scholarly journals Equilibrium uniqueness in aggregative games: very practical conditions

2021 ◽  
Author(s):  
Jun-ichi Itaya ◽  
Pierre von Mouche
Keyword(s):  
Nash Equilibrium ◽  
Broad Class ◽  
Economic Games ◽  
Equilibrium Uniqueness ◽  
Aggregative Games ◽  
Payoff Functions

AbstractVarious Nash equilibrium results for a broad class of aggregative games are presented. The main ones concern equilibrium uniqueness. The setting presupposes that each player has $$\mathbb {R}_+$$ R + as strategy set, makes smoothness assumptions but allows for a discontinuity of stand-alone payoff functions at 0; this possibility is especially important for various contest and oligopolistic games. Conditions are completely in terms of marginal reductions which may be considered as primitives of the game. For many games in the literature they can easily be checked. They automatically imply that conditional payoff functions are strictly quasi-concave. The results are proved by means of the Szidarovszky variant of the Selten–Szidarovszky technique. Their power is illustrated by reproducing quickly and improving upon various results for economic games.

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 Designing Strategic Games with Preestablished Nash Equilibrium through Artificial Inference and Global Learning

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hime A. e Oliveira Jr.
Keyword(s):  
Global Optimization ◽  
Nash Equilibrium ◽  
Solution Concept ◽  
Optimization Method ◽  
Optimization Techniques ◽  
Strategic Games ◽  
Global Learning ◽  
Finite State ◽  
Payoff Functions ◽  
Game Designer

Abstract This work presents novel results obtained by the application of global optimization techniques to the design of finite, normal form games with mixed strategies. To that end, the Fuzzy ASA global optimization method is applied to several design examples of strategic games, demonstrating its effectiveness in obtaining payoff functions whose corresponding games present a previously established Nash equilibrium. In other words, the game designer becomes able to choose a convenient Nash equilibrium for a generic finite state strategic game and the proposed method computes payoff functions that will realize the desired equilibrium, making it possible for the players to reach the favorable conditions represented by the chosen equilibrium. Considering that game theory is a very useful approach for modeling interactions between competing agents and Nash equilibrium represents a powerful solution concept, it is natural to infer that the proposed method may be very useful for strategists in general. In summary, it is a genuine instance of artificial inference of payoff functions after a process of global machine learning, applied to their numerical components.

HADAMARD AND TYKHONOV WELL-POSEDNESS IN TWO PLAYER GAMES

2003 ◽  
Vol 05 (04) ◽  
pp. 375-384 ◽  
Author(s):  
GRAZIANO PIERI ◽  
ANNA TORRE
Keyword(s):  
Nash Equilibrium ◽  
Equilibrium Point ◽  
Nash Equilibria ◽  
Well Posedness ◽  
Nash Equilibrium Point ◽  
Suitable Definition ◽  
Payoff Functions ◽  
The Stability ◽  
Definition Of ◽  
Zero Sum

We give a suitable definition of Hadamard well-posedness for Nash equilibria of a game, that is, the stability of Nash equilibrium point with respect to perturbations of payoff functions. Our definition generalizes the analogous notion for minimum problems. For a game with continuous payoff functions, we restrict ourselves to Hadamard well-posedness with respect to uniform convergence and compare this notion with Tykhonov well-posedness of the same game. The main results are: Hadamard implies Tykhonov well-posedness and the converse is true if the payoff functions are bounded. For a zero-sum game the two notions are equivalent.

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