Distributed Power Allocation With Rate Constraints in Gaussian Parallel Interference Channels

I consider games with ambiguous payoffs played by non-Expected Utility decision makers. Three equilibrium solutions are studied. Nash equilibrium in which equilibrium mixed strategies must be best responses, Crawford equilibrium in beliefs and pure equilibrium in beliefs in which equilibrium strategies are mixtures of best responses, with the latter restricting best responses to pure actions. I study the interactions between ambiguity preferences on one side and equilibrium properties on the other. I show how the equilibrium concepts differ, computing necessary and sufficient conditions for existence and equivalence. I also show how these solution concepts fare against two fundamental principles of Nash equilibrium in standard games: the principle of indifference and the reduction principle. Given both are central to the computation of Nash equilibrium in games with Expected Utility players, their failure indicates how relaxing the Expected Utility hypothesis may disrupt standard game theoretic results such as the characterization of equilibria in two-player games.

Learning in Games by Random Sampling

Among other solution concepts, the notion of the pure Nash equilibrium plays a central role in Game Theory. Pure Nash equilibria in a game characterize situations with non-cooperative deterministic players in which no player has any incentive to unilaterally deviate from the current situation in order to achieve a higher payoff. Unfortunately, it is well known that there are games that do not have pure Nash equilibria. Furhermore, even in games where the existence of equilibria is guaranteed, their computation can be a computationally hard task. Such negative results significantly question the importance of pure Nash equilibria as solution concepts that characterize the behavior of rational players. Approximate pure Nash equilibria, which characterize situations where no player can significantly improve her payoff by unilaterally deviating from her current strategy, could serve as alternative solution concepts provided that they exist and can be computed efficiently. In this letter, we discuss recent positive algorithmic results for approximate pure Nash equilibria in congestion games.

Abstract: Nash equilibrium is an important concept for studying human behavior in group decision making process. Given the complexity of finding Nash equilibria, computational tools are necessary to find them. Several programs were developed for this task. However, available programs are either not comprehensive or might be of difficult installation and handling, creating a “barrier of entry” to non-specialists. The aims of this research are twofold: (i) firstly, it was to identify and to discuss about the available programs for finding Nash equilibria; and (ii) secondly, based on the theoretical proprieties of a Nash equilibrium, to develop a program capable of finding all pure Nash equilibria in games with “n” players and “m” strategies (“n” and “m” being finite numbers) as a Macro tool for Microsoft Excel®. It is expected that the program can contributed to the area of Operations Research by providing a new tool that facilitate the use of game theory concepts within group decision-making problem-solving scenarios enabling practical applications using a widespread software.

This paper generalises the approach taken by Dasgupta & Maskin (1986) and Simon (1989) and provides necessary and sufficient conditions for the existence of pure and mixed strategy Nash equilibrium in games with continuous strategy spaces and discontinuous payoff functions. The conditions can be applied widely, and examples for existence of pure strategy and monotonic equilibria in First-Price auctions are provided. The conditions are also appropriate for ensuring that computer generated equilibrium solutions can be extended to continuous strategy spaces.

Global stability of unique Nash equilibrium in Cournot oligopoly and rent-seeking game

In cognitive radio network (CRN), the utilities results in Nash equilibrium of power control game without using pricing are inefficient. In this paper, a distributed power control algorithm is proposed to improve the utilities of both primary user (PU) and secondary users (SUs) in the CRN based on game theoretic framework. A distributed power control is a non-cooperative power control game, and the quality of service (QoS) received by PU and SUs terminals are referred to as the utility function. PU and SUs act as decision makers in the game and they maximize their utilities in a distributed fashion. We introduce a new pricing function for SUs as a function of transmit power and square amount of interference in order to guide SUs to an efficient Nash equilibrium point. Analysis of the existence and uniqueness of Nash equilibrium for the proposed power control game with pricing is presented. Simulation results show that the proposed power control algorithm via a new pricing function maximizes the number of SUs access the unused spectrum, and improves the utilities of PU and SUs.

In this thesis, we study the problem of computing approximate equilibria in several classes of games. In particular, we study approximate Nash equilibria and approximate well-supported Nash equilibria in polymatrix and bimatrix games and approximate equilibria in Lipschitz games, penalty games and biased games. We construct algorithms for computing approximate equilibria that beat the cur- rent best algorithms for these problems. In Chapter 3, we present a distributed method to compute approximate Nash equilibria in bimatrix games. In contrast to previous approaches that analyze the two payoff matrices at the same time (for example, by solving a single LP that combines the two players’ payoffs), our algorithm first solves two independent LPs, each of which is derived from one of the two payoff matrices, and then computes an approximate Nash equilibrium using only limited communication between the players. In Chapter 4, we present an algorithm that, for every δ in the range 0 < δ ≤ 0.5, finds a (0.5+δ)-Nash equilibrium of a polymatrix game in time polynomial in the input size and 1 . Note that our approximation guarantee does not depend on δ the number of players, a property that was not previously known to be achievable for polymatrix games, and still cannot be achieved for general strategic-form games. In Chapter 5, we present an approximation-preserving reduction from the problem of computing an approximate Bayesian Nash equilibrium (e-BNE) for a two-player Bayesian game to the problem of computing an e-NE of a polymatrix game and thus show that the algorithm of Chapter 4 can be applied to two-player Bayesian games. Furthermore, we provide a simple polynomial-time algorithm for computing a 0.5-BNE. In Chapter 5, we study games with non-linear utility functions for the players. Our key insight is that Lipschitz continuity of the utility function allows us to provide algorithms for finding approximate equilibria in these games. We begin by studying Lipschitz games, which encompass, for example, all concave games with Lipschitz continuous payoff functions. We provide an efficient algorithm for computing approximate equilibria in these games. Then we turn our attention to penalty games, which encompass biased games and games in which players take risk into account. Here we show that if the penalty function is Lipschitz continuous, then we can provide a quasi-polynomial time approximation scheme. Finally, we study distance biased games, where we present simple strongly poly- nomial time algorithms for finding best responses in L1, L2, and L∞ biased games, and then use these algorithms to provide strongly polynomial algorithms that find 2/3, 5/7, and 2/3 approximations for these norms, respectively.

Flow and congestion control allow the users of a telecommunication network to regulate the traffic that they send into the network in accordance with the quality of service that they require. Flow control may be performed by the network, as is the case in asynchronous transfer mode (ATM) networks (the available bit rate (ABR) transfer capacity), or by the users themselves, as is the case in the Internet [transmission control protocol/Internet protocol (TCP/IP)]. We study both situations using optimal control and dynamic game techniques. The first situation leads to the formulation of a dynamic team problem, while the second one leads to a dynamic noncooperative game, for which we establish the existence and uniqueness of a linear Nash equilibrium and obtain a characterization of the corresponding equilibrium policies along with the performance costs. We further show that when the users update their policies in a greedy manner, not knowing a priori the utilities of the other players, the sequence of policies thus generated converges to the Nash equilibrium. Finally, we study an extension of the model that accommodates multiple traffic types for each user, with the switching from one type of traffic to another being governed by a Markov jump process. Presentation of some numerical results complements this study.

Multi-person stopping games with players’ priorities are considered. Players observe sequentially offers Y1,Y2,… at jump times T1,T2,… of a Poisson process. Y1,Y2,… are independent identically distributed random variables. Each accepted offer Yn results in a reward Gn=Ynr(Tn), where r is a non-increasing discount function. If more than one player wants to accept an offer, then the player with the highest priority (the lowest ordering) gets the reward. We construct Nash equilibrium in the multi-person stopping game using the solution of a multiple optimal stopping time problem with structure of rewards {Gn}. We compare rewards and stopping times of the players in Nash equilibrium in the game with the optimal rewards and optimal stopping times in the multiple stopping time problem. It is also proved that presented Nash equilibrium is a Pareto optimum of the game. The game is a generalization of the Elfving stopping time problem to multi-person stopping games with priorities.

For multi-antenna Orthogonal Frequency-Division Multiple Access (OFDMA) based multicell networks, the channel state information (CSI) of all users is required to share among base stations in order to perform distributed power allocation. However, the amount of feedback increases with the number of users, base stations, subcarriers and antennas. Therefore, it is important to perform a selection at the user side to reduce the feedback load and the complexity of resource allocation. In this letter, we propose adaptive reduced feedback links by choosing the users based on their approximate signal to interference noise ratio (SINR) and their locations in the cell to satisfy users' rate constraints. We illustrate the performance results of reduced feedback links by employing distributed resource allocation with link adaptation.

Given a map whose roots are the Nash equilibria of a game, each component of the equilibrium set has an associated index, defined as the local degree of the map. This note shows that for a two-player game, every map with the same roots induces the same index. Moreover, this index agrees with the Shapley index constructed from the Lemke-Howson algorithm.

Generic determinacy of Nash equilibrium in network-formation games

Roughly speaking, Interdependent Defense (IDD) games, previously proposed, model the situation where an attacker wants to cause as much damage as possible to a network by attacking one of the sites in the network. Each site must make an investment decision regarding security to protect itself against a direct or indirect attack, the latter due to potential transfer-risk from an unprotected neighboring site. The work introducing IDD games discusses potential applications to model the essence of real-world scenarios such as the 2006 transatlantic aircraft plot. In this paper, our focus is the study of the problem of computing a Nash Equilibrium (NE) in IDD games. We show that an efficient algorithm to determine whether some attacker’s strategy can be a part of a NE in an instance of IDD games is unlikely to exist. Yet, we provide a dynamic programming algorithm to compute an approximate NE when the graph/network structure of the game is a directed tree with a single source, and show that it is an FPTAS. We also introduce an improved heuristic to compute an approximate NE on arbitrary graph structures. Our experiments show that our heuristic is more efficient, and provides better approximations, than best-response-gradient dynamics for the case of Internet games, a class of games introduced and studied in the original work on IDD games.

The privacy leakage resulting from location-based service (LBS) has become a critical issue. To preserve user privacy, many previous studies have investigated to prevent LBS servers from user privacy theft. However, they only consider whether the peers are innocent or malicious but ignore the relationship between the peers, whereas such a relationship between each pairwise of users affects the privacy leakage tremendously. For instance, a user has less concern of privacy leakage from a social friend than a stranger. In this paper, we study cyber-physical-social (CPS) aware method to address the privacy preserving in the case that not only LBS servers but also every other participant in the network has the probability to be malicious. Furthermore, by exploring the physical coupling and social ties among users, we construct CPS-aware privacy utility maximization (CPUM) game. We then study the potential Nash equilibrium of the game and show the existence of Nash equilibrium of CPUM game. Finally, we design a CPS-aware algorithm to find the Nash equilibrium for the maximization of privacy utility. Extensive evaluation results show that the proposed approach reduces privacy leakage by 50% in the case that malicious servers and users exist in the network.