Homotopy methods to compute equilibria in game theory

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.

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.

Learning in Games by Random Sampling

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.

Equilibrium in spatial voting: The median voter result is an artifact

Linear Dynamic Logic on finite traces (LDL_f) is a powerful logic for reasoning about the behaviour of concurrent and multi-agent systems. In this paper, we investigate techniques for both the characterisation and verification of equilibria in multi-player games with goals/objectives expressed using logics based on LDL_f. This study builds upon a generalisation of Boolean games, a logic-based game model of multi-agent systems where players have goals succinctly represented in a logical way. Because LDL_f goals are considered, in the setting we study---iterated Boolean games with goals over finite traces (iBG_f)---players' goals can be defined to be regular properties while achieved in a finite, but arbitrarily large, trace. In particular, using alternating automata, the paper investigates automata-theoretic approaches to the characterisation and verification of (pure strategy Nash) equilibria, shows that the set of Nash equilibria in games with LDL_f objectives is regular, and provides complexity results for the associated automata constructions.

The notion of perfect equilibrium was formulated by Selten (Int J Game Theory 4(1):25–55, 1975) as a strict refinement of Nash equilibrium. For an extensive-form game with perfect recall, every perfect equilibrium of its agent normal-form game yields a perfect equilibrium of the extensive-form game. This paper aims to develop a differentiable homotopy method for computing perfect equilibria of normal-form games. To accomplish this objective, we constitute an artificial game by introducing a continuously differentiable function of an extra variable. The artificial game defines a differentiable homotopy mapping and establishes the existence of a smooth path to a perfect equilibrium. For numerical comparison, we also describe a simplicial homotopy method. Numerical results show that the differentiable homotopy method is numerically stable and efficient and significantly outperforms the simplicial homotopy method especially when the problem is large.

We propose a solution concept, called interperceptional equilibrium, in games with interperception. Two sufficient conditions for an outcome to be an interperceptional equilibrium are given. It is also shown that, under some realistic conditions, the concept of interperceptional equilibrium can be regarded as a generalization of the concept of Nash equilibrium. The relations between the framework given in the paper and those of "soft" game theory and drama theory are also discussed.

In the rapid development of wireless communication network technology, the application of Internet of Things (IoT) is increasing day by day. Because the number of end devices is increasing, there is one of important issues about gateway selection for end devices. Therefore, it is important to select the most effective gateway for the application of Internet of Things. Gateway Selection strategy will be a more significant topic at the moment. In recent years, game theory in the network resource allocation or simulation of resource competition has a wide range of applications, not only can clearly understand the game model, but also that the game's equilibrium. In this paper, we focus on non-cooperative dynamic game with end devices for the game players and IoT gateways for the selection strategy. The utility function is derived from the characteristics of players and IoT gateways. Finally, we compare with the maximum received signal strength indicator to meet the overall demands of the end devices and the IoT gateway maximum load difference are superior to the traditional way.

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.

We study convergence of ?-Nash dynamics in congestion games when delay functions of all resources are similar. Delay functions are said to be similar if their values are within a polynomial factor at every congestion level. We show that for any ?> 0, an ?-Nash dynamics in symmetric congestion games with similar resources converge in steps polynomial in 1/? and the number of players are resources, yielding an FPTAS. Our result can be contrasted with that of Chien and Sinclair [3], which showed polynomial convergence result for symmetric congestion games where the delay functions have polynomially bounded jumps. Our assumption of similar delay functions is orthogonal to that of bounded jumps in that neither assumption implies the other. Our convergence result also hold for several natural variants of ?-Nash dynamics, including the most general polynomial liveness dynamics, where each player is given a chance to move frequently enough. We also extend our positive results to give an FPTAS for computing equilibrium in asymmetric games with similar resources, in which players share k distinct strategy spaces for any constant k. We complement our positive results by showing that computing an exact pure Nash equilibrium in symmetric congestion game with similar resources is PLS-complete. Furthermore, we show that for any ?> 0, all sequences of ?-Nash dynamics takes exponential steps to reach an approximate equilibrium in general congestion games with similar resources, as well as in symmetric congestion games with two groups of similar resources.

Correlated relaxed equilibria in nonzero-sum linear differential games

This paper aims to contribute to bridge the gap between existing theoretical results in distributed radio resource allocation policies based on equilibria in games (assuming complete information and rational players) and practical design of signal processing algorithms for self-configuring wireless networks. For this purpose, the framework of learning theory in games is exploited. Here, a new learning algorithm based on mild information assumptions at the transmitters is presented. This algorithm possesses attractive convergence properties not available for standard reinforcement learning algorithms and in addition, it allows each transmitter to learn both its optimal strategy and the values of its expected utility for all its actions. A detailed convergence analysis is conducted. In particular, a framework for studying heterogeneous wireless networks where transmitters do not learn at the same rate is provided. The proposed algorithm, which can be applied to any wireless network verifying the information assumptions stated, is applied to the case of multiple access channels in order to provide some numerical results.

FPGA based acceleration of game theory algorithm in edge computing for autonomous driving

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.