<p>I have a question regarding the uniqueness of preference functionals under risky and dynamic settings. Two well known models to represent preferences for both settings are the Expected Utility Model and the Discounted Time-Separable Utility respectively. Formally</p>

<p><strong>Risk:</strong></p>

<p>$$V(f)=\sum_{\forall s \in S} p(s) u(f(s))$$</p>

<p>Where $S$ is the set of states and $p(s)$ is the probability of state $s$, and $u(\cdot)$ a cardinal utility function.</p>

<p><strong>Dynamics:</strong></p>

<p>$$W(c)=\sum_{t=0}^{T} \mathcal{D}(t) v(c_t)$$
Where $\mathcal{D}(t)$ is a discount factor and $v(\cdot)$ is a cardinal utility function.</p>

<p>We know that both $u(\cdot)$ and $v(\cdot)$ are <em>cardinal</em> utility functions, and I understand why. But while every representation theorem of this sort of preferences mention the <em>uniqueness</em> of the functions $u(\cdot)$ and $v(\cdot)$(i.e <em>unique up to positive affine transformations</em>), they are silent about the uniqueness of $V(\cdot)$ and $W(\cdot)$. My question is: <em>are the functions $V(\cdot)$ and $W(\cdot)$ ordinal or cardinal?</em> .</p>

<p>My intuition tells me that they should be ordinal, as they only rank objects of choice and the attitudes towards risk and time are captured respectively by the function $u(\cdot)$ in the case of risk, and the <em>preference for consumption smoothing</em> is given by the function $v(\cdot)$. </p>

<p>For example if $V(f)&gt;V(g)$ then, it should also be that for every increasing function $H(\cdot)$, $H(V(f))&gt;H(V(g))$. </p>

<p>Nevertheless by the linearity of the Expected Utility Functional, i.e,</p>

<p>$$h\Bigg(\sum_{\forall s \in S} p(s) u(f(s)\Bigg)=\sum_{\forall s \in S} p(s) h\big( u(f(s)\big)$$</p>

<p>[So if this is correct, $h(\cdot)$ should be affine, making $V(f)$ <em>cardinal</em>. ]</p>

<p>I suspect that my intuition might be wrong (same applies to the dynamic setting), so I was wondering if you guys could help me clarify this question. </p>

<p>Thank you very much! </p>

CC BY-SA 3.0
2016-12-28T10:59:20.770Z
@TheoreticalEconomist This observation is not correct. For instance, the functional $HV(f)=(V(f))^3$ preserves the ordering between risky acts in spite of the fact that $x \rightarrow x^3$ is not affine. The functional form has no cardinal properties, only ordinal ones.
Oliv
CC BY-SA 3.0
2016-12-28T11:02:29.573Z
@Oliv Of course; I stand corrected.
Theoretical Economist
CC BY-SA 3.0
2016-12-28T13:39:49.430Z
@Oliv TheoreticalEconomist thank you both. Im still struggling to see how the ordinality of $V(f)$ coexists with the linearity property of the Expected Utility Functional that made me think in the first place that $h$ should be affine. If anyone could enlighten me on this would be great! On the same line I suspect Oliv that you also think that $W(\cdot)$ is ordinal aswell?
thekiciminister
CC BY-SA 3.0
2016-12-28T17:17:39.973Z
You are right, $W$ is not cardinal. In the expected utility theory, the relative valuation of $A$ vs $B$, equal to $u(A)-u(B)$, is identified (in a cardinal sense) by trade-offs of the form "how does the decision-maker value a 1% increase in the probability of receiving good A vs good B". For the complete functional, no such thought experiment is possible since we cannot increase by 1% the probability of receiving the act $f$ vs the act $g$.
Oliv
<mathematical-economics><expected-utility>
On the uniqueness of utility functions for both risk and time
thekiciminister
# On the uniqueness of utility functions for both risk and time

(Asked by: username_0 on 2016-12-27)

I have a question regarding the uniqueness of preference functionals under risky and dynamic settings. Two well known models to represent preferences for both settings are the Expected Utility Model and the Discounted Time-Separable Utility respectively. Formally

**Risk:**

$$V(f)=\sum\_{\forall s \in S} p(s) u(f(s))$$

Where $S$ is the set of states and $p(s)$ is the probability of state $s$, and $u(\cdot)$ a cardinal utility function.

**Dynamics:**

$$W(c)=\sum\_{t=0}^{T} \mathcal{D}(t) v(c\_t)$$
Where $\mathcal{D}(t)$ is a discount factor and $v(\cdot)$ is a cardinal utility function.

We know that both $u(\cdot)$ and $v(\cdot)$ are *cardinal* utility functions, and I understand why. But while every representation theorem of this sort of preferences mention the *uniqueness* of the functions $u(\cdot)$ and $v(\cdot)$(i.e *unique up to positive affine transformations*), they are silent about the uniqueness of $V(\cdot)$ and $W(\cdot)$. My question is: *are the functions $V(\cdot)$ and $W(\cdot)$ ordinal or cardinal?* .

My intuition tells me that they should be ordinal, as they only rank objects of choice and the attitudes towards risk and time are captured respectively by the function $u(\cdot)$ in the case of risk, and the *preference for consumption smoothing* is given by the function $v(\cdot)$.

For example if $V(f)>V(g)$ then, it should also be that for every increasing function $H(\cdot)$, $H(V(f))>H(V(g))$.

Nevertheless by the linearity of the Expected Utility Functional, i.e,

$$h\Bigg(\sum\_{\forall s \in S} p(s) u(f(s)\Bigg)=\sum\_{\forall s \in S} p(s) h\big( u(f(s)\big)$$

[So if this is correct, $h(\cdot)$ should be affine, making $V(f)$ *cardinal*. ]

I suspect that my intuition might be wrong (same applies to the dynamic setting), so I was wondering if you guys could help me clarify this question.

Thank you very much!

**Tags:** mathematical-economics, expected-utility

> **Comments:**
> **username_1**: @TheoreticalEconomist This observation is not correct. For instance, the functional $HV(f)=(V(f))^3$ preserves the ordering between risky acts in spite of the fact that $x \rightarrow x^3$ is not affine. The functional form has no cardinal properties, only ordinal ones.  
> **username_2**: @username_1 Of course; I stand corrected.  
> **username_0**: @username_1 TheoreticalEconomist thank you both. Im still struggling to see how the ordinality of $V(f)$ coexists with the linearity property of the Expected Utility Functional that made me think in the first place that $h$ should be affine. If anyone could enlighten me on this would be great! On the same line I suspect username_1 that you also think that $W(\cdot)$ is ordinal aswell?  
> **username_1**: You are right, $W$ is not cardinal. In the expected utility theory, the relative valuation of $A$ vs $B$, equal to $u(A)-u(B)$, is identified (in a cardinal sense) by trade-offs of the form "how does the decision-maker value a 1% increase in the probability of receiving good A vs good B". For the complete functional, no such thought experiment is possible since we cannot increase by 1% the probability of receiving the act $f$ vs the act $g$.  

---

## 0 answers

