We consider a situation where wireless service providers compete for heterogenous wireless users. The users differ in their willingness to pay as well as in their individual channel gains. We prove existence and uniqueness of the Nash equilibrium for the competition of two service providers, for a generic channel model. Interestingly, the competition of two providers leads to a globally optimal outcome. We extend some of the results to the case where more than two providers are competing. Finally, we provide numerical examples that illustrate the effects of various parameters on the Nash equilibrium.
The past ten years have seen an ever increasing number of research that focuses on the application of game theory and pricing in analyzing network service providers, initiated by the work of Kelly @cite_6 . The majority of research in the wireless setting focuses on resource allocation within the same service provider (see for example @cite_14 , @cite_12 , @cite_7 ) and on the interaction between the users of one provider ( @cite_5 , @cite_9 , @cite_15 ). Perhaps surprisingly, only a few works focus on the competition between providers in a wireless setting. We can distinguish two types of competitions between wireless service providers: competing on behalf of the users (such as @cite_8 and @cite_1 ) and price competition to attract users (such as @cite_11 , @cite_16 and @cite_0 Another related work is @cite_3 which considers two cellular providers competing for users by changing the strength of their pilot signals. ). The latter is the subject of our work.
@cite_14 @cite_7 @cite_8 @cite_9 @cite_1 @cite_6 @cite_3 @cite_0 @cite_5 @cite_15 @cite_16 @cite_12 @cite_11
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We study competition between wireless devices with incomplete information about their opponents. We model such interactions as Bayesian interference games. Each wireless device selects a power profile over the entire available bandwidth to maximize its data rate. Such competitive models represent situations in which several wireless devices share spectrum without any central authority or coordinated protocol. In contrast to games where devices have complete information about their opponents, we consider scenarios where the devices are unaware of the interference they cause to other devices. Such games, which are modeled as Bayesian games, can exhibit significantly different equilibria. We first consider a simple scenario of simultaneous move games, where we show that the unique Bayes-Nash equilibrium is where both devices spread their power equally across the entire bandwidth. We then extend this model to a two-tiered spectrum sharing case where users act sequentially. Here one of the devices, called the primary user, is the owner of the spectrum and it selects its power profile first. The second device (called the secondary user) then responds by choosing a power profile to maximize its Shannon capacity. In such sequential move games, we show that there exist equilibria in which the primary user obtains a higher data rate by using only a part of the bandwidth. In a repeated Bayesian interference game, we show the existence of reputation effects: an informed primary user can bluff to prevent spectrum usage by a secondary user who suffers from lack of information about the channel gains. The resulting equilibrium can be highly inefficient, suggesting that competitive spectrum sharing is highly suboptimal. In each country today, cellular networks operate on carefully separated frequency bands. This separation is imposed by the regulators of the given country to avoid interference between these networks. But, the separation is only valid within the borders of a country, hence the operators are left on their own to resolve cross-border interference of their cellular networks. In this paper, we focus on the scenario of two operators, each located on one side of the border. We assume that they want to fine-tune the emitting power of the pilot signals (i.e., beacon signals) of their base stations. This operation is crucial, because the pilot signal power determines the number of users they can attract and hence the revenue they can obtain. In the case of no power costs, we show that there exists a motivation for the operators to be strategic, meaning to fine-tune the pilot signal powers of their base stations. In addition, we study Nash equilibrium conditions in an empirical model and investigate the efficiency of the Nash equilibria for different user densities. Finally, we modify our game model to take power costs into account. The game with power costs corresponds to the well-known prisoner's dilemma: The players are still motivated to adjust their pilot powers, but their strategic behavior leads to a sub-optimal Nash equilibrium. * The first tutorial-style book that gives all the relevant theory at the right level of rigor, for the wireless communications engineer. * Bridges the gap between theory and practice by giving examples and case studies showing how game theory can solve real-word problems. * Contains algorithms and techniques to implement game theory in wireless terminals. Written by leading experts in the field, Game Theory and Learning for Wireless Networks Covers how theory can be used to solve prevalent problems in wireless networks such as power control, resource allocation or medium access control. With the emphasis now on promoting 'green' solutions in the wireless field where power consumption is minimized, there is an added focus on developing network solutions that maximizes the use of the spectrum available. With the growth of distributed wireless networks such as Wi-Fi and the Internet; the push to develop ad hoc and cognitive networks has led to a considerable interest in applying game theory to wireless communication systems. Game Theory and Learning for Wireless Networks is the first comprehensive resource of its kind, and is ideal for wireless communications R&D engineers and graduate students. Samson Lasaulce is a senior CNRS researcher at the Laboratory of Signals and Systems (LSS) at Suplec, Gif-sur-Yvette, France. He is also a part-time professor in the Department of Physics at cole Polytechnique, Palaiseau, France. Hamidou Tembine is a professor in the Department of Telecommunications at Suplec, Gif-sur-Yvette, France. Merouane Debbah is a professor at Suplec, Gif-sur-Yvette, France. He is the holder of the Alcatel-Lucent chair in flexible radio since 2007. The first tutorial style book that gives all the relevant theory, at the right level of rigour, for the wireless communications engineer Bridges the gap between theory and practice by giving examples and case studies showing how game theory can solve real world resource allocation problems Contains algorithms and techniques to implement game theory in wireless terminals We address the problem of spectrum pricing in a cognitive radio network where multiple primary service providers compete with each other to offer spectrum access opportunities to the secondary users. By using an equilibrium pricing scheme, each of the primary service providers aims to maximize its profit under quality of service (QoS) constraint for primary users. We formulate this situation as an oligopoly market consisting of a few firms and a consumer. The QoS degradation of the primary services is considered as the cost in offering spectrum access to the secondary users. For the secondary users, we adopt a utility function to obtain the demand function. With a Bertrand game model, we analyze the impacts of several system parameters such as spectrum substitutability and channel quality on the Nash equilibrium (i.e., equilibrium pricing adopted by the primary services). We present distributed algorithms to obtain the solution for this dynamic game. The stability of the proposed dynamic game algorithms in terms of convergence to the Nash equilibrium is studied. However, the Nash equilibrium is not efficient in the sense that the total profit of the primary service providers is not maximized. An optimal solution to gain the highest total profit can be obtained. A collusion can be established among the primary services so that they gain higher profit than that for the Nash equilibrium. However, since one or more of the primary service providers may deviate from the optimal solution, a punishment mechanism may be applied to the deviating primary service provider. A repeated game among primary service providers is formulated to show that the collusion can be maintained if all of the primary service providers are aware of this punishment mechanism, and therefore, properly weight their profits to be obtained in the future.