Nash Equilibrium In Non-cooperative Game Research Articles

Norms2: Norms About Norms

In this paper, I outline and defend the view that variations in compliance levels with one and the same norm represent different norms about following norms. In support of this claim, I first argue that classic game-theoretic accounts, which define norms as Nash equilibria of noncooperative games, typically consider variations in compliance levels as separate norms. After that, I suggest a more fine-grained, game-theoretic distinction that accounts for degrees of compliance with the same norm and I show how to incorporate such an account into a psychological framework. Finally, the paper examines what given degrees of compliance can reveal about the dynamics underlying the process of norm change. I will argue that they are indicators of different reactions to the introduction of new norms.

Nonsmooth Continuous-Time Distributed Algorithms for Seeking Generalized Nash Equilibria of Noncooperative Games via Digraphs.

In this article, the problem of distributed generalized Nash equilibrium (GNE) seeking in noncooperative games is investigated via multiagent networks, where each player aims to minimize his or her own cost function with a nonsmooth term. Each player's cost function and feasible action set in the noncooperative game are both determined by actions of others who may not be neighbors, as well as his/her own action. Particularly, feasible action sets are constrained by private convex inequalities and shared linear equations. Each player can only have access to his or her own cost function, private constraint, and a local block of shared constraints, and can only communicate with his or her neighbours via a digraph. To address this problem, a novel continuous-time distributed primal-dual algorithm involving Clarke's generalized gradient is proposed based on consensus algorithms and the primal-dual algorithm. Under mild assumptions on cost functions and graph, we prove that players' actions asymptotically converge to a GNE. Finally, a simulation is presented to demonstrate the effectiveness of our theoretical results.

Differential evolution particle swarm optimization algorithm based on good point set for computing Nash equilibrium of finite noncooperative game

In this paper, a hybrid differential evolution particle swarm optimization (PSO) method based on a good point set (GPDEPSO) is proposed to compute a finite noncooperative game among N people. Stochastic functional analysis is used to prove the convergence of this algorithm. First, an ergodic initial population is generated by using a good point set. Second, PSO is proposed and utilized as the variation operator to perform variation crossover selection with differential evolution (DE). Finally, the experimental results show that the proposed algorithm has a better convergence speed, accuracy, and global optimization ability than other existing algorithms in computing the Nash equilibrium of noncooperative games among N people. In particular, the efficiency of the algorithm is higher for determining the Nash equilibrium of a high-dimensional payoff matrix game.

Game Theory-Based Framework for Modeling Human–Vehicle Interactions on the Road

New application domains have faded the barriers between humans and robots, introducing a new set of complexities to robotic systems. The major impediment is the uncertainties associated with human decision making, which makes it challenging to predict human behavior. A realistic model of human behavior is thus vital to capture humans’ interactive behavior with their surroundings and provide robots with reliable estimates on what is most likely to happen. Focusing on operations of connected and automated vehicles (CAVs) in areas with a high presence of human actors (i.e., pedestrians), this study creates an interactive decision-making framework to predict pedestrians’ trajectories when walking in a shared environment with vehicles and other pedestrians. It develops a game theoretical structure to approximate the movement and directional components of pedestrian motion using the theory of Nash equilibria in non-cooperative games. It also introduces a novel payoff structure to address the inherent uncertainties in human behavior. Ground truth pedestrian trajectories are then used to calibrate the game parameters and evaluate the model’s performance in approximating the motion decisions of human agents in interaction with interfering vehicles and pedestrians. The main contribution of the study is to develop an interactive human–vehicle decision-making framework toward realizing human–vehicle coexistence by capturing the effect of pedestrian–vehicle and pedestrian–pedestrian interactions on choice of walking strategies. The derived knowledge could be used in CAV navigation algorithms to provide the vehicle with more accurate predictions of pedestrian behavior, and in turn, improve CAV motion planning in human-populated areas.

A Necessary and Sufficient Condition for Partial Monotonicity of Noncooperative Games

It is a classical and interesting problem to find a Nash equilibrium of noncooperative games in the strategic form. It is well known that the game always has a mixed-strategy Nash equilibrium, but it does not necessarily have a pure-strategy Nash equilibrium. Takeshita and Kawasaki proved that every noncooperative partially monotone game has a pure-strategy Nash equilibrium, that is, the partial monotonicity is a sufficient condition for a noncooperative game to have a pure-strategy Nash equilibrium. In this paper, we prove the necessary and sufficient condition for a noncooperative [Formula: see text]-person game with [Formula: see text] to be partially monotone. This result is an improvement of Takeshita and Kawasaki’s result.

A decentralized strategy for generalized Nash equilibrium with linear coupling constraints

In this paper, we proposed a continuous-time distributed algorithm based on the operator splitting method to search the generalized Nash equilibrium in non-cooperative game. For the considered problem, the local objective function of each player is effected by other players’ strategies within a local area network, meanwhile the local feasible strategy sets of players are coupled with each other over an undirected connected graph. The global linear inequality and equality constraints are employed to describe this relationship. With variational inequality theory, projection operator theory and Lyapunov stability theory, the convergence of the proposed decentralized algorithm is analyzed. Finally, a numerical example is formulated to verify the effectiveness of the proposed algorithm.

The Myopic Stable Set for Social Environments

We introduce a new solution concept for models of coalition formation, called the myopic stable set. The myopic stable set is defined for a very general class of social environments and allows for an infinite state space. We show that the myopic stable set exists and is non-empty. Under minor continuity conditions, we also demonstrate uniqueness. Furthermore, the myopic stable set is a superset of the core and of the set of pure strategy Nash equilibria in noncooperative games. Additionally, the myopic stable set generalizes and unifies various results from more specific environments. In particular, the myopic stable set coincides with the coalition structure core in coalition function form games if the coalition structure core is non-empty; with the set of stable matchings in the standard one-to-one matching model; with the set of pairwise stable networks and closed cycles in models of network formation; and with the set of pure strategy Nash equilibria in finite supermodular games, finite potential games, and aggregative games. We illustrate the versatility of our concept by characterizing the myopic stable set in a model of Bertrand competition with asymmetric costs, for which the literature so far has not been able to fully characterize the set of all (mixed) Nash equilibria.

Distributed Algorithms for Searching Generalized Nash Equilibrium of Noncooperative Games.

In this paper, the distributed Nash equilibrium (NE) searching problem is investigated, where the feasible action sets are constrained by nonlinear inequalities and linear equations. Different from most of the existing investigations on distributed NE searching problems, we consider the case where both cost functions and feasible action sets depend on actions of all players, and each player can only have access to the information of its neighbors. To address this problem, a continuous-time distributed gradient-based projected algorithm is proposed, where a leader-following consensus algorithm is employed for each player to estimate actions of others. Under mild assumptions on cost functions and graphs, it is shown that players' actions asymptotically converge to a generalized NE. Simulation examples are presented to demonstrate the effectiveness of the theoretical results.

The Myopic Stable Set for Social Environments

We introduce a new solution concept for models of coalition formation, called the myopic stable set. The myopic stable set is defined for a very general class of social environments and allows for an infinite state space. We show that the myopic stable set exists and is non-empty. Under minor continuity conditions, we also demonstrate uniqueness. Furthermore, the myopic stable set is a superset of the core and of the set of pure strategy Nash equilibria in noncooperative games. Additionally, the myopic stable set generalizes and unifies various results from more specific environments. In particular, the myopic stable set coincides with the coalition structure core in coalition function form games if the coalition structure core is non-empty; with the set of stable matchings in the standard one-to-one matching model; with the set of pairwise stable networks and closed cycles in models of network formation; and with the set of pure strategy Nash equilibria in finite supermodular games, finite potential games, and aggregative games. We illustrate the versatility of our concept by characterizing the myopic stable set in a model of Bertrand competition with asymmetric costs, for which the literature so far has not been able to fully characterize the set of all (mixed) Nash equilibria.

Distributed Seeking of Time-Varying Nash Equilibrium for Non-Cooperative Games

In this note, we address a Nash equilibrium seeking problem for non-cooperative games. In contrast to previous works on Nash equilibrium seeking, the Nash equilibrium under consideration can be time-varying. A non-model-based seeking scheme is proposed to achieve time-varying Nash equilibrium seeking, where each player updates its strategy by employing an extremum seeking method. The proposed Nash seeking scheme consists of a gradient estimation algorithm and a gradient search algorithm, which can be designed in a modular fashion. For symmetric quadratic games, the proposed Nash equilibrium seeking method enables the estimated strategy to globally asymptotically converge to the Nash equilibrium. For general quadratic games that are not necessarily symmetric, the estimated strategy converges to a neighborhood of the Nash equilibrium. For more general non-quadratic games that may admit multiple equilibria, local convergence to the Nash equilibrium is proven.

Topologically-based characterizations of the existence of solutions of optimization-related problems

Necessary and sufficient conditions for the existence of solutions of optimization-related problems, defined on general sets, are established by using topologically-based structures, without the usual linear structure. First, we prove that the existence of solutions to a variational relation problem is equivalent to the existence of either a KKM-structure or a connectedness structure, satisfying additionally some verifiable conditions. Then, applying these results, we obtain such full (two-way) characterizations for the existence of invariant points, solutions of quasiequilibrium problems of the Stampacchia-Minty type, saddle points, and Nash equilibria for noncooperative games. The main advantages of our scheme with the mentioned two structures are that full characterizations for existence are obtained in a unified way, with no convexity assumptions, and also that one has flexibility to choose a suitable structure in investigations. (This flexibility is decisive when the natural underlying structure of the given problem is scarcely employed.)

Extended Nash Equilibria of Nonmonetized Noncooperative Games on Preordered Sets

A noncooperative game is said to be nonmonetized if the ranges of the utilities (payoffs) of the players are preordered sets. In this paper, we examine some nonmonetized noncooperative games in which both of the collection of strategies and the ranges of the utilities for the players are preordered sets. Then, we spread the concept of extended Nash equilibria of noncooperative games from posets to preordered sets. By applying some fixed point theorems on preordered sets and by using the order preserving property of the utilities, we prove an existence theorem of extended Nash equilibria for nonmonetized noncooperative games.

Generalized Nash Equilibria of Nonmonetized Noncooperative Games on Lattices

In this paper, we examine noncooperative games with both of the collection of strategies and the ranges of the utilities for the players are lattices, which are said to be nonmonetized. Then we extend the concept of Nash equilibria of noncooperative games to generalized Nash equilibria of nonmonetized noncooperative games. By applying some fixed point theorems in posets and by using the order preserving property of mappings, we prove an existence theorem of generalized Nash equilibria for nonmonetized noncooperative games.

Computing Nash Equilibria in Non-Cooperative Games

Game Theory has recently drawn attention in new fields which go from algorithmic mechanism design to cybernetics. However, a fundamental problem to solve for effectively applying Game Theory in real word applications is the definition of well-founded solution concepts of a game and the design of efficient algorithms for their computation. A widely accepted solution concept for games in which any cooperation among the players must be self-enforcing (non-cooperative games) is represented by the Nash equilibrium. However, even in the two players case, the best algorithm known for computing Nash equilibria has an exponential worst-case running time; furthermore, if the computation of equilibria with simple additional properties is required, the problem becomes NP-hard. The paper aims to provide a solution for efficiently computing the Nash equilibria of a game as the result of the evolution of a system composed by interacting agents playing the game.

A Unified Iterative Treatment for Solutions of Problems of Split Feasibility and Equilibrium in Hilbert Spaces

We at first raise the so called split feasibility fixed point problem which covers the problems of split feasibility, convex feasibility, and equilibrium as special cases and then give two types of algorithms for finding solutions of this problem and establish the corresponding strong convergence theorems for the sequences generated by our algorithms. As a consequence, we apply them to study the split feasibility problem, the zero point problem of maximal monotone operators, and the equilibrium problem and to show that the unique minimum norm solutions of these problems can be obtained through our algorithms. Since the variational inequalities, convex differentiable optimization, and Nash equilibria in noncooperative games can be formulated as equilibrium problems, each type of our algorithms can be considered as a generalized methodology for solving the aforementioned problems.

Generalized Browder-type fixed point theorem with strongly geodesic convexity on Hadamard manifolds with applications

In this paper, a generalized Browder-type fixed point theorem on Hadamard manifolds is introduced, which can be regarded as a generalization of the Browder-type fixed point theorem for the set-valued mapping on an Euclidean space to a Hadamard manifold. As applications, a maximal element theorem, a section theorem, a Ky Fan-type Minimax Inequality and an existence theorem of Nash equilibrium for non-cooperative games on Hadamard manifolds are established.

Noncompact Equilibrium Points and Applications

We prove an equilibrium existence result for vector functions defined on noncompact domain and we give some applications in optimization and Nash equilibrium in noncooperative game.

Sampled fictitious play for approximate dynamic programming

Sampled fictitious play (SFP) is a recently proposed iterative learning mechanism for computing Nash equilibria of non-cooperative games. For games of identical interests, every limit point of the sequence of mixed strategies induced by the empirical frequencies of best response actions that players in SFP play is a Nash equilibrium. Because discrete optimization problems can be viewed as games of identical interests wherein Nash equilibria define a type of local optimum, SFP has recently been employed as a heuristic optimization algorithm with promising empirical performance. However, there have been no guarantees of convergence to a globally optimal Nash equilibrium established for any of the problem classes considered to date. In this paper, we introduce a variant of SFP and show that it converges almost surely to optimal policies in model-free, finite-horizon stochastic dynamic programs. The key idea is to view the dynamic programming states as players, whose common interest is to maximize the total multi-period expected reward starting in a fixed initial state. We also offer empirical results suggesting that our SFP variant is effective in practice for small to moderate sized model-free problems.

Nash Equilibrium Seeking with Finitely-and Infinitely-Many Players

We introduce a non-model based approach for stable attainment of Nash equilibria in noncooperative games. Unlike classical game theory, which requires some amount of modeling information, this approach employs deterministic extremum seeking to enable the players to maximize their payoff functions without knowing the underlying model of the game. The players only need to measure their own payoff values. We present results for games for some basic models of economic competition with two players, N players, and infinitely-many players.

Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects

In the setting of a real Hilbert space H, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order evolution equations ¨(t )+ γ ˙ u(t )+ ∇φ(u(t)) + A(u(t)) = 0, where ∇φ is the gradient operator of a convex differentiable potential function φ : H→ R, A : H→H is a maximal monotone operator which is assumed to be λ-cocoercive, and γ> 0i s a damping parameter. Potential and non-potential effects are associated respectively to ∇φ and A. Under condition λγ 2 > 1, it is proved that each trajectory asymptotically weakly converges to a zero of ∇φ + A. This condition, which only involves the non-potential operator and the damping parameter, is sharp and consistent with time rescaling. Passing from weak to strong convergence of the trajectories is obtained by introducing an asymptotically vanishing Tikhonov-like regularizing term. As special cases, we recover the asymptotic analysis of the heavy ball with friction dynamic attached to a convex potential, the second-order gradient-projection dynamic, and the second-order dynamic governed by the Yosida approximation of a general maximal monotone operator. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of constrained optimization, dynamical approach to Nash equilibria for noncooperative games, and asymptotic stabilization in the case of a continuum of equilibria.