Social choice models and topological dynamics

Abstract

In this paper, we propose a topological social choice model in an equivariant setting. We generalize the Chichilnisky–Heal impossibility theorem to the realm of proper actions of almost-connected locally compact groups. We also improve one result stated in Juárez-Anguiano (Topol Appl 279:107246, 2020) about the preservation of equivariant ANR’s by symmetric powers.

1 Introduction

Topological social choice theory is a field in economic theory which studies the possibility of fair aggregation maps of individual preferences when the set of preferences is a topological space. Chichilnisky [15] was the first researcher in studying social choice rules by topological methods. According to [18], a topological social choice rule in the sense of Chichilnisky–Heal is a function $p:X^n \rightarrow X$, where the preferences set X is a topological space, which satisfies the following conditions:

• a) p is a continuous function;

• b) Unanimity:  $p(x,x,\ldots ,x)=x$ for each $x\in X$;

• c) Anonymity:  $p(x_1,\ldots ,x_n)=p(y_1,\ldots ,y_n)$ for every rearrangement $(y_1,\ldots ,y_n)$ of the n-tuple $(x_1,\ldots ,x_n)$.

In a subclass of CW-complexes, Chichilnisky and Heal [18] found necessary and sufficient conditions for the existence of such topological social choice rule. The aggregation rule exists if and only if the preferences space is contractible. Many preferences spaces in economy are spheres and such spaces are not contractible, for this reason the result is known as Chichilnisky–Heal Impossibility Theorem. However, Eckmann [21] tackled and solved this problem many years before with different terminology and context.

The preferences spaces in [18] and [22] are CW-complexes and polyhedra, respectively. In the appendix of the book [29, Theorem 5, p. 317] is proved the well-known result that the category of polyhedra and homotopy classes of continuous functions is equivalent to the category of ANR-spaces and homotopy classes of continuous functions. In this paper, we shall use ANR-spaces because the equivariant version is more convenient in this context than equivariant polyhedra or G-CW-complexes.

In [27], we obtained an equivariant version of the Impossibility Theorem in the class of G-ANR’s assuming that the acting group G is compact.

In some cases, for example in evolutionary game theory (see [33, Chapter 1] or [37]), the preferences are changing with respect to the time; for this reason, the effect of the time should be present in the definition of social choice functions.

Following [23] and motivated by the results in [27], we change the category where a social choice function is defined. We use the theory of topological transformation groups, in particular the theory of proper actions of locally compact groups, to give a treatment of the social choice functions when the action of such group is present in the preferences space.

In Sect. 3, we prove Theorem 3.4 which states that $SP^n(X)$ is a proper G-space with a G-invariant metric whenever X has this properties. Also, we prove Theorem 3.6, which gives the more general result known until now about preservation of G-ANR’s by symmetric powers. A particular case (in the non-equivariant sense) of this result was implicitly used in [16] (see also [27]). Finally, in Sect. 4, we prove Theorem 4.5 which yields a characterization of the existence of topological social choice functions on the class of proper G-spaces.

Topological social choice theory has shown its usefulness not only in economic theory, but also in other areas of social sciences. In [17, 19], its methods were applied to game theory, and in [32, Section 5.4] to psychology.

2 Preliminaries

By a G-space, we mean a completely regular Hausdorff space together with a fixed continuous action $\alpha :G \times X \rightarrow X$ of a locally compact Hausdorff group G on it.

We shall denote by gx the image of the pair $(g, x) \in G \times X$ under the action $\alpha$. If X is a G-space, then for a subset $S \subset X$ and a subgroup $H < G$, the H-hull (or H-saturation) of S is defined by $H(S) = \{hs: h \in H, s \in S \}$. If S is the one point set $\{x\}$, then the H-hull H(S) is denoted by H(x) and called the H-orbit of x. The H-orbit space X/H is always considered with the quotient topology with respect to the orbit projection $\pi :X\rightarrow X/H$. It is well known that $\pi$ is an open map ([28, Proposition 1.54]) and when H is a compact group, it is a perfect map ([14, Ch. I, Theorem 3.1]).

A subset $S \subset X$ is called H-invariant if it coincides with its H-hull, i.e., $S =H(S)$. The H-fixed point set $X ^H$ is defined to be the set $\{x \in X: hx = x, h \in H\}$.

If X and Y are two G-spaces, then a continuous map $f: X \rightarrow Y$ is called a G-map or an equivariant map, if $f (gx) = g f (x)$ for every $x \in X$ and $g \in G$. If G acts trivially on Y then we use the term “invariant map” (or “G-invariant map”) instead of “equivariant map”. Any G-map $f: X \rightarrow Y$ sends the H-fixed point set $X^ H$ in the H-fixed point set $Y^H$. We shall denote by $f^H$ the restriction $f |_{X ^H}$. In [26, Chapter IV], it is shown that the operation of taking fixed-point sets is functorial.

The notion of a proper G-space in question was introduced in 1961 by R. Palais [31] with the purpose to extend a substantial portion of the theory of compact Lie group actions to the case of noncompact ones.

Recall that a G-space X is called proper (in the sense of Palais [31, Definition 1.2.2]), if each point of X has a, so called, small neighborhood, i.e., a neighborhood V such that for every point of X, there is a neighborhood U with the property that the set $\langle U,V\rangle =\{g\in G \ | \ gU\cap V\not = \emptyset \}$ has compact closure in G. Clearly, if G is compact, then every G-space is proper.

In [2, Proposition 1.2], it is proved that a finite union of small sets is a small set and if V is a small set and $U\subset V$ then U is a small set. It is well known [31, Proposition 1.2.8] that for each proper G-space X, its orbit space X/G is Tychonoff.

A compatible metric $\rho$ on a metrizable G-space X is called invariant or G-invariant, if $\rho (gx, gy)=\rho (x, y)$ for all $g\in G$ and $x, y\in X$. If $\rho$ is a G-invariant metric on a G-space X, then it is easy to verify that the formula

$$\begin{aligned} \widetilde{\rho }\bigl (G(x), G(y)\bigr )=\inf \{\rho (x', y') \ | \ \ x'\in G(x), \ y'\in G(y)\} \end{aligned}$$

(1)

defines a pseudometric $\widetilde{\rho }$, compatible with the quotient topology of X/G. If, in addition, X is a proper G-space then $\widetilde{\rho }$ is, in fact, a metric on X/G (see [10, Theorem B]).

A locally compact group G is called almost-connected if the quotient group $G/G_0$ is compact, where $G_0$ is the connected component of the identity. Such a group has a maximal compact subgroup K, i.e., every compact subgroup of G is conjugate to a subgroup of K [1, Theorem A.5].

We are specially interested in the class G-$\mathcal {M}$, which consists of all metrizable proper G-spaces X that are metrizable by a G-invariant metric. In [9, Theorem 3.4], it is proved that if G is almost-connected, then every strongly metrizable proper G-space X admits a compatible invariant metric. This class includes the separable, locally separable and locally compact metrizable proper G-spaces.

For each $n\ge 2$ and a topological space X, the symmetric group $S_n$ acts on $X^n$ permuting the coordinates, the space $SP^n(X)=X^n/S_n$ is the n-th symmetric power of X. Moreover, if Y is another topological space and $f:X\rightarrow Y$ is a continuous function, there exists a induced continuous function $SP^n(f):SP^n(X)\rightarrow SP^n(Y)$ given by $SP^n(f)([x_1,\ldots , x_n])=[f(x_1),\ldots ,f(x_n)]$. It is easy to see that $SP^n(-)$ is a functor in the category of topological spaces and continuous functions.

It is clear that the mapping $\Delta : X\hookrightarrow SP^n(X)$, where $\Delta (x)=(x,x,\ldots ,x)$ for each $x\in X$, is an embedding.

If X is a metrizable G-space with the action $\alpha : G\times X\rightarrow X$, then for each $n\ge 2$, $SP^n(X)$ is a metrizable G-space with the metric induced by its $S_n$-orbit space topology and with the action $\overline{\alpha }:G\times SP^n(X)\rightarrow SP^n(X)$ defined by

$$\begin{aligned} g[x_1,\ldots , x_n]=[gx_1,\ldots , gx_n]. \end{aligned}$$

Indeed, since $S_n$ is a finite group, the orbit space $X^n/{S_n}$ is metrizable, with the metric given by the formula (1). Now, consider the following commutative diagram

The diagonal action $\alpha ^n$ in the product space $X^n$ is continuous and the orbit projection $\pi$ is continuous and open. These facts give the continuity of $\overline{\alpha }$.

Basmanov [12] proved this affirmation for a compact G-space X (and a compact group G) in G-$\mathcal {M}$. Later, in Lemma 3.1, we shall prove that the induced G-action on the symmetric power is proper whenever X is a proper G-space.

Let I be the unit interval with its usual topology. A G-homotopy $F: X \times I \rightarrow Y$ is a homotopy such that $F(gx, t) = gF(x, t)$ for every $x \in X$, $g \in G$ and $t \in I$. Two G-maps $f_0, f_1: X \rightarrow Y$ are G-homotopic, if there exists a G-homotopy $F: X \times I \rightarrow Y$ such that $F(x, 0) = f_0(x)$ and $F(x, 1) = f_1(x)$ for all $x \in X$. We write $f_0 \simeq _G f_1$ if $f_0$ and $f_1$ are G-homotopic. Two G-spaces X and Y are G-homotopy equivalent if there exist G-maps $f: X \rightarrow Y$ and $g: Y \rightarrow X$ such that $g \circ f \simeq _G 1_X$ and $f \circ g \cong _G 1_Y$. A space X is G-contractible if it is G-homotopy equivalent to a space with one point. For $G=\{e\}$, we obtain the class of contractible spaces.

For homology groups of ANR-spaces see [25].

A G-space $Y\in G$-$\mathcal {M}$ is called an equivariant absolute neighborhood retract for the class G-$\mathcal {M}$ (notation: $Y \in G$-ANR or Y is a G-ANR-space), provided that for any closed G-embedding of Y in a G-space $X \in G$-$\mathcal {M}$, there exists a G-retraction $r:U \rightarrow Y$, where U is an invariant neighborhood of Y in X. If, in addition, one can always take $U = X$, then we say that Y is an equivariant absolute retract for the class G-$\mathcal {M}$ (notation: $Y \in G$-AR or Y is a G-AR-space). For $G=\{e\}$ we obtain the classes of ANR-spaces and AR-spaces, respectively.

Similarly, a G-space $Y\in G$-$\mathcal {M}$ is called an equivariant absolute neighborhood extensor for the class G-$\mathcal {M}$ (notation: $Y \in G$-ANE or Y is a G-ANE-space), if for each $X\in G$-$\mathcal {M}$ and every invariant closed subset A of X, every equivariant map $f:A\rightarrow Y$ can be extended to an equivariant map $f:U\rightarrow Y$, where U is an invariant neighborhood of A in X. If one can always take $U=X$, then we say that Y is an equivariant absolute extensor for the class G-$\mathcal {M}$ (notation: $Y \in G$-AE or Y is a G-AE-space). For $G=\{e\}$ we obtain the classes of ANE-spaces and AE-spaces.

In [3, Corollary 6.3], it is proved the following affirmation: a G-space $X\in G$-$\mathcal {M}$ is a G-ANR-space (resp., a G-AR-space) if and only if X is a G-ANE-space (resp., a G-AE-space).

In [25, Theorem 7.1] it is stated that contractible ANR-spaces are precisely AR-spaces. So the existence of a topological social choice rule, in the Chichilnisky–Heal’s sense, is equivalent to the preferences space being an AR-space.

Definition 2.1

Let X be a metrizable space and $n\ge 2$. A topological social choice rule $p:X^n\rightarrow X$ is a map which satisfies the following properties

• a) Unanimity:  $p(x,x,\ldots ,x)=x$ for each $x\in X$;

• b) Anonymity:  p is $S_n$-invariant with respect to the natural action of the symmetric group $S_n$ on $X^n$ and the trivial action on X.

According to [18, 21, 22], this is the topological social choice rule in the sense of Eckmann–Chichilnisky–Heal. Since contractible ANR-spaces are AR-spaces, we have the following result whose proof can be found in [18, 21, 23].

Theorem 2.2

(Eckmann–Chichilnisky–Heal Impossibility Theorem) Let X be a connected ANR-space with finitely generated homology groups and almost all vanish, then the following statements are equivalent

• a) there exists a topological social choice rule $p:X^n\rightarrow X$ for every $n\ge 2$;

• b) there exists a topological social choice rule $p:X^n\rightarrow X$ for some $n\ge 2$;

• c) X is an AR-space.

Remark 2.3

The equivalence $a)\Leftrightarrow c)$ in Theorem 2.2 can be consulted in [22, Theorem 3] (the hypothesis that almost all homology groups vanish is not necessary here). Implication $a)\Rightarrow b)$ is trivial. The most important part is implication $b\Rightarrow c)$. For spaces with the homotopy type of compact polyhedra, a proof can revised in [23, Corollary 3.2] (see also [35, Theorem 1.1]). But, as is remarked in the third paragraph of [23, p. 90], their proof is valid for spaces with the homotopy type of polyhedra (non necessary compact) with finitely generated homology groups and almost all vanish.

Basic facts in the realms of transformation groups and topological dynamics can be found in [14, 20, 36]. For the terms of general topology the reader can consult [24].

3 Preservation of proper G-ANR-spaces by symmetric powers

In this section, we are going to prove some properties of the functor $SP^n(-)$. First of all, we shall prove that this functor preserves properness of the action.

Lemma 3.1

Let X be a proper G-space in G-$\mathcal {M}$. Then for each $n\ge 2$, $SP^n(X)$ is a proper G-space.

Proof

We know that the class of proper G-spaces is closed under finite products, which implies that $X^n$ is a proper G-space. Take $\tilde{x}\in SP^n(X)$ and let $\pi :X^n\rightarrow X^n/{S_n}\cong SP^n(X)$ be the $S_n$-orbit projection. Consider the fiber of $\tilde{x}$, $\pi ^{-1}(\tilde{x})=\{x_1,\ldots , x_k\}$, then for each $i\in \{1,\ldots ,k\}$, there exists a small neighborhood $V_i$ of $x_i$ in $X^n$. Let $O=\cup _{i=1}^k V_i$. Since $S_n$ is a finite group and $\pi ^{-1}(\tilde{x})\subset O$, there exists an $S_n$-invariant neighborhood W of $\pi ^{-1}(\tilde{x})$ such that $W\subset O$. Hence, W is a small $S_n$-invariant neighborhood of $\pi ^{-1}(\tilde{x})$.

We put $\tilde{V}=\pi (W)$ and prove that $\tilde{V}$ is a small neighborhood of $\tilde{x}$ in $SP^n(X)$. Indeed, for each $\tilde{y} \in SP^n(X)$, consider $\pi ^{-1}(\tilde{y})=\{y_1,\ldots ,y_m\}$, hence for each $i\in \{1,\ldots ,m\}$, there exists a neighborhood $W_i$ of $y_i$ such that the set $\langle W,W_i\rangle$ has compact closure in G. Since $\pi ^{-1}(y)\subset \cup _{i=1}^m W_i$, there exists a $S_n$-invariant neighborhood U of $\pi ^{-1}(y)$ with $U\subset \cup _{i=1}^m W_i$. Thus, $\tilde{U}= \pi (U)$ is a neighborhood of $\tilde{y}$ and

$$\begin{aligned} \langle \tilde{V},\tilde{U}\rangle \subset \bigcup _{i=1}^m \langle W,W_i\rangle . \end{aligned}$$

Hence, $\langle \tilde{V},\tilde{U}\rangle$ has compact closure in G. This finishes the proof. $\square$

Now, we are going to prove some properties concerning the dynamics of the actions of $G\times S_n$ and G in the spaces $X^n$ and $SP^n(X)$, respectively.

Lemma 3.2

Let X be a proper G-space in G-$\mathcal {M}$. Then for each $n\ge 2$, $X^n$ is a proper $G\times S_n$-space with the induced action given by

$$\begin{aligned} (g,\sigma )(x_1,x_2\ldots ,x_n)=(gx_{\sigma ^{-1}(1)},gx_{\sigma ^{-1}(2)},\ldots , gx_{\sigma ^{-1}(n)}). \end{aligned}$$

Proof

Let $\pi :X^n\rightarrow SP^n(X)$ be the $S^n$-orbit projection. Since $S_n$ acts trivially in $SP^n(X)$, it is a proper $G\times S_n$-space by Lemma 3.1. Now [2, Proposition 1.2 (f)] gives that $X^n$ is a proper $G\times S_n$-space. $\square$

Lemma 3.3

Let X be a proper G-space in G-$\mathcal {M}$ such that for each $n\ge 2$, $X^n$ is equipped with the induced $G\times S_n$-action defined in Lemma 3.2. Then,

• i) $X^n$ has a $G\times S_n$-invariant metric compatible with its topology;

• ii) $SP^n(X)$ has a G-invariant metric compatible with its topology.

Proof

• i) We first choose a G-invariant metric, say $\rho$, on X. Then, the function $\rho ':X^n\times X^n \rightarrow \mathbb {R}$ given by

  $$\begin{aligned} \rho '((x_1\ldots , x_n), (y_1,\ldots , y_n))=\rho (x_1,y_1)+\cdots + \rho (x_n,y_n) \end{aligned}$$

  is $G\times S_n$-invariant metric on $X^n$.

• ii) Consider $\rho '$ defined en part (i). Then, the formula

  $$\begin{aligned} d(\tilde{x},\tilde{y})=\min \{\rho '(x', y'): x' \in \tilde{x}, y'\in \tilde{y}\} \end{aligned}$$

  defines a G-invariant metric in $SP^n(X)$ compatible with its topology.

$\square$

Combining Lemmas 3.1 and 3.3(ii), we have the following result.

Theorem 3.4

Let X be a proper G-space in G-$\mathcal {M}$. Then for each $n\ge 2$, $SP^n(X)\in G$-$\mathcal {M}$.

We will use the following theorem due to S. Antonyan [7].

Theorem 3.5

[7, Theorem 3(1)] Let G be an almost-connected locally compact group and X be a proper G-ANE-space(respectively, any G-AE). Then for each compact normal subgroup H of G, X/H is a proper G/H-ANE-space(respectively, a proper G-AE-space).

This theorem and some ideas in the proof of [5, Theorem 1] give the following result.

Theorem 3.6

Let G be an almost-connected locally compact group and X be a metrizable proper G-space. If $X\in G$-ANR (resp., G-AR) then $SP^n(X)\in G$-ANR (resp., G-AR) for each $n\ge 2$.

Proof

Let X be a G-ANR-space and $n\ge 2$ (for G-AR-spaces the proof is similar). We consider $X^n\cong C_\kappa ({\textbf {n}},X)$, the space of continuous functions from the discrete space ${\textbf {n}}=\{1,2,\ldots ,n\}$ into X with the compact-open topology. It is a $G\times S_n$-space with the action defined in Lemma 3.2.

We are going to prove that $X^n$ is a proper $G\times S_n$-ANE-space. Lemmas 3.2 and 3.3 (i) imply that $X^n$ is a proper $G\times S_n$-space with a $G\times S_n$-invariant metric.

Let (Y, A) be a $G\times S_n$-pair and $f:A\rightarrow C_\kappa ({\textbf {n}},X)$ a $G\times S_n$-map. Consider the map

$$\begin{aligned}f^\star :A\times {\textbf {n}}\rightarrow X \end{aligned}$$

defined by $f^\star (a,k)=f(a)(k)$. It is continuous by [24, Theorem 3.4.8]. In $A\times {\textbf {n}}$ we have the action $(g,\sigma )(a,k)=((g,\sigma )a, \sigma k))$. Take $( a,k)\in A\times {\textbf {n}}$, then

$$\begin{aligned} f^\star ((e,\sigma )(a,k))=f((e,\sigma )a)(\sigma k)=(e,\sigma )f(a)(\sigma k)&= \\ f(a)(\sigma ^{-1}\sigma k)=f(a)(k)&=f^\star (a,k). \end{aligned}$$

Therefore, this map is $\{e\}\times S_n$-invariant. Hence, the induced map $f^\star /S_n:(A\times {\textbf {n}})/{S_n}\rightarrow X$ is G-equivariant.

By hypothesis, there exist a G-invariant neighborhood U of $(A\times {\textbf {n}})/{S_n}$ in $(Y\times {\textbf {n}})/{S_n}$ and a G-map $\varphi :U\rightarrow X$ which extends $f^\star /S_n$. Let $q:Y\times {\textbf {n}}\rightarrow (Y\times {\textbf {n}})/S_n$ be the $S_n$-orbit projection. We define $V=q^{-1}(U)$ and $F^\star =\varphi \circ q$. V is an invariant neighborhood of $A\times {\textbf {n}}$ in $Y\times {\textbf {n}}$ and $F^\star$ is a continuous mapping. Since ${\textbf {n}}$ is finite, there exists an invariant neighborhood W of A in X such that $A\times {\textbf {n}}\subset W\times {\textbf {n}}\subset V$. Now, we consider the mapping $F:W\rightarrow C_\kappa ({\textbf {n}},X)\cong X^n$ defined by $F(x)(k)=F^\star (x,k)$. It is continuous by [24, Theorem 3.4.1] and satisfies that

$$\begin{aligned} F((g,\sigma )x)(k)=F^\star ((g,\sigma )x,k)&=F^\star (x,\sigma ^{-1}k)\\&=F(x)(\sigma ^{-1}k)=[(g,\sigma )F(x)](k). \end{aligned}$$

Therefore, F is a $G\times S_n$-mapping. It implies that $X^n$ is a proper $G\times S_n$-ANE-space. Hence by Theorem 3.5, $SP^n(X)\cong X^n/{S_n}$ is a proper G-ANR-space. $\square$

Remark 3.7

Results related to Theorem 3.6 have been stated in [6, 12, 27, 38].

The hypothesis about almost-connected of the acting group relies in Theorem 3.5. So, we state the following question.

Problem 3.8

For an arbitrary locally compact group G and a proper G-A(N)R-space X, is it true that for each $n\ge 2$, $SP^n(X)$ is a proper G-A(N)R-space?.

4 The equivariant social choice model

In this section, we shall define the main notion in this paper and prove the equivariant version of the Eckmann–Chichilnisky–Heal theorem.

The following result generalizes [8, Corollary 1.2] to the realm of proper actions of almost-connected groups.

Theorem 4.1

Let G be an almost-connected locally compact group and X be a proper G-space with an invariant metric. If $X\in G$-ANR and for each compact subgroup H of G, $X^H\in AR$, then $X\in G$-AR.

Proof

Let K be the maximal compact subgroup of G. By [7, Proposition 4(i)], X is a K-ANR. Then for each compact subgroup H of K, $X^H$ is AR. Therefore by [8, Corollary 1.2], X is a K-AR. Hence by [7, Proposition 4(i)], X is a G-AR. $\square$

Preservation of absolute extensors by fixed-point sets was proved by Smirnov [34]. For absolute neighborhood extensors, the reader can see [26, Proposition 8.45]. Another formulation in terms of G-ANR-spaces can be consulted in [4].

Proposition 4.2

Let G be an almost-connected locally compact group and X be a proper G-space which is a G-ANR (resp., G-AR). Then for each compact subgroup H of G, $X^H\in ANR$ (resp. AR).

Proof

Let X be a G-ANR-space (for G-AR-spaces the proof is similar) and H be a compact subgroup of G. Since G has a maximal compact subgroup K, then there exits $g\in G$ such that $H'=gHg^{-1}\subset K$. By [7, Proposition 4(i)], X is a K-ANR. Now, by [26, Proposition 8.45], $X^{H'}$ is an ANE-space. Besides by [28, Proposition 1.52], $X^H\cong X^{H'}$. Hence $X^H\in ANR$. $\square$

Remark 4.3

In Definition 2.1, properties a) and b) are equivalent to the existence of a continuous function $\tilde{p}:SP^n(X)\rightarrow X$ such that the following diagram is commutative

where $\pi : X^n\rightarrow SP^n(X)$ is the orbit projection corresponding to the action of $S_n$ in $X^n$. The purpose of presenting the conditions a) and b) in the definition of the topological social choice functions in terms of a commutative diagram, is that we can change the category where these functions are defined.

Definition 4.4

Let X be a G-space. An equivariant social choice function is a map $p:X^n\rightarrow X$, with $n\ge 2$, such that all arrows in the commutative diagram of Remark 4.3 are G-functions.

Now, we are ready to give an equivariant version of the Theorem 2.2.

Theorem 4.5

Let G be an almost-connected group and X be a proper G-ANR-space such that for each compact subgroup H of G, $X^H$ is a connected space with finitely generated homology groups and almost all vanish. Then the following conditions are equivalent

• a) there exists an equivariant topological social choice rule $p:X^n\rightarrow X$ for every $n\ge 2$;

• b) there exists an equivariant topological social choice rule $p:X^n\rightarrow X$ for some $n\ge 2$;

• c) X is a proper G-AR-space.

Proof

First, we show that $c)\Rightarrow a)$. Let X be a proper G-AR-space. By [3, Theorem 6.1] there exists a linear G-space L such that X is embedded as an invariant closed subspace of a convex set V of L and V is a proper G-space. Define the G-map $q:X^n\rightarrow V$ by the formula

$$\begin{aligned} q((x_1,x_2,\ldots ,x_n))=\frac{1}{n}\sum _{i=1}^{n} x_i. \end{aligned}$$

Since X is a G-AR-space, there exists a G-retraction $r:V\rightarrow X$. Hence, the map $p=q\circ r$ is an equivariant topological social choice rule.

Implication $a)\Rightarrow b)$ is trivial.

Now, let us show $b)\Rightarrow c)$. Suppose that for some $n\ge 2$, there exists an equivariant topological social choice rule $p:X^n\rightarrow X$. Then, by Remark 4.3, we have a G-map $\tilde{p}:SP^n(X)\rightarrow X$ which satisfies the following commutative diagram

Let H be a compact subgroup of G. Since taking fixed points is a functor, we have that the following diagram commutes

Now, let us show that $SP^n(X^H)\subseteq (SP^n(X))^H$. Pick $[x_1,\ldots , x_n]\in SP^n(X^H)$ and $h\in H$. Then,

$$\begin{aligned}{}[x_1,\ldots , x_n]=[hx_1,\ldots , hx_n]=h[x_1,\ldots , x_n]. \end{aligned}$$

Hence, $[x_1,\ldots , x_n]\in (SP^n(X))^H$. Let $i_H:SP^n(X^H)\hookrightarrow (SP^n(X))^H$ be the inclusion. Consider the orbit map $\pi _H:(X^H)^n\rightarrow SP^n(X^H)$ and the embedding $\Delta _H:X^H\rightarrow SP^n(X^H)$. Since $(X^H)^n=(X^n)^H$, we have the following commutative diagram

Hence, if we put $\tilde{p}_H=\tilde{p}^H\circ i_H$, we get the following commutative diagram

Therefore, by Remark 4.3, $p^H$ is a topological social choice rule. By Proposition 4.2$X^H$ is an ANR-space and by hypothesis it is a connected space with finitely generated homology groups and almost all vanish. Then, by virtue of Theorem 2.2, $X^H$ is an AR-space. Hence, by Theorem 4.1, we have that X is a G-AR-space. This finishes the proof. $\square$

From Theorems 4.5 and 3.6, we obtain the following result.

Theorem 4.6

Let G be an almost-connected group and X be a proper G-ANR-space such that for each compact subgroup H of G, $X^H$ is a connected space with finitely generated homology groups and almost all vanish. If there exists an equivariant topological social choice rule $p:X^n\rightarrow X$, then $SP^n(X)$ is a G-AR-space.

The following problem arises naturally.

Problem 4.7

Is Theorem 4.5 valid without the hypothesis of almost-connectedness in the acting group?.

Remark 4.8

The results stated in this section offer a way to study evolutionary game theory (or more general dynamic games) with equivariant social choice rules in the same manner as topological social choice rules were related with game theory in [17, 19, 32].

In the language of topological dynamics, the dynamical systems (i.e., $\mathbb {R}$-spaces) satisfying the condition of properness are called dispersive [13, Ch. IV]. So, we finish with the following problem.

Problem 4.9

Find a characterization of the existence of equivariant topological social choice rules in other subclasses of dynamical systems.