Common Knowledge as a Coinductive Modality
Common Knowledge as a Coinductive Modality
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作为共归纳模态的常识
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发表时间：
2007
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通讯作者：
Venanzio Capretta
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作者：
Venanzio Capretta
在Coq Aumann定理中，我证明了：在完全信息博弈中，理性的常识蕴含着逆向归纳均衡。共同知识的概念是形式化的，使用共归纳的定义，作为一个模态包含了无限的信息量。1.在亚马逊人的土地上，有一个奇怪的禁忌，他们中没有一个人被允许知道她自己眼睛的颜色。每个人都能看到别人眼睛的颜色，但是没有镜子或其他反射表面来检查自己的外表。也严禁要求别人透露自己眼睛的颜色。具有讽刺意味的是，他们每个人都有绿色的眼睛。他们每个人都能看到她的同伴们都嫉妒，但她对自己感到怀疑。亚马逊人的另一个特点是他们都是精通逻辑的人：他们可以毫无错误地进行最复杂的推理。有一天，女神阿耳忒弥斯来到这个村庄，宣布：“我决定给你们每一个有绿色眼睛的人一个金弓。要申请奖品，你需要在任何一天的午夜前把前门漆成绿色。然后，第二天早上，我会来检查你的眼睛是否确实是绿色。如果是的话，你会得到价格。但要注意，对骗子是不会容忍的：如果你的眼睛变成了另一种颜色，我会用我的一支箭把你射倒。我敢肯定我没有白来，因为我可以看到你们中至少有一个人有绿色的眼睛。我的问题是：你能预测亚马逊人的行为吗？有没有人把她的门漆成绿色了，什么时候漆的？解决方案在文章的最后公布。[1]这是一个变种的欺骗妻子[5]（或丈夫[6]）的难题，也被称为泥泞的孩子难题[2]。关于类型理论、λ-演算和心智的思考--献给亨克·巴伦德雷格特的文章，纪念他60岁生日
I prove in Coq Aumann’s Theorem: In perfect information games, common knowledge of rationality implies backward induction equilibrium. The notion of common knowledge is formalized, using a coinductive definition, as a modality containing an infinite amount of information. 1. The Green-Eyed Amazons In the land of the Amazons a strange taboo was enforced.1 None of them was allowed to know the colour of her own eyes. Everybody could see the colour of the eyes of the others, but there were no mirrors or other reflective surfaces on which to check one’s own appearance. It was also strictly forbidden to ask somebody else to reveal the colour of one’s eyes. The ironic fact is that every one of them had green eyes. Each of them could see that all her companions were green-eyed, but she was in doubt about herself. Another peculiarity of the Amazons was that they were all proficient logicians: They could make the most complex reasoning without error. One day the goddess Artemis came to the village and announced: ‘I decided to give a golden bow to every one of you who has green eyes. To apply for the prize you will need to paint your front door green before midnight of any day. Then, the following morning, I will come to check whether your eyes are indeed green. If they are, you will receive the price. But be warned, there will be no tolerance for cheaters: If your eyes turn out to be of a different colour, I will strike you down with one of my arrows. I am quite sure that I didn’t come here in vain, because I can see that at least one of you has green eyes.’ My question to you is: Can you predict how the Amazons behaved. Did any of them paint her door green, and when did they do it? The solution is revealed at the end of the article. 1This is a variant of the cheating wives [5] (or husbands [6]) puzzle, also known as the muddy children puzzle [2]. REFLECTIONS ON TYPE THEORY, λ-CALCULUS, AND THE MIND Essays Dedicated to Henk Barendregt on the Occasion of his 60th Birthday Copyright c © 2007 by Venanzio Capretta