<p>I presume that by &quot;fair gamble&quot; <span class="math-container">$G$</span> we mean a gamble whose expected value is zero, <span class="math-container">$E(G) = 0$</span>.</p>
<p>Risk-aversion can be expressed mathematically by showing that the utility function is strictly concave, which by Jensen's Inequality imples
<span class="math-container">$$U[E(G)] &gt; E[U(G)]$$</span></p>
<p>Given the utility function in our case this means that we have to show</p>
<p><span class="math-container">$$U(0) &gt; E[U(G)] \implies 0 &gt; E[U(G)].$$</span></p>
<p>So we need a way to show that, for the given utility function, the expected utility from <em>any</em> &quot;fair gamble&quot; will be negative.</p>
<p>Intuitively, we should expect that since we see that negative outcomes weigh more heavily than positive ones (due to the <span class="math-container">$2.5$</span> mutliplicative factor of what happens to utility when the outcome is negative).</p>
<p><strong>How to show?</strong></p>
<p>Define the fair Gamble providing a set of <span class="math-container">$m$</span> payoffs <span class="math-container">$\{x^G_1,..., x^G_m\}$</span> and a probability distribution over them <span class="math-container">$P_G = \{p_1,...p_m\}$</span>. To be a fair gamble it must be the case that</p>
<p><span class="math-container">$$E(G) = \sum_{i=1}^n p_ix^G_i = 0.$$</span></p>
<p>Write the utility function using the indicator function</p>
<p><span class="math-container">$$U(x) = [1-I\{x&lt; 0\}]\cdot x + 2.5 \cdot I\{x&lt; 0\})]\cdot x = x + 1.5\cdot I\{x&lt; 0\})]\cdot x $$</span></p>
<p>Then
<span class="math-container">$$E[U(G)] = \sum_{i=1}^m p_iu(x^G_i) = ...$$</span></p>
<p>I guess you can take it from here.</p>

CC BY-SA 4.0
2024-01-23T09:34:05.540Z
2024-01-23T09:34:05.540Z
Alecos Papadopoulos
<p>The 1997 Quarterly Journal of Economics (QJE) paper titled &quot;The Effect of Myopia and Loss Aversion on Risk Taking: An Experimental Test&quot; by Thaler, Tversky, Kahneman and Schwartz, says that an individual with the following utility function</p>
<p><span class="math-container">$$U(x)=\begin{cases} x, x\geq0\\ 2.5x, x&lt;0\end{cases}$$</span> is risk neutral for gains and losses. <strong>How can I prove that this individual is risk-averse in mixed gambles?</strong> A mixed gamble is a gamble with some positive and negative outcomes.</p>

CC BY-SA 4.0
2024-01-23T06:47:57.023Z
Do you know the definition of risk-averse?
Giskard
CC BY-SA 4.0
2024-01-23T06:54:44.890Z
It means that an individual gets a higher utility from choosing a sure outcome than a fair gamble/lottery.
Ramandeep
CC BY-SA 4.0
2024-01-23T07:02:18.617Z
Any sure outcome? Like I choose getting $-100\$$ with 100% instead of losing or winning 10\$ with 50-50%?
Giskard
CC BY-SA 4.0
2024-01-23T07:39:08.557Z
No, it must be that the sure outcome is something greater than the lowest payoff from the gamble considered. So, in your example, the sure outcome has to be something greater than the lowest payoff from the gamble with a 50-50% chance of winning $10.
Ramandeep
CC BY-SA 4.0
2024-01-23T08:02:35.517Z
I recommend looking up the exact definition, it uses a very specific sure outcome. Afterwards I think you can solve this problem by applying the definition.
Giskard
<utility><theory><loss-aversion>
Mixed gambles and risk aversion
Ramandeep
# Mixed gambles and risk aversion

(Asked by: username_0 on 2024-01-23)

The 1997 Quarterly Journal of Economics (QJE) paper titled "The Effect of Myopia and Loss Aversion on Risk Taking: An Experimental Test" by Thaler, Tversky, Kahneman and Schwartz, says that an individual with the following utility function

$$U(x)=\begin{cases} x, x\geq0\\ 2.5x, x<0\end{cases}$$ is risk neutral for gains and losses. **How can I prove that this individual is risk-averse in mixed gambles?** A mixed gamble is a gamble with some positive and negative outcomes.

**Tags:** utility, theory, loss-aversion
> **Comments:**
> **username_1**: Do you know the definition of risk-averse?  
> **username_0**: It means that an individual gets a higher utility from choosing a sure outcome than a fair gamble/lottery.  
> **username_1**: Any sure outcome? Like I choose getting $-100\$$ with 100% instead of losing or winning 10\$ with 50-50%?  
> **username_0**: No, it must be that the sure outcome is something greater than the lowest payoff from the gamble considered. So, in your example, the sure outcome has to be something greater than the lowest payoff from the gamble with a 50-50% chance of winning $10.  
> **username_1**: I recommend looking up the exact definition, it uses a very specific sure outcome. Afterwards I think you can solve this problem by applying the definition.  

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## 1 answers

### Answer by username_2 on 2024-01-23. Score: 2

I presume that by "fair gamble" $G$ we mean a gamble whose expected value is zero, $E(G) = 0$.

Risk-aversion can be expressed mathematically by showing that the utility function is strictly concave, which by Jensen's Inequality imples
$$U[E(G)] > E[U(G)]$$

Given the utility function in our case this means that we have to show

$$U(0) > E[U(G)] \implies 0 > E[U(G)].$$

So we need a way to show that, for the given utility function, the expected utility from *any* "fair gamble" will be negative.

Intuitively, we should expect that since we see that negative outcomes weigh more heavily than positive ones (due to the $2.5$ mutliplicative factor of what happens to utility when the outcome is negative).

**How to show?**

Define the fair Gamble providing a set of $m$ payoffs $\{x^G\_1,..., x^G\_m\}$ and a probability distribution over them $P\_G = \{p\_1,...p\_m\}$. To be a fair gamble it must be the case that

$$E(G) = \sum\_{i=1}^n p\_ix^G\_i = 0.$$

Write the utility function using the indicator function

$$U(x) = [1-I\{x< 0\}]\cdot x + 2.5 \cdot I\{x< 0\})]\cdot x = x + 1.5\cdot I\{x< 0\})]\cdot x $$

Then
$$E[U(G)] = \sum\_{i=1}^m p\_iu(x^G\_i) = ...$$

I guess you can take it from here.


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