我們研究在複製方程式下償付矩陣為反對稱的族群演化，研究結果顯示存活的族數皆是奇數且符合Large-Dimensional Replicator Equations with Antisymmetric Random Interactions文中提到的，大致以組合數NCk的機率呈現。而存活的族群人口比正是存活族群的償付矩陣本徵值為零的本徵向量。在全部分作用下，存活族群數在約10%部分作用下，開始顯著與全作用不同，這意味著看似無直接關係的物種，卻彼此間接地影響。若調變償付矩陣的元素(絕對值非1的元素)，在特定值之下，能有偶數族存活的情形，但人口數目與初始值有關，非恆定不變。從奇數族到偶數族的相變，在靠近臨界點時，其鬆弛時間T∝|(γ- γ c)/γc|-1。

We study the evolutionary of species under the replicator equation with antisymmetric payoff . All of out results show odd number of species survived and its probability distritbution agrees with the combination NCk which is reported by “Large-Dimensional Replicator Equations with Antisymmetric Random Interactions” . Further , the population of survival species is identical with the eigenvector for eigenvalue zero of a reduced payoff for the survived species .Our results for the partly interaction cases show strong resemblance to the global one till the portion of interaction is roughly less than 10%. That means all species are sort of correlated even though most of them are not directly interacting. Some of results show even number of species survived on specific value since there are isolated points ,we conjecture the measure of even number of survived species in parameter space,i.e, we change continuously a parameter in the payoff table . The relaxation time near the even species is a power law with critical exponent ~1 ,|(γ- γ c)/γc|-1.

論文審定書----------------------------------------------------------------------------i
誌謝-------------------------------------------------------------------------------------ii
中文摘要------------------------------------------------------------------------------iii
Abstract------------------------------------------------------------------------------iv
第一章 緒論--------------------------------------------------------------------1
1-1 賽局理論---------------------------------------------------------------------1
1-2 複製方程式------------------------------------------------------------------2
1-3 分析賽局的方法-----------------------------------------------------------3
1-4 複製方程式下的零和賽局----------------------------------------------4
第二章 全域作用下眾族群的演化--------------------------------------6
2-1 全域作用下眾族群存活的情形---------------------------------------6
2-2 償付矩陣對結果的影響-------------------------------------------------8
2-3 演化結果的臨界點------------------------------------------------------12
第三章 非全域作用下眾族群的演化---------------------------------16
3-1 非全域作用下眾族群的存活與償付矩陣的關係---------------16
3-2 非全域作用下眾族群平衡時的存活與定量分析---------------17
第四章 結果與結論-------------------------------------------------------19
參考文獻---------------------------------------------------------------------------21

[1]Josef Hofbauer and Karl Sigmund, May 1998, Evolutionary Games and population dynamics, United Kingdom, Cambridge University Press
[2] T. Chawanya, K. Tokita, February 2002, Large-Dimensional Replicator Equations with Antisymmetric Random Interactions, Journal of the Physical Society of Japan, Volume 71, Issue 2, pp. 429
[3]Armen E. Allahverdyan and Chin-Kun Hu, “Replicators in a Fine-Grained Environment: Adaptation and Polymorphism”, Phys. Rev. Lett. 102, 058102 (2009)
[4] Yu. M. Svirezhev and V. P. Passekov, Fundamentals of Mathematical Evolutionary Genetics (Kluwer, Dordrecht, 1990).
[5] P.J. Kim, H. Jeong, 2005, Spatio-temporal dynamics in the origin of genetic information, physics D, 203:88-99
[6]羅智耀、馬美蘭、陳品璋。2009。複雜策略下協議之最佳化研究。第三屆管理與決策學術研討會特刊。17-40
[7] William Poundstone, 2012, 囚犯的兩難 :賽局理論與數學天才馮紐曼的故事, 再版, 台北, 左岸文化