scholarly journals Convergence method, properties and computational complexity for Lyapunov games

2011 ◽  
Vol 21 (2) ◽  
pp. 349-361 ◽  
Author(s):  
Julio Clempner ◽  
Alexander Poznyak
Keyword(s):  
Computational Complexity ◽  
Equilibrium Point ◽  
Repeated Games ◽  
Common Knowledge ◽  
Repeated Game ◽  
Graph Structure ◽  
Natural Evolution ◽  
Nash Equilibrium Point ◽  
Convergence Method ◽  
Differentiable Vector

Convergence method, properties and computational complexity for Lyapunov gamesWe introduce the concept of a Lyapunov game as a subclass of strictly dominated games and potential games. The advantage of this approach is that every ergodic system (repeated game) can be represented by a Lyapunov-like function. A direct acyclic graph is associated with a game. The graph structure represents the dependencies existing between the strategy profiles. By definition, a Lyapunov-like function monotonically decreases and converges to a single Lyapunov equilibrium point identified by the sink of the game graph. It is important to note that in previous works this convergence has not been guaranteed even if the Nash equilibrium point exists. The best reply dynamics result in a natural implementation of the behavior of a Lyapunov-like function. Therefore, a Lyapunov game has also the benefit that it is common knowledge of the players that only best replies are chosen. By the natural evolution of a Lyapunov-like function, no matter what, a strategy played once is not played again. As a construction example, we show that, for repeated games with bounded nonnegative cost functions within the class of differentiable vector functions whose derivatives satisfy the Lipschitz condition, a complex vector-function can be built, where each component is a function of the corresponding cost value and satisfies the condition of the Lyapunov-like function. The resulting vector Lyapunov-like function is a monotonic function which can only decrease over time. Then, a repeated game can be represented by a one-shot game. The functionality of the suggested method is successfully demonstrated by a simulated experiment.

scholarly journals A Type of Time-Symmetric Stochastic System and Related Games

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 118
Author(s):  
Qingfeng Zhu ◽  
Yufeng Shi ◽  
Jiaqiang Wen ◽  
Hui Zhang
Keyword(s):  
Nash Equilibrium ◽  
Time Delay ◽  
Differential Equations ◽  
Equilibrium Point ◽  
Stochastic Differential Equations ◽  
Stochastic System ◽  
Stochastic Systems ◽  
Necessary Condition ◽  
Nash Equilibrium Point ◽  
Doubly Stochastic

This paper is concerned with a type of time-symmetric stochastic system, namely the so-called forward–backward doubly stochastic differential equations (FBDSDEs), in which the forward equations are delayed doubly stochastic differential equations (SDEs) and the backward equations are anticipated backward doubly SDEs. Under some monotonicity assumptions, the existence and uniqueness of measurable solutions to FBDSDEs are obtained. The future development of many processes depends on both their current state and historical state, and these processes can usually be represented by stochastic differential systems with time delay. Therefore, a class of nonzero sum differential game for doubly stochastic systems with time delay is studied in this paper. A necessary condition for the open-loop Nash equilibrium point of the Pontriagin-type maximum principle are established, and a sufficient condition for the Nash equilibrium point is obtained. Furthermore, the above results are applied to the study of nonzero sum differential games for linear quadratic backward doubly stochastic systems with delay. Based on the solution of FBDSDEs, an explicit expression of Nash equilibrium points for such game problems is established.

scholarly journals What Can You Do with a Rock? Affordance Extraction via Word Embeddings

2017 ◽  
Author(s):  
Nancy Fulda ◽  
Daniel Ricks ◽  
Ben Murdoch ◽  
David Wingate
Keyword(s):  
Reinforcement Learning ◽  
Computational Complexity ◽  
Linear Algebra ◽  
Autonomous Agents ◽  
Common Knowledge ◽  
Search Space ◽  
Word Embeddings ◽  
Knowledge Database ◽  
Learning Agent ◽  
Action Spaces

Autonomous agents must often detect affordances: the set of behaviors enabled by a situation. Affordance extraction is particularly helpful in domains with large action spaces, allowing the agent to prune its search space by avoiding futile behaviors. This paper presents a method for affordance extraction via word embeddings trained on a tagged Wikipedia corpus. The resulting word vectors are treated as a common knowledge database which can be queried using linear algebra. We apply this method to a reinforcement learning agent in a text-only environment and show that affordance-based action selection improves performance in most cases. Our method increases the computational complexity of each learning step but significantly reduces the total number of steps needed. In addition, the agent's action selections begin to resemble those a human would choose.

scholarly journals Application of the Game Theory with Perfect Information to an agricultural company

2013 ◽  
Vol 59 (No. 1) ◽  
pp. 1-7 ◽  
Author(s):  
S. Cabrera García ◽  
J.E. Imbert Tamayo ◽  
J. Carbonell-Olivares ◽  
Y. Pacheco Cabrera
Keyword(s):  
Game Theory ◽  
Linear Programming ◽  
Equilibrium Point ◽  
Perfect Information ◽  
Agricultural Economics ◽  
Nash Equilibrium Point ◽  
The Game Theory ◽  
The Hierarchical Structure ◽  
Linear Programming Problems ◽  
The One

This paper deals with the application of Game Theory with Perfect Information to an agricultural economics problem. The goal of this analysis is demonstrating the possibility of obtaining an equilibrium point, as proposed by Nash, in the case of an agricultural company that is considered together with its three sub-units in developing a game with perfect information. Production results in terms of several crops will be considered in this game, together with the necessary parameters to implement different linear programming problems. In the game with perfect information with the hierarchical structure established between the four considered players (a management center and three production units), a Nash equilibrium point is reached, since once the strategies of the rest of the players are known, if any of them would use a strategy different to the one proposed, their earnings would be less than the ones obtained by using the proposed strategies. When the four linear programming problems are solved, a particular case of equilibrium point is reached.

scholarly journals Dynamic Cournot Duopoly Game with Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
A. A. Elsadany ◽  
A. E. Matouk
Keyword(s):  
Nash Equilibrium ◽  
Equilibrium Point ◽  
Local Stability ◽  
Stability Region ◽  
Equilibrium Points ◽  
Cournot Duopoly ◽  
Nash Equilibrium Point ◽  
Dynamical Behaviors ◽  
Delayed System

The delay Cournot duopoly game is studied. Dynamical behaviors of the game are studied. Equilibrium points and their stability are studied. The results show that the delayed system has the same Nash equilibrium point and the delay can increase the local stability region.

On the Strategic Equivalence of Linear Dynamic and Repeated Games

2015 ◽  
Vol 17 (03) ◽  
pp. 1550006
Author(s):  
Joachim Hubmer
Keyword(s):  
Repeated Games ◽  
Dynamic Game ◽  
Repeated Game ◽  
Transition Function ◽  
The State ◽  
State Variables ◽  
Equivalent Representation ◽  
Deterministic Dynamic ◽  
Subgame Perfect Equilibria ◽  
Perfect Equilibria

Dynamic (or stochastic) games are, in general, considerably more complicated to analyze than repeated games. This paper shows that for every deterministic dynamic game that is linear in the state, there exists a strategically equivalent representation as a repeated game. A dynamic game is said to be linear in the state if it holds for both the state transition function as well as for the one-period payoff function that (i) they are additively separable in action profiles and states and (ii) the state variables enter linearly. Strategic equivalence refers to the observation that the two sets of subgame perfect equilibria coincide, up to a natural projection of dynamic game strategy profiles on the much smaller set of repeated game histories. Furthermore, it is shown that the strategic equivalence result still holds for certain stochastic elements in the transition function if one allows for additional signals in the repeated game or in the presence of a public correlating device.

Reciprocal Social Strategy in Social Repeated Games and Emergence of Social Norms

2017 ◽  
Vol 26 (01) ◽  
pp. 1760007
Author(s):  
Chi-Kong Chan ◽  
Jianye Hao ◽  
Ho-Fung Leung
Keyword(s):  
Social Norms ◽  
Repeated Games ◽  
Repeated Game ◽  
Social Utility ◽  
Learning Mechanism ◽  
Artificial Society ◽  
Social Strategy ◽  
Long Run ◽  
The Individual ◽  
High Level

In an artificial society where agents repeatedly interact with one another, effective coordination among agents is generally a challenge. This is especially true when the participating agents are self-interested, and that there is no central authority to coordinate, and direct communication or negotiation are not possible. Recently, the problem was studied in a paper by Hao and Leung, where a new repeated game mechanism for modeling multi-agent interactions as well as a new reinforcement learning based agent learning method were proposed. In particular, the game mechanism differs from traditional repeated games in that the agents are anonymous, and the agents interact with randomly chosen opponents during each iteration. Their learning mechanism allows agents to coordinate without negotiations. The initial results had been promising. However, extended simulation also reveals that the outcomes are not stable in the long run in some cases, as the high level of cooperation is eventually not sustainable. In this work, we revisit he problem and propose a new learning mechanism as follows. First, we propose an enhanced Q-learning-based framework that allows the agents to better capture both the individual and social utilities that they have learned through observations. Second, we propose a new concept of \social attitude" for determining the action of the agents throughout the game. Simulation results reveal that this approach can achieve higher social utility, including close-to-optimal results in some scenarios, and more importantly, the results are sustainable with social norms emerging.

COMPLEXITY OF A REAL ESTATE GAME MODEL WITH A NONLINEAR DEMAND FUNCTION

2011 ◽  
Vol 21 (11) ◽  
pp. 3171-3179 ◽  
Author(s):  
LINGLING MU ◽  
PING LIU ◽  
YANYAN LI ◽  
JINZHU ZHANG
Keyword(s):  
Nash Equilibrium ◽  
Real Estate ◽  
Equilibrium Point ◽  
Demand Function ◽  
Control Method ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Game Model ◽  
Nonlinear Feedback ◽  
Nash Equilibrium Point

In this paper, a real estate game model with nonlinear demand function is proposed. And an analysis of the game's local stability is carried out. It is shown that the stability of Nash equilibrium point is lost through period-doubling bifurcation as some parameters are varied. With numerical simulations method, the results of bifurcation diagrams, maximal Lyapunov exponents and strange attractors are presented. It is found that the chaotic behavior of the model has been stabilized on the Nash equilibrium point by using of nonlinear feedback control method.

Determining Optimal Strategy of a Micro-Grid through Hybrid Method of Nash Equilibrium –Genetic Algorithm

2019 ◽  
Vol 20 (3) ◽  
Author(s):  
Ehsan Jafari
Keyword(s):  
Genetic Algorithm ◽  
Nash Equilibrium ◽  
Equilibrium Point ◽  
Hybrid Method ◽  
Optimal Strategy ◽  
Energy Resource ◽  
Energy Resources ◽  
Optimal Planning ◽  
Nash Equilibrium Point ◽  
Micro Grid

Abstract Increasing the fossil fuels consumption, pollution and rising prices of these fuels have led to the expansion of renewable resources and their replacement with conventional sources. In this paper, a robust algorithm for a micro-grid (MG) planning with the goal of maximizing profits is presented in day-ahead market. The energy resources in MG are wind farms (WFs), photovoltaic (PV), fuel cell (FC), combined heat and power(CHP) units, tidal steam turbine (TST) and energy storage devices (ESDs). This algorithm is divided into two main parts: (1) Optimal planning of each energy resource; (2) Using the Nash equilibrium –genetic algorithm (NE-GA) hybrid method to determine the optimal MG strategy. In energy resources optimal planning, using a stochastic formulation, the generation bids of each energy resource is determined in such a way that the profit of each one is maximized. Also, the constraints of renewable and load demands and selection the best method of demand response (DR) program are investigated. Then the Nash equilibrium point is determined using the primary population produced in the previous step using the NE-GA hybrid method to determine the optimal MG strategy. Thus, using the ability of the genetic algorithm method, the Nash equilibrium point of the generation units is obtained at an acceptable time, and This means that none of the units are willing to change their strategy and that the optimal strategy is extracted. Comparison of results with previous studies shows that the expected profit in the proposed method is more than other method.

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