IEEE Signal Processing - July 2018 - 94

overview of the basic game-theoretic concept and a discussion
of the different game- and auction-theoretic approaches that
have been used to design various past energy-management
schemes in the aforementioned three domains. It is important
to note that, while the literature on game theory in energy
management is extensive, only some of the key studies in each
domain are discussed.

Basic game-theoretic concepts
Game theory is a mathematical and signal processing tool [16]
that analyzes strategies in competitive situations where the outcome of a participant's choice of action depends on the actions
of other participants. It can be divided into two main branches:
noncooperative game theory and cooperative game theory.

Noncooperative games
A noncooperative game analyzes the strategic decision-making
process of a number of independent players that have partially or
totally conflicting interests in the outcome of a decision-making
process influenced by their actions. Such games allow players to
take necessary action, e.g., making optimal decisions, without
any coordination or communication. Note that the term noncooperative does not mean that the players do not cooperate; rather,
it refers to the fact that any cooperation that may arise in the noncooperative game must not be the result of either communication
or coordination of strategic choices among the players [5].
In general, a noncooperative game can be divided into one
of two categories: static or dynamic.
■ Static game: In a static game, the players take action only
once, either simultaneously or at different times. A static
game can be defined in its strategic form as {N, ^S nhn ! N ,
^U n hn ! N }, where N is the set of all of the participating
players in the game and each player n ! N has a strategy
set S n from which it chooses an action s n ! S n to optimize
its utility function U n . The utility that a player n attains is
affected by the choices of action S -n of the players in the
set N = {n}.
■ Dynamic game: In contrast, players in a dynamic game act
more than once and have some input regarding the choices
of other players. In dynamic games, time plays a central
role in the decision-making process of each player. Dynamic
games can also be formulated as static games; however,
there is a need for some additional information, including
time and information sets, which are usually reflected in the
utility functions.
For both static and dynamic noncooperative games, the players make their decisions either in a deterministic manner (pure
strategies) or in a probabilistic manner (mixed strategies).
The most popular solution concept of the noncooperative game {N, ^S n hn ! N , ^U n hn ! N } is the Nash equilibrium. A
Nash equilibrium is a vector of actions s ) such that U n ^s )h $
U n (s n, s )-n), 6n ! N, where s = 6s n, s n@ . Thus, a Nash equilibrium refers to a stable state of a noncooperative game in which
no player n ! N can improve its utility by unilaterally altering
its action s n from s *n when the actions of the other participating players N = {n} are fixed at s *-n . While a Nash equilibrium
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always exists in a noncooperative game with mixed strategies, the
existence is not guaranteed in a game with pure strategies. Furthermore, a noncooperative game may also have multiple Nash
equilibria, and, in such cases, it is important to select an efficient
and desirable Nash equilibrium as the solution of the game.

Cooperative games
In contrast, with cooperative games, the focus is on how one
can provide incentives to independent decision makers to act
together as one entity to improve their position in the game.
Essentially, both Nash bargaining and the coalitional game can
be considered under the same umbrella of a cooperative game.
Nash bargaining is the study of terms and conditions under
which a number of players may agree to form a coalition,
while coalitional games deal with the formation of coalitions
[5]. In general, a coalitional game can be expressed by the pair
^N c, o h, which involves a set of players N c that seek to form
cooperative groups. o is the value function associated with
each coalition S 3 N c and is expressed by a real number to
quantify the value of the respective coalition. The most common form of a coalitional game is the characteristic form [17]
in which the value of the coalition is determined by the members of that coalition, regardless of how the players in the coalition are structured. A coalitional game can be classified into
one of three types: a canonical coalitional game, a coalition
formation game, or a coalitional graph game.
■ A canonical coalitional game: This game can be expressed
with a transferable or nontransferrable utility. In this type,
the formation of the grand coalition (i.e., the coalition of all
of the players in the game) is never detrimental to the players, which pertains to the mathematical property known as
superadditivity. The main objective of a canonical coalitional game is to study the properties and stability of the grand
coalition, the gains resulting from the coalition, and the distribution of these gains in a fair manner to the players. The
most commonly considered solution concept for the coalitional game is the core, which is directly related to the stability of the grand coalition. Essentially, the core is defined
as the set of revenues x for which no coalition S 1 N c has
any incentive to reject the grand coalition for the proposed
revenue allocation x.
■ A coalition formation game: In this game, the network
structure and the cost of cooperation play a major role. In
general, a coalition formation game is not superadditive,
and, although forming a coalition brings gains to its members, the gains are limited by the cost associated with a
coalition formation. As a result, the formation of a grand
coalition is very rare; therefore, the objective of a static
coalition formation game is to study the network coalitional
structure. In a dynamic coalitional game, however, the game
is subject to environmental changes, including a change in
the number of players or a variation in network topology.
Hence, the main objective is to analyze the formation of a
coalitional structure through players' interactions and study
the properties of the structure and its adaptability to environmental variations.

IEEE Signal Processing Magazine

July 2018

Table of Contents for the Digital Edition of IEEE Signal Processing - July 2018