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Can any social choice rule that is not complete said to be violating the unrestricted domain condition? Could you provide an example of SCR other than Pareto dominance that is not complete or violates unrestricted domain property?

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    $\begingroup$ Either I misunderstand your question, or it is incredibly trivial. Let $f$ be a social choice function. No matter what the individual preferences are, $f$ does not return a social preference. (Imagine a broken program.) $\endgroup$ – Giskard Nov 16 '19 at 19:42
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Can any social choice rule that is not complete said to be violating the unrestricted domain condition?

Yes.

Could you provide an example of SCR other than Pareto dominance that is not complete or violates unrestricted domain property?

Let society consist of two individuals $i$ and $j$. Let there be exactly two alternatives $A$ and $B$.

Each individual has 3 possible preferences: strictly prefer $A$ to $B$, indifferent between $A$ and $B$, or strictly prefer $B$ to $A$.

Hence, there are 3 × 3 = 9 possible social preference profiles:

  1. $i$ strictly prefers $A$ to $B$ and $j$ strictly prefers $A$ to $B$.
  2. $i$ strictly prefers $A$ to $B$ and $j$ is indifferent between $A$ and $B$.
  3. $i$ strictly prefers $A$ to $B$ and $j$ strictly prefers $B$ to $A$.
  4. $i$ is indifferent between $A$ to $B$ and $j$ strictly prefers $A$ to $B$.
  5. $i$ is indifferent between $A$ to $B$ and $j$ is indifferent between $A$ and $B$.
  6. $i$ is indifferent between $A$ to $B$ and $j$ strictly prefers $B$ to $A$.
  7. $i$ strictly prefers $B$ to $A$ and $j$ strictly prefers $A$ to $B$.
  8. $i$ strictly prefers $B$ to $A$ and $j$ is indifferent between $A$ and $B$.
  9. $i$ strictly prefers $B$ to $A$ and $j$ strictly prefers $B$ to $A$.

We say that a social choice rule (SCR) is complete or satisfies the unrestricted domain condition if it maps (or assigns) each of the above 9 profiles to a preference. (A trivial example of such a SCR is that which maps each of the above 9 profiles to the preference "strictly prefer $B$ to $A$".)

A trivial example of a SCR that is incomplete or violates the unrestricted domain condition is that which maps each of profiles 1–8 to the preference "strictly prefer $B$ to $A$", but fails to map profile 9 to any preference. (Another trivial example is @Giskard's: The SCR which fails to map any of the 9 profiles to any preference.)

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Your first question seems to be terminological. The standard definition of "unrestricted domain" says that the social choice rule produces a complete, transitive preference ranking for every profile of individual preferences. But this is just a definition, and anyone is free to modify it as they see fit for their own purposes, as long as they are clear and consistent in their use of terminology. There are papers that propose social choice rules whose output is an incomplete social preference relation (or even an intransitive social preference relation), and their authors might still describe these rules as satisfying "unrestricted domain" because all they ever wanted was an incomplete (or even intransitive) relation. So I don't think much hangs on this definition one way or the other.

As for your second question: yes, there are examples. Suppose there is a social choice rule F that is "well-behaved" within a certain domain D of preference profiles (e.g. it produces a transitive social preference order), but is very badly behaved outside of the domain D. In that case, it is a common move to think of F as a rule which is only "defined" on the domain D, and hence violates the "unrestricted domain" assumption.

The most obvious example of this is majority rule. In certain domains, majority rule produces transitive social preferences. Outside of these domains, it is badly behaved. Thus, it seems reasonable to think of majority rule as a social choice rule which is only "defined" inside its "domain of transitivity". For example, Eric Maskin and Partha Dasgupta took this approach in their 2008 paper "On the Robustness of Majority Rule" (Journal of the European Economic Association 6 (5) pp.949-973). In this paper, Dasgupta and Maskin showed that majority rule satisfies five desirable properties on a certain domain D of profiles; furthermore, it satisfies these properties on a larger domain than any other social choice rule. (And hence, in this sense, is more "robust" than other rules.)

Another place where a similar strategy is used is in the analysis of strategy-proof social choice rules. Many social choice rules can be shown to be strategy-proof, but only within a certain domain of preferences. For example, the Groves-Clarke pivotal voting rule is only strategy-proof if we assume that all voters have preferences which are quasilinear between money and public goods.

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