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.
@cite_14: A major challenge in the operation of wireless communications systems is the efficient use of radio resources. One important component of radio resource management is power control, which has been studied extensively in the context of voice communications. With the increasing demand for wireless data services, it is necessary to establish power control algorithms for information sources other than voice. We present a power control solution for wireless data in the analytical setting of a game theoretic framework. In this context, the quality of service (QoS) a wireless terminal receives is referred to as the utility and distributed power control is a noncooperative power control game where users maximize their utility. The outcome of the game results in a Nash (1951) equilibrium that is inefficient. We introduce pricing of transmit powers in order to obtain Pareto improvement of the noncooperative power control game, i.e., to obtain improvements in user utilities relative to the case with no pricing. Specifically, we consider a pricing function that is a linear function of the transmit power. The simplicity of the pricing function allows a distributed implementation where the price can be broadcast by the base station to all the terminals. We see that pricing is especially helpful in a heavily loaded system.
@cite_7: In wireless cellular networks and wireless local area networks, nonlinear network utility maximization need to be conducted over both user rates and transmit powers. For each of the three cases considered in this paper, we present an algorithm that converges to the jointly optimal pair of rate vector and power vector. For the simple case when data rates are not limited by interferences, for example in single-cell downlink transmissions, we propose algorithm 1, which is an iterative bidding mechanism between the base station and mobile users, where knowledge about channel conditions and individual user utility functions is only needed locally at each user but not needed at the base station. In the case when data rates are limited by interferences, the utility maximization problem is complicated both by nonlinear coupling between powers and rates, and by interference among powers. Through centralized iterative steps, we propose algorithm 2, which converges to a joint and global optimum over the solution space of rates and powers. We then consider end-to-end transmissions in cellular networks, which traverse both wireless fading channels and many hops of wired links shared by other traffic. There is a tradeoff between attaining air-interface capacity in the wireless hop and controlling congestion in the wired backbone wide area network. We formulate this end-to-end resource allocation problem in such hybrid networks, and present a solution to obtain the Pareto optimal tradeoff between attaining wireless multi-access fading channel capacity and maximizing global network utility.
@cite_8: Power allocation across users in two adjacent cells is studied for a code-division multiple access (CDMA) data service. The forward link is considered and cells are modeled as one-dimensional with uniformly distributed users and orthogonal signatures within each cell. Each user is assumed to have a utility function that describes the user's received utility, or willingness to pay, for a received signal-to-interference-plus-noise ratio (SINR). The objective is to allocate the transmitted power to maximize the total utility summed over all users subject to power constraints in each cell. It is first shown that this optimization can be achieved by a pricing scheme in which each base station announces a price per unit transmitted power to the users, and each user requests power to maximize individual surplus (utility minus cost). Setting prices to maximize total revenue over both cells is also considered, and it is shown that, in general, the solution is different from the one obtained by maximizing total utility. Conditions are given for which independent optimization in each cell, which leads to a Nash equilibrium (NE), is globally optimal. It is shown that, in general, coordination between the two cells is needed to achieve the maximum utility or revenue.
@cite_9: We study a spectrum sharing problem in an unlicensed band where multiple systems coexist and interfere with each other. We first analyze a cooperative setting where all the systems collaborate to achieve a common goal. Under the assumptions that the systems communicate with Gaussian signals and treat interference as noise, we study the structure of the optimal power allocations. We show that any Pareto efficient vector of rates can be achieved with piece-wise constant power allocations. Moreover, if a strong interference condition among all the systems is satisfied, we show that frequency division multiplexing is optimal. We then consider a non-cooperative situation, where the systems act in a selfish and rational way, and investigate how the lack of cooperation can affect performance. Using game theory, we first analyze the possible outcomes of a one shot game, and observe that in many cases an inefficient solution results. We show that by extending the game definition to that of a repeated game, the possibility of building reputations and applying punishments allows to enlarge the set of achievable rates. We present examples that show that in many cases, the performance loss due to lack of cooperation is small. We also provide a converse theorem that proves that our results are tight and quantify the best achievable performance in a non-cooperative scenario
@cite_1: We consider the problem of two wireless networks operating on the same (presumably unlicensed) frequency band. Pairs within a given network cooperate to schedule transmissions, but between networks there is competition for spectrum. To make the problem tractable, we assume transmissions are scheduled according to a random access protocol where each network chooses an access probability for its users. A game between the two networks is defined. We characterize the Nash Equilibrium behavior of the system. Three regimes are identified: one in which both networks simultaneously schedule all transmissions, one in which the denser network schedules all transmissions and the sparser only schedules a fraction, and one in which both networks schedule only a fraction of their transmissions. The regime of operation depends on the path loss exponent α, the latter regime being desirable but attainable only for α > 4. This suggests that in certain environments, rival wireless networks may end up naturally cooperating. To substantiate our analytical results, we simulate a system where networks iteratively optimize their access probabilities in a greedy manner. We also discuss a distributed scheduling protocol that employs carrier sensing and demonstrate via simulations that again a near cooperative equilibrium exists for sufficiently large α.
@cite_6: This paper addresses the issues of charging, rate control and routing for a communication network carrying elastic traffic, such as an ATM network offering an available bit rate service. A model is described from which max-min fairness of rates emerges as a limiting special case; more generally, the charges users are prepared to pay influence their allocated rates. In the preferred version of the model, a user chooses the charge per unit time that the user will pay; thereafter the user's rate is determined by the network according to a proportional fairness criterion applied to the rate per unit charge. A system optimum is achieved when users' choices of charges and the network's choice of allocated rates are in equilibrium.
@cite_3: 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.
@cite_0: Dynamic spectrum access can significantly improve the spectrum utilization. For wireless service providers, the emergence of dynamic spectrum access brings new opportunities and challenges. The flexible spectrum acquisition gives a particular provider the chance to easily adapt its system capacity to fit end users' demand. However, the competition among several providers for both spectrum and end users complicates the situation. In this paper, we propose a general three-layer spectrum market model for the future dynamic spectrum access system, in which the interaction among spectrum holder, wireless service providers and end users are considered. We study a duopoly situation, where two wireless service providers participate in bandwidth competition in spectrum purchasing and price competition to attract end users, with the aim of maximizing their own profit. We believe we are the first one to explicitly study the relation of these two competitions in dynamic spectrum market. We formulate the wireless service providers' competition as a non-cooperative two-stage game. We first analyze the static game when full information is available for providers. Under general assumptions about the price and demand functions, a unique pure Nash equilibrium is identified as the outcome of the game, which shows the stability of the market. We further evaluate the market efficiency of the equilibrium in a symmetric case, and show that the gap with the social optimal is bounded within a small constant ratio. When the market information is limited, we provide myopically optimal adjustment algorithms for the providers. With such strategies, short term price updating converges to the Nash equilibrium of the given subgame, while long term bandwidth updating converges to a point close to the Nash equilibrium of the full game.
@cite_5: 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.
@cite_15: We consider a distributed power control scheme for wireless ad hoc networks, in which each user announces a price that reflects compensation paid by other users for their interference. We present an asynchronous distributed algorithm for updating power levels and prices. By relating this algorithm to myopic best response updates in a fictitious game, we are able to characterize convergence using supermodular game theory. Extensions of this algorithm to a multichannel network are also presented, in which users can allocate their power across multiple frequency bands.
@cite_16: In the future, we can expect to see more dynamic service offerings and profiles, as users move from long-term service provider agreements to more opportunistic service models. Moreover, when the radio spectrum is itself traded in a market- based scenario, wireless service providers (WSPs) will likely require new strategies to deploy services, define service profiles, and price them. Currently, there is little understanding on how such a dynamic trading system will operate so as to make the system feasible under economic terms. From an economic point of view, we analyze two main components of this overall trading system: (i) spectrum allocation to WSPs and (ii) interaction of end users with the WSPs. For this two-tier trading system, we present a winner determining sealed-bid knapsack auction mechanism that dynamically allocates spectrum to the WSPs based on their bids. We propose a dynamic pricing strategy based on game theory to capture the conflict of interest between WSPs and end users, both of whom try to maximize their respective net utilities. We show that even in such a greedy and non-cooperative behavioral game model, it is in the best interest of the WSPs to adhere to a price threshold which is a consequence of a price equilibrium in an oligopoly situation. Through simulation results, we show that the proposed auction entices the WSPs to participate in the auction, makes optimal use of the common spectrum pool, and avoids collusion among WSPs. Moreover, numerical results demonstrate how pricing can be used as an effective tool for providing incentives to the WSPs to upgrade their network resources and offer better services.
@cite_12: This paper considers resource allocation and pricing for the downlink of a wireless network. We describe a model that applies to either a time-slotted system (e.g. Qualcomm's HDR proposal) or a CDMA system; the main feature of this model is that the channel quality varies across the users. We study using a pricing scheme for the allocation of radio resources. We show that to maximize revenue in such a system, the base station should allocate resources in a discriminatory manner, where different users are charged different prices based in part on their channel quality. However, optimally allocating resources in this way is shown to require knowledge about each user's utility function. We consider a suboptimal scheme which does not require knowledge of the users' utility functions, and show that this scheme is asymptotically optimal, in the limit of large demand. Moreover, such a scheme is shown to maximize social welfare. We also consider a heuristic scheme for the case of small demand, which does not require perfect knowledge about the users' utility functions. We provide numerical results that illustrate the performance of this heuristic.
@cite_11: In this paper we investigate network design for a wireless service provider using two orthogonal technologies: a WAN technology with uniform spatial coverage and set of LAN access points each with limited coverage. We assume that the system is designed so that users (or their agents) independently and greedily select among the two options based on maximizing a specified utility function which may be a function of the quality of the wireless link, distance to the access points, and or congestion on system resources. We focus on two complementary aspects of this problem. On the one hand we study system performance under such decision-making strategies. We show convergence of decision-making process to an equilibrium, and that a congestion-sensitive utility can provide substantial (300 ) performance improvements over natural proximity-based criterion. On the other hand, we consider various problems associated with dimensioning typically expensive backhaul links, for the WAN and set of LAN hotspots. Our results show how to best jointly exploit technologies with different coverage scales so as to statistically multiplex spatial load fluctuations in order to reduce backhaul costs.
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.