Pensamiento filosófico en los años de vida colonial.

In many cooperative scenarios, superadditivity can be quite a restrictive concepts. For instance, it is quite natural to consider that any cooperation is accompanied with an inherent cost that can limit the benefits of this cooperation. In consequence, the formation of a grand coalition is seldom guaranteed. In such cases, canonical coalitional games are not suited for modeling the cooperative behavior of the players. In this regards, coalition formation games encompass coalitional games where, unlike the canon- ical class, network structure and cost for cooperation play a major role. The characteristics of a coalition formation game can be summarized as follows:

1. Forming a coalition brings gains to its members, but the gains are limited by a cost for forming the coalition. Thus, the game is non- superadditive and the formation of a grand coalition is not guaran- teed.

2. The game can be in characteristic or partition form.

3. The objective is to study the network coalitional structure, i.e., an- swering questions like which coalitions will form, what is the optimal coalition size and how can we assess the structure’s characteristics, and so on.

In many problems, forming a coalition requires a negotiation process or an information exchange process which can incur a cost, thus, reducing the gains from forming the coalition. Therefore, in such scenarios, coalition formation games prove to be quite a solid tool. In contrast to canonical games, where formal rules and analytical concepts exist, solving a coalition formation game, is more difficult, and application-specific. In any coalition formation game, the following definition is useful:

Coalitional Game Theory

Definition 3 Given a coalition formation game among a set of playersN , a

collection of coalitions, denoted byS, is defined as the set S = {S1, . . . , Sl} of

mutually disjoint non-empty coalitions Si ⊂ N . In other words, a collection is any arbitrary group of disjoint coalitions SiofN not necessarily spanning all players ofN . If the collection spans all the players of N ; that is∪l_j=1Sj =N ,

the collection is apartition of N or a coalitional structure.

In the presence of a coalitional structure, the solution concepts dis- cussed in the previous subsection need substantial changes in their defi- nition for applying them in a coalition formation setting. Even by changing the definition, finding these solutions is by no means straightforward and can be cumbersome. In [30], it was shown that, in the presence of a coalitional structure, the core and the nucleolus, as defined in canonical coalitional games, are inapplicable and an alternative definition is provided instead. In contrast, by a slight modification of its definition, the Shapley value can be found by computing the Shapley value over each coalition present in the partition [30]. Hence, finding optimal coalitional structure and characterizing their formation is quite a challenging process, and, un- like canonical coalitional games, no unified or formal solution concepts exist. In fact, a majority of the literature dealing with coalition formation games, such as [31–34] or others, usually re-defines the solution concepts or presents alternatives that are specific to the game being studied.

For coalition formation games, the most important aspect is the forma- tion of the coalitions, i.e., answering the question of “how to form a coali- tional structure that is suitable to the studied game”. In practice, coalition formation entails finding a coalitional structure which either maximizes the total utility (social welfare) if the game is TU, or finding a structure with Pareto optimal payoff distribution for the players if the game is NTU. For achieving such a goal, a centralized approach can be used; however, such an approach is generally NP-complete [31–34]. The reason is that, finding an optimal partition in a general case, requires iterating over all the partitions of the player set N . The number of partitions of a set N grows exponentially with the number of players in N and is given by a value known as the Bell number [31]. For example, for a game whereN has only 10 elements, the number of partitions that a centralized approach must go through is 115975 (computed through the Bell number). Hence, finding an optimal partition from a centralized approach is, in general, compu- tationally complex and impractical. In some cases, it may be possible to explore the properties of the game, notably of the value v, for reducing the centralized complexity. Nonetheless, in many practical applications, it is

desirable that the coalition formation process takes place in a distributed manner, whereby the players have an autonomy on the decision as to whether or not they join a coalition. In fact, the complexity of the central- ized approach as well as the need for distributed solutions have sparked a huge growth in the coalition formation literature that aims to find low complexity and distributed algorithms for forming coalitions [31–34].

The approaches used for distributed coalition formation are quite var- ied and range from heuristic approaches [31], Markov chain-based meth- ods [32], to set theory based methods [33] as well as approaches that use bargaining theory or other negotiation techniques from economics [34]. Clearly, constructing coalition formation algorithms is application-specific, however, some work, such as [33] provides generic rules that can be used to derive coalition formation algorithms in different scenarios. In this re- gards, the work in [33] does not provide an algorithm for coalition forma- tion, but it presents a framework that can be tailored for developing such an algorithm. The main ingredients presented in [33] that are presented in [33] are the following:

1. Well-defined orders suitable to compare collections of coalitions. 2. Two simple operations for forming or breaking coalitions.

3. Stability notions that can be suited in a coalition formation context. By using the guidelines in [33], one can devise different coalition formation algorithms. Moreover, many of the algorithms in the literature can also be tailored to new applications through adequate modifications.

Further, we note that coalition formation approaches can be either fully reversible, partially reversible or irreversible [32]. An irreversible coalition formation approach implies that, whenever a coalition forms, its members are not allowed to leave it. In contrast, in a fully reversible approach, the players can join and leave coalitions with no restrictions. On one hand, a fully reversible approach is practical and flexible, however, deriving such an approach can be complicated. On the other hand, although irreversible approaches are easy to construct, their practicality is limited as the players are bound to remain in a coalition they join with no possibility of break- ing the agreement. For this purpose, partially reversible approaches have been recently sought as they provide a balance between practicality and complexity. In partially reversible coalition formation approaches, once the players form a coalition, they can break that coalition under certain

Coalitional Game Theory

conditions. Under different applications, one can carefully select the most practical and suited approach.

In summary, coalition formation games are diverse, and, in addition to the previously mentioned approaches, numerous schemes and rules exist. For example, a type of coalition formation games, known as hedonic coali-

tion formation games has been widely studied in game theory. Hedonic games are interesting since they allow the formation of coalitions (whether dynamic or static) based on the individual preferences of the players. Fur- ther, hedonic games admit different stability concepts that are extensions to well known concepts such as the core or the Nash equilibrium used in a coalition formation setting [35]. In this regard, hedonic games constitute a very useful analytical framework which has a very strong potential to be adopted in modeling problems in wireless and communication networks (only few contributions such as [36] used this framework in a communi- cation/wireless model). Further, a multitude of algorithms and concepts pertaining to coalition formation games can be found in [31–34] and many others.

Coalition Formation Games in Wireless Networks

While canonical coalitional games have had several applications in wire- less networks, surprisingly, coalition formation games applications are still scarce (e.g., in [37, 38]). This is mainly due to the fact that, un- like canonical coalitional games, no unified reference or formal rules exist for solving coalition formation games. In addition, most existing tutorial or references on coalitional game theory mainly focus on canonical coalitional games, with little mention of coalition formation.

However, one can see that coalition formation games have a huge po- tential of applications within wireless networks. For instance, in a wireless or communications environment, cooperation always entails costs such as energy, power, time, or others. In most wireless problems, cooperating, i.e., forming a coalition, is preceded by a negotiation process or an infor- mation exchange process which incurs costs that can significantly reduce the gains from forming a coalition. Hence, in these scenarios, canonical coalitional games are inapplicable and one must revert to formulating a coalition formation approach. In addition, next generation wireless net- works are large-scale, heterogeneous, and characterized by a dynamically varying environment. In such a setting, it is restrictive to assume that a grand coalition would form and it is imperative to study how the network structure would be affected by the presence of cooperative nodes. Further,

with the recent interest in cooperation as well as the need for next gener- ation wireless users to learn and adapt to their environment (changes in topology, technologies, service demands, application context, etc), coali- tion formation game models are bound to be ubiquitous in future wireless communication networks. In brief, any problem involving the study of co- operative wireless nodes behavior when a cost is present, is a candidate for modeling using coalition formation games. Thus, the potential applica- tions of coalition formation games in wireless networks are numerous and diverse.

In Section 8 of this dissertation, we provide numerous coalition forma- tion models, algorithms and applications suited for wireless and commu- nication networks.