Motivated by this paper and its economics motivations, we recall that a social choice among $n$ objects is a continuous function $$f:\overbrace{M\times M\times,\ldots \times M}^{n\;\; times}\to M$$

which satisfy the following conditions:

1)$f(x,x,\ldots,x)=x$

2)f is unchanged under all permutation $\sigma \in S_{n},$ the symmetric group on $n$ elements.

This is a mathematical modeling of the following economic situation:

A client has to choose one item among $n$ items $(x_{1},x_{2},\ldots,x_{n})$. His preference function is denoted by the above $f$.

Now it is natural that we assume that the client faces with n different items, so $x_{i} \neq x_{j},\;\forall i \neq j$. So we consider the ordered configuration space $$F_{n}(M)=\{(x_{1},x_{2},\ldots,x_{n})\in M^{n}\mid x_{i} \neq x_{j},\;\forall i \neq j\} $$.

There is an obvious action of the symmetric group on $F_{n}(M)$.

Now the following question can be counted as an equivariant analogy of the social choice problem:

Assume that $M$ is a manifold which is acted by the symetric group $S_{n}$. Is there always an equivariant continuous map $f:F_{n}(M)\to M$?If not, for what type of manifolds the answer is affirmative?What type of algebraic topological obstructions would appear?

Can we find an economics interpretation for this equivariant version?