Solution concepts for games with ambiguous payoffs

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

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.

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.

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.

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.

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.

In part I of this study we established two existence theorems for mixed-strategy equilibrium in games with discontinuous payoff functions (see Theorems 4(4*) and 5(5*) in Dasgupta and Maskin (1986); hereafter D-M (1986)). In this second part we use the second of these two theorems to prove the existence of mixed-strategy equilibrium in some wellknown discontinuous economic games that fail to have pure-strategy equilibria. The models we analyse are the Bertrand-Edgeworth example of price setting duopolists with capacity constraints (see Section 2.2); price competition among firms producing differentiated products (Hotelling (1929) and d'Aspremont, Gabszewicz and Thisse (1979); see Section 2.3); spatial competition (Eaton and Lipsey (1975) and Shaked (1975); see Section 3); and models of market dependent information (Rothschild and Stiglitz (1976) and Wilson (1977); Section 4). The paper concludes (Section 5) with a classification of these models.

Investigating the Interaction Between Traffic Flow and Vehicle Platooning Using a Congestion Game

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.

Wireless networks implementing dynamic channel assignment mechanisms are vulnerable to stealthy decoy attacks that aim to disrupt natural network operation by creating cascading channel conflicts. This paper develops a game-theoretic defense strategy in which a network administrator makes judicious adjustments of the transmission footprint of the various nodes, thereby continuously adapting the underlying network topology. The defense strategy is a mixed-strategy Nash equilibrium of a formulated 2-player zero-sum game between the admin and the attacker. As the space of strategies of the admin grows exponentially with the size of the network, a scalable decomposition-based approach is developed yielding a defense strategy whose performance is shown to closely approach the Nash equilibrium of the non-decomposed game. Numerical results demonstrate the effectiveness of the proposed strategies against various attack policies under different attack costs and initial conflict sizes.

Generic determinacy of Nash equilibrium in network-formation 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.

We establish the existence of fixed points for set-valued maps defined on metric spaces and satisfying a pointwise or a local version of Banach’s contraction property. As an application, we demonstrate the existence of Nash equilibrium in a general class of strategic games played on metric spaces of strategies.

In this paper, we first introduce the new notions of $$\overline{{\mathbb {R}}}_+$$-local-intersection property, $$\overline{{\mathbb {R}}}_+$$-local-inclusion property, $$\overline{{\mathbb {R}}}_+$$-strongly transfer lower semicontinuity, $${\mathbb {R}}_-$$-weakly transfer lower semicontinuity and $${\mathbb {R}}_-$$-strongly transfer lower semicontinuity for a set-valued mapping $$\Phi $$ in topological vector spaces. Then, using these notions, we characterize the existence of set-valued equilibrium without assuming any form of convexity of function and/or convexity and compactness of sets. Furthermore, we apply the basic results obtained in the paper to Browder variational inclusion, with weekend conditions on the involved set-valued operators and provide an affirmative answer to some open question posed by Alleche and Rǎdulescu in their final remark of (Optim Lett, 2018. https://doi.org/10.1007/s11590-1233-2). Some application of our results to characterize the existence of set-valued pure strategy Nash equilibrium in games with discontinuous and non-convex payoff functions and nonconvex and/or noncompact strategy spaces is presented. Finally, we characterize the existence of single-valued equilibrium problem without assuming any form of convexity of function and/or convexity and compactness of sets in the setting of topological vector spaces. Our results improve and generalize many known results in the current literature.