scholarly journals Uniform Sampling of the Infinite Noncooperative Game on Unit Hypercube and Reshaping Ultimately Multidimensional Matrices of Player’s Payoff Values

2015 ◽  
Vol 8 (1) ◽  
pp. 13-19 ◽  
Author(s):  
Vadim Romanuke
Keyword(s):  
Approximate Solution ◽  
Payoff Function ◽  
Noncooperative Game ◽  
Sampling Step ◽  
Uniform Sampling ◽  
Relative Independence ◽  
Multidimensional Matrices ◽  
Unit Hypercube ◽  
Multidimensional Matrix ◽  
Payoff Functions

Abstract The paper suggests a method of obtaining an approximate solution of the infinite noncooperative game on the unit hypercube. The method is based on sampling uniformly the players’ payoff functions with the constant step along each of the hypercube dimensions. The author states the conditions for a sufficiently accurate sampling and suggests the method of reshaping the multidimensional matrix of the player’s payoff values, being the former player’s payoff function before its sampling, into a matrix with minimally possible number of dimensions, where also maintenance of one-to-one indexing has been provided. Requirements for finite NE-strategy from NE (Nash equilibrium) solution of the finite game as the initial infinite game approximation are given as definitions of the approximate solution consistency. The approximate solution consistency ensures its relative independence upon the sampling step within its minimal neighborhood or the minimally decreased sampling step. The ultimate reshaping of multidimensional matrices of players’ payoff values to the minimal number of dimensions, being equal to the number of players, stimulates shortened computations.

scholarly journals Approximation of Isomorphic Infinite Two-Person Non-Cooperative Games by Variously Sampling the Players’ Payoff Functions and Reshaping Payoff Matrices into Bimatrix Game

2016 ◽  
Vol 20 (1) ◽  
pp. 5-14 ◽  
Author(s):  
Vadim V. Romanuke ◽  
Vladimir V. Kamburg
Keyword(s):  
Cooperative Games ◽  
Equilibrium Strategy ◽  
Finite Dimensional Space ◽  
Sampling Step ◽  
Bimatrix Game ◽  
Infinite Game ◽  
Consistent Solution ◽  
Unit Hypercube ◽  
Payoff Functions ◽  
Non Cooperative Games

Abstract Approximation in solving the infinite two-person non-cooperative games is studied in the paper. An approximation approach with conversion of infinite game into finite one is suggested. The conversion is fulfilled in three stages. Primarily the players’ payoff functions are sampled variously according to the stated requirements to the sampling. These functions are defined on unit hypercube of the appropriate Euclidean finite-dimensional space. The sampling step along each of hypercube dimensions is constant. At the second stage, the players’ payoff multidimensional matrices are reshaped into ordinary two-dimensional matrices, using the reversible index-to-index reshaping. Thus, a bimatrix game as an initial infinite game approximation is obtained. At the third stage of the conversion, the player’s finite equilibrium strategy support is checked out for its weak consistency, defined by five types of inequalities within minimal neighbourhood of every specified sampling step. If necessary, the weakly consistent solution of the bimatrix game is checked out for its consistency, strengthened in that the cardinality of every player’s equilibrium strategy support and their densities shall be non-decreasing within minimal neighbourhood of the sampling steps. Eventually, the consistent solution certifies the game approximation acceptability, letting solve even games without any equilibrium situations, including isomorphic ones to the unit hypercube game. A case of the consistency light check is stated for the completely mixed Nash equilibrium situation.

scholarly journals Dynamics of Strategy Distributions in a One-Dimensional Continuous Trait Space for Games with a Quadratic Payoff Function

Games ◽  
2020 ◽  
Vol 11 (1) ◽  
pp. 14
Author(s):  
Georgiy Karev
Keyword(s):  
Broad Class ◽  
Limit State ◽  
Single Point ◽  
Initial Distribution ◽  
Payoff Function ◽  
Quadratic Term ◽  
Uniform Distributions ◽  
Entire Line ◽  
Payoff Functions ◽  
Initial Distributions

Evolution of distribution of strategies in game theory is an interesting question that has been studied only for specific cases. Here I develop a general method to extend analysis of the evolution of continuous strategy distributions given a quadratic payoff function for any initial distribution in order to answer the following question—given the initial distribution of strategies in a game, how will it evolve over time? I look at several specific examples, including normal distribution on the entire line, normal truncated distribution, as well as exponential and uniform distributions. I show that in the case of a negative quadratic term of the payoff function, regardless of the initial distribution, the current distribution of strategies becomes normal, full or truncated, and it tends to a distribution concentrated in a single point so that the limit state of the population is monomorphic. In the case of a positive quadratic term, the limit state of the population may be dimorphic. The developed method can now be applied to a broad class of questions pertaining to evolution of strategies in games with different payoff functions and different initial distributions.

scholarly journals Lipschitz Continuity and Approximate Equilibria

Algorithmica ◽  
2020 ◽  
Vol 82 (10) ◽  
pp. 2927-2954
Author(s):  
Argyrios Deligkas ◽  
John Fearnley ◽  
Paul Spirakis
Keyword(s):  
Polynomial Time ◽  
Lipschitz Continuity ◽  
Payoff Function ◽  
Polynomial Algorithms ◽  
Continuous Action ◽  
Lipschitz Continuous ◽  
Payoff Functions ◽  
Action Spaces ◽  
Best Responses ◽  
Strongly Polynomial

Abstract In this paper, we study games with continuous action spaces and non-linear payoff functions. Our key insight is that Lipschitz continuity of the payoff 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 polynomial time algorithms for finding best responses in $$L_1$$ L 1 and $$L_2^2$$ L 2 2 biased games, and then use these algorithms to provide strongly polynomial algorithms that find 2/3 and 5/7 approximate equilibria for these norms, respectively.

scholarly journals Acceptable-and-attractive Approximate Solution of a Continuous Non-Cooperative Game on a Product of Sinusoidal Strategy Functional Spaces

2021 ◽  
Vol 46 (2) ◽  
pp. 173-197
Author(s):  
Vadim Romanuke
Keyword(s):  
Noncooperative Game ◽  
Phase Lag ◽  
Acceptable Solution ◽  
Uniform Sampling ◽  
Finite Games ◽  
Phase Lags ◽  
Pure Strategies ◽  
Sinusoidal Functions ◽  
Payoff Matrices ◽  
Positive Coefficients

Abstract A problem of solving a continuous noncooperative game is considered, where the player’s pure strategies are sinusoidal functions of time. In order to reduce issues of practical computability, certainty, and realizability, a method of solving the game approximately is presented. The method is based on mapping the product of the functional spaces into a hyperparallelepiped of the players’ phase lags. The hyperparallelepiped is then substituted with a hypercubic grid due to a uniform sampling. Thus, the initial game is mapped into a finite one, in which the players’ payoff matrices are hypercubic. The approximation is an iterative procedure. The number of intervals along the player’s phase lag is gradually increased, and the respective finite games are solved until an acceptable solution of the finite game becomes sufficiently close to the same-type solutions at the preceding iterations. The sufficient closeness implies that the player’s strategies at the succeeding iterations should be not farther from each other than at the preceding iterations. In a more feasible form, it implies that the respective distance polylines are required to be decreasing on average once they are smoothed with respective polynomials of degree 2, where the parabolas must be having positive coefficients at the squared variable.

Age-Dependent Payoffs and Assortative Matching by Age in a Market with Search

2017 ◽  
Vol 9 (2) ◽  
pp. 263-294
Author(s):  
Anja Sautmann
Keyword(s):  
Sufficient Conditions ◽  
Payoff Function ◽  
Assortative Matching ◽  
Constant Flow ◽  
Age At Marriage ◽  
Matching Market ◽  
Age Dependent ◽  
Finite Period ◽  
Payoff Functions ◽  
Increasing Functions

This paper considers a matching market with two-sided search and transferable utility where match payoffs depend on age at marriage (time until match) and search is finite. We define and prove existence of equilibrium, and provide sufficient conditions for positive assortative matching that build on restricting the slope and curvature of the marriage payoff function to generate single-peaked preferences in age and therefore convex matching sets. Payoff functions that are incompatible with positive sorting by age include all strictly increasing functions and constant flow payoffs enjoyed for some finite period. (JEL C78, D83, J12)

ON A CLASS OF NASH EQUILIBRIA WITH MEMORY STRATEGIES FOR NONZERO-SUM DIFFERENTIAL GAMES

2000 ◽  
Vol 02 (02n03) ◽  
pp. 173-192 ◽  
Author(s):  
JEAN MICHEL COULOMB ◽  
VLADIMIR GAITSGORY
Keyword(s):  
Nash Equilibrium ◽  
Feedback Control ◽  
Differential Games ◽  
Differential Game ◽  
Nash Equilibria ◽  
Payoff Function ◽  
Feedback Controls ◽  
Memory Strategies ◽  
Payoff Functions ◽  
With Memory

A two-player nonzero-sum differential game is considered. Given a pair of threat payoff functions, we characterise a set of pairs of acceptable feedback controls. Any such pair induces a history-dependent Nash δ-equilibrium as follows: the players agree to use the acceptable controls unless one of them deviates. If this happens, a feedback control punishment is implemented. The problem of finding a pair of "acceptable" controls is significantly simpler than the problem of finding a feedback control Nash equilibrium. Moreover, the former may have a solution in case the latter does not. In addition, if there is a feedback control Nash equilibrium, then our technique gives a subgame perfect Nash δ-equilibrium that might improve the payoff function for at least one player.

scholarly journals Bi-Objective Optimization Method and Application of Mechanism Design Based on Pigs' Payoff Game Behavior

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Lu Wang ◽  
Jian-gang Wang ◽  
Rui Meng ◽  
Neng-gang Xie
Keyword(s):  
Optimization Problems ◽  
Optimization Method ◽  
Payoff Function ◽  
Strategy Space ◽  
Design Variables ◽  
Game Player ◽  
Payoff Functions ◽  
Game Players ◽  
Behavior Characteristics ◽  
The Impact

It takes two design goals as different game players and design variables are divided into strategy spaces owned by corresponding game player by calculating the impact factor and fuzzy clustering. By the analysis of behavior characteristics of two kinds of intelligent pigs, the big pig's behavior is cooperative and collective, but the small pig's behavior is noncooperative, which are endowed with corresponding game player. Two game players establish the mapping relationship between game players payoff functions and objective functions. In their own strategy space, each game player takes their payoff function as monoobjective for optimization. It gives the best strategy upon other players. All the best strategies are combined to be a game strategy set. With convergence and multiround game, the final game solution is obtained. Taking bi-objective optimization of luffing mechanism of compensative shave block, for example, the results show that the method can effectively solve bi-objective optimization problems with preferred target and the efficiency and accuracy are also well.

scholarly journals Fact, Fiction, and Fitness

Entropy ◽  
2020 ◽  
Vol 22 (5) ◽  
pp. 514 ◽  
Author(s):  
Chetan Prakash ◽  
Chris Fields ◽  
Donald D. Hoffman ◽  
Robert Prentner ◽  
Manish Singh
Keyword(s):  
Natural Selection ◽  
Payoff Function ◽  
Cyclic Groups ◽  
Interface Theory ◽  
As Species ◽  
Payoff Functions ◽  
Evolution By Natural Selection ◽  
Species Specific ◽  
Perceptual Systems ◽  
Total Orders

A theory of consciousness, whatever else it may do, must address the structure of experience. Our perceptual experiences are richly structured. Simply seeing a red apple, swaying between green leaves on a stout tree, involves symmetries, geometries, orders, topologies, and algebras of events. Are these structures also present in the world, fully independent of their observation? Perceptual theorists of many persuasions—from computational to radical embodied—say yes: perception veridically presents to observers structures that exist in an observer-independent world; and it does so because natural selection shapes perceptual systems to be increasingly veridical. Here we study four structures: total orders, permutation groups, cyclic groups, and measurable spaces. We ask whether the payoff functions that drive evolution by natural selection are homomorphisms of these structures. We prove, in each case, that generically the answer is no: as the number of world states and payoff values go to infinity, the probability that a payoff function is a homomorphism goes to zero. We conclude that natural selection almost surely shapes perceptions of these structures to be non-veridical. This is consistent with the interface theory of perception, which claims that natural selection shapes perceptual systems not to provide veridical perceptions, but to serve as species-specific interfaces that guide adaptive behavior. Our results present a constraint for any theory of consciousness which assumes that structure in perceptual experience is shaped by natural selection.

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