The main two components of a coalitional game are the players set, nor- mally denoted by N = {1, . . . , N}, and the coalition value. The set N rep- resents the players that interact in order to form cooperative groups, i.e., coalitions, in order to improve their position in the game. Thus, a coalition
S⊆ N represents an agreement between the members of S to act as a sin-
gle entity in a given game. Forming coalitions or alliances is pervasive in many disciplines such as politics, economics, and, more recently, wireless
Coalitional Game Theory
networks. Further, the coalition value, usually denoted as v, quantifies the worth or utility of a coalition in a game. Depending on the definition of the value, a coalitional game can have different properties. Nonethe- less, independent of the definition of the value, any coalitional game can be uniquely defined by the pair (N , v)1.
In general, a coalition value can be in three different forms:
• Characteristic form. • Partition form. • Graph form.
The characteristic form is the most common in game theory literature, and it was first introduced by Von Neuman and Morgenstern [21]. In char-
acteristic form, the worth or utility of any coalition S ⊆ N is independent of the coalitions/structure formed among the players outside S, i.e., the players in N \ S. Thus, a value of a coalition S in characteristic form de- pends solely on the members of that coalition. In contrast, a game is in
partition formif, for any coalition S ⊆ N , the coalitional value depends both on the members of S as well as on the coalitions formed by the members in inN \ S. The concept of the partition form was introduced by Thrall and Lucas [22] with the characteristic form as a particular case. Further, in many coalitional games, the connection between the players in the coali- tion, i.e., “who is connected to whom”, strongly impacts the value of the game. In such coalitional games, the interconnection between the players is usually captured through a graph structure. Subsequently, for model- ing such coalitional games, the value is considered in graph form, i.e., for each graph structure, a different utility can be yielded. The idea of captur- ing the interconnection graph within coalitions started with the pioneering work of Myerson [23]. In [23], Myerson started with a game in charac- teristic form, and layered on top of that a network structure, represented by a graph that indicated which players can communicate. Consequently, the value function became dependent on the communication graph which led to the idea of a game in graph form (this work was further generalized in [24]).
In any coalitional game (independent of its form), it is always important to distinguish between two entities: the value of a coalition and the payoff received by a player. The value of a coalition represents the amount of
1
Since, for a given players’ setN , the value v completely describe the coalitional game,
utility that a coalition, as a whole, can obtain. In contrast, the payoff of a player, represents the amount of utility that a player, member of a certain coalition, will obtain. For instance, depending on how the value is mapped into payoffs, the coalitional game can be either with transferable utility (TU) or with non-transferable utility (NTU). A TU game implies that the total utility received by any coalition S ⊆ N can be apportioned in any manner between the members of S. A prominent example of TU type games is when the value represents an amount of money, which can be distributed in any way between coalition members. In particular, when considering a TU game in characteristic form, the value is a function2 over the real line defined as v : 2N → R. In such a setting, for every coalition, the value function associates a real number which represents the overall utility or worth of this coalition. Further, due to the TU property, this real number can be divided in any manner (e.g., using some fairness rule), in order to obtain each player’s payoff from the value received by any coalition S. The amount of utility that a player i∈ S receives from the division of v(S) constitutes the user’s payoff and is denoted by xi hereafter. The vector
x ∈ RS with each element xi being the payoff of user i ∈ S constitutes a payoff allocation.
Although TU models are quite popular and useful, many scenarios exist where the coalition value cannot be assigned a single real number or rigid restrictions exist on the division of the utility. These games are known as coalitional games with non-transferable utility (NTU) and were first derived using non-cooperative strategic games as a basis [12]. The idea is that, in a NTU game, the payoff that each user in a coalition S receives is dependent on the joint actions that the players of coalition S select3. Hence, in a NTU
game, the value of a coalition S is no longer a function over the real line but a set of payoff vectors. For example, in an NTU game in characteristic form, the value of a coalition S would be given by the set v(S)⊆ RS, where
each element xi of a vector x∈ v(S) represents a payoff that user i ∈ S can
obtain when acting within coalition S given a certain strategy. Moreover, given this definition, a TU game can be seen as a particular case of the NTU framework [12].
In general, the most well studied aspect of coalitional game theory is that pertaining to games in characteristic form with TU or NTU which are widely spread in game theory literature. Different properties and solution concepts can be defined for these games, as will be also seen in the re-
2In these games, the value is commonly known as the characteristic function.
Coalitional Game Theory
mainder of this section. In particular, given a game in characteristic form with TU, we define the following property:
Definition 1 A coalitional game (N , v), in characteristic form with trans- ferable utility, is said to be superadditive if for any two disjoint coalitions S1, S2 ⊂ N , S1∪ S2=∅, v(S1∪ S2)≥ v(S1) + v(S2).
Superadditivity implies that, given any two disjoint coalitions S1 and S2, if
coalition S1 ∪ S2 forms, then it can give its members any allocations they
can achieve when acting in S1 and S2 separately. In other words, a game
is superadditive, if cooperation, i.e., the formation of a large coalition out of disjoint coalitions, guarantees at least the value that is obtained by the disjoint coalitions separately. The rationale behind the superadditivity property is that within a coalition, the players can always revert back to their non-cooperative behavior and, thus, achieving their non-cooperative payoffs. Consequently, in a superadditive game, cooperation is always beneficial. Note that, an analogous definition of a superadditive game also exists in the NTU framework [12].
For superadditive games, it is to the joint benefit of the players to al- ways form the grand coalition N , i.e, the coalitions of all the users in N , since the payoff received from v(N ) is at least as large as the amount re- ceived by the players in any disjoint set of coalitions they could form. As a result, determining whether a game is superadditive or not strongly im- pacts the approach that must be used to solve the game.
Having laid out the basic concepts of coalitional game theory, in the next subsection, using these properties and concepts, we provide a novel engineering-oriented classification of coalitional game theory.