<p>Judging from the reference you provide, this refers to whether the set <span class="math-container">$\Theta$</span> is <a href="https://en.wikipedia.org/wiki/Total_order" rel="nofollow noreferrer">ordered</a> or not. For example, natural numbers or the alphabet are ordered sets. In the context of moral hazard problems, examples could be effort, or ability. Since this is a numerical variable, it is an ordered set.</p>
<p>In the paper you mention, the phrase &quot;one dimensional, ordered type...&quot; is not found. However, the paper states that:</p>
<blockquote>
<p>... differences among sellers are assumed to be parameterized by a <strong>single unobservable characteristic</strong> <span class="math-container">$\theta \in \Theta$</span>. Then the family of utility functions can be written in the alternative form,</p>
<p><span class="math-container">$U_i=U(\theta_i;y,p) \text{, } i \in I$</span></p>
<p>If seller <span class="math-container">$i$</span> has <span class="math-container">$\theta_i$</span> <strong>units of the characteristic</strong> we may therefore refer to him as being of &quot;type <span class="math-container">$\theta_i$</span>.</p>
</blockquote>
<p>The first bold statement refers to the one dimensionality of the unobserved type, whereas the second refers to it being a numerical variable, which means <span class="math-container">$\Theta$</span> is an ordered set.</p>
<p>Conversely, a non-ordered set contains elements which have no intrinsic order. For instance, you can think of a case where agents do not know the city where they are to be located. The set &quot;cities&quot; is unordered because cities do not necessarily hold a particular order in a one-dimension set. You can add a second dimension (e.g. population), where you could order them based on city size.</p>
<p>I would say that having ordered sets facilitates algebraic analysis, since it enables the introduction of monotonic strategies, whereby an important outcome variable is a monotonic function of the agent's <strong>type</strong>. In effect, this seems to be important in the paper, when the author states that:</p>
<blockquote>
<p>A seller of type <span class="math-container">$\theta_i$</span> brings to market a product that is valued equally by each of the set of potential buyers, <span class="math-container">$J$</span>. This valuation, <span class="math-container">$V_i$</span>, measured in dollars, is an <strong>increasing function</strong> of <span class="math-container">$\theta$</span> and (possibly) also of the intensity of sales-related activity, <span class="math-container">$y$</span>:</p>
<p><span class="math-container">$V_i = V(\theta_i;y)$</span></p>
</blockquote>
<p>The bold emphasis highlights precisely the monotonicity enabled by the order property of <span class="math-container">$\Theta$</span>. Another example is given in <a href="http://www.jstor.org/stable/2692247?seq=1#page_scan_tab_contents" rel="nofollow noreferrer">Athey (2001)</a>:</p>
<p><span class="math-container">$\hskip1in$</span><a href="https://i.sstatic.net/tUPg5.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/tUPg5.png" alt="enter image description here" /></a></p>
<hr />
<p>In response to the second reference you give (<a href="https://sites.wustl.edu/gottlieb/files/2016/11/NonExistenceofRiley-t05znz.pdf" rel="nofollow noreferrer">here</a>), in the phrase</p>
<blockquote>
<p>We now show that Riley equilibria may not exist <strong>when types are not ordered</strong>.</p>
</blockquote>
<p>the &quot;ordered&quot; adjective refers to the second interpretation given above (i.e. of monotonicity). First, they assume that</p>
<blockquote>
<p>Consumers are uniformly distributed in the unit interval, <span class="math-container">$\theta in [0,1]$</span></p>
</blockquote>
<p>which is an ordered set. However, because both the utility function and the cost function are quadratic, it results that</p>
<blockquote>
<p>... the types in the extreme points have the strongest preference for one of the contracts and are the most costly to serve.</p>
</blockquote>
<p>which means that the utility and cost functions are not monotonic over <span class="math-container">$\theta$</span>.</p>

CC BY-SA 3.0
2017-07-19T15:10:35.277Z
So loosely speaking, let $U(\theta_i)$ denotes the utility of ordered type, then $dU(\theta)/d\theta>0$? If it not ordered type, $dU(\theta)/d\theta>0$ might not hold?
High GPA
CC BY-SA 3.0
2017-07-19T15:24:13.107Z
@HighGPA Yes and no. A unidimensional, numerical variable is, _I think_, always ordered. My point is that an ordered set **facilitates** the algebraic treatment of monotonicity, but does not **require it** (as in your second example). Think in the variable "effort". We can imagine situations whereby exerting so much effort ends up being less productive than by having a bit of time off (to rest or etc). Here, the variable effort its still ordered, yet its outcome (productivity) might not be monotonic. Meanwhile, if the set is nor ordered, the derivative makes less (no?) sense (e.g. cities).
luchonacho
CC BY-SA 3.0
2017-07-19T07:16:26.423Z
2017-07-20T12:25:21.723Z
2020-06-17T09:40:54.643Z
luchonacho
<p>I am reading papers about moral hazard. What is one dimensional, ordered type $\theta\in\Theta$?</p>

<p>What is one dimensional, not ordered type $\theta\in\Theta$?</p>

<p>Could you please give an example?</p>

<hr>

<p>References: </p>

<p><a href="https://www.jstor.org/stable/1914187" rel="nofollow noreferrer">Riley (1979)</a></p>

<p><a href="https://sites.wustl.edu/gottlieb/files/2016/11/NonExistenceofRiley-t05znz.pdf" rel="nofollow noreferrer">Azevedo and Gottlieb (2016)</a></p>

<microeconomics><utility><supply-and-demand><choice-theory><asymmetric-information>
What is one dimensional, ordered type?
High GPA
# What is one dimensional, ordered type?

(Asked by: username_0 on 2017-07-19)

I am reading papers about moral hazard. What is one dimensional, ordered type $\theta\in\Theta$?

What is one dimensional, not ordered type $\theta\in\Theta$?

Could you please give an example?

---

References:

[Riley (1979)](https://www.jstor.org/stable/1914187)

[Azevedo and Gottlieb (2016)](https://sites.wustl.edu/gottlieb/files/2016/11/NonExistenceofRiley-t05znz.pdf)

**Tags:** microeconomics, utility, supply-and-demand, choice-theory, asymmetric-information


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## 1 answers

### Answer by username_1 on 2017-07-19. Score: 5 | Accepted answer

Judging from the reference you provide, this refers to whether the set $\Theta$ is [ordered](https://en.wikipedia.org/wiki/Total_order) or not. For example, natural numbers or the alphabet are ordered sets. In the context of moral hazard problems, examples could be effort, or ability. Since this is a numerical variable, it is an ordered set.

In the paper you mention, the phrase "one dimensional, ordered type..." is not found. However, the paper states that:

> ... differences among sellers are assumed to be parameterized by a **single unobservable characteristic** $\theta \in \Theta$. Then the family of utility functions can be written in the alternative form,
>
> $U\_i=U(\theta\_i;y,p) \text{, } i \in I$
>
> If seller $i$ has $\theta\_i$ **units of the characteristic** we may therefore refer to him as being of "type $\theta\_i$.

The first bold statement refers to the one dimensionality of the unobserved type, whereas the second refers to it being a numerical variable, which means $\Theta$ is an ordered set.

Conversely, a non-ordered set contains elements which have no intrinsic order. For instance, you can think of a case where agents do not know the city where they are to be located. The set "cities" is unordered because cities do not necessarily hold a particular order in a one-dimension set. You can add a second dimension (e.g. population), where you could order them based on city size.

I would say that having ordered sets facilitates algebraic analysis, since it enables the introduction of monotonic strategies, whereby an important outcome variable is a monotonic function of the agent's **type**. In effect, this seems to be important in the paper, when the author states that:

> A seller of type $\theta\_i$ brings to market a product that is valued equally by each of the set of potential buyers, $J$. This valuation, $V\_i$, measured in dollars, is an **increasing function** of $\theta$ and (possibly) also of the intensity of sales-related activity, $y$:
>
> $V\_i = V(\theta\_i;y)$

The bold emphasis highlights precisely the monotonicity enabled by the order property of $\Theta$. Another example is given in [Athey (2001)](http://www.jstor.org/stable/2692247?seq=1#page_scan_tab_contents):

$\hskip1in$[![enter image description here](https://i.sstatic.net/tUPg5.png)](https://i.sstatic.net/tUPg5.png)

---

In response to the second reference you give ([here](https://sites.wustl.edu/gottlieb/files/2016/11/NonExistenceofRiley-t05znz.pdf)), in the phrase

> We now show that Riley equilibria may not exist **when types are not ordered**.

the "ordered" adjective refers to the second interpretation given above (i.e. of monotonicity). First, they assume that

> Consumers are uniformly distributed in the unit interval, $\theta in [0,1]$

which is an ordered set. However, because both the utility function and the cost function are quadratic, it results that

> ... the types in the extreme points have the strongest preference for one of the contracts and are the most costly to serve.

which means that the utility and cost functions are not monotonic over $\theta$.

> **Comments:**
> **username_0**: So loosely speaking, let $U(\theta_i)$ denotes the utility of ordered type, then $dU(\theta)/d\theta>0$? If it not ordered type, $dU(\theta)/d\theta>0$ might not hold?  
> **username_1**: @HighGPA Yes and no. A unidimensional, numerical variable is, _I think_, always ordered. My point is that an ordered set **facilitates** the algebraic treatment of monotonicity, but does not **require it** (as in your second example). Think in the variable "effort". We can imagine situations whereby exerting so much effort ends up being less productive than by having a bit of time off (to rest or etc). Here, the variable effort its still ordered, yet its outcome (productivity) might not be monotonic. Meanwhile, if the set is nor ordered, the derivative makes less (no?) sense (e.g. cities).  

---

