<p>First of all, I don't know if there is a convention as I never use weak preferences myself. But I can give you some suggestion that should work:</p>

<p>Do you have a <strong>finite number of goods</strong>, lets say $n$? Then $\mathbb{N}$ is clearly enough:
Just map the best good to 1, the second best to $n$, third best to $n^2$,... .
If you are indifferent between 2 goods (lets say they would be ranked third overall) assign them to $n^2$ and $n^2+1$. As @MettaWorldPeace already pointed out: it is kind of "wasteful" to use $\mathbb{N}$, so can we use less? And clearly we can never need more than $(n-1)^n+1$ numbers, so a mapping on $\{1,2, ..., (n-1)^n+1 \}$ would be sufficient (you can use even less numbers, but it is annoying to write down, so I just leave it at that).</p>

<p>What if we have a <strong>(countably) infinite number</strong> of goods? Here we can use the same trick, using $\mathbb{R}$ (or, if you like to, use $\mathbb{Q}$) just assign the best good $1$, the second best $2$, ... . If there are indifferences (i.e., you have three second best goods, the first one is assigned $2$, the second one $2.1$, the third $2.11$(or whatever you like)). Again, as before it is kind of wasteful, what else can you use? You could use a mapping on $\mathbb{N}\times \mathbb{N}$.</p>

<p>The same idea clearly works for every other number of goods, but I guess you can figure that out yourself.</p>

<p><em>Remark: This is not more than a different way to write preferences. You can not use this, e.g. as a utility function. However, all of these examples have the property that you can compare the ranking by simple mathematical operations (i.e., comparing $\lfloor \frac{x}{n}\rfloor$ for the first example).</em></p>

CC BY-SA 3.0
2015-04-10T23:55:51.287Z
According to Michael Carter's Foundations of Mathematical Economics, Economists are averse to this term, because they don't want the foundation of economics is on something weak.
Metta World Peace
CC BY-SA 3.0
2015-04-10T08:23:58.997Z
2015-04-10T08:23:58.997Z
The Almighty Bob
<p>Suppose I have $$\text{apples} \succ \text{pears} \succ \text{pineapples}$$</p>

<p>I can map this onto the natural numbers as follows </p>

<p>$$\text{apples}\mapsto 1$$
$$\text{pears}\mapsto 2$$
$$\text{pineapple}\mapsto 3$$</p>

<p>But for weak preferences we might have $$\text{apples} \sim \text{pears}$$
But this wouldn't map bijectively onto the natural numbers because then for example 
$$\text{apples} \mapsto 2$$
$$\text{pears} \mapsto 2$$</p>

<p><strong>My Question:</strong> </p>

<p>What do we map weak preferences on to? Or do we relax the assumption the map has to be bijective and allow it to be a surjective mapping onto $\Bbb{N}$? </p>

CC BY-SA 3.0
2015-04-10T05:46:35.273Z
Map three different kinds of fruits ONTO $\mathbb N$?!
Metta World Peace
CC BY-SA 3.0
2015-04-10T05:56:45.050Z
is mapping different from indexing? I thought that's what an order was....it was like assigning an index to each one
Stan Shunpike
CC BY-SA 3.0
2015-04-10T07:56:59.797Z
What is the purpose of the mapping? Are you trying to construct a utility function that represents the preferences?
Ubiquitous
CC BY-SA 3.0
2015-04-10T08:03:25.207Z
@Ubiquitous No, I just want a preference relation that orders them and to define a notion of ordering. In my mind $1,2,3$ were such an ordering.
Stan Shunpike
<microeconomics>
If strict preferences map onto $\Bbb{N}$ what do weak preferences map onto?
Stan Shunpike
# If strict preferences map onto $\Bbb{N}$ what do weak preferences map onto?

(Asked by: username_0 on 2015-04-10)

Suppose I have $$\text{apples} \succ \text{pears} \succ \text{pineapples}$$

I can map this onto the natural numbers as follows

$$\text{apples}\mapsto 1$$
$$\text{pears}\mapsto 2$$
$$\text{pineapple}\mapsto 3$$

But for weak preferences we might have $$\text{apples} \sim \text{pears}$$
But this wouldn't map bijectively onto the natural numbers because then for example
$$\text{apples} \mapsto 2$$
$$\text{pears} \mapsto 2$$

**My Question:**

What do we map weak preferences on to? Or do we relax the assumption the map has to be bijective and allow it to be a surjective mapping onto $\Bbb{N}$?

**Tags:** microeconomics
> **Comments:**
> **username_1**: Map three different kinds of fruits ONTO $\mathbb N$?!  
> **username_0**: is mapping different from indexing? I thought that's what an order was....it was like assigning an index to each one  
> **username_2**: What is the purpose of the mapping? Are you trying to construct a utility function that represents the preferences?  
> **username_0**: @Ubiquitous No, I just want a preference relation that orders them and to define a notion of ordering. In my mind $1,2,3$ were such an ordering.  

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## 1 answers

### Answer by username_3 on 2015-04-10. Score: 2

First of all, I don't know if there is a convention as I never use weak preferences myself. But I can give you some suggestion that should work:

Do you have a **finite number of goods**, lets say $n$? Then $\mathbb{N}$ is clearly enough:
Just map the best good to 1, the second best to $n$, third best to $n^2$,... .
If you are indifferent between 2 goods (lets say they would be ranked third overall) assign them to $n^2$ and $n^2+1$. As @MettaWorldPeace already pointed out: it is kind of "wasteful" to use $\mathbb{N}$, so can we use less? And clearly we can never need more than $(n-1)^n+1$ numbers, so a mapping on $\{1,2, ..., (n-1)^n+1 \}$ would be sufficient (you can use even less numbers, but it is annoying to write down, so I just leave it at that).

What if we have a **(countably) infinite number** of goods? Here we can use the same trick, using $\mathbb{R}$ (or, if you like to, use $\mathbb{Q}$) just assign the best good $1$, the second best $2$, ... . If there are indifferences (i.e., you have three second best goods, the first one is assigned $2$, the second one $2.1$, the third $2.11$(or whatever you like)). Again, as before it is kind of wasteful, what else can you use? You could use a mapping on $\mathbb{N}\times \mathbb{N}$.

The same idea clearly works for every other number of goods, but I guess you can figure that out yourself.

*Remark: This is not more than a different way to write preferences. You can not use this, e.g. as a utility function. However, all of these examples have the property that you can compare the ranking by simple mathematical operations (i.e., comparing $\lfloor \frac{x}{n}\rfloor$ for the first example).*

> **Comments:**
> **username_1**: According to Michael Carter's Foundations of Mathematical Economics, Economists are averse to this term, because they don't want the foundation of economics is on something weak.  

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