stability consider only one-step “deviations” initiated by individuals. We can introduce a farsighted stability notion as an alternative for each of these three concepts. 2.3.1
Farsighted Nash stability
To de…ne farsighted Nash stability, we …rst formalized that if i 2 N leaves a coalition in P to join another coalition also in P , the resulting coalition 12
structure, P 0; is de…ned as follows: (1) SP (i) n fig 2 P 0 if SP (i) n fig 6= ;; (2) 9 T 2 P [ f;g such that T [ fig 2 P 0 ; (3) S 2 P 0 for all S 2 P such fig that S 6= SP (i) and S 6= T: We shall write P * P 0 in this case. Now, we can adapt the notions of farsighted stability in Section 2.1, observing that only fig individuals can move according to *. The following example illustrates the myopia embedded in Nash stability. Example 3 (An undesired guest ) 2 Consider a game with 3 players and the following preference orderings: f1; 2g  f1; 2g  f1; 2; 3g Â
1f1g 2f2g
 1 f1; 2; 3g  1 f1; 3g  2 f1; 2; 3g  2 f2; 3g
3f2; 3g
 3 f1; 3g  3 f3g
No partition is this game is Nash stable. For example, ff1; 2g; f3gg3 is not Nash stable because 3 prefers to joint f1; 2g than staying alone. However, ff1; 2; 3gg is not Nash stable either. In fact, players 1 and 2 will eventually bring back ff1; 2g; f3gg: If players are farsighted, then ff1; 2g; f3gg is stable. 2.3.2
Farsighted individual and contractual stability
Recall, in the previous subsection, player i 2 N can change P 2 P to another fig P 0 2 P such that P * P 0. However, if player i need the permission of SP 0 (i) n fig as in the case of individual stability, then i alone cannot change fig fig P to P 0 if P * P 0: We write P * P 0 to denote that i needs the permission SP 0 (i)
fig
of SP 0 (i) n fig to change P to P 0. Thus, P is individually stable if P *
SP 0 (i)
P 0 ; P 0 Âfig P; and, P 0 &SP 0 (i) P: If players are farsighted, they should consider that P 0 may be replaced by another coalition structure and it is the “…nal” coalition structure that matters. For this we can de…ne the following indirect dominance relation.
Indirect individual dominance For P; P 0 2 P, P 0 indirectly individually dominates P , or P 0 m P; if there exists a sequence of coalition structures P 1; P 2; : : : ; P k ; with P 1 = P and P k = P 0 , and a sequence of 2 3
See Bogomolnaia and Jackson (2000). This partition is, however, core stable.
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individuals i 1; i2 ; : : : ; ik¡1 such that P j
fij g
*
SP j +1 (i)
P 0 &SP j +1 (i) P j for all j = 1; 2; : : : ; k ¡ 1:
P j+1 and P 0 Âij P j and
Individually conservative stable set (under foresight): Q is conservatively internally stable if Q 2 Q implies that there do not exist P 2 P fig and i 2 N such that Q * P; Qj P;m 6= ;; P 0 Â i Q; and P 0 &SP (i) Q SP (i)
for all P 0 2 Qj P;m : Q is conservatively externally stable if Q 2 P n Q fig
implies that there exist P 2 P and i 2 N such that Q * P; QjP;m 6= SP (i)
;; P 0 Âi Q; and P 0 & SP (i) Q for all P 0 2 Qj P;m Q is conservatively stable if it is both conservatively internally and externally stable. Note that for Q ½ P and P 2 P, Qj P;m = fP 0 2 Q j P 0 = P or P 0 m P g : We can also de…ne a similar notion of contractual stable set under foresight by modifying the above de…nitions to capture the fact that a deviating player also needs the permission of the coalition from which he departs. The following example illustrates the implication of foresight. Example 4 (Switching partners) Consider a game with 4 players. The following is part of the preference ordering of the players. The rest of the relevant coalitions are ranked strictly below these coalitions. f1; 4g Â
1f1; 2g
f2; 3g Â
2f2; 3; 4g
f1; 4g Â
4f3; 4g
f2; 3g Â
3f3; 4g
 2 f1; 2g
ff1; 2g; f3; 4gg is individually stable and contractually individually stable: For example, if 2 leaves his current partner 1, then 1 will be worse o¤; moreover, if 1 joins the coalition of 3 and 4, both 3 and 4 will be worse o¤. If players are farsighted, however, they will anticipate that once 2 leaves 1 and joints f3; 4g, 4 will subsequently leave 2 and 3 to join 4. That is, f2g
f4g
f:g
f:g
ff1; 2g ; f3; 4gg * ff1g ; f2; 3; 4gg * ff1; 4g ; f2; 3gg : 14
Thus, not only players 2 and 4 have incentive to initiate their moves but also other relevant coalitions have incentive to approve their moves. Note that the …nal coalition structure ff1; 4g ; f2; 3gg Pareto dominates ff1; 2g ; f3; 4gg.
3
Results
In this section, we …rst establish the existence of the notions we propose and then proceed to analyze their properties.
3.1
Existence
The following theorem shows that the conservative stable set always exists; this is in contrast to the fact that the core may be empty and the vN-M stable set may fail to exist. Theorem 1 There exists a conservative stable set and a maximal one with respect to set inclusion. Proof. Let Q 0 = P and for k = 0; 1; : : : de…ne recursively, ( ) S k k k S ½ N s.t. Q ! P; Q jP;À 6= ;; : Qk+1 = Q 2 Q j @P 2 Q and and P 0 ÂS Q 8P 0 2 Qk jP;À ¤
¤
¤
We shall show that there exits k¤ such that Qk +1 = Qk and Qk is conservatively stable. First we claim that for all k and P 2 P, there exist P 0 2 Qk such that Qk jP;À ¾ Qk jP 0 ;À 3 P 0: Obviously, this is true for k = 0: Assume the claim is true for k. Let P 1 2 Qk n Qk+1; then there exist S Q 2 Qk and S ½ N such that P 1 ! Q; Qk jQ;À 6= ;; and P 0 Â S P 1 for all P 0 2 Qkj Q;À: Thus, P 0 À P 1 for all P 0 2 Qk j Q;À: Given that À is irre‡exive, Qk jP 1 ;À ) Qk j Q;À. Since the claim is true for k, then there exists P 2 2 Qk such that Qkj Q;À ¾ Qk+1j P 2 ;À 3 P 2. Thus, we Qk j P 1 ;À ) Qk j P 2 ;À 3 P 2: Obviously, Qk+1 jP 1;À ¾ Qk+1(P 2): If P 2 2 Qk+1jP 2 ;À; we are done. Otherwise, there is P 3 2 Qk such that Qk j P 2 ;À ) Qk j P 3 ;À 3 P 3: Since Qk is …nite, there exist ` such that Qk j P 1 ;À ¾ Qk j P 2 ;À ¾ ¢ ¢ ¢ ¾ Qk jP ` ;À 3 P `. Moreover, ¤ ¤ it follows that there exits k¤ such that Qk +1 = Qk : 15
¤
¤
¤
Qk +1 = Qk implies that Qk is conservatively internally stable. To ¤ show conservative external stability, suppose P 2 P n Qk : Then there exists k < k ¤ such that P 2 Qk nQk+1: Thus, there exist Q 2 Qk and S ½ N S such that P ! Q; Qk j Q;À 6= ;;and P 0 Â S P for all P 0 2 Qk jQ;À: Since ¤ ¤ ¤ Qk jQ;À ¾ Qk jQ;À 6= ;; we have P 0 ÂS P for all P 0 2 Qk j Q;À: Thus, Qk is conservatively external stable. ¤ To show that Qk is the largest conservative stable set, assume to the contrary that there exists another conservative stable set Q0 such that there exists P 2 Q0 n Q. There exists k such that Qk ¾ Q0 : Since Q0 is stable, Q` ¾ Q0 for all ` ¸ k. Contradiction. Existence of other farsighted notions can be established in the same fashion.
3.2
Farsighted Coalitional Stability and the Core
As mentioned in the introduction, the Core of hedonic games with strict preferences exhibits a surprising robustness. Although it may be expected that core partitions would be excluded from the conservative stable set on the grounds that coalitions may form (without any immediate improvement) looking ahead to future (improving) restructuring, it is actually not the case. In fact, the following result formalized by theorem 2, shows that core partitions are always included in the maximal conservative stable set, when preferences are strict. Concluding, therefore, that core partitions survive farsighted players. However, this in not the case when preferences are not strict as illustrated in Example 8. The critical element that collapses the intuitive “expectations” expressed above is the conservative aspect embedded in the players’ behavior. Indeed, it may be the case that such a sequence of events exists that improves an initial acting coalition, departing from a core partition, some steps after its action. However, along such a sequence of events many other sequences may originate, at least one them leading back to the starting core partition, which is not improving, discouraging therefore, due to the conservative behavior the initial acting coalition. Exactly this feature of core partitions is captured by
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lemma 3. Theorem 2 Consider a game G with strict preferences and let Q¤ be the maximal conservative stable set. Then Core(P ; >) ½ Q¤: ¹ = fP g such that P 2 Core(P ; >): We will show that Proof. Consider Q ¹ is conservatively stable. Q Internal stability is trivial. External stability can also be established ¹ is with the direct use of Lemma 3, which asserts that any partition P 0 2 =Q indirectly dominated by P: Finally, according to Theorem 1 the maximal conservative stable set Q¤ ¹ and therefore, P 2 Q¤: ¾Q Lemma 3 Consider a game G with strict preferences and assume that the Core(P ; >) 6= ;. If partition P ¤ 2 Core(P; >); then P ¤ indirectly dominates any partition P 2 P nfP ¤g: Proof. We will show that P ¤ À P by constructing a sequence of moves, departing from P and ending at P ¤, where every acting coalition prefers P ¤ to its status quo. The sequence consists of two main parts: in the …rst part P is decomposed into singletons and in the second part P ¤ is constructed from the singletons. We start with the …rst part. For convenience let SP denote an arbitrary coalition in P: Since P ¤ 2 Core(P; >); @ S 2 P such that P ÂS P ¤: That is, for any SP there exists i 2 SP such that P ¤ Â i P: Let i initiate the sequence by breaking away from SP . That is, T1 = fig induces partition P1 such that T1 2 P1 and (SP nT1 ) 2 P1 : Again, since P ¤ 2 Core(P ; >); @ S 2 P1 such that P1 Â S P ¤: Any SP1 contains j such that P ¤ Â j P1 : Let T2 = fjg induce partition P2 such that T1 ; T2; (SP1 nT2 ) 2 P2: We can continue in this manner until we reach partition Pt such that every jSPt j = 1. Obviously, since P ¤ 2 Core(P; >); for every SPt = fig 2 Pt we have P ¤ Âfig Pt : Next we proceed with second part of the sequence which involves constructing P ¤ : 17
Let P ¤ = fS1; S2; :::; Sk g: Then, let the next step of the sequence be the formation of S1 ; since all of its members would rather be together (which is the case under P ¤ ) than alone (which is the case under Pt ): Denote the new partition by Pt1 . Let the next step of the sequence be the formation of S2 : Again all of S2 ’s members would rather be together (which is the case under P ¤ ) than alone, which is the case under Pt 1 : Denote the new partition by Pt2 : Continue in this manner, until P ¤ is formed. Note that along the way, when Sj forms from Pt(j ¡1) ; all of its members would rather be together (under P ¤) than alone (under Pt(j ¡1) ). In summarizing: 1st part : 2nd part :
T
T
T
T
T P !1 P1 !2 P2 !3 ::: ! Pt
S
S
S
Sk¡1
S
Pt !1 Pt1 !2 P t2 !3 ::: ! Pt(k¡1) !k P ¤:
While P ¤ ÂTi+1 Pi for i = 0; :::; T where P = P0 and P ¤ Â Sj+1 Ptj for j = 0; :::; k ¡ 1; where Pt0 = Pt . Thus, P ¤ À P: The following example asserts that theorem 2 holds only in the case of strict preferences, by illustrating a case (involving indi¤erences) where core partitions are actually excluded from the maximal conservative stable set. Example 5 (Core Re…nement ) Consider a game with 4 players and the following preference ordering: f1; 2g Â
1 f1; 3g
Â1 f1; 4g Â1 f1g  1 ¢ ¢ ¢
f2; 4g Â
2 f1; 2g
»2 f2; 3g Â2 f2g  2 ¢ ¢ ¢
f1; 3g Â
3 f3; 4g
»3 f2; 3g »3 f3g  3 ¢ ¢ ¢
f3; 4g Â
4 f2; 4g
Â4 f1; 4g Â4 f4g  4 ¢ ¢ ¢
The rest of relevant coalitions are ranked strictly below the above coalitions. The core contains the two partitions ff1; 2g; f3; 4gg and ff1; 3g; f2; 4gg:
18
Given ff1; 2g; f3; 4gg, however, players 2 and 3 will formed a coalition, expecting the following chain of events ff1; 2g; f3; 4gg
f2;3g
f1;3g
¡! ff1g; f2; 3g; f4gg ¡! ff1; 3g; f2g; f4gg f2;4g
¡! ff1; 3g; f2; 4gg:
More formally, the unique stable set contains only ff1; 3g; f2; 4g Remark 1 It is easy to see that if the jCore(P ; >)j > 1 then Core(P; À) is always empty, since any one core outcome indirectly dominates all the others. The reverse however, is not true. As is demonstrated by example 2, jCore(P ; >)j = 1, yet Core(P; À) = ; since the unique core outcome is indirectly dominated by another (non-core) outcome and vice versa, ruling each other out. Banerjee, Konishi and Sönmez (1998) …rst introduce the top-coalition property as a relaxation of the common ranking property which is due to Farell and Scotchmer (1988). It requires that there exists a coalition that is preferred the most by its members compared to any other coalition each member could possibly join. Moreover, the condition holds for any subset of players. Top-coalition property Given a non-empty set of players V ½ N, a coalition S ½ V is a top-coalition of V if for every i 2 S and any T (i) ½ V we have S ºi T (i): A hedonic game G satis…es the top-coalition property if for any non-empty set of players V ½ N, there exists a top-coalition of V . The authors show that under the top-coalition property and strict preferences the Core always contains a unique partition, related to top-coalitions, that we formally de…ne as a top-coalition partition: Top-coalition partition Given a game G that satis…es the top-coalition property, a partition P ¤ 2 P is a top-coalition partition if P ¤ is such that P ¤ = fS1; S2; :::; Sj ; :::Sk g where S1 is a top-coalition of N, S2 is a top-coalition of NnS1; S3 is a top-coalition of N nfS1 [ S2g,..., Sk is a top-coalition of Nnf[j): And since there is only one element in Q¤; even if there are more sequences stemming from Q 0; their terminal coalition structure is bound to be P ¤. However, we …nd the construction of the following sequence of coalition structures leading from any arbitrary coalition structure Q0 to P ¤ more intuitive due to the direct use of the top coalitions. In particular, the sequence proposed here omits the 1st part -the dissolution of the starting partition into singletons- of the sequence proposed in lemma 3 and starts immediately from the 2nd part, that is, the formation of the terminal sequence. Step 1: Consider an arbitrary partition Q0 6= P ¤: If S1 2 Q0 proceed to step 2; otherwise let the …rst step of the sequence be the formation of S1 since, by de…nition, Q0 ÁS1 P ¤ no matter what Q 0 is as long as S1 2 = Q0: Let the new coalition structure be denoted by Q1 ; and note that S1 2 Q1: 4 The
authors also de…ne a weaker notion of the top-coalition property, appropriately named weak-top-coalition property and they show that the Core is always non-empty under the weak-top-coalition property, yet they do not o¤er any characterization of the solution apart from the fact that it contains at least all the weak-top-coalition partitions. It is easy to construct examples where the Core along with the conservative stable set may contain partitions that are not weak-top-coalition.
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Step 2: If S2 2 Q1 proceed to step 3, otherwise let the second step of the sequence be the formation of S2 , since by de…nition Q1 ÁS2 P ¤: More speci…cally, we know that all the members of S2 prefer to be in S2 than in any other coalition involving members of NnS1, which is exactly their situation under Q1; since S1 has already formed: Let the new coalition structure be denoted by Q2; and note that fS1 ; S2 g 2 Q2 : .. .
Step k: If Sk 2 Qk¡1 , then Qk¡1 is already P ¤; otherwise let the last step of the sequence be the formation of Sk , since be de…nition Qk¡1 ÁSk P ¤: Note that if Sk is not formed then it must partitioned in smaller coalitions, whose members would rather merge since Sk is a top-coalition of Nnf[j
