基于异质认知种群的演化博弈模型与均衡研究
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摘要
经典博弈论作为一种策略性决策分析方法，在许多学科都获得过成功的应用，但由于其过于严格的基础假设——“完全理性假设”，致使理论解释力受到了极大限制。近年来发展起来的演化博弈虽然将有限理性引入到博弈分析过程中，但这种分析都是在同质行为模式假设下进行的，即假设所有参与人的行为模式和认知能力都相同。但是，认知理论最新研究成果表明：对于不同的有限理性参与人，其在博弈过程中的行为模式或认知能力可能不同。因此，考虑参与人在博弈过程中的差异性，将参与人的异质认知能力纳入到演化博弈模型中，研究基于异质认知种群的演化博弈模型与均衡，对于完善现有博弈理论乃至经典经济理论、突破博弈理论研究脱离实际的瓶颈，都具有重要的理论意义和实际价值。
论文将传统演化博弈模型中参与人的有限理性假设从“行为模式”扩展到“认知能力”方面，构建异质认知种群的虚拟博弈和Logit模型，分析异质认知种群虚拟博弈、Logit模型及其反应动态的均衡，重点探讨参与人认知能力异质性对博弈均衡和收敛速度的影响。
论文将传统虚拟博弈中参与人的认知层次进行扩展，构建异质认知种群随机匹配的虚拟博弈模型，分析异质认知种群虚拟博弈的均衡，以及收敛时间与博弈结构、初始信念和异质认知种群分布的关系，重点分析认知能力的异质性对虚拟博弈均衡结果的影响。所得结论表明：在异质认知种群虚拟博弈中，两人对称博弈如果只存在一个Nash均衡则高阶参与人的虚拟博弈均衡收敛到该Nash均衡策略上去；当存在多个Nash均衡时，无论是协调博弈还是斗鸡博弈，高阶参与人的虚拟博弈均衡最终会收敛到某个Nash均衡策略上，但具体收敛到哪个均衡取决于高阶参与人对低阶参与人选择的初始信念。收敛速度除了和各认知层次参与人的初始信念有关外，还和博弈结构相关。
论文将认知层次选择内生化，构建异质认知种群虚拟博弈的认知层次自主选择模型。在0~2阶的特定异质认知种群中，比较任意时刻各阶参与人主观期望支付的关系，探讨具有不同认知禀赋的自利参与人在不同博弈结构下如何选择推理方式，并以社会总支付衡量社会福利，考察不同博弈结构和初始信念下高阶参与人认知层次的策略选择行为对社会福利的影响。所得结论表明：在协调博弈结构下，不存在认知层次的策略性选择行为，各阶参与人都主观认为自己的推理方式最优；而在斗鸡博弈结构下，2阶参与人推理方式的策略性选择行为只可能发生在极端情况下。
论文研究非对称博弈结构Logit随机反应动态过程的解特征，给出不同势能博弈结构下动态过程达到唯一均衡（快速选择均衡或非快速选择均衡）的条件。针对对称支付的典型博弈结构，探讨Logit反应动态收敛速度，给出收敛时间的上、下界；针对非对称博弈结构，利用博弈仿真对其收敛速度进行分析。
论文还将传统Logit模型的参与人认知层次进行扩展，构建了异质认知种群随机匹配Logit模型，在对称博弈结构下，按照Nash均衡结构和Logit反应函数性状划分博弈结构类型，用数值分析方法比较异质认知种群Logit模型与原Logit模型均衡结果的差异，及认知差异对Logit模型均衡的影响。所得结论表明：在有限的认知层次下，当整个种群具有一致初始信念时，各阶参与人均衡选择不仅仅取决于博弈结构的差异，更大程度上依赖于人们的认知水平和初始信念。
最后，论文讨论异质认知种群的Logit随机反应动态过程，对有限认知能力下的Logit反应动态进行一般性描述。在对称博弈结构情形下分析异质认知种群Logit反应动态的均衡存在性和唯一性条件，并给出异质认知种群Logit反应动态收敛时间的上、下界。
论文对全文进行了总结并提出了有待进一步研究的问题。
As an analytical method of strategic decision-making, classical game theory is widelyused in many fields. For the perfectly rational hypothesis is overly strict, the explanationof this theory is great limited. Evolutionary game theory which is developed in recentyears introduces the bounded rationality into game analysis. However, most studies are onthe basis of homogenous behavioral pattern, which means that the behavioral pattern andcognitive ability for every one in the population are identical. The latest research results ofcognitive theory show that the behavioral pattern or cognitive ability of player ininteractive process could be different from the ones of opponent. Therefore, it is necessaryto take into account the behavioral agents’ heterogeneity and introduce heterogeneouscognitive ability into evolutionary game theory to construct evolutionary game model andanalyze its equilbrium in heterogenous cognitive population, which has great significancein theory and application to improve game theory and even economic theory, breaches thecurrent bottleneck of game theory which is deviated from reality.
This dissertation extends the hypothesis of bounded rationality in the classical gametheory from “behavioral pattern” to “cognitive ability”, which is based on Simon’s “Intendto be rational, but can only reach limited range”. It constructs models of fictitious play andLogit quantal response dynamic in heterogeneous cognitive population and analyzes theirequilibrium respectively. It mainly explores how the heterogeneity of players’ cognitiveability influences game equilibrium and its stability.
The cognitive hierarchy of players in traditional fictitious play is extended in thedissertation, and a new fictitious play model is constructed, in which players are randomlyselected and matched within the mixed cognitive hierarchy population. It analyzes theequilibrium of fictitious play in mixed cognitive hierarchy population and the relationsbetween convergence time and game structure, initial beliefs and distributions of cognitivehierarchies. It focuses on the impact that heterogeneous cognitive ability has on theequilibrium of fictitious play. Results obtained shows: for fictitious play withinheterogeneous cognitive population, belief of high level players about others will convergeto the Nash equilibrium if there is only one of it in2-person symmetric game; when thereare several Nash equilibriums, belief of high level players about others will converge to Nash equilibrium both in coordinative and chicken games and it depends on initial beliefsof players’ choice. The convergence rates do not only relate to the initial beliefs of players’choice of players with different cognitive hierarchy, but also to the game structure.
It constructs a general model in which the cognitive hierarchy of players isendogenous and they can strategically choose their cognitive hierarchy, and each player inthe mixed cognitive hierarchy population has a strategic behavior to choose his cognitivehierarchy in the fictitious play. In population with heterogenous cognitive hierarchy fromonly level-0~2the relationship among the subjective expected payoffs of players at eachlevel at any moment is discussed, and how the self-interested players owning differentcognitive gift choose the inference way in different game structures is explored. Then itregards the total social payoff as social welfare function to measure social welfare, andstudies how High level players’ strategic behavior influences social welfare under differentgame structures and initial beliefs. Results obtained shows: there is no strategic choice ofcognitive hierarchy in coordinative game. Every player takes his own reasoning approachas the best; however, the strategic choice of cognitive hierarchy of2-level player couldappear only in some extreme cases.
The dissertation explores the characteristic of equilibra in Logit quantal responsedynamic process with asymmetric games. It proofs the conditions on which dynamicprocess has the unique equilibrium (fast selection equilibrium or non-fast selectionequilibrium) in different potential functions. For the game structure with symmetric payoff,the convergence rate of Logit response dynamic and relative upper and lower bounds of itshitting time are explored. And for the game structure with asymmetric payoff, it givessimulations to discuss the convergence of the process.
It improves the traditional Logit model by random matching within mixed cognitivehierarchy population. It classifies symmetric games according to the structure of Nashequilibirum and the character of Logit response function. Then it applies numerical analysismethod to identify the differences between the outcome of Logit model and original Logitmodel under bounded cognitive ability, and explores how cognitive diversity influencesthe Logit equilibrium. Results obtained shows: under bounded cognitive hierarchy, whenplayers in population have the same initial beliefs, the equilibrium choices of players withdifferent cognitive hierarchy depend not only on the game structure, but more on players’cognitive ability and initial beliefs.
Finally, the dissertation discusses the Logit quantal response dynamic process in themixed cognitive hierarchy population, and provides the general description of Logitresponse dynamic with bounded cognitive ability. The equilibrium existence anduniqueness on Logit response dynamic in the mixed cognitive hierarchy population areanalyzed in symmetric games. What’s more, upper and down bound of the hitting time aregiven.
The summary and problems further studied have been put forward at the end.
论文将传统演化博弈模型中参与人的有限理性假设从“行为模式”扩展到“认知能力”方面，构建异质认知种群的虚拟博弈和Logit模型，分析异质认知种群虚拟博弈、Logit模型及其反应动态的均衡，重点探讨参与人认知能力异质性对博弈均衡和收敛速度的影响。
论文将传统虚拟博弈中参与人的认知层次进行扩展，构建异质认知种群随机匹配的虚拟博弈模型，分析异质认知种群虚拟博弈的均衡，以及收敛时间与博弈结构、初始信念和异质认知种群分布的关系，重点分析认知能力的异质性对虚拟博弈均衡结果的影响。所得结论表明：在异质认知种群虚拟博弈中，两人对称博弈如果只存在一个Nash均衡则高阶参与人的虚拟博弈均衡收敛到该Nash均衡策略上去；当存在多个Nash均衡时，无论是协调博弈还是斗鸡博弈，高阶参与人的虚拟博弈均衡最终会收敛到某个Nash均衡策略上，但具体收敛到哪个均衡取决于高阶参与人对低阶参与人选择的初始信念。收敛速度除了和各认知层次参与人的初始信念有关外，还和博弈结构相关。
论文将认知层次选择内生化，构建异质认知种群虚拟博弈的认知层次自主选择模型。在0~2阶的特定异质认知种群中，比较任意时刻各阶参与人主观期望支付的关系，探讨具有不同认知禀赋的自利参与人在不同博弈结构下如何选择推理方式，并以社会总支付衡量社会福利，考察不同博弈结构和初始信念下高阶参与人认知层次的策略选择行为对社会福利的影响。所得结论表明：在协调博弈结构下，不存在认知层次的策略性选择行为，各阶参与人都主观认为自己的推理方式最优；而在斗鸡博弈结构下，2阶参与人推理方式的策略性选择行为只可能发生在极端情况下。
论文研究非对称博弈结构Logit随机反应动态过程的解特征，给出不同势能博弈结构下动态过程达到唯一均衡（快速选择均衡或非快速选择均衡）的条件。针对对称支付的典型博弈结构，探讨Logit反应动态收敛速度，给出收敛时间的上、下界；针对非对称博弈结构，利用博弈仿真对其收敛速度进行分析。
论文还将传统Logit模型的参与人认知层次进行扩展，构建了异质认知种群随机匹配Logit模型，在对称博弈结构下，按照Nash均衡结构和Logit反应函数性状划分博弈结构类型，用数值分析方法比较异质认知种群Logit模型与原Logit模型均衡结果的差异，及认知差异对Logit模型均衡的影响。所得结论表明：在有限的认知层次下，当整个种群具有一致初始信念时，各阶参与人均衡选择不仅仅取决于博弈结构的差异，更大程度上依赖于人们的认知水平和初始信念。
最后，论文讨论异质认知种群的Logit随机反应动态过程，对有限认知能力下的Logit反应动态进行一般性描述。在对称博弈结构情形下分析异质认知种群Logit反应动态的均衡存在性和唯一性条件，并给出异质认知种群Logit反应动态收敛时间的上、下界。
论文对全文进行了总结并提出了有待进一步研究的问题。
As an analytical method of strategic decision-making, classical game theory is widelyused in many fields. For the perfectly rational hypothesis is overly strict, the explanationof this theory is great limited. Evolutionary game theory which is developed in recentyears introduces the bounded rationality into game analysis. However, most studies are onthe basis of homogenous behavioral pattern, which means that the behavioral pattern andcognitive ability for every one in the population are identical. The latest research results ofcognitive theory show that the behavioral pattern or cognitive ability of player ininteractive process could be different from the ones of opponent. Therefore, it is necessaryto take into account the behavioral agents’ heterogeneity and introduce heterogeneouscognitive ability into evolutionary game theory to construct evolutionary game model andanalyze its equilbrium in heterogenous cognitive population, which has great significancein theory and application to improve game theory and even economic theory, breaches thecurrent bottleneck of game theory which is deviated from reality.
This dissertation extends the hypothesis of bounded rationality in the classical gametheory from “behavioral pattern” to “cognitive ability”, which is based on Simon’s “Intendto be rational, but can only reach limited range”. It constructs models of fictitious play andLogit quantal response dynamic in heterogeneous cognitive population and analyzes theirequilibrium respectively. It mainly explores how the heterogeneity of players’ cognitiveability influences game equilibrium and its stability.
The cognitive hierarchy of players in traditional fictitious play is extended in thedissertation, and a new fictitious play model is constructed, in which players are randomlyselected and matched within the mixed cognitive hierarchy population. It analyzes theequilibrium of fictitious play in mixed cognitive hierarchy population and the relationsbetween convergence time and game structure, initial beliefs and distributions of cognitivehierarchies. It focuses on the impact that heterogeneous cognitive ability has on theequilibrium of fictitious play. Results obtained shows: for fictitious play withinheterogeneous cognitive population, belief of high level players about others will convergeto the Nash equilibrium if there is only one of it in2-person symmetric game; when thereare several Nash equilibriums, belief of high level players about others will converge to Nash equilibrium both in coordinative and chicken games and it depends on initial beliefsof players’ choice. The convergence rates do not only relate to the initial beliefs of players’choice of players with different cognitive hierarchy, but also to the game structure.
It constructs a general model in which the cognitive hierarchy of players isendogenous and they can strategically choose their cognitive hierarchy, and each player inthe mixed cognitive hierarchy population has a strategic behavior to choose his cognitivehierarchy in the fictitious play. In population with heterogenous cognitive hierarchy fromonly level-0~2the relationship among the subjective expected payoffs of players at eachlevel at any moment is discussed, and how the self-interested players owning differentcognitive gift choose the inference way in different game structures is explored. Then itregards the total social payoff as social welfare function to measure social welfare, andstudies how High level players’ strategic behavior influences social welfare under differentgame structures and initial beliefs. Results obtained shows: there is no strategic choice ofcognitive hierarchy in coordinative game. Every player takes his own reasoning approachas the best; however, the strategic choice of cognitive hierarchy of2-level player couldappear only in some extreme cases.
The dissertation explores the characteristic of equilibra in Logit quantal responsedynamic process with asymmetric games. It proofs the conditions on which dynamicprocess has the unique equilibrium (fast selection equilibrium or non-fast selectionequilibrium) in different potential functions. For the game structure with symmetric payoff,the convergence rate of Logit response dynamic and relative upper and lower bounds of itshitting time are explored. And for the game structure with asymmetric payoff, it givessimulations to discuss the convergence of the process.
It improves the traditional Logit model by random matching within mixed cognitivehierarchy population. It classifies symmetric games according to the structure of Nashequilibirum and the character of Logit response function. Then it applies numerical analysismethod to identify the differences between the outcome of Logit model and original Logitmodel under bounded cognitive ability, and explores how cognitive diversity influencesthe Logit equilibrium. Results obtained shows: under bounded cognitive hierarchy, whenplayers in population have the same initial beliefs, the equilibrium choices of players withdifferent cognitive hierarchy depend not only on the game structure, but more on players’cognitive ability and initial beliefs.
Finally, the dissertation discusses the Logit quantal response dynamic process in themixed cognitive hierarchy population, and provides the general description of Logitresponse dynamic with bounded cognitive ability. The equilibrium existence anduniqueness on Logit response dynamic in the mixed cognitive hierarchy population areanalyzed in symmetric games. What’s more, upper and down bound of the hitting time aregiven.
The summary and problems further studied have been put forward at the end.
引文
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