The Prisoners' Dilemma

TNC · 发表于 2005-7-17 17:51

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Some of you have known about this classical example in mathematics nowdays, some may have not. But you probably did watch the movie "beautiful mind" depicted a trendary life of a well-known mathematician, Nash. The following words were extracted from mitbbs.com. 《美丽心灵》里略带有经济学意味的地方是在酒吧“抠女”（泡MM）的一段戏：一个金发 美人和她的几个女性朋友东张西望，几个朋友在调侃纳什：要亲近那个金发美人必须通过 她的朋友（you can lead the blonde water but you can't make a drink）。而我们都 想上那个美人，经济学始祖亚当斯密说，在竞争的情况下，每个人在私人动机的驱动下做 到自己最好（发挥自己的最高水平），就一定会造成最好的局面（社会效果）。用中国人 都熟悉的话说，主观为自己客观为大家比主观为大家做得还更多。（in competition .individuals and vision serve the common good：every man for himself） 金发美人的流光溢彩使纳什猛然开窍，纳什发表了自己的高论，亚当斯密错了，因为 他需要一只鸽子（传递信息，adam smith needs for pigeon）。我们都想去泡金发妹妹 ，但是我们相持不下，没有单独一个人可以得到她。于是为了增加自己的情感砝码，我们 必须转过头去与她带来女性朋友沟通（让她们说好话？我不清楚电影台词的意思），而她 们如果知道我们的意图会羞愤兼嗤之以鼻（they will all give us a cold-shoulder） ，我们会再度失落（cause no body like to be a second choice，no one goes to the blonde）。那么怎么办？我们和这些二流女人货色接触时一定要隐瞒自己的意图，装做很 有诚意，就是相中她们的。（我突然想到，我和我的好朋友小学时候喜欢一个女孩，但我 和他都急不可耐地找女孩的同桌女孩讲话，以致于这个相貌平平同桌女孩自信指数陡然上 升，以为我和我的朋友会为她打上一架。）所以，亚当斯密错了，不是为自己做到最好也 就得到最多，而是在他人的存在下，努力使自己得到最多。（adam smith said the best result comes from everyone in the group doing what is best for himself，incompleted，because the best result would come …… every one in the group doing what is best for himself and the group） 老实说，《美丽心灵》里的“抠女”博弈并不是一个非常好的设计。因为它反映出的 纳什均衡的思想很稀薄。而纳什均衡指的是这样一种结果：大家都不要乱动，谁动谁倒霉 （不动的人可能得益）。这种局面对每个人来说都是最优的，因为这是在他人的战略影响 下做的最好的了。 很多人说纳什的博弈论最鲜明的两点，一个是进入非零合博弈领域。当然把纳什的理 论推理到零合博弈也是适用的，与冯诺伊曼的零合博弈论是等价的。冯诺伊曼所考察的是 系统间的相等问题（两个系统是等价的，但是内部会存在某种量的位移，这边多了那边就 少了，总的不变没有增良），而纳什考虑的是个体策略演进的问题。理解这些理论的精髓 其实一点也不难，中央电视台的著名解说员、博弈论大师韩乔生两句著名的解说词就深刻 地表达了它们的差异。解释零合博弈，他说：“我统计了一下意甲前八轮的进球和失球总 数，惊奇的发现一个巧合，那就是它们刚好一样多。”，在解释非零合博弈，他说，“今 天斯托克顿6 投9 中”。而此时，一边听讲的张五常同学用韩大师的名言粗犷地表达了他 对博弈论感觉相当不爽认为不可能有用的看法：“乔丹又习惯性的舔舔自己的舌头。”

TNC · 发表于 2005-7-17 17:51

Hooking up girls in the bar is definitely not the best example in describing game theory. The prisoners' dilemma is a better one. Two thieves are captured. Each has to choose whether or not to confess the other. If neither man confesses, then both will serve one year. If each confesses the other, both will go to prison for 10 years. However, if one thief confesses the other, and the other one does not confess, the one who has collaborated with the police will go free, while the other thief will go to prison for 20 years on the maximum charge. The strategies in this case are: confess or don't confess. The penalties are the sentences served. We can express all this compactly in a "payoff table" of a kind that has become pretty standard in game theory. Here is the payoff table for the Prisoners' Dilemma game: T2 confess don't confess (10, 10) (0,20) T1 don't (20,0) (1,1) T1 stands for theif A, who is row player here; T2 thief B, column player; The rational solution here: they both confess and go to prison for 10 years each. Of course this is the simpliest example on game theory, which was first researched by John Von Neuman (hehe, you should know this genius) and Oscar Morgenstern (a economist to my knowledge), and improved , perfected by Nash, who won NObel prize in economics for his efforts and contribution in game theory.

TNC · 发表于 2005-7-17 17:52

Game theory is now everywhere, mathematics, economics, military strategies... as long as to deal with conflicting situations, such as "worst case". I could name an application in my research of robust adaptive control. In this research, basically, optimization on a cost function is realized. Simply, we expect less control effort, less tracking error despite the disturbance input and the initial estimation error in the system. We could view the disturbance and error message as the maximizer and the desired performance the minimizer in dynamic game theory. Along the dynamic game-theoretical approach, we can formulate the original robust control problem into H-infinity optimal control problem. Of course, the above is just a simply rough depiction too short to understand. Anyway, it's an application of dynamical game theory in control theory.

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[分形与混沌] The Prisoners' Dilemma

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