本文選題：集值映射不動(dòng)點(diǎn)　切入點(diǎn)：Nash平衡　出處：《貴州大學(xué)》2016年碩士論文

【摘要】：Brouwer不動(dòng)點(diǎn)定理和Kakutani不動(dòng)點(diǎn)定理被成功的運(yùn)用于博弈論、數(shù)理經(jīng)濟(jì)等領(lǐng)域,尤其是Nash用Brouwer不動(dòng)點(diǎn)定理建立了非合作博弈理論,Arrow和Debreu用Kakutani不動(dòng)點(diǎn)定理證明了一般均衡的存在性后,不動(dòng)點(diǎn)理論引起了更加廣泛的關(guān)注.博弈論是運(yùn)籌學(xué)的一個(gè)分支,非合作博弈在博弈論中處于基礎(chǔ)和核心的地位,Nash平衡是非合作博弈理論中最主要、最核心的概念.Nash平衡與非線性問題的解有著某種等價(jià)關(guān)系,例如與集值映射不動(dòng)點(diǎn)、變分不等式解的等價(jià)性.Nash平衡的存在性與穩(wěn)定性可以轉(zhuǎn)化為相關(guān)非線性問題解的存在性與穩(wěn)定性.本文正是從這些等價(jià)性出發(fā),通過集值映射不動(dòng)點(diǎn)、Ky-Fan引理及變分不等式的解的穩(wěn)定性來研究Nash平衡的穩(wěn)定性.主要是通過改進(jìn)集值映射和函數(shù)的擾動(dòng),即引入幾種更強(qiáng)的擾動(dòng)來研究這些問題解的穩(wěn)定性.本文研究了強(qiáng)?-擾動(dòng)意義下集值映射不動(dòng)點(diǎn)的強(qiáng)本質(zhì)性,并由此給出了Nash平衡的強(qiáng)穩(wěn)定集,研究了Ky-Fan截口問題、Ky-Fan不等式的解集在強(qiáng)?-擾動(dòng)下本質(zhì)連通區(qū)的存在性,研究了非合作向量博弈Pareto-Nash平衡的穩(wěn)定性.第一章:基礎(chǔ)知識(shí).主要有:集值映射的上、下半連續(xù)性、Fan-Glicksberg不動(dòng)點(diǎn)定理、Fort引理、稠密剩余集、本質(zhì)不動(dòng)點(diǎn)、不動(dòng)點(diǎn)的本質(zhì)集、不動(dòng)點(diǎn)的本質(zhì)連通區(qū)、Hausdorff距離等.第二章:研究了強(qiáng)?-擾動(dòng)下不動(dòng)點(diǎn)的穩(wěn)定性和Nash平衡策略集的穩(wěn)定性.強(qiáng)?-擾動(dòng)下得到的不動(dòng)點(diǎn)穩(wěn)定性最強(qiáng),對(duì)應(yīng)Nash平衡策略的穩(wěn)定集——SBR-穩(wěn)定集,具有關(guān)于映射擾動(dòng)、映射和定義域擾動(dòng)的穩(wěn)定性.SBR-穩(wěn)定集對(duì)應(yīng)的強(qiáng)?-擾動(dòng)是用Hausdorff距離來定義的,沒有附加其他任何距離上的條件,所以這種方法可以用來研究任何非合作有限博弈.第三章:運(yùn)用強(qiáng)?-擾動(dòng)的方法,本節(jié)給出了Ky-Fan截口問題解的更強(qiáng)的連續(xù)性和穩(wěn)定性,給出了在強(qiáng)?-擾動(dòng)擾動(dòng)下的本質(zhì)連通區(qū)和在圖像?-擾動(dòng)下的本質(zhì)連通區(qū).并由Ky-Fan截口問題解的強(qiáng)本質(zhì)性,得到了Ky-Fan不等式解集的強(qiáng)本質(zhì)連通區(qū).再由Ky-Fan點(diǎn)集的強(qiáng)本質(zhì)連通區(qū),得到了Nash平衡的強(qiáng)本質(zhì)連通區(qū).第3.5節(jié)針對(duì)Ky-Fan點(diǎn)和Nash平衡點(diǎn)的穩(wěn)定性,考慮不等式函數(shù)或者支付函數(shù)有可能滿足Ky-Fan點(diǎn)或Nash平衡點(diǎn)條件的點(diǎn)處所產(chǎn)生的擾動(dòng),進(jìn)一步拓展Ky-Fan點(diǎn)集和Nash平衡點(diǎn)集穩(wěn)定性的擾動(dòng)范圍.第四章:從優(yōu)化的角度,研究了非合作向量博弈Pareto-Nash平衡的穩(wěn)定性.向量博弈的理想-Nash平衡點(diǎn)、滿意-Nash平衡點(diǎn)和分層次Pareto-Nash平衡點(diǎn),都是Pareto有效點(diǎn).它們都具有Pareto-Nash平衡點(diǎn)的連續(xù)性(上半連續(xù)性需要條件)、通有穩(wěn)定性和本質(zhì)連通區(qū)性質(zhì).
[Abstract]:Brouwer fixed point theorem and Kakutani fixed point theorem have been successfully applied in the fields of game theory, mathematical economics and so on, especially after Nash established the non-cooperative game theory by Brouwer fixed point theorem and Debreu proved the existence of general equilibrium by Kakutani fixed point theorem. Game theory is a branch of operational research, and non-cooperative game plays an important role in non-cooperative game theory. The most important concept, the Nash equilibrium, has some equivalent relationship with the solution of nonlinear problems, such as fixed points with set-valued mappings, Equivalence of Solutions of variational inequalities. The existence and stability of Nash equilibrium can be transformed into the existence and stability of solutions for related nonlinear problems. The stability of Nash equilibrium is studied by means of Ky-Fan Lemma of set-valued mapping and stability of solutions of variational inequalities, mainly by improving set-valued mappings and perturbation of functions. In this paper, we study the stability of the solutions of these problems by introducing several stronger perturbations. The strong property of fixed point of set-valued mapping in perturbed sense is given, and the strong stable set of Nash equilibrium is obtained. The solution set of Ky-Fan inequality for Ky-Fan cut problem is studied. -existence of essential connected region under perturbation, the stability of Pareto-Nash equilibrium in non-cooperative vector game is studied. Chapter 1: basic knowledge. The main contents are as follows: upper, lower semi-continuous Fan-Glicksberg fixed point theorem Fort Lemma, dense residual set. Essential fixed points, essential set of fixed points, essential connected regions of fixed points, Hausdorff distance, etc. -the stability of fixed points under perturbation and the stability of the set of Nash equilibrium strategies. The stable set corresponding to the Nash equilibrium strategy is the stable set of SBR-stable set, which has the strong stability of the map perturbation, mapping and domain perturbation. SBR-stable set corresponding to the Nash equilibrium strategy, and the stable set corresponding to the Nash equilibrium strategy, are strongly correlated with the stability of the map perturbation, the mapping and the perturbation of the defined domain. -perturbation is defined by Hausdorff distance without any other conditions on the distance, so this method can be used to study any non-cooperative finite game. In this section, we give the stronger continuity and stability of the solution of the Ky-Fan section problem, and give the stronger continuity and stability of the solution. -intrinsically connected regions and images under perturbed perturbations? By using the strong nature of the solution of the Ky-Fan cut problem, the strongly essential connected region of the solution set of the Ky-Fan inequality is obtained, and then the strongly essential connected region of the Ky-Fan point set is obtained. The strongly essential connected region of Nash equilibrium is obtained. For the stability of Ky-Fan point and Nash equilibrium point, the perturbation at the point where the inequality function or the payment function may satisfy the Ky-Fan point or Nash equilibrium condition is considered. Chapter 4: from the point of view of optimization, we study the stability of Pareto-Nash equilibrium in non-cooperative vector game. Both satisfied -Nash equilibrium points and hierarchical Pareto-Nash equilibrium points are Pareto efficient points. They all have the continuity of Pareto-Nash equilibrium points (the upper semicontinence needs the condition), and they all have the properties of stability and essential connected region.
【學(xué)位授予單位】：貴州大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2016
【分類號(hào)】：O225;O177.91

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