Existence, Uniqueness and Stability of Market Equilibrium in Oligopoly Markets

In this paper we build a pragmatic model on competition in oligopoly markets. To achieve this goal, we use an approach based on studying the response functions of each market participant, thus making it possible to address both Cournot and Bertrand industrial structures with a unified formal method. In contrast to the restrictive theoretical constructs of duopoly equilibrium, our study is able to account for real-world limitations like minimal sustainable production levels and exclusive access to certain resources. We prove and demonstrate that by using carefully constructed response functions it is possible to build and calibrate a model that reflects different competitive strategies used in extremely concentrated markets. The response functions approach makes it also possible to take into consideration different barriers to entry. By fitting to the response functions rather than the profit maximization of the payoff functions problem we alter the classical optimization problem to a problem of coupled fixed points, which has the benefit that considering corner optimum, corner equilibria and convexity condition of the payoff function can be skipped.

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A variational principle, coupled fixed points and market equilibrium

Nonlinear Analysis Modelling and Control ◽

10.15388/namc.2021.26.21413 ◽

2021 ◽

Vol 26 (1) ◽

pp. 169-185

Author(s):

Stanimir Kabaivanov ◽

Boyan Zlatanov

Keyword(s):

Fixed Points ◽

Metric Spaces ◽

Sufficient Conditions ◽

Profit Maximization ◽

Payoff Function ◽

Response Functions ◽

Barriers To Entry ◽

Complete Metric Spaces ◽

Oligopoly Markets ◽

Coupled Fixed Points

We present a possible kind of generalization of the notion of ordered pairs of cyclic maps and coupled fixed points and its application in modelling of equilibrium in oligopoly markets. We have obtained sufficient conditions for the existence and uniqueness of coupled fixed in complete metric spaces. We illustrate one possible application of the results by building a pragmatic model on competition in oligopoly markets. To achieve this goal, we use an approach based on studying the response functions of each market participant, thus making it possible to address both Cournot and Bertrand industrial structures with unified formal method.We show that whenever the response functions of the two players are identical, then the equilibrium will be attained at equal levels of production and equal prices. The response functions approach makes it also possible to take into consideration different barriers to entry. By fitting to the response functions rather than the profit maximization of the payoff functions problem we alter the classical optimization problem to a problem of coupled fixed points, which has the benefit that considering corner optimum, corner equilibrium and convexity condition of the payoff function can be skipped.

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

Games ◽

10.3390/g11010014 ◽

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.

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Lipschitz Continuity and Approximate Equilibria

Algorithmica ◽

10.1007/s00453-020-00709-3 ◽

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.

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

American Economic Journal Microeconomics ◽

10.1257/mic.20150237 ◽

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)

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

International Game Theory Review ◽

10.1142/s021919890000010x ◽

2000 ◽

Vol 02 (02n03) ◽

pp. 173-192 ◽

Cited By ~ 1

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.

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

Mathematical Problems in Engineering ◽

10.1155/2012/808161 ◽

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.

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Barriers to entry and competitive strategies

Strategic Management Journal ◽

10.1002/smj.4250020407 ◽

1981 ◽

Vol 2 (4) ◽

pp. 395-412 ◽

Cited By ~ 82

Author(s):

Kathryn Rudie Harrigan

Keyword(s):

Barriers To Entry ◽

Competitive Strategies

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Fact, Fiction, and Fitness

Entropy ◽

10.3390/e22050514 ◽

2020 ◽

Vol 22 (5) ◽

pp. 514 ◽

Cited By ~ 3

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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The determinacy of Blackwell games

Journal of Symbolic Logic ◽

10.2307/2586667 ◽

1998 ◽

Vol 63 (4) ◽

pp. 1565-1581 ◽

Cited By ~ 128

Author(s):

Donald A. Martin

Keyword(s):

Imperfect Information ◽

Pure Strategy ◽

Substantial Improvement ◽

Payoff Function ◽

Perfect Information ◽

Infinite Length ◽

Perfect Information Games ◽

Payoff Functions ◽

Borel Measurable ◽

Information Games

Games of infinite length and perfect information have been studied for many years. There are numerous determinacy results for these games, and there is a wide body of work on consequences of their determinacy.Except for games with very special payoff functions, games of infinite length and imperfect information have been little studied. In 1969, David Blackwell [1] introduced a class of such games and proved a determinacy theorem for a subclass. During the intervening time, there has not been much progress in proving the determinacy of Blackwell's games. Orkin [17] extended Blackwell's result to a slightly wider class. Blackwell [2] found a new proof of his own result. Maitra and Sudderth [9, 10] improved Blackwell's result in a different direction from that of Orkin and also generalized to the case of stochastic games. Recently Vervoort [18] has obtained a substantial improvement. Nevertheless, almost all the basic questions have remained open.In this paper we associate with each Blackwell game a family of perfect information games, and we show that the (mixed strategy) determinacy of the former follows from the (pure strategy) determinacy of the latter. The complexity of the payoff function for the Blackwell game is approximately the same as the complexity of the payoff sets for the perfect information games. In particular, this means that the determinacy of Blackwell games with Borel measurable payoff functions follows from the known determinacy of perfect information games with Borel payoff sets.

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Sleep Control Game for Wireless Sensor Networks

Mobile Information Systems ◽

10.1155/2016/3085408 ◽

2016 ◽

Vol 2016 ◽

pp. 1-9

Author(s):

Sang Hoon Lee ◽

Hyeokman Kim ◽

Lynn Choi

Keyword(s):

Wireless Sensor Networks ◽

Energy Consumption ◽

Sensor Networks ◽

Packet Delay ◽

Payoff Function ◽

Sensor Nodes ◽

Wireless Sensor ◽

The Game Theory ◽

Payoff Functions ◽

Sleep Control

In wireless sensor networks (WSNs), each node controls its sleep to reduce energy consumption without sacrificing message latency. In this paper we apply the game theory, which is a powerful tool that explains how each individual acts for his or her own economic benefit, to analyze the optimal sleep schedule for sensor nodes. We redefine this sleep control game as a modified version of the Prisoner’s Dilemma. In the sleep control game, each node decides whether or not it wakes up for the cycle. Payoff functions of the sleep control game consider the expected traffic volume, network conditions, and the expected packet delay. According to the payoff function, each node selects the best wake-up strategy that may minimize the energy consumption and maintain the latency performance. To investigate the performance of our algorithm, we apply the sleep control game to X-MAC, which is one of the recent WSN MAC protocols. Our detailed packet level simulations confirm that the proposed algorithm can effectively reduce the energy consumption by removing unnecessary wake-up operations without loss of the latency performance.

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