Publication: Resolución numérica de juegos diferenciales deterministas y estocásticos en equilibrios de Nash
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2021-07-27
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Universidad Complutense de Madrid
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Abstract
El objetivo de esta tesis es el diseño e implementación de un algoritmo numérico para la solución de juegos diferenciales, tanto deterministas como estocásticos, en equilibrios de Nash en bucle cerrado ya horizonte infinito. Estos juegos son de especial relevancia puesto que, en la literatura, suelen ser los más comunes debido a sus amplias aplicaciones. Este algoritmo se denomina RaBVItG (Radial Basis and Value Iteration for Games) dadas las principales características de su diseño. Desde nuestro punto de vista, este algoritmo puede considerarse innovador, puesto que en la literatura no hemos encontrado un competidor que resuelva tanto juegos deterministas como estocásticos. Además, cuenta con un diseño "sin mallado", de tal forma que pueden resolverse juegos con N jugadores, donde N es mucho mayor que 2 (en la literatura suelen plantearse algoritmos para mallados con 2 jugadores). En este trabajo, utilizaremos los artículos [13] (en el caso determinista) y [76] (en el caso estocástico) como dos referencias con las que poder compararnos siempre que sea posible. Los principales resultados se enmarcan, por un lado, en la eficiencia del algoritmo. Comparado, por ejemplo, con [13], RaBVItG presenta menor tiempo de cálculo y menor error, en general, debido a la ventaja que supone no tener que usar un mallado. Otro conjunto de resultados de interés son las aplicaciones del algoritmo a dos casos prácticos de dos áreas de conocimiento: marketing y psicología matemática. Estos dos campos, a su vez, están abriendo líneas futuras de investigación donde poder usar el algoritmo y sus sucesivas mejoras...
The goal of this thesis is the design and implementation of a numerical algorithm for solving deterministic and stochastic infinite horizon differential games in closed-loop Nash equilibria. These games are specially relevant since, in the Literature, are the most commonly used due to their wide range of applications. We call the algorithm RaBVItG (Radial Basis and Value Iteration for Games) trying to specify the main characteristics of its design.The algorithm introduced here can be considered innovative because, as far as we know, we have not found a competitor solving both deterministic and stochastic cases. Additionally, the algorithm has a "mesh-free" design, so that it is possible to solve games with N players, where N is considerably greater than 2. We use the reference papers [13] in the deterministic case and [76] in the stochastic, as two main works to compare with our results.Our remarkable results are twofold. On the one hand, the algorithm’s efficiency: Comparing with [13], RaBVItG outperforms needing less computational time and lower errors. This is possible, in general, due to the advantages of the "mesh-free" computational design. On the other hand, we apply our algorithm to two problems borrowed from two scientific fields such as Marketing and Mathematical Psychology. Both contributions are opening future researching lines to use the algorithm and its future improvements...
The goal of this thesis is the design and implementation of a numerical algorithm for solving deterministic and stochastic infinite horizon differential games in closed-loop Nash equilibria. These games are specially relevant since, in the Literature, are the most commonly used due to their wide range of applications. We call the algorithm RaBVItG (Radial Basis and Value Iteration for Games) trying to specify the main characteristics of its design.The algorithm introduced here can be considered innovative because, as far as we know, we have not found a competitor solving both deterministic and stochastic cases. Additionally, the algorithm has a "mesh-free" design, so that it is possible to solve games with N players, where N is considerably greater than 2. We use the reference papers [13] in the deterministic case and [76] in the stochastic, as two main works to compare with our results.Our remarkable results are twofold. On the one hand, the algorithm’s efficiency: Comparing with [13], RaBVItG outperforms needing less computational time and lower errors. This is possible, in general, due to the advantages of the "mesh-free" computational design. On the other hand, we apply our algorithm to two problems borrowed from two scientific fields such as Marketing and Mathematical Psychology. Both contributions are opening future researching lines to use the algorithm and its future improvements...
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Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 05/11/2020