本文關(guān)鍵詞： 網(wǎng)絡(luò)演化博弈 納什均衡 策略?xún)?yōu)化 協(xié)同演化 傳染病動(dòng)態(tài) 矩陣半張量積　出處：《山東大學(xué)》2016年博士論文　論文類(lèi)型：學(xué)位論文

【摘要】：近年來(lái)隨著復(fù)雜網(wǎng)絡(luò)的快速發(fā)展,網(wǎng)絡(luò)化博弈已經(jīng)成為博弈論研究中的一個(gè)熱門(mén)課題.與傳統(tǒng)博弈相比,網(wǎng)絡(luò)化博弈用網(wǎng)絡(luò)拓?fù)渲械捻旤c(diǎn)表示玩家,用邊表示相鄰?fù)婕议g存在博弈關(guān)系,每個(gè)玩家僅與周?chē)泥従舆M(jìn)行博弈并獲得累積收益,這與社會(huì)實(shí)際更加相符.目前在生物學(xué)、經(jīng)濟(jì)學(xué)、國(guó)際關(guān)系、計(jì)算機(jī)科學(xué)、軍事戰(zhàn)略和其他諸多領(lǐng)域都得到了廣泛的應(yīng)用,有著極其重要的理論價(jià)值和研究意義.在有限網(wǎng)絡(luò)演化博弈中,每個(gè)玩家有有限個(gè)策略可以選擇,其動(dòng)態(tài)演化過(guò)程與邏輯網(wǎng)絡(luò)動(dòng)態(tài)有著天然的聯(lián)系.而矩陣半張量積作為處理有限集上動(dòng)態(tài)的有力工具,也可以用來(lái)研究有限網(wǎng)絡(luò)演化博弈問(wèn)題.利用該方法可以將博弈動(dòng)態(tài)轉(zhuǎn)化為代數(shù)形式,為其建立嚴(yán)格的數(shù)學(xué)框架,這有利于人們對(duì)實(shí)際生活中的博弈現(xiàn)象產(chǎn)生更精確更深入的認(rèn)識(shí),進(jìn)而指導(dǎo)人們對(duì)博弈進(jìn)行預(yù)測(cè)和控制.鑒于網(wǎng)絡(luò)演化博弈問(wèn)題的多樣性,半張量積理論在網(wǎng)絡(luò)演化博弈領(lǐng)域還有廣闊的應(yīng)用空間,值得進(jìn)一步研究.本文利用矩陣半張量積方法研究博弈納什均衡點(diǎn)的存在性,固定拓?fù)浜蜁r(shí)變拓?fù)湎戮W(wǎng)絡(luò)演化博弈的建模、分析及控制問(wèn)題,并將所得結(jié)果應(yīng)用于時(shí)尚策略決定、傳染病傳播和免疫控制中.主要研究?jī)?nèi)容如下：1.研究了靜態(tài)博弈純策略納什均衡點(diǎn)的存在性問(wèn)題.通過(guò)構(gòu)建結(jié)構(gòu)矩陣,將玩家的收益函數(shù)轉(zhuǎn)化為代數(shù)形式.利用偽布爾函數(shù)的導(dǎo)數(shù)的代數(shù)形式,分別給出了完全信息和非完全信息雙選擇靜態(tài)博弈納什均衡點(diǎn)存在的充要條件.利用玩家收益函數(shù)的結(jié)構(gòu)矩陣,建立了多玩家多選擇靜態(tài)博弈納什均衡點(diǎn)的求解算法.并且針對(duì)雙選擇和多選擇靜態(tài)博弈給出了納什均衡點(diǎn)存在的一個(gè)統(tǒng)一的充要條件.將所得的尋找納什均衡點(diǎn)的結(jié)果應(yīng)用到了時(shí)尚博弈納什均衡策略選擇問(wèn)題中,并研究了時(shí)尚博弈中社會(huì)福利最優(yōu)化問(wèn)題和規(guī)范化的滿(mǎn)意度總和最優(yōu)化問(wèn)題.2.研究了一類(lèi)固定拓?fù)湎戮W(wǎng)絡(luò)演化博弈的代數(shù)描述和策略?xún)?yōu)化問(wèn)題.為基于短視最優(yōu)響應(yīng)策略升級(jí)規(guī)則的網(wǎng)絡(luò)演化博弈動(dòng)態(tài)建立了構(gòu)造其代數(shù)形式的算法,并分析博弈的演化結(jié)果.利用偽玩家的控制作用研究博弈的策略?xún)?yōu)化選擇問(wèn)題,設(shè)計(jì)控制策略使得玩家的長(zhǎng)期平均收益最大化.3.研究了固定拓?fù)湎戮W(wǎng)絡(luò)演化博弈中玩家策略組合的穩(wěn)定度問(wèn)題.基于最優(yōu)模仿策略升級(jí)規(guī)則,為博弈演化動(dòng)態(tài)建立了代數(shù)方程.根據(jù)提出的策略組合穩(wěn)定度的概念,建立了策略組合為k度穩(wěn)定的充要條件,并給出了受擾策略組合還原所需要的暫態(tài)時(shí)間的計(jì)算方法.建立了一類(lèi)事件觸發(fā)控制的設(shè)計(jì)方法,以使得給定策略組合達(dá)到期望的穩(wěn)定度.4.為時(shí)變拓?fù)湎碌木W(wǎng)絡(luò)演化博弈動(dòng)態(tài)建立代數(shù)描述,并基于此分析博弈的演化規(guī)律.將所得的結(jié)果用來(lái)研究動(dòng)態(tài)網(wǎng)絡(luò)上的傳染病傳播問(wèn)題.根據(jù)一類(lèi)確定性的共演化規(guī)則,分別建立了個(gè)體狀態(tài)和網(wǎng)絡(luò)拓?fù)鋭?dòng)態(tài)的矩陣表示.對(duì)于所有的初始個(gè)體狀態(tài)和網(wǎng)絡(luò)拓?fù)浣Y(jié)構(gòu),分析最終可能的傳播平衡點(diǎn).研究了控制疾病傳播的疫苗控制問(wèn)題,給出了動(dòng)態(tài)網(wǎng)絡(luò)上的傳染病傳播能夠在狀態(tài)反饋疫苗控制作用下使得所有個(gè)體恢復(fù)易感狀態(tài)的充分必要條件.
[Abstract]:In recent years, with the rapid development of complex network, network game has become a hot topic in game theory research. Compared with the traditional game, the network game network topology of vertices with edges indicate the presence of game player, game relationship between each adjacent game player, game player only and the neighbors for the game and get accumulated earnings this, more consistent with social reality. In biology, economics, computer science, international relations, military strategy and many other fields have been widely used, has extremely important theoretical value and significance of the research. In the limited network evolution game, each game player has a finite strategy choice, its dynamic evolution the process and logic network has a natural link. And the semi tensor product of matrices as a powerful tool to deal with a finite set of dynamic, can also be used to study the limited network evolution game The problem. Using this method can be transformed into a dynamic game algebra, a rigorous mathematical framework for it, which is conducive to the people of the game phenomenon in real life to produce more accurate and in-depth understanding, and guide people to predict and control the game. In view of the network game play diversity problem, semi tensor product the evolution of space field and the wide application of game theory in the network, it is worthy of further study. The existence of game matrix semi tensor product of Nash equilibrium, variable modeling evolutionary game under fixed topology and network topology, analysis and control problems, and the results are applied to a strategic decision, the spread of infectious diseases and immune control. The main research contents are as follows: 1. the existence problem of static game of pure strategy Nash equilibrium point. By constructing the structure matrix, the game player gain function Into algebraic form. Using algebraic form derivative pseudo Boolean functions, are given complete information and incomplete information to choose two necessary and sufficient conditions for the existence of static game Nash equilibrium point. The game player payoff function structure matrix is established, many game player selection algorithm to solve the static game Nash equilibrium. And based on the double selection and how to choose the static game to give a unified necessary and sufficient conditions for the existence of Nash equilibrium. The search for the Nash equilibrium results of the game are Nash equilibrium strategy application problems, and study the Shangbo when social welfare optimization problems and the satisfaction of the sum of the game code optimization problem is studied for a class of.2. the evolutionary game under fixed topology network algebraic description and policy optimization problem. As the game dynamic evolution myopic best response strategy to build rules based network upgrade The structure of the algebraic form of the algorithm, and analyze the evolution results of the game. The optimization selection strategy based on the pseudo control game game player, control strategy is designed to maximize the long-term average.3. game player of game playing strategy combination of stability problem of fixed topology network evolution. The optimal strategies of imitation upgrade based on the rules for the game evolution established algebraic equations. According to the proposed strategy combination stability concept, establish a strategic combination of necessary and sufficient conditions for K stability, and gives the calculation method of transient disturbance by the time required for the reduction strategy. A design method for a kind of event triggered control, so that a given strategy combination to achieve the desired stability.4. time-varying topology under dynamic network evolution game algebraic description, and based on the analysis of the evolution of the game. The results for the spread of infectious diseases research on dynamic network. According to the evolution rules of a class of deterministic, respectively established matrix individual condition and dynamic network topology. For the initial state of all individual and network topology, communication equilibrium analysis may eventually. The vaccine control is studied to control the spread of the disease and given the spread of infectious diseases on the network dynamics in state feedback control under the action of vaccine which every individual to restore the necessary and sufficient conditions for vulnerable state.

【學(xué)位授予單位】：山東大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O157.5;O183.2

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