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经济代写|行为金融学代写Behavioral Finance代考|Kahneman and Tversky’s prospect theory
As we have seen above, Kahneman and Tversky’s (1979) experimental results do not fit easily with the predictions of EUT and Kahneman and Tversky construct prospect theory to reconcile some common behavioural anomalies. They argue that choices are made as the outcome of two separate, sequential processes: an editing phase and an evaluation phase.
The editing phase is about simplifying the representation of prospects and there are a number of ways in which this is done including coding, combination and cancellation.
- Coding: When prospects are coded as either gains or losses relative to a reference point, this reference point is not necessarily set at zero. In fact, the reference point is more often set by the status quo, for example a person’s current asset position.
- Combination: Probabiliies associated with identical outcomes are combined. If a set of prospects includes a $25 \%$ chance of 200 , another $25 \%$ chance of 200 and a $50 \%$ chance of zero, the 200 payoff will be combined into a $50 \%$ chance of 200 .
- Cancellation: Occurs when people disregard common elements in a set of choices set, for example ignoring first stages in sequentional decisions, or ignoring a common bonus as outlined in the isolation effect examples above. Other editing operations include simplifications such as rounding-up probabilities and payoffs to approximate amounts and discarding outcomes that are very unlikely. Editing will also involve the deletion of dominated prospects – that is prospects for which there is always a better alternative.
Editing does create the possibility of inconsistency and intransitivity because differences in prospects which are eliminated in the editing process may change the preference ordering of prospects, especially as the outcome of simplification will depend on the editing sequence and context. Kahneman and Tversky give the example of a choice between one prospect involving a 500 payoff with $20 \%$ probability versus a 101 payoff with $49 \%$ probability and another prospect involving a choice between 500 with probability $15 \%$ and 99 with probability $51 \%$. The second choices in both prospects might be simplified to a $50 \%$ chance of a 100 payoff and then the first prospect will appear to dominate the second whereas it would not have dominated if the choices had not been simplified in the editing phase.
经济代写|行为金融学代写Behavioral Finance代考|The weighting function
Kahneman and Tversky assign weights to the probabilities of prospects, as captured by the weighting function and the mathematics are outlined in this chapter’s Mathematical Appendix A4.3. Building on empirical evidence cited in Kahneman and Tversky (1979, p. 280) five independent studies of 30 decision-makers identified concave utility functions for gains and convex utility functions for losses, with utility functions usually steeper for losses than for gains.
The weighting function is steep at its extremes and is discontinuous when probabilities are close to 0 or 1 because people do not know how to comprehend extreme events. They do not know how to weight extreme probabilities or even if they should weight them at all. This can be captured by recognizing that, in prospect theory, the scaling of the value function is complicated by the introduction of the weighting function. Decision weights can capture complexity of decision-making. They can transform linear value functions into nonlinear ones to capture risk aversion and risk-seeking. Kahneman and Tversky emphasize that their weighting function is not about degrees of belief, as is the focus in some other studies, for example, Keynes (1921), Ellsberg (1961) and Fellner (1961). Instead, decision weights measure relationship between likelihood of events alongside their probability. In EUT, the focus is on simple problems but in prospect theory, other factors beyond simple probability, such as ambiguity, determine desirability and this reflects decision weights.
The weighting function has a number of properties including overweighting, subcertainty and sub-proportionality. The sub-certainty property captures the fact that probabilistic outcomes are given less weight than certain outcomes, and this feature captures the Allais paradox. Sub-certainty will be more pronounced for vague probabilities than for clear probabilities. There is overweighting of very low probabilities and subsmall probabilities than for large probabilities.
Once the weighting of probabilities is incorporated into prospect theory, in contrast to the nonlinear utility functions from Markowitz, the expectation principle of EUT no longer holds. There will be violations of dominance reflecting the nonlinearity of the prospect theory weighting function. The editing phase has significant implications here: simplification of prospects during editing leads to very low probabilities being treated as if they are impossible and very high probabilities being treated as if they are certain.
经济代写|行为金融学代写Behavioral Finance代考|Kahneman and Tversky’s prospect theory
正如我们在上面看到的，Kahneman 和 Tversky（1979）的实验结果并不容易与 EUT 的预测相吻合，并且 Kahneman 和 Tversky 构建了前景理论来调和一些常见的行为异常。他们认为，选择是两个独立的连续过程的结果：编辑阶段和评估阶段。
编辑阶段是关于简化潜在客户的表示，有多种方法可以做到这一点，包括编码、组合和取消。
- 编码：当前景被编码为相对于参考点的收益或损失时，该参考点不一定设置为零。事实上，参考点更多地是由现状设定的，例如一个人当前的资产位置。
- 组合：组合与相同结果相关的概率。如果一组潜在客户包括25%200的机会，另一个25%机会 200 和50%机会为零，200 的收益将合并为50%200 的机会。
- 取消：当人们忽略一组选择集中的共同元素时发生，例如忽略顺序决策中的第一阶段，或忽略上面隔离效应示例中概述的共同奖励。其他编辑操作包括简化，例如将概率和收益舍入到近似数量以及丢弃不太可能的结果。编辑还将涉及删除占主导地位的前景——即总有更好选择的前景。
编辑确实会产生不一致和不及物的可能性，因为在编辑过程中消除的前景差异可能会改变前景的偏好顺序，特别是因为简化的结果将取决于编辑顺序和上下文。Kahneman 和 Tversky 举了一个例子，一个前景涉及 500 的回报和20%概率与 101 的回报49%概率和另一个前景涉及在 500 与概率之间进行选择15%和 99 概率51%. 两个前景中的第二个选择可能会简化为50%如果有 100 的回报机会，那么第一个潜在客户似乎会主导第二个潜在客户，而如果在编辑阶段没有简化选择，它就不会占主导地位。
经济代写|行为金融学代写Behavioral Finance代考|The weighting function
Kahneman 和 Tversky 为前景概率分配权重，由加权函数捕获，本章的数学附录 A4.3 中概述了数学。基于 Kahneman 和 Tversky (1979, p. 280) 中引用的经验证据，对 30 位决策者进行的五项独立研究确定了收益的凹效用函数和损失的凸效用函数，损失的效用函数通常比收益更陡峭。
加权函数在其极端处是陡峭的，并且在概率接近 0 或 1 时是不连续的，因为人们不知道如何理解极端事件。他们不知道如何衡量极端概率，甚至根本不知道应该衡量它们。这可以通过认识到，在前景理论中，价值函数的缩放因引入加权函数而变得复杂。决策权重可以捕捉决策的复杂性。他们可以将线性价值函数转换为非线性价值函数，以捕捉风险厌恶和风险寻求。Kahneman 和 Tversky 强调，他们的权重函数与信念程度无关，正如其他一些研究的重点，例如 Keynes (1921)、Ellsberg (1961) 和 Fellner (1961)。反而，决策权重衡量事件的可能性及其概率之间的关系。在 EUT 中，重点是简单的问题，但在前景理论中，除了简单概率之外的其他因素，例如模糊性，决定了可取性，这反映了决策权重。
加权函数具有许多属性，包括超加权、次确定性和次比例性。次确定性属性反映了概率结果的权重低于某些结果的事实，并且该特征捕获了阿莱悖论。模糊概率的次确定性将比明确的概率更明显。与大概率相比，极低概率和次小概率的权重过高。
一旦概率权重被纳入前景理论，与 Markowitz 的非线性效用函数相比，EUT 的期望原则不再成立。将存在反映前景理论加权函数的非线性的支配性侵犯。编辑阶段在这里具有重要意义：在编辑过程中简化前景导致非常低的概率被视为不可能，而非常高的概率被视为确定。
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