【英语生活】从游戏看人性

Monty Hall, the quiz show host who bemused many FT readers a year ago by trying to persuade them to switch boxes in the hope of winning a car, has a new game. There are only two boxes and one contains twice as much money as the other. When you choose one, he shows you that it contains £100. Will you stick with your original choice, or switch to the other box?

This problem is real. Anyone who has changed jobs, bought a house or planned a merger has encountered a version of the two-box game; keep what you know, or go for an uncertain alternative. But familiar problems are not necessarily easy to model. Monty points out that you can lose only £50 but might gain £100. He says he does not know which box has the bigger prize. If untrue, is he trying to save his employer money, or trying to help you win? You have no way of judging whether the £50 loss is more or less likely than the £100 gain.



Decision theory tells you there is an expected gain of £25 from an equal chance of winning £100 or losing £50. But many people do not see the problem this way. They dislike the prospect of losing £50 more than they like the prospect of gaining £100. This, decision theorists claim, is irrational. If you accepted 100 gambles like this, you are virtually certain to end up with a substantial gain.



But, you may say, I am not playing this game 100 times. I am only playing it once and you cannot guarantee a gain in a single trial. That is true, but it illustrates “the fallacy of large numbers”. On the 100th trial, you are in the same position as someone who is offered the chance to do it once. So you should not do it the 100th time. But then you should not do it the 99th time, or the 98th – or the first.



Even if it is irrational to be more depressed by losses than elated by equivalent gains, this is how many people behave. That may be the best explanation of why the equity premium is so high – volatile assets need to show much higher returns to compensate for the pain of frequently seeing small losses. The argument also suggests a strategy for benefiting from this irrationality. Stop looking at share prices so often and in the long run you will get the benefit of the higher return without the pain of observing volatility.



So perhaps you can steel yourself to follow decision theory and go for the higher expected value. But then you encounter another, deeper, difficulty in Monty’s argument. It cannot make sense that, if you choose the first box, you should always switch to the second and if you choose the second, you should always switch to the first. True, you get the information that your chosen box contains £100, but Monty could have used his argument to persuade you to switch even before he opened the first box: your potential gain is twice your potential loss however much the first box holds.



So the probabilities of gain and loss cannot be equal. Then what are they? You can show that you cannot define a probability distribution for this problem, but this is not a satisfying answer; in several decades, no one has come up with a simple and compelling explanation of why Monty’s characterisation of the two-box problem is wrong.



The message of both the original Monty Hall problem and of this one is that, even in very simple cases, it is impossible to be certain that a particular mathematical representation of a real problem is a correct description. For people in business who rely on models and for people in financial services who must choose between boxes with uncertain contents every day, that is a disturbing conclusion. Next week, however, I will describe a successful strategy for the two-box problem.

一年前，益智游戏节目主持人蒙蒂•霍尔(Monty Hall)试图说服参赛者改变自己选择的盒子，以赢取汽车大奖，这令英国《金融时报》的许多读者们深感困惑。如今他推出了一档新游戏，其中只有两个盒子，一个盒子里装的钱是另外一个的两倍。当你选定一个盒子时，他向你展示其中装有100英镑。那么你是会坚持选这个盒子呢，还是转而选择另外一个盒子？



这个问题颇为实际。任何一个换过工作、买过房子或计划进行兼并的人，都曾面临过这种“两个盒子游戏”式的选择；要么维持你已知的选择，要么换另外一个不确定的选择。但是，熟悉的问题并不一定容易建立模型。蒙蒂指出，你只会损失50英镑，但或许能赢得100英镑。他表示，自己并不知道哪个盒子的钱更多。如果他说的是假话，他是在设法给自己的老板省钱呢，还是想帮你赢取大奖？你无从判断，损失50英镑的几率，是大于还是小于赢取100英镑的几率。

决策论(Decision Theory)告诉我们，在赢取100英镑和损失50英镑的几率相等的情况下，存在25英镑的预期收益。但许多人并不这样来看待这个问题。他们对于可能损失50英镑的讨厌程度，超过了对于可能赢得100英镑的欢喜程度。研究决策论的人声称，这是非理性的。如果你进行100次类似的赌博，实际上你最终肯定会赢得一笔可观的收益。



但你可能会说，我不是在玩100次这种游戏。我只玩一次，而你并不能保证我只试一次就能赢。此话不假，但它体现了“大数谬论”(fallacy of large numbers)。在尝试第100次时，你与只得到一次尝试机会的人处于相同的境地。因此，你不应该进行第100次尝试。但以此类推，你也不应该试第99次，或者第98次——或者第一次。



因损失而沮丧的程度，超过获得相等收益的欣喜程度，即便不理性，但许多人正是如此行事的。这或许是对于股票溢价如此之高的最佳诠释——价格波动的资产，需要有高得多的回报率，才能补偿因频频目睹小额损失而造成的痛苦。这种观点同时也揭示出一种得益于这种非理性的战略：别再那么频繁地关注股价，长期而言，你将得到更高回报的收益，而无须承受关注股价波动带来的痛苦。





因此，或许你可以让自己坚强起来，遵循决策论，追逐更高的预期价值。不过，接着你就会在蒙蒂的主张中遇到另一个更深层次的难题。如果你选了第一个盒子，却总是转而选第二个盒子，选择第二个盒子，又总是转而选第一个盒子，那么决策论就不会有任何意义。不错，你得到的信息是自己选中的盒子里有100英镑，但是，甚至在打开第一个盒子之前，蒙蒂就可能已经利用自己的论点，来说服你改变选择，即：不管第一个盒子里有多少钱，你可能得到的钱都是你可能失去的两倍之多。



所以，得失的几率不可能均等。那么，几率会是怎样的呢？你可以表示，自己无法确切描述这个问题的几率分布情况，但是，这并不是令人满意的回答；在数十年的时间里，都没有人给出一个简单而令人信服的解释，来说明蒙蒂对两个盒子问题的描述为何是错的。



蒙蒂•霍尔当初的问题和这个问题，都传递了这样的信息：即使是在非常简单的情况下，都不可能确定对某一实际问题的特定数学表述是一种正确描述。对依赖模型的商界人士，以及每天必须在内容不确定的“盒子”之间进行选择的金融界人士而言，这是个烦人的结论。不过，下周，我会讲述一个关于两个盒子问题的成功战略。



译者/刘彦 徐柳