<p>The value function used in Kahneman's prospect theory (which your plot shows) is supposed to capture empirically observed behavior of people's attitudes towards gains/losses as well as to risks in those domains.</p>

<p>In the domain for gains, people are usually risk averse. However, in the domain for losses, people do tend to take larger risks. A nice illustration is the observation that when a gambler is losing money (relative to the amount he came to the casino with), he tends to wager on higher stakes (larger risks), perhaps in an attempt to "win back" the losses. </p>

<p>So the convex curvature in the domain of losses is meant to capture this type of behavior. </p>

<hr>

<p>In fact, prospect theory's treatment of people's risk attitude is even more nuanced than the above. In addition to the S-shaped value function you showed, it also has a component that allows for non-linear probability weighting. Together, the value function and non-linear probability weighting generate the following four-fold classification of risk attitudes: 
\begin{array}{c|cc}
&amp;\text{gains}&amp;\text{losses}\\\hline
\text{low probability}&amp;\text{risk loving}&amp;\text{risk averse}\\
\text{high probability}&amp;\text{risk averse}&amp;\text{risk loving}
\end{array}</p>

<p>See <a href="http://cemi.ehess.fr/docannexe/file/2780/tversjy_kahneman_advances.pdf" rel="nofollow noreferrer">Kahneman and Tversky (1992)</a> for more detail.</p>

CC BY-SA 3.0
2018-03-01T00:08:11.773Z
Doesn't your table imply that the shape of the value function depends on the quantity of the probabilities involved?
GambitSquared
CC BY-SA 3.0
2018-03-01T00:12:33.633Z
@GambitSquared: Yes, it does. As I said, *the value function* **and** *non-linear probability weighting* generate the four-fold classification of risk attitudes. Prospect theory consists of both of these components. What you show in the question is only about the value function, which is only part of PT.
Herr K.
CC BY-SA 3.0
2018-03-01T00:16:20.963Z
Furthermore the fact that this plot is supposed to be empirical doesn't connect it to my intuition and personal preference (see examples in my question). I just don't understand why someone would take such a bet. We aren't all addicted to gambling, are we?
GambitSquared
CC BY-SA 3.0
2018-03-01T00:19:10.050Z
Would it be correct that the shape of the value function represents the high-probability case? What would the shape of the value function be in the low-probability case?
GambitSquared
CC BY-SA 3.0
2018-03-01T00:27:15.770Z
@GambitSquared: The gambler's example in my answer is well-documented. I'm not making it up. Also, the paper linked at the end reports experimental evidence that a convex curvature in the domain of losses is justified. Regarding your last two questions: it may appear the way you're suggesting, but when the S-shaped value function and non-linear probability weighting are put together, it's hard to separate the effects of individual components, since the combo is highly non-linear.
Herr K.
CC BY-SA 3.0
2018-03-05T12:10:27.233Z
This article shows that the Value-curve can also have a different shape in different circumstances. In this case the circumstance is time-pressure: https://www.sciencedirect.com/science/article/pii/S0749597815000722
GambitSquared
CC BY-SA 3.0
2018-03-05T12:17:25.807Z
@GambitSquared Sure. But how is this related to your original question, why the value function is convex over losses? If you have a different question, ask it in a different post.
Herr K.
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2018-03-05T12:19:46.613Z
Because it shows that the value-function over losses is not convex in all situations.
GambitSquared
CC BY-SA 3.0
2018-03-05T12:36:56.900Z
@GambitSquared I don't dispute the finding you post. But your question was *why* utility over loss is convex. Also, your question suggested that you're interested in why Kahneman formulated the function as such. I addressed these two aspects in my answer.
Herr K.
CC BY-SA 3.0
2018-03-05T12:42:01.773Z
As far as I understand it now, the answer to my question should be something like: "the marginal utililty of losses is not always diminishing, it depends on the situation, therefore the assumption in your question is wrong. The Value-plot can have a different shape such as in the situations you describe." Or something similar. Or am I missing something?
GambitSquared
CC BY-SA 3.0
2018-03-05T12:45:04.683Z
@GambitSquared you're always welcome to post that as an answer
Herr K.
CC BY-SA 3.0
2018-03-05T12:48:08.907Z
Of course, but since I am not an expert, I am not sure about the validity of that answer. It is just my tentative conclusion.
GambitSquared
CC BY-SA 4.0
2018-12-01T22:09:15.830Z
Can you add a short explanation on what you meant by "low probability" and "high probability" in the table?
xuhdev
CC BY-SA 4.0
2018-12-01T22:26:01.420Z
@xuhdev: "Low" and "high" are in reference to KT's original experiment. At the risk of quoting them out of context, think of "low probability" as "$p\le0.1$" and "high probability" as "$p\ge0.9$". But please, if you're really interested in this, read KT's paper linked in my answer.
Herr K.
CC BY-SA 4.0
2018-12-02T02:06:06.447Z
@HerrK. Thanks, I'm just looking for a summary, but this has really confused me. Actually, what confused me is really what the probability means: The probability of what?
xuhdev
CC BY-SA 4.0
2018-12-02T02:07:15.313Z
@xuhdev: probability of gains or losses
Herr K.
CC BY-SA 3.0
2018-03-01T00:03:04.450Z
2018-03-01T00:03:04.450Z
Herr K.
<p>This is Kahneman's value-plot on prospect theory:</p>
<p><a href="https://i.sstatic.net/zqmCA.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/zqmCA.jpg" alt="enter image description here" /></a></p>
<p>QUESTION: Why is the Marginal Utility of <strong>losses</strong> deminishing?</p>
<p>CONTEXT:</p>
<ul>
<li><p>I fully understand that the Marginal Utility of gains deminishes: 100 dollar gains has less utility to me when I am rich than when I am poor.</p>
</li>
<li><p>But to me it is different with losses: If I can choose to lose 100 dollars for sure or a 50% chance to lose 200 dollars I would definitely not go for the bet! Why would I risk losing even more? After all, like most people, <strong>I am risk-averse</strong>. My reference point would be that I already lost 100 euro's.</p>
</li>
<li><p>Furthermore: If all I own should be 200 dollars, I might regret losing 100 dollars very much, but I will regret losing a second 100 dollars even more because then I would be left with nothing and I can't buy any food.</p>
</li>
<li><p>In other words: I guess my reference point is always zero and compared to that I will always be risk-averse... but isn't that what most people do or should do?</p>
</li>
</ul>
<hr />
<p>UPDATE: This article shows that the Value-curve can also be shaped differently, for example under time pressure: <a href="https://www.sciencedirect.com/science/article/pii/S0749597815000722" rel="nofollow noreferrer">https://www.sciencedirect.com/science/article/pii/S0749597815000722</a></p>

<decision-theory><risk-aversion>
Why is the Marginal Utility of losses diminishing in Prospect Theory?
GambitSquared
# Why is the Marginal Utility of losses diminishing in Prospect Theory?

(Asked by: username_0 on 2018-02-28)

This is Kahneman's value-plot on prospect theory:

[![enter image description here](https://i.sstatic.net/zqmCA.jpg)](https://i.sstatic.net/zqmCA.jpg)

QUESTION: Why is the Marginal Utility of **losses** deminishing?

CONTEXT:

* I fully understand that the Marginal Utility of gains deminishes: 100 dollar gains has less utility to me when I am rich than when I am poor.
* But to me it is different with losses: If I can choose to lose 100 dollars for sure or a 50% chance to lose 200 dollars I would definitely not go for the bet! Why would I risk losing even more? After all, like most people, **I am risk-averse**. My reference point would be that I already lost 100 euro's.
* Furthermore: If all I own should be 200 dollars, I might regret losing 100 dollars very much, but I will regret losing a second 100 dollars even more because then I would be left with nothing and I can't buy any food.
* In other words: I guess my reference point is always zero and compared to that I will always be risk-averse... but isn't that what most people do or should do?

---

UPDATE: This article shows that the Value-curve can also be shaped differently, for example under time pressure: <https://www.sciencedirect.com/science/article/pii/S0749597815000722>

**Tags:** decision-theory, risk-aversion

---

## 1 answers

### Answer by username_1 on 2018-03-01. Score: 4

The value function used in Kahneman's prospect theory (which your plot shows) is supposed to capture empirically observed behavior of people's attitudes towards gains/losses as well as to risks in those domains.

In the domain for gains, people are usually risk averse. However, in the domain for losses, people do tend to take larger risks. A nice illustration is the observation that when a gambler is losing money (relative to the amount he came to the casino with), he tends to wager on higher stakes (larger risks), perhaps in an attempt to "win back" the losses.

So the convex curvature in the domain of losses is meant to capture this type of behavior.

---

In fact, prospect theory's treatment of people's risk attitude is even more nuanced than the above. In addition to the S-shaped value function you showed, it also has a component that allows for non-linear probability weighting. Together, the value function and non-linear probability weighting generate the following four-fold classification of risk attitudes:
\begin{array}{c|cc}
&\text{gains}&\text{losses}\\\hline
\text{low probability}&\text{risk loving}&\text{risk averse}\\
\text{high probability}&\text{risk averse}&\text{risk loving}
\end{array}

See [Kahneman and Tversky (1992)](http://cemi.ehess.fr/docannexe/file/2780/tversjy_kahneman_advances.pdf) for more detail.

> **Comments:**
> **username_0**: Doesn't your table imply that the shape of the value function depends on the quantity of the probabilities involved?  
> **username_1**: @GambitSquared: Yes, it does. As I said, *the value function* **and** *non-linear probability weighting* generate the four-fold classification of risk attitudes. Prospect theory consists of both of these components. What you show in the question is only about the value function, which is only part of PT.  
> **username_0**: Furthermore the fact that this plot is supposed to be empirical doesn't connect it to my intuition and personal preference (see examples in my question). I just don't understand why someone would take such a bet. We aren't all addicted to gambling, are we?  
> **username_0**: Would it be correct that the shape of the value function represents the high-probability case? What would the shape of the value function be in the low-probability case?  
> **username_1**: @GambitSquared: The gambler's example in my answer is well-documented. I'm not making it up. Also, the paper linked at the end reports experimental evidence that a convex curvature in the domain of losses is justified. Regarding your last two questions: it may appear the way you're suggesting, but when the S-shaped value function and non-linear probability weighting are put together, it's hard to separate the effects of individual components, since the combo is highly non-linear.  
> **username_0**: This article shows that the Value-curve can also have a different shape in different circumstances. In this case the circumstance is time-pressure: https://www.sciencedirect.com/science/article/pii/S0749597815000722  
> **username_1**: @GambitSquared Sure. But how is this related to your original question, why the value function is convex over losses? If you have a different question, ask it in a different post.  
> **username_0**: Because it shows that the value-function over losses is not convex in all situations.  
> **username_1**: @GambitSquared I don't dispute the finding you post. But your question was *why* utility over loss is convex. Also, your question suggested that you're interested in why Kahneman formulated the function as such. I addressed these two aspects in my answer.  
> **username_0**: As far as I understand it now, the answer to my question should be something like: "the marginal utililty of losses is not always diminishing, it depends on the situation, therefore the assumption in your question is wrong. The Value-plot can have a different shape such as in the situations you describe." Or something similar. Or am I missing something?  
> **username_1**: @GambitSquared you're always welcome to post that as an answer  
> **username_0**: Of course, but since I am not an expert, I am not sure about the validity of that answer. It is just my tentative conclusion.  
> **username_2**: Can you add a short explanation on what you meant by "low probability" and "high probability" in the table?  
> **username_1**: @xuhdev: "Low" and "high" are in reference to KT's original experiment. At the risk of quoting them out of context, think of "low probability" as "$p\le0.1$" and "high probability" as "$p\ge0.9$". But please, if you're really interested in this, read KT's paper linked in my answer.  
> **username_2**: @HerrK. Thanks, I'm just looking for a summary, but this has really confused me. Actually, what confused me is really what the probability means: The probability of what?  
> **username_1**: @xuhdev: probability of gains or losses  

---

