Spatial evolutionary game theory: Deterministic approximations, decompositions, and hierarchical multi-scale models

Evolutionary game theory has recently emerged as a key paradigm in various behavioral science disciplines. In particular it provides powerful tools and a conceptual framework for the analysis of the time evolution of strategic interdependence among players and its consequences, especially when the p...

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Main Author: Hwang, Sungha
Language:ENG
Published: ScholarWorks@UMass Amherst 2011
Subjects:
Online Access:https://scholarworks.umass.edu/dissertations/AAI3482707
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spelling ndltd-UMASS-oai-scholarworks.umass.edu-dissertations-63312020-12-02T14:32:23Z Spatial evolutionary game theory: Deterministic approximations, decompositions, and hierarchical multi-scale models Hwang, Sungha Evolutionary game theory has recently emerged as a key paradigm in various behavioral science disciplines. In particular it provides powerful tools and a conceptual framework for the analysis of the time evolution of strategic interdependence among players and its consequences, especially when the players are spatially distributed and linked in a complex social network. We develop various evolutionary game models, analyze these models using appropriate techniques, and study their applications to complex phenomena. In the second chapter, we derive integro-differential equations as deterministic approximations of the microscopic updating stochastic processes. These generalize the known mean-field ordinary differential equations and provide powerful tools to investigate the spatial effects on the time evolutions of the agents' strategy choices. The deterministic equations allow us to identify many interesting features of the evolution of strategy profiles in a population, such as standing and traveling waves, and pattern formation, especially in replicator-type evolutions. We introduce several methods of decomposition of two player normal form games in the third chapter. Viewing the set of all games as a vector space, we exhibit explicit orthonormal bases for the subspaces of potential games, zero-sum games, and their orthogonal complements which we call anti-potential games and anti-zero-sum games, respectively. Perhaps surprisingly, every anti-potential game comes either from Rock-paper-scissors type games (in the case of symmetric games) or from Matching Pennies type games (in the case of asymmetric games). Using these decompositions, we prove old (and some new) cycle criteria for potential and zero-sum games (as orthogonality relations between subspaces). We illustrate the usefulness of our decompositions by (a) analyzing the generalized Rock-Paper-Scissors game, (b) completely characterizing the set of all null-stable games, (c) providing a large class of strict stable games, (d) relating the game decomposition to the Hodge decomposition of vector fields for the replicator equations, (e) constructing Lyapunov functions for some replicator dynamics, (f) constructing Zeeman games—games with an interior asymptotically stable Nash equilibrium and a pure strategy ESS. The hierarchical modeling of evolutionary games provides flexibility in addressing the complex nature of social interactions as well as systematic frameworks in which one can keep track of the interplay of within-group dynamics and between-group competitions. For example, it can model husbands and wives' interactions, playing an asymmetric game with each other, while engaging coordination problems with the likes in other families. In the fourth chapter, we provide hierarchical stochastic models of evolutionary games and approximations of these processes, and study their applications. 2011-01-01T08:00:00Z text https://scholarworks.umass.edu/dissertations/AAI3482707 Doctoral Dissertations Available from Proquest ENG ScholarWorks@UMass Amherst Applied Mathematics|Mathematics
collection NDLTD
language ENG
sources NDLTD
topic Applied Mathematics|Mathematics
spellingShingle Applied Mathematics|Mathematics
Hwang, Sungha
Spatial evolutionary game theory: Deterministic approximations, decompositions, and hierarchical multi-scale models
description Evolutionary game theory has recently emerged as a key paradigm in various behavioral science disciplines. In particular it provides powerful tools and a conceptual framework for the analysis of the time evolution of strategic interdependence among players and its consequences, especially when the players are spatially distributed and linked in a complex social network. We develop various evolutionary game models, analyze these models using appropriate techniques, and study their applications to complex phenomena. In the second chapter, we derive integro-differential equations as deterministic approximations of the microscopic updating stochastic processes. These generalize the known mean-field ordinary differential equations and provide powerful tools to investigate the spatial effects on the time evolutions of the agents' strategy choices. The deterministic equations allow us to identify many interesting features of the evolution of strategy profiles in a population, such as standing and traveling waves, and pattern formation, especially in replicator-type evolutions. We introduce several methods of decomposition of two player normal form games in the third chapter. Viewing the set of all games as a vector space, we exhibit explicit orthonormal bases for the subspaces of potential games, zero-sum games, and their orthogonal complements which we call anti-potential games and anti-zero-sum games, respectively. Perhaps surprisingly, every anti-potential game comes either from Rock-paper-scissors type games (in the case of symmetric games) or from Matching Pennies type games (in the case of asymmetric games). Using these decompositions, we prove old (and some new) cycle criteria for potential and zero-sum games (as orthogonality relations between subspaces). We illustrate the usefulness of our decompositions by (a) analyzing the generalized Rock-Paper-Scissors game, (b) completely characterizing the set of all null-stable games, (c) providing a large class of strict stable games, (d) relating the game decomposition to the Hodge decomposition of vector fields for the replicator equations, (e) constructing Lyapunov functions for some replicator dynamics, (f) constructing Zeeman games—games with an interior asymptotically stable Nash equilibrium and a pure strategy ESS. The hierarchical modeling of evolutionary games provides flexibility in addressing the complex nature of social interactions as well as systematic frameworks in which one can keep track of the interplay of within-group dynamics and between-group competitions. For example, it can model husbands and wives' interactions, playing an asymmetric game with each other, while engaging coordination problems with the likes in other families. In the fourth chapter, we provide hierarchical stochastic models of evolutionary games and approximations of these processes, and study their applications.
author Hwang, Sungha
author_facet Hwang, Sungha
author_sort Hwang, Sungha
title Spatial evolutionary game theory: Deterministic approximations, decompositions, and hierarchical multi-scale models
title_short Spatial evolutionary game theory: Deterministic approximations, decompositions, and hierarchical multi-scale models
title_full Spatial evolutionary game theory: Deterministic approximations, decompositions, and hierarchical multi-scale models
title_fullStr Spatial evolutionary game theory: Deterministic approximations, decompositions, and hierarchical multi-scale models
title_full_unstemmed Spatial evolutionary game theory: Deterministic approximations, decompositions, and hierarchical multi-scale models
title_sort spatial evolutionary game theory: deterministic approximations, decompositions, and hierarchical multi-scale models
publisher ScholarWorks@UMass Amherst
publishDate 2011
url https://scholarworks.umass.edu/dissertations/AAI3482707
work_keys_str_mv AT hwangsungha spatialevolutionarygametheorydeterministicapproximationsdecompositionsandhierarchicalmultiscalemodels
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