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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?

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    $\begingroup$ "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. $\endgroup$
    – 1muflon1
    Jul 6 at 16:45
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    $\begingroup$ @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. $\endgroup$
    – afreelunch
    Jul 6 at 16:48
  • $\begingroup$ @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? $\endgroup$
    – afreelunch
    Jul 6 at 16:49
  • $\begingroup$ 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 $\endgroup$
    – 1muflon1
    Jul 6 at 16:56
  • $\begingroup$ @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). $\endgroup$
    – afreelunch
    Jul 6 at 17:03
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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, here for the paper of Gibbard and here for Satterthwaite's paper). Another impossibility theorem for social choice functions is the Muller-Satterthwaite Theorem (see here for the paper).

I guess there are many more...

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  • $\begingroup$ 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’. $\endgroup$
    – afreelunch
    Jul 7 at 9:06
  • $\begingroup$ 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). $\endgroup$
    – afreelunch
    Jul 7 at 11:05
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    $\begingroup$ @afreelunch I think you are correct. A better name would be to call it a social choice function. See also here. I will change it in my answer. $\endgroup$
    – tdm
    Jul 7 at 11:16
  • $\begingroup$ Agreed! See also Def 7.3 here: resources.mpi-inf.mpg.de/departments/d1/teaching/ss15/AGT/… $\endgroup$
    – afreelunch
    Jul 7 at 11:39

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