In cognitive networks, how to stimulate cooperation among nodes is very important. However, most existing game-theoretic cooperation stimulation approaches rely on the assumption that the interactions between any pair of players are long-lasting. When this assumption is not true, such as in the well-known Prisoner’s dilemma and the backward induction principle, the unique Nash equilibrium is to always play noncooperatively. In this chapter, we discuss a cooperation stimulation scheme for the scenario in which the number of interactions is finite. This scheme is based on indirect reciprocity game modeling where the key concept is “I help you not because you have helped me but because you have helped others.” The problem of finding the optimal action rule is formulated as a Markov decision process, and a modified value-iteration algorithm is utilized to find the optimal action rule. Using the packet forwarding game as an example, it is shown that with an appropriate cost-to-gain ratio, the strategy of forwarding the number of packets that is equal to the reputation level of the receiver is an evolutionarily stable strategy.

In the third part of this book, the third branch of modern game theory – sequential decision-making – is presented. The important components in sequential decision-making, such as network externality, information asymmetry, and user rationality, are presented and defined. The limitations of the existing approaches, such as social learning and multiarm bandit problems, are also presented.

In this chapter we discuss how multiple radios interact. We introduce the concept of duplex and various approaches to enable radios to perform bidirectional communications. We also introduce the concept of network topologies such as star and mesh approaches. We discuss multiple media access control techniques. We introduce aloha, carrier-sense multiple access, time-division multiple access, frequency-division multiple access, and code-division multiple access.

Yes, mathematics may be difficult on occasion, but doing anything technically interesting without math is impossible. Learning mathematics is akin to learning a language, so you should expect that it will take significant practice to become accomplished.

The viability of cooperative communications depends on the willingness of users to help. Therefore, it is important to study incentive issues when designing such systems. In this chapter, we discuss a cooperation stimulation scheme for multiuser cooperative communications using an indirect reciprocity game. By introducing the notion of reputation and social norms, rational users who care about their future utility are incentivized to cooperate with others. Differently from existing works on reputation-based schemes that mainly rely on experimental verification, the effectiveness of the scheme is demonstrated in two steps. First, we conduct a steady-state analysis of the game and show that cooperating with users who have a good reputation can be sustained as an equilibrium when the cost-to-gain ratio is below a certain threshold. Then, by modeling the action spreading at transient states as an evolutionary game, we show that the equilibria we found in the steady-state analysis are stable and can be reached with proper initial conditions. Moreover, we introduce energy detection to handle the possible cheating behaviors of users and study its impact on the indirect reciprocity game.

In this chapter, we discuss the fundamental limit – the Shannon limit – on data rate for a communications link. We motivate this limit by providing a sketch of the derivation. To construct this sketch, we discuss the idea of the ratio of hypersphere volumes and how the radius of Gaussian vectors converges to a known radius as the dimensionality goes to infinity. We provide a second approach to think about the Shannon limit, or equivalently channel capacity, of a data link by considering mutual information and entropy. By using a simple line-of-sight channel, we discuss the resulting link theoretical capacity and an approach to estimating the practical limit on data rate. Finally, we provide an overview of source coding approaches.

A huge amount of information, created and forwarded by millions of people with various characteristics, propagates through online social networks every day. Understanding the mechanisms of information diffusion over social networks is critical to various applications, including online advertisements and website management. Differently from most existing works in this area, we investigate information diffusion from an evolutionary game-theoretic perspective and try to reveal the underlying principles dominating the complex information diffusion process over heterogeneous social networks. Modeling the interactions among the heterogeneous users as a graphical evolutionary game, we derive the evolutionary dynamics and the evolutionarily stable states (ESSs) of the diffusion. The different payoffs of the heterogeneous users lead to different diffusion dynamics and ESSs among them, in accordance with the heterogeneity observed in real-world data sets. The theoretical results are confirmed by simulations. We also test the theory on the Twitter hashtag data set. We observe that the evolutionary dynamics fit the data well and can predict future diffusion data.

In this chapter, we discuss the ideas of signal acquisition and radio-to-radio synchronization in both time and frequency. We address the critical question: Is anyone out there? We discuss the uncertainty in time and frequency alignment between radios. We introduce and analyze the performance of multiple signal acquisition techniques: energy, cross-correlation, normalized inner product, and autocovariance detectors. We develop the maximum likelihood estimators for temporal and spectral synchronization for single-carrier approaches. Finally, we also introduce a temporal synchronization approach for an OFDM symbol.

Network service acquisition in a wireless environment requires the selection of a wireless access network. A key problem in wireless access network selection is studying rational strategies that consider negative network externality. In this chapter, we formulate the wireless network selection problem as a stochastic game with negative network externality and show that finding the optimal decision rule can be modeled as a multidimensional Markov decision process. A modified value-iteration algorithm is utilized to efficiently obtain the optimal decision rule with a simple threshold structure. We further investigate the mechanism design problem with incentive compatibility constraints, which force the networks to reveal truthful state information. The formulated problem is a mixed-integer programming problem that, in general, lacks an efficient solution. Exploiting the optimality of substructures, we introduce a dynamic programming algorithm that can optimally solve the problem in the two-network scenario. For the multinetwork scenario, the dynamic programming algorithm can outperform the heuristic greedy approach in polynomial-time complexity.

In a social network, agents are intelligent and have the capacity to make decisions so as to maximize their utility. They can either make wise decisions by taking advantages of other agents’ experiences through learning or make decisions earlier to avoid competition from huge crowds. Both of these effects – social learning and negative network externality – play important roles in the decision-making process of an agent. In this chapter, a new game called the Chinese restaurant game is introduced to formulate the social learning problem with negative network externality. Through analyzing the Chinese restaurant game, we derive the optimal strategy of each agent and provide a recursive method to achieve the optimal strategy. How social learning and negative network externality influence each other under various settings is studied through simulations. We also illustrate the spectrum access problem in cognitive radio networks as one application of the Chinese restaurant game. We find that the Chinese restaurant game-theoretic approach indeed helps users make better decisions and improves overall system performance.

In this chapter we discuss the use of multiple antennas by radios. While the radio links that we discussed up to this point in the text have assumed single-input single-output (SISO) channels, we now consider the use of multiple antennas at both the transmitter (source) and receiver (destination), as indicated in Figure 11.1. We introduce the channel model for a multiple-antenna receiver. We discuss channel estimation and spatial receive beamforming techniques. We introduce the multiple-input multiple-output (MIMO) channel model, define the capacity of this system under the assumptions that the transmitter is uninformed and the transmitter is informed of the channel matrix. Finally, we discuss the concept of space–time coding and present various approaches, including Alamouti’s space–time block code.

In this chapter we discuss Fourier analysis. We categorize signals into energy or power signals. We introduce the foundational concept of the complex tone. We define the Fourier transform and identify it as a linear operator. We review energy and the power spectral densities. We survey a set of useful Fourier transform relationships such as time shift, frequency shift, scaling, time reversal, and conjugation. We evaluate the Fourier transform of the top hat function, real Gaussian function, convolution, functional derivative, and autocorrelation. We introduce the Fourier series and evaluate series coefficients for sawtooth functions and impulse train. We discuss the discrete-time Fourier transform and the discrete Fourier transform and related fast Fourier transform. Finally, we review digital filters.

Many spectrum sensing methods and dynamic access algorithms have been proposed to improve secondary users’ access opportunities. However, few of them have considered integrating the design of spectrum sensing and access algorithms together by taking into account the mutual influence between them. In this chapter, we focus on jointly analyzing the spectrum sensing and access problem. Due to their selfish nature, secondary users tend to act selfishly to access the channel without contributing to spectrum sensing. Moreover, they may employ out-of-equilibrium strategies because of the uncertainty of others’ strategies. To model the complicated interactions among secondary users, the joint spectrum sensing and access problem is formulated as an evolutionary game and the evolutionarily stable strategy (ESS) that no one will deviate from is studied. Furthermore, a distributed learning algorithm for the secondary users to converge to the ESS is introduced. Simulation results shows that the system can quickly converge to the ESS and such an ESS is robust to the sudden unfavorable deviations of the selfish secondary users.

Cooperation is a promising approach to simultaneously achieving efficient spectrum resource use and improving the quality of service of primary users in dynamic spectrum access networks. However, due to their selfish nature, how to stimulate secondary users to play cooperatively is an important issue. In this chapter, we discuss a reputation-based spectrum access framework where the cooperation stimulation problem is modeled as an indirect reciprocity game. In this game, secondary users choose how to help primary users relay information and gain reputation, based on which they can access a certain amount of vacant licensed channels in the future. By formulating a secondary user's decision-making as a Markov decision process, the optimal action rule can be obtained, according to which the secondary user will use maximal power to help the primary user relay data and thus greatly improve the primary user's quality of service as well as the spectrum utilization efficiency. Moreover, the uniqueness of the stationary reputation distribution is proved, and the conditions under which the optimal action rule is evolutionarily stable are theoretically derived.

In this chapter, we discuss the concepts of amplification that we use to overcome noise. We review the idea of power amplifiers that are used by the transmitter and introduce metrics for nonlinear contributions in signal amplification. We discuss the concept of the low-noise amplifier (LNA) that is typically the first amplifier in the receiver chain. We motivate this type of amplifier by introducing the idea of the noise figure. Finally, we review the idea of automatic gain control.