scholarly journals TROPICAL POLYHEDRA ARE EQUIVALENT TO MEAN PAYOFF GAMES

2012 ◽  
Vol 22 (01) ◽  
pp. 1250001 ◽  
Author(s):  
MARIANNE AKIAN ◽  
STÉPHANE GAUBERT ◽  
ALEXANDER GUTERMAN
Keyword(s):  
Convex Sets ◽  
External Representation ◽  
The Matrix ◽  
Tropical Convexity ◽  
Finite Action ◽  
Player Game ◽  
Set Up ◽  
Zero Sum ◽  
Action Spaces ◽  
Mean Payoff

We show that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum two player game problems. In particular, we set up an equivalence between the external representation of tropical convex sets and zero-sum stochastic games, in which tropical polyhedra correspond to deterministic games with finite action spaces. Then, we show that the winning initial positions can be determined from the associated tropical polyhedron. We obtain as a corollary a game theoretical proof of the fact that the tropical rank of a matrix, defined as the maximal size of a submatrix for which the optimal assignment problem has a unique solution, coincides with the maximal number of rows (or columns) of the matrix which are linearly independent in the tropical sense. Our proofs rely on techniques from non-linear Perron–Frobenius theory.

scholarly journals Subgame Maxmin Strategies in Zero-Sum Stochastic Games with Tolerance Levels

2021 ◽  
Author(s):  
János Flesch ◽  
P. Jean-Jacques Herings ◽  
Jasmine Maes ◽  
Arkadi Predtetchinski
Keyword(s):  
Stochastic Games ◽  
Sufficient Conditions ◽  
Payoff Function ◽  
Surprising Result ◽  
Countable State ◽  
Finite Action ◽  
Zero Sum ◽  
Necessary And Sufficient ◽  
Action Spaces ◽  
Special Case

AbstractWe study subgame $$\phi $$ ϕ -maxmin strategies in two-player zero-sum stochastic games with a countable state space, finite action spaces, and a bounded and universally measurable payoff function. Here, $$\phi $$ ϕ denotes the tolerance function that assigns a nonnegative tolerated error level to every subgame. Subgame $$\phi $$ ϕ -maxmin strategies are strategies of the maximizing player that guarantee the lower value in every subgame within the subgame-dependent tolerance level as given by $$\phi $$ ϕ . First, we provide necessary and sufficient conditions for a strategy to be a subgame $$\phi $$ ϕ -maxmin strategy. As a special case, we obtain a characterization for subgame maxmin strategies, i.e., strategies that exactly guarantee the lower value at every subgame. Secondly, we present sufficient conditions for the existence of a subgame $$\phi $$ ϕ -maxmin strategy. Finally, we show the possibly surprising result that each game admits a strictly positive tolerance function $$\phi ^*$$ ϕ ∗ with the following property: if a player has a subgame $$\phi ^*$$ ϕ ∗ -maxmin strategy, then he has a subgame maxmin strategy too. As a consequence, the existence of a subgame $$\phi $$ ϕ -maxmin strategy for every positive tolerance function $$\phi $$ ϕ is equivalent to the existence of a subgame maxmin strategy.

Two-Player Non-Zero-Sum Games

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Nash Equilibrium ◽  
Security Policy ◽  
The Other ◽  
Bimatrix Games ◽  
Zero Sum Games ◽  
Best Response ◽  
Key Concepts ◽  
Player Game ◽  
Zero Sum ◽  
Action Spaces

This chapter defines a number of key concepts for non-zero-sum games involving two players. It begins by considering a two-player game G in which two players P₁ and P₂ are allowed to select policies within action spaces Γ‎₁ and Γ‎₂, respectively. Each player wants to minimize their own outcome, and does not care about the outcome of the other player. The chapter proceeds by discussing the security policy and Nash equilibrium for two-player non-zero-sum games, bimatrix games, admissible Nash equilibrium, and mixed policy. It also explores the order interchangeability property for Nash equilibria in best-response equivalent games before concluding with practice exercises and their corresponding solutions, along with additional exercises.

Position Estimation of Single Camera Visual System Based on Total Least Squares Algorithm

2012 ◽  
Vol 239-240 ◽  
pp. 1352-1355
Author(s):  
Jing Zhou ◽  
Yin Han Gao ◽  
Chang Yin Liu ◽  
Ji Zhi Li
Keyword(s):  
Visual System ◽  
Least Squares ◽  
Total Least Squares ◽  
Position Estimation ◽  
Feature Points ◽  
Perspective Theory ◽  
Optical Feature ◽  
The Matrix ◽  
Least Squares Algorithm ◽  
Set Up

The position estimation of optical feature points of visual system is the focus factor of the precision of system. For this problem , to present the Total Least Squares Algorithm . Firstly , set up the measurement coordinate system and 3D model between optical feature points, image points and the position of camera according to the position relation ; Second , build the matrix equations between optical feature points and image points ; Then apply in the total least squares to have an optimization calculation ; Finally apply in the coordinate measuring machining to have a simulation comparison experiment , the results indicate that the standard tolerance of attitude coordinate calculated by total least squares is 0.043mm, it validates the effectiveness; Compare with the traditional method based on three points perspective theory, measure the standard gauge of 500mm; the standard tolerance of traditional measurement system is 0.0641mm, the standard tolerance of Total Least Squares Algorithm is 0.0593mm; The experiment proves the Total Least Squares Algorithm is effective and has high precision.

scholarly journals Research and Practice of individualized talent Training Mode for rail transit majors from the perspective of new engineering

2020 ◽  
Vol 218 ◽  
pp. 02003
Author(s):  
Zhao Wu ◽  
Hai Xiang Li ◽  
Jun Ying Qi
Keyword(s):  
Evaluation System ◽  
Teaching Practice ◽  
Implementation Process ◽  
Training Model ◽  
Urban Rail Transit ◽  
Rail Transit ◽  
Existing Problems ◽  
Training Mode ◽  
The Matrix ◽  
Set Up

In order to cultivate application-oriented talents of urban rail transit, individualized talent training mode is an important measure. In view of the existing problems in the training of rail transit professionals, the research group proposed the framework of individualized talent training under the background of new engineering, planned the matrix corresponding to graduation requirements and knowledge, ability and quality, and then set up the curriculum system and built the multi-evaluation system in the implementation process. The developed solution has been put into practice and will be tested in the future teaching practice activities in order to constantly improve the personalized talent training model.

Games in Extensive Form

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Saddle Point ◽  
Matrix Game ◽  
Extensive Form ◽  
Multi Stage ◽  
Form Representation ◽  
Other Information ◽  
The Matrix ◽  
Recursive Computation ◽  
Zero Sum ◽  
Computation Of Equilibria

This chapter discusses a number of key concepts for extensive form game representation. It first considers a matrix that defines a zero-sum matrix game for which the minimizer has two actions and the maximizer has three actions and shows that the matrix description, by itself, does not capture the information structure of the game and, in fact, other information structures are possible. It then describes an extensive form representation of a zero-sum two-person game, which is a decision tree, the extensive form representation of multi-stage games, and the notions of security policy, security level, and saddle-point equilibrium for a game in extensive form. It also explores the matrix form for games in extensive form, recursive computation of equilibria for single-stage games, feedback games, feedback saddle-point for multi-stage games, and recursive computation of equilibria for multi-stage games. It concludes with a practice exercise with the corresponding solution, along with additional exercises.

Mixed Policies

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Probability Distribution ◽  
Saddle Point ◽  
Security Policy ◽  
General Type ◽  
Zero Sum Games ◽  
Security Levels ◽  
Mixed Action ◽  
Zero Sum ◽  
Saddle Point Equilibrium ◽  
Action Spaces

This chapter explores the concept of mixed policies and how the notions for pure policies can be adapted to this more general type of policies. A pure policy consists of choices of particular actions (perhaps based on some observation), whereas a mixed policy involves choosing a probability distribution to select actions (perhaps as a function of observations). The idea behind mixed policies is that the players select their actions randomly according to a previously selected probability distribution. The chapter first considers the rock-paper-scissors game as an example of mixed policy before discussing mixed action spaces, mixed security policy and saddle-point equilibrium, mixed saddle-point equilibrium vs. average security levels, and general zero-sum games. It concludes with practice exercises with corresponding solutions and an additional exercise.

Causal amplitudes and the Yang-Feldman formalism

1957 ◽  
Vol 53 (4) ◽  
pp. 843-847 ◽  
Author(s):  
J. C. Polkinghorne
Keyword(s):  
Field Theory ◽  
Green’S Functions ◽  
Free Field ◽  
Dispersion Relations ◽  
Matrix Elements ◽  
Field Equations ◽  
Commutation Relations ◽  
The Matrix ◽  
Free Fields ◽  
Set Up

ABSTRACTThe Yang-Feldman formalism vising the Feynman-like Green's functions is set up. The corresponding free fields have non-trivial commutation relations and contain information about the scattering. S-matrix elements are simply the matrix elements of anti-normal products of the field φF′(x). These are evaluated, and they give directly expressions used in the theory of causality and dispersion relations. It is possible to formulate field theory in a form in which the fields obey free field equations and the effects of interaction are contained in their commutation relations.

scholarly journals Computational Procedures for a Class of GI/D/kSystems in Discrete Time

2009 ◽  
Vol 2009 ◽  
pp. 1-18
Author(s):  
Md. Mostafizur Rahman ◽  
Attahiru Sule Alfa
Keyword(s):  
Discrete Time ◽  
Queue Length ◽  
Computation Time ◽  
Single Server ◽  
The Matrix ◽  
Interarrival Times ◽  
Computational Procedures ◽  
Set Up ◽  
First In First Out ◽  
Service Duration

A class of discrete time GI/D/ksystems is considered for which the interarrival times have finite support and customers are served in first-in first-out (FIFO) order. The system is formulated as a single server queue with new general independent interarrival times and constant service duration by assuming cyclic assignment of customers to the identical servers. Then the queue length is set up as a quasi-birth-death (QBD) type Markov chain. It is shown that this transformed GI/D/1 system has special structures which make the computation of the matrixRsimple and efficient, thereby reducing the number of multiplications in each iteration significantly. As a result we were able to keep the computation time very low. Moreover, use of the resulting structural properties makes the computation of the distribution of queue length of the transformed system efficient. The computation of the distribution of waiting time is also shown to be simple by exploiting the special structures.

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