<p>Arrows impossibility theorem deals with what we call a social welfare ordering. For any profile of preferences, it provides a ranking over all alternatives.</p>
<p>What you refer to is called a social choice function (or voting rule). This gives, for any profile of preferences, an &quot;most preferred&quot; option.</p>
<p>The most famous impossibilty theorem for social welfare functions is the Gibbard–Satterthwaite (<a href="https://en.wikipedia.org/wiki/Gibbard%E2%80%93Satterthwaite_theorem" rel="nofollow noreferrer">see here fore the wiki</a>, here for the <a href="http://darp.lse.ac.uk/papersdb/Gibbard_(Econometrica_73).pdf" rel="nofollow noreferrer">paper of Gibbard</a> and here <a href="https://www.sciencedirect.com/science/article/abs/pii/0022053175900502" rel="nofollow noreferrer">for Satterthwaite's paper</a>). Another impossibility theorem for social choice functions is the Muller-Satterthwaite Theorem (<a href="http://www.eecs.harvard.edu/cs286r/courses/fall11/papers/MS77.pdf" rel="nofollow noreferrer">see here for the paper</a>).</p>
<p>I guess there are many more...</p>

CC BY-SA 4.0
2021-07-07T09:06:11.703Z
Thanks, this is useful — Gibbard/Satterthwaite is about choosing best alternatives, not best orderings over alternatives. However, 1) Is this really an analogue of Arrow’s theorem? (I understand that this question is a bit vague!) 2. I think your use of ‘social welfare function’ is a bit off: a SWF generates a complete ranking of alternatives (not just a most preferred option). From the Wiki, I see that the terminology for what I have in mind is a ‘voting rule’.
afreelunch
CC BY-SA 4.0
2021-07-07T11:05:37.867Z
To further elaborate on 2: just as a utility function represents an individual's preference ordering over alternatives, a social welfare function represents a "social ordering" over alternatives. So I am definitely *not* referring to a SWF; in the language of the Wiki article you linked, I am referring to a "voting rule" (i.e. a map from the set of profiles of individual orderings into the set of alternatives).
afreelunch
CC BY-SA 4.0
2021-07-07T11:16:26.943Z
@afreelunch I think you are correct. A better name would be to call it a *social choice function*. See also [here](https://math.stackexchange.com/questions/349446/what-is-the-difference-between-social-choice-function-and-social-welfare-functio). I will change it in my answer.
tdm
CC BY-SA 4.0
2021-07-07T11:39:24.560Z
Agreed! See also Def 7.3 here: http://resources.mpi-inf.mpg.de/departments/d1/teaching/ss15/AGT/lecturenotes7.pdf
afreelunch
CC BY-SA 4.0
2021-07-06T17:29:13.100Z
2021-07-07T11:16:53.277Z
2021-07-07T11:16:53.277Z
tdm
<p>Arrow's impossibility theorem states that, assuming that there are at least 3 alternatives to choose from, it is impossible to produce a 'collective ordering' from individual orderings in a way that satisfies a set of intuitively plausible conditions. In real life, however, we rarely need to generate a collective ordering. For example, when voting for a President, we are not after a full ranking of the candidates -- a selection of one candidate for the role will suffice.</p>
<p><strong>Question:</strong> Does Arrow's impossibility theorem also apply in situations where we are simply looking for a map that specifies a &quot;best alternative&quot; for every possible set of individual orderings?</p>

CC BY-SA 4.0
2021-07-06T16:45:45.427Z
"In real life, however, we rarely need to generate a collective ordering". This is not really accurate, IRL our society has to collectively make economic decisions that require ordering of various choices constantly. We just typically do not use voting systems for it and that's partially thanks to insights by Arrow.
1muflon1
CC BY-SA 4.0
2021-07-06T16:48:21.370Z
@1muflon1 I don't agree -- even when we are making "economic decisions", the main question is to find the BEST choice (that's the choice we're going to implement!) It's much less important to decide on which choice is 5th best, although I can imagine situations where we would also want to agree on that.
afreelunch
CC BY-SA 4.0
2021-07-06T16:49:40.863Z
@1muflon1 But in any case, this is a side issue: my question is about whether Arrow extends to situations when you are just looking for a best choice. Do you have any thoughts on that?
afreelunch
CC BY-SA 4.0
2021-07-06T16:56:01.667Z
1. Arrow impossibility theorem (AIT) is about selecting one option preferred by the community from more than 2 choices, thats what the choice ordering refers to if you have 3 options left center right AIT says that democratically under certain conditions it is impossible to find the best option. 2. Even if we forget about A, our society has more than enough resources typically to select more than one option, most economic optimization problems are about how many different options you can squeeze into the budget constraint
1muflon1
CC BY-SA 4.0
2021-07-06T17:03:27.683Z
@1muflon1 I'm afraid you are just wrong about this. The theorem is about finding a social *ordering* given a set of individual orderings (*not* a social *choice* given a set of individual orderings). Formally, if there are $n \geq 2$ individuals and $O$ is the set of orderings over $k \geq 3$ alternatives (there are $k!$ possible orderings), Arrow's problem is to find a map $O^n \rightarrow O$. The output of this map is an *ordering* in $O$, *not* a "best choice" (as you wrongly assert).
afreelunch
CC BY-SA 4.0
2021-07-06T17:08:41.203Z
I was imprecise with my language the theorem itself is just about the mapping preference but the theorem is derived and used in the context of social choice which is what I meant. When I talk about best option I mean the most preferred
1muflon1
CC BY-SA 4.0
2021-07-06T17:08:51.780Z
Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/127250/discussion-between-afreelunch-and-1muflon1).
afreelunch
<microeconomics>
Does Arrovian impossibility apply when selecting a best alternative?
afreelunch
# Does Arrovian impossibility apply when selecting a best alternative?

(Asked by: username_0 on 2021-07-06)

Arrow's impossibility theorem states that, assuming that there are at least 3 alternatives to choose from, it is impossible to produce a 'collective ordering' from individual orderings in a way that satisfies a set of intuitively plausible conditions. In real life, however, we rarely need to generate a collective ordering. For example, when voting for a President, we are not after a full ranking of the candidates -- a selection of one candidate for the role will suffice.

**Question:** Does Arrow's impossibility theorem also apply in situations where we are simply looking for a map that specifies a "best alternative" for every possible set of individual orderings?

**Tags:** microeconomics
> **Comments:**
> **username_1**: "In real life, however, we rarely need to generate a collective ordering". This is not really accurate, IRL our society has to collectively make economic decisions that require ordering of various choices constantly. We just typically do not use voting systems for it and that's partially thanks to insights by Arrow.  
> **username_0**: @1muflon1 I don't agree -- even when we are making "economic decisions", the main question is to find the BEST choice (that's the choice we're going to implement!) It's much less important to decide on which choice is 5th best, although I can imagine situations where we would also want to agree on that.  
> **username_0**: @1muflon1 But in any case, this is a side issue: my question is about whether Arrow extends to situations when you are just looking for a best choice. Do you have any thoughts on that?  
> **username_1**: 1. Arrow impossibility theorem (AIT) is about selecting one option preferred by the community from more than 2 choices, thats what the choice ordering refers to if you have 3 options left center right AIT says that democratically under certain conditions it is impossible to find the best option. 2. Even if we forget about A, our society has more than enough resources typically to select more than one option, most economic optimization problems are about how many different options you can squeeze into the budget constraint  
> **username_0**: @1muflon1 I'm afraid you are just wrong about this. The theorem is about finding a social *ordering* given a set of individual orderings (*not* a social *choice* given a set of individual orderings). Formally, if there are $n \geq 2$ individuals and $O$ is the set of orderings over $k \geq 3$ alternatives (there are $k!$ possible orderings), Arrow's problem is to find a map $O^n \rightarrow O$. The output of this map is an *ordering* in $O$, *not* a "best choice" (as you wrongly assert).  
> **username_1**: I was imprecise with my language the theorem itself is just about the mapping preference but the theorem is derived and used in the context of social choice which is what I meant. When I talk about best option I mean the most preferred  
> **username_0**: Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/127250/discussion-between-afreelunch-and-1muflon1).  

---

## 1 answers

### Answer by username_2 on 2021-07-06. Score: 4 | Accepted answer

Arrows impossibility theorem deals with what we call a social welfare ordering. For any profile of preferences, it provides a ranking over all alternatives.

What you refer to is called a social choice function (or voting rule). This gives, for any profile of preferences, an "most preferred" option.

The most famous impossibilty theorem for social welfare functions is the Gibbard–Satterthwaite ([see here fore the wiki](https://en.wikipedia.org/wiki/Gibbard%E2%80%93Satterthwaite_theorem), here for the [paper of Gibbard](http://darp.lse.ac.uk/papersdb/Gibbard_(Econometrica_73).pdf) and here [for Satterthwaite's paper](https://www.sciencedirect.com/science/article/abs/pii/0022053175900502)). Another impossibility theorem for social choice functions is the Muller-Satterthwaite Theorem ([see here for the paper](http://www.eecs.harvard.edu/cs286r/courses/fall11/papers/MS77.pdf)).

I guess there are many more...

> **Comments:**
> **username_0**: Thanks, this is useful — Gibbard/Satterthwaite is about choosing best alternatives, not best orderings over alternatives. However, 1) Is this really an analogue of Arrow’s theorem? (I understand that this question is a bit vague!) 2. I think your use of ‘social welfare function’ is a bit off: a SWF generates a complete ranking of alternatives (not just a most preferred option). From the Wiki, I see that the terminology for what I have in mind is a ‘voting rule’.  
> **username_0**: To further elaborate on 2: just as a utility function represents an individual's preference ordering over alternatives, a social welfare function represents a "social ordering" over alternatives. So I am definitely *not* referring to a SWF; in the language of the Wiki article you linked, I am referring to a "voting rule" (i.e. a map from the set of profiles of individual orderings into the set of alternatives).  
> **username_2**: @afreelunch I think you are correct. A better name would be to call it a *social choice function*. See also [here](https://math.stackexchange.com/questions/349446/what-is-the-difference-between-social-choice-function-and-social-welfare-functio). I will change it in my answer.  
> **username_0**: Agreed! See also Def 7.3 here: http://resources.mpi-inf.mpg.de/departments/d1/teaching/ss15/AGT/lecturenotes7.pdf  

---

