Very often mathematics provides powerful instruments to deal with complex problems that appear unsolvable at first sight. In a cognitive radio operative environment, the following situation can be found: a certain number of unoccupied frequency bands, i.e. spectrum holes, potentially available for cognitive terminals. Assuming the existence of an infrastructure based communication system with centralized resource allocation for cognitive users, it would solve most problem related to the occupation of spectrum holes. Actually, the former solution is not so convenient since a high number of CR terminals would imply a huge amount of data exchange with the system base station, resulting in corresponding decrease of data throughput. Furthermore each cognitive terminal senses the operating environment which differs from the other terminals one, e.g., a cognitive radio could detect several spectrum holes while a CR terminal nearby could detect only one available hole because it is surrounded by a number of licensed users.
It is clear that in such a case, the problem must be approached differently. Instead of an infrastructure based system managing the radio resources as well, it could be better to distribute the occupation of the spectrum holes. Therefore, there is a need for a tool able to analyze and model the contention of radio resources. Mathematics can help us with the former issue providing a really suitable instrument: the Game Theory. At the beginning it was created mainly to analyze economic situations in order to study what would be the result of interaction between many users, e.g. a bid for a good, or the battle of prices between two or more industries. The games dealt with were only cooperative games. Afterwards, Game Theory developed through the concept of Nash Equilibrium as solution for non-cooperative games defined by the mathematician J.F. Nash. The Nash Equilibrium idea made the whole theory interesting for various applications: mainly economics but also social sciences, politics, evolutionary biology, and in the last decade also wireless communications.
The idea of a distributed management of spectrum holes makes the Game Theory particularly suitable for its analysis. The attractiveness of the Game Theory is because, differently from the decision theory, it deals with several agent interactions instead of considering only one. In particular, the main difference with the decision theory is the dependence of the outcome on the interaction under study. In fact, while in decision theory a situation where the choice of a unique agent directly affects its outcome, the Game Theory analyzes situations with multiple agents where the outcome achieved by one agent (also referred to as player) strictly depends on both its own choices and the other players actions. The game theory finds a natural application in cognitive radio as a modeling tool to analyze how the resources and the channel access are shared between the terminals.
Considering the original cognitive radio tasks, the game theory role is to translate the observations, orientations and preferences of the cognitive terminals into decisions. These decisions can be taken by cognitive radios opportunistically or in a cooperative way. This consideration marks the difference between application of non-cooperative or cooperative game theory approach in a cognitive radio based network. Game theory therefore is particularly useful framework for investigating cognitive networks, since it models the decision-making processes of groups of rational agents. Game theory can give some insight into the convergence properties, fixed points and system behaviors of the network. It also provides a convenient notational framework for describing the goals, behaviors and action spaces of the multiple autonomous cognitive elements of the network.
