40. Which Tire Was Flat?

Two guys were taking a Game-Theory class from a very boring Economics professor. They did pretty well on all of the quizzes and the midterms and assignments, and felt that they had a solid A's going into the final exam.

These two friends were so confident that they decided to go partying with some friends the weekend before the early Monday morning exam. So they did this and had a great time. However, with their hangovers and everything, they overslept all day Sunday and didn't make it 【1】 back to school until Monday afternoon.

They contacted the professor and explained to him why they missed the final. They told him that they had planned to come back in time 【2】 to take exam, but that they had a flat tire on the way back and didn't have a spare and couldn't get help for a long time and so were late getting back to campus. They asked to take the final the following morning. The professor thought this over and then agreed that they could make up 【3】 the final on the following day. The two guys were elated and relieved.

So, they studied that night and went in the next day at the time that the professor had told them. He placed them in separate rooms and handed each of them a test booklet and told them to begin. They looked at the first problem, which was something simple about Prisoner's Dilemma and was worth 5 points. "Cool!" they thought, "this is going to be easy." They did that problem and then turned the page. They were unprepared, however, for what they saw on the next page.

It said: (95 points) Which tire?

哪只轮胎爆胎？

两个小伙子跟随一位讲课非常单调的经济学教授学习博弈论。他们都很出色地完成了所有的小测试、中期考试以及课外作业，他们觉得在马上就要进行的期末考试中准能得优。

这两位朋友信心十足，于是决定在这个周末出去和朋友们聚会一番，而期末考试将在下周一一大早举行。他们周末出去聚了会，玩得很开心。但是由于喝醉了酒，就在外面住了一晚，并且星期天一整天都在睡觉，直到周一下午才赶回了学校。

他们联系上了教授，解释了错过期末考试的原因。他们告诉教授原本计划能够赶回来参加考试的，但是在回来的路上汽车爆胎了，又没有备用轮胎，而且很长时间得不到帮助，因此很晚才赶回校园。他们请求教授让他们再考一次。教授想了一会，然后同意让他们第二天上午补考。两个小伙子兴高采烈，如释重负。

于是他们挑灯夜战，加紧复习，第二天在教授说好的时间找到了他。教授把他们安排在不同的房间里，发给每人一套卷子，告诉他们开始作答。他们看到了第一个问题，那是一道关于囚徒困境的简单问题，只有5分。“太棒了！”他们暗想，“这次试卷不会很难。”他们做完了第一道题，然后翻到试卷的第二页。第二页上的试题内容出乎他们的意料之外。

上面写道：（本题95分）请问爆胎的是哪一只轮子？

背景知识与点评

·博弈论中最经典的案例就是“囚徒困境”，指的是两个共犯被抓，但是警察并没有抓住他们犯罪的证据，于是把他们分别关在不同的房间里进行审讯。审讯规则是，如果他们两人同时承认了犯罪，则犯罪事实清楚，他们被判有罪且被判入狱5年；如果两人都不坦白，则警察证据不足，只能对他们各判2年；如果一方坦白而另外一方不坦白，则坦白从宽，坦白者被判1年，而不坦白者被重判7年。当然，博弈的结果是两人都坦白，尽管这个结果对两人都不利。

·本笑话中的博弈论教授显然受到了“囚徒困境”的启发，于是也用类似“囚徒困境”的方法来考他的这两位博弈论高材生，非常有意思。

·按常规他们开的汽车有四只轮子。答案一致，他们在第二道题上就都会得到95分，得优一点问题没有。答案不一致，他们就每人只得5分，期末考试不及格。那么他们得优的概率是多少呢？只有四分之一。

·更深入地探讨一下，其实对于两个学过博弈论的人来说，两人得优的概率还可以提高，这就用到了博弈论中“聚点均衡”（focus equilibrium）的概念：假如汽车四只轮子中最容易爆胎的是前面左边的那只轮子（就像自行车的后胎比前胎更容易爆胎一样），那么两人就应该都选择前面左边的那只轮子作为答案。

英语知识链接

【1】 make it：及时到达

【2】 in time：及时；适时

【3】 make up：弥补；偿还