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经济代写|行为金融学代写Behavioral Finance代考|Representativeness and the conjunction fallacy
Another illustration of the representativeness illustrates another way in which people misapply statistical rules when judging the probability of a conjunctive event. Conjunctive events are events that are not independent of each other. For example, if an urn contains three red balls two white balls, a red ball is drawn on the second draw, and is not returned to the urn, then this will affect the chance that a red ball will be drawn on the third draw. However, if the red ball is replaced, then the draw of a red ball on the second and third draws are disjunctive events – the chances of one occurring does not affect the chances of another occurring. One behavioural bias identified by Kahenman and Tversky links to a set of conjunctive events and is labelled the conjunction fallacy. There are many illustrations of the conjunction fallacy and a classic is the “Linda problem” (Tversky and Kahneman, 1983):
Linda is 31 years old, single, outspoken and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations. Please check off the most likely alternative:
- Linda is a bank teller.
- Linda is a bank teller and is active in the feminist movement.
In this, there is an interaction between representativeness and the conjunction fallacy. In experiments, a large proportion of people will judge it to be more likely that Linda is a social worker active in the feminist movement, than that she is an unspecified sort of social worker even though the former is a subset of the latter and therefore statistically must be at most equally probable. If this problem is framed as a statistical question then most people with a basic knowledge of probability would realize that the conjunction of two events is less probable than each event alone.
经济代写|行为金融学代写Behavioral Finance代考|Bayes’ rule and the Monty Hall problem
Nalebuff (1987) describes a number of other paradoxes driven by people’s updating of probabilities – including a paradox often referred to as the Monty Hall problem. This paradox is interesting because its counter-intuitive natures has triggered much debate. Even some statisticians struggled to understand why the correct answer is correct because the answer is counterintuitive. To understand the correct decision, we need to use a statistical rule known as Bayes’ rule – named after its inventor, the Reverend Thomas Bayes. Bayes’ rule sets out how to adjust our estimates of the chances of an event when new information comes along. We start with a prior probability – based on all the information we currently have. Then some new information comes along and we update our prior probability using this new information to form a posterior probability – which takes into account the new information.
The Monty Hall version of this paradox is named in honour of Monty Hall who was the host of a TV show Let’s Make a Deal. Contestants are shown three curtains. A large prize is hidden behind one curtain, and small prizes are hidden behind the other two. The contestant makes a choice and then Monty Hall opens the curtain to reveal what’s behind one of the curtains not chosen and asks the contestant if they want to change their mind.
If people are reasoning in a Bayes rational way then they should decide to change their mind. If they don’t then they are making a decision without properly updating their prior probabilities using Bayes” rule.
Bar-Hillel and Falk (1982) show why this is the case in describing essentially the same problem set in a different context – the Three Prisoners problem. Tom, Dick and Harry are held in a jail. The next day, one of them will be executed and the other two will be set free. Tom, Dick and Harry’s prior probabilities of being executed are $1 / 3$ but Dick is anxious and asks the jailor to tell him just whether Tom or Harry will be set free. No new information has been revealed about the probability that Dick will be executed and so his overall chance of execution remains $1 / 3$. The fact that this result seems counterintuitive to most people reflects the fact that human intuition is not probabilistic; often people struggle intuitively to understand statistical problems. Intuitive responses to this question reflect confusion, at least in terms of Bayes’ rule, about the conditional probabilities. BarHillel and Falk explain the correct answer using Bayes’ rule, as outlined in the Mathematical Appendix A4.1.
经济代写|行为金融学代写Behavioral Finance代考|Representativeness and the conjunction fallacy
代表性的另一个例子说明了人们在判断联合事件的概率时误用统计规则的另一种方式。联合事件是不相互独立的事件。例如，如果一个瓮中包含三个红球和两个白球，第二次抽出一个红球，并且没有返回到瓮中，那么这将影响第三次抽出一个红球的机会。然而，如果红球被替换，那么在第二次和第三次抽签中抽出一个红球是分离事件——一个发生的机会不会影响另一个发生的机会。Kahenman 和 Tversky 确定的一种行为偏差与一组合取事件有关，并被称为合取谬误。
琳达 31 岁，单身，直言不讳，非常聪明。她主修哲学。学生时代，她深切关注歧视和社会正义问题，还参加了反核示威活动。请检查最可能的替代方案：
- 琳达是一名银行出纳员。
- 琳达是一名银行出纳员，积极参与女权运动。
在此，代表性和合取谬误之间存在相互作用。在实验中，大部分人会判断琳达是活跃于女权运动中的社会工作者，而不是她是未指定类型的社会工作者，尽管前者是后者的一个子集，因此在统计上最多应该是等概率的。如果这个问题是一个统计问题，那么大多数具有概率基本知识的人都会意识到，两个事件的结合比每个事件单独发生的概率更小。
经济代写|行为金融学代写Behavioral Finance代考|Bayes’ rule and the Monty Hall problem
Nalebuff (1987) 描述了由人们更新概率驱动的许多其他悖论——包括一个通常被称为蒙蒂霍尔问题的悖论。这个悖论很有趣，因为它的反直觉性质引发了很多争论。甚至一些统计学家也难以理解为什么正确答案是正确的，因为答案是违反直觉的。为了理解正确的决定，我们需要使用称为贝叶斯规则的统计规则——以其发明者托马斯贝叶斯牧师的名字命名。贝叶斯规则规定了当新信息出现时如何调整我们对事件发生概率的估计。我们从一个先验概率开始——基于我们目前拥有的所有信息。
这个悖论的蒙蒂霍尔版本是为了纪念电视节目让我们做一笔交易的主持人蒙蒂霍尔而命名的。参赛者被展示了三个窗帘。一个大的奖品藏在一个窗帘后面，小奖品藏在另外两个的后面。参赛者做出选择，然后蒙蒂·霍尔打开窗帘，揭示其中一个未选择的窗帘背后的内容，并询问参赛者是否想改变主意。
如果人们以贝叶斯理性的方式进行推理，那么他们应该决定改变主意。如果他们不这样做，那么他们在没有使用贝叶斯规则正确更新他们的先验概率的情况下做出决定。
Bar-Hillel 和 Falk (1982) 在描述不同背景下的基本相同问题时说明了为什么会出现这种情况——三个囚徒问题。汤姆、迪克和哈利被关在监狱里。第二天，其中一人将被处决，另外两人将被释放。汤姆、迪克和哈利被处决的先验概率是1/3但迪克很着急，要求狱卒告诉他汤姆或哈利是否会被释放。关于迪克将被处决的可能性没有透露任何新信息，因此他的整体被处决机会仍然存在1/3. 这个结果对大多数人来说似乎违反直觉这一事实反映了人类直觉不是概率的事实。人们常常难以直观地理解统计问题。对这个问题的直观回答反映了对条件概率的混淆，至少在贝叶斯规则方面。BarHillel 和 Falk 使用贝叶斯规则解释正确答案，如数学附录 A4.1 中所述。
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