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.

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.

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.

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.

We examine the long-run behavior of multiagent online learning in games that evolve over time. Specifically, we focus on a wide class of policies based on mirror descent, and we show that the induced sequence of play (a) converges to a Nash equilibrium in time-varying games that stabilize in the long run to a strictly monotone limit, and (b) it stays asymptotically close to the evolving equilibrium of the sequence of stage games (assuming they are strongly monotone). Our results apply to both gradient- and payoff-based feedback—that is, when players only get to observe the payoffs of their chosen actions. Funding: This research was partially supported by the European Cooperation in Science and Technology COST Action [Grant CA16228] “European Network for Game Theory” (GAMENET). P. Mertikopoulos is grateful for financial support by the French National Research Agency (ANR) in the framework of the “Investissements d’avenir” program [Grant ANR-15-IDEX-02], the LabEx PERSYVAL [Grant ANR-11-LABX-0025-01], MIAI@Grenoble Alpes [Grant ANR-19-P3IA-0003], and the ALIAS [Grant ANR-19-CE48-0018-01].

Generic determinacy of Nash equilibrium in network-formation games

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.

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.

Semi-mixed algebraic systems are those where the equations can be partitioned into subsets with common Newton polytopes. We are interested in counting roots of semi-mixed multihomogeneous systems, where both variables and equations can be partitioned into blocks, and each block of equations has a given degree in each block of variables. The motivating example is counting the number of totally mixed Nash equilibria in games of several players. Firstly, this paper relates and unifies the BKK and multivariate Bézout bounds for semi-mixed systems, through mixed volumes and matrix permanents. Permanent expressions for BKK bounds hold for all multihomogeneous systems, without any requirement of semi-mixed structure, as well as systems whose Newton polytopes are products of polytopes in complementary subspaces. Secondly, by means of a novel asymptotic analysis, the complexity of a combinatorial geometric algorithm for semi-mixed volumes (i.e., mixed volumes of semi-mixed systems) is explored and juxtaposed to the complexities of computing permanents, or using generating functions (via MacMahon's Master theorem), or orthogonal polynomials.

Weak Monotone Comparative Statics

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

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.