Quasi-stationary social welfare functions

Abstract

This paper examines collective decision-making with an infinite-time horizon setting. First, we establish a result on the collection of decisive sets: if there are at least four alternatives and Arrow’s axioms are satisfied on the selfish domain, then the collection of decisive sets forms an ultrafilter. Second, we impose generalized versions of stationarity axiom for social preferences, which are substantially weaker than the standard version. We show that if any of our generalized versions are satisfied in addition to Arrow’s axioms, then some generation is dictatorial. Moreover, we specify a very weak stationarity axiom that guarantees a possibility result.

Appendix

1.1 Proof of Theorem 2

Lemma 1

Suppose that$|X| \ge 4$and a social welfare functionfsatisfies selfish domain, weak Pareto, and independence of irrelevant alternatives. If a set$T \subseteq {\mathbb {N}}$is decisive over some ordered pairs$({\mathbf{x}},{\mathbf{y}})$, thenTis decisive over all distinct pairs.

Proof

Suppose that a set $T \subseteq {\mathbb {N}}$ is decisive over $({\mathbf{x}},{\mathbf{y}})$. We will show that T is decisive over all distinct pairs $({\mathbf{w}},{\mathbf{z}})$.

(i) We show $T \subseteq {\mathbb {N}}$ is decisive over $(\mathbf{y},{\mathbf{z}})$ (${\mathbf{z}} \ne {\mathbf{x}},{\mathbf{y}}$). Note that it can be the case that $y_t=z_t$ or $x_t=z_t$ for some $t \in {\mathbb {N}}$. Consider ${\mathbf{w}} \in X^{\infty }$ such that

$$\begin{aligned} w_t \in X {\setminus } \{ x_t,y_t,z_t \} \text{ for } \text{ all } t \in {\mathbb {N}}. \end{aligned}$$

Let ${\mathbf {R}} \in {\mathcal {R}}_S^\infty$ be such that

$$\begin{aligned} {\mathbf{x}} P(R_t){\mathbf{y}} P(R_t) {\mathbf{w}}&\text{ for } \text{ all } t \in T,\\ {\mathbf{y}}P(R_t) {\mathbf{w}}&\text{ for } \text{ all } t \in {\mathbb {N}}{\setminus } T. \end{aligned}$$

Since T is decisive over $({\mathbf{x}},{\mathbf{y}})$, it follows that ${\mathbf{x}}P(f({\mathbf {R}})) {\mathbf{y}}$. Weak Pareto implies that $\mathbf{y} P(f({\mathbf {R}})) {\mathbf{w}}$. By the transitivity of $f({\mathbf {R}})$, we have ${\mathbf{x}} P(f({\mathbf {R}})) {\mathbf{w}}$. Since the preferences of individuals outside of T over ${\mathbf{x}}$ and ${\mathbf{w}}$ are not specified, independence of irrelevant alternatives implies that T is decisive over $({\mathbf{x}},{\mathbf{w}})$.

Let ${\mathbf {R}}' \in {\mathcal {R}}_S^\infty$ be such that

$$\begin{aligned} {\mathbf{x}} P(R'_t){\mathbf{w}} P(R'_t) {\mathbf{z}}&\text{ for } \text{ all } t \in T,\\ {\mathbf{x}}P(R'_t) {\mathbf{w}}&\text{ for } \text{ all } t \in {\mathbb {N}}{\setminus } T. \end{aligned}$$

Since T is decisive over $({\mathbf{x}},{\mathbf{w}})$, it follows that ${\mathbf{x}}P(f({\mathbf {R}}')) {\mathbf{w}}$. Weak Pareto implies that $\mathbf{w} P(f({\mathbf {R}}')) {\mathbf{z}}$. By the transitivity of $f({\mathbf {R}}')$, we have ${\mathbf{x}} P(f({\mathbf {R}}')) {\mathbf{z}}$. Since the preferences of individuals outside of T over ${\mathbf{x}}$ and ${\mathbf{z}}$ are not specified, independence of irrelevant alternatives implies that T is decisive over $({\mathbf{x}},{\mathbf{z}})$.

Let ${\mathbf {R}}'' \in {\mathcal {R}}_S^\infty$ be such that

$$\begin{aligned} {\mathbf{w}} P(R''_t){\mathbf{x}} P(R''_t) {\mathbf{z}}&\text{ for } \text{ all } t \in T,\\ {\mathbf{w}}P(R''_t) {\mathbf{x}}&\text{ for } \text{ all } t \in {\mathbb {N}}{\setminus } T. \end{aligned}$$

Since T is decisive over $({\mathbf{x}},{\mathbf{z}})$, it follows that ${\mathbf{x}}P(f({\mathbf {R}}'')) {\mathbf{z}}$. Weak Pareto implies that ${\mathbf{w}} P(f({\mathbf {R}}'')) {\mathbf{x}}$. By the transitivity of $f({\mathbf {R}}'')$, we have ${\mathbf{w}} P(f({\mathbf {R}}'')) {\mathbf{z}}$. Since the preferences of individuals outside of T over ${\mathbf{w}}$ and ${\mathbf{z}}$ are not specified, independence of irrelevant alternatives implies that T is decisive over $({\mathbf{w}},{\mathbf{z}})$.

Let ${\mathbf {R}}''' \in {\mathcal {R}}_S^\infty$ be such that

$$\begin{aligned} {\mathbf{y}} P(R''_t){\mathbf{w}} P(R''_t) {\mathbf{z}}&\text{ for } \text{ all } t \in T,\\ {\mathbf{y}}P(R''_t) {\mathbf{w}}&\text{ for } \text{ all } t \in {\mathbb {N}}{\setminus } T. \end{aligned}$$

Since T is decisive over $({\mathbf{w}},{\mathbf{z}})$, it follows that ${\mathbf{w}}P(f({\mathbf {R}}'')) {\mathbf{z}}$. Weak Pareto implies that ${\mathbf{y}} P(f({\mathbf {R}}'')) {\mathbf{w}}$. By the transitivity of $f({\mathbf {R}}'')$, we have ${\mathbf{y}} P(f({\mathbf {R}}'')) {\mathbf{z}}$. Since the preferences of individuals outside of T over ${\mathbf{y}}$ and ${\mathbf{z}}$ are not specified, independence of irrelevant alternatives implies that T is decisive over $({\mathbf{y}},{\mathbf{z}})$.

(ii) We show $T \subseteq {\mathbb {N}}$ is decisive over $(\mathbf{y},{\mathbf{x}})$. Take ${\mathbf{z}} \ne {\mathbf{x}},{\mathbf{y}}$. From (i), $T \subseteq {\mathbb {N}}$ is decisive over $({\mathbf{y}},{\mathbf{z}})$. Consider ${\mathbf{w}} \in X^{\infty }$ such that

$$\begin{aligned} w_t \in X {\setminus } \{ x_t,y_t,z_t \} \text{ for } \text{ all } t \in {\mathbb {N}}. \end{aligned}$$

Let ${\mathbf {R}} \in {\mathcal {R}}_S^\infty$ be such that

$$\begin{aligned} {\mathbf{y}} P(R_t){\mathbf{z}} P(R_t) {\mathbf{w}}&\text{ for } \text{ all } t \in T,\\ {\mathbf{z}}P(R_t) {\mathbf{w}}&\text{ for } \text{ all } t \in {\mathbb {N}}{\setminus } T. \end{aligned}$$

Since T is decisive over $({\mathbf{y}},{\mathbf{z}})$, it follows that ${\mathbf{y}}P(f({\mathbf {R}})) {\mathbf{z}}$. Weak Pareto implies that $\mathbf{z} P(f({\mathbf {R}})) {\mathbf{w}}$. By the transitivity of $f({\mathbf {R}})$, we have ${\mathbf{y}} P(f({\mathbf {R}})) {\mathbf{w}}$. Since the preferences of individuals outside of T over ${\mathbf{y}}$ and ${\mathbf{w}}$ are not specified, independence of irrelevant alternatives implies that T is decisive over $({\mathbf{y}},{\mathbf{w}})$.

Let ${\mathbf {R}}' \in {\mathcal {R}}_S^\infty$ be such that

$$\begin{aligned} {\mathbf{y}} P(R'_t){\mathbf{w}} P(R'_t) {\mathbf{x}}&\text{ for } \text{ all } t \in T,\\ {\mathbf{w}}P(R'_t) {\mathbf{x}}&\text{ for } \text{ all } t \in {\mathbb {N}}{\setminus } T. \end{aligned}$$

Since T is decisive over $({\mathbf{y}},{\mathbf{w}})$, it follows that ${\mathbf{y}}P(f({\mathbf {R}}')) {\mathbf{w}}$. Weak Pareto implies that $\mathbf{w} P(f({\mathbf {R}}')) {\mathbf{x}}$. By the transitivity of $f({\mathbf {R}}')$, we have ${\mathbf{y}} P(f({\mathbf {R}}')) {\mathbf{x}}$. Since the preferences of individuals outside of T over ${\mathbf{x}}$ and ${\mathbf{y}}$ are not specified, independence of irrelevant alternatives implies that T is decisive over $({\mathbf{y}},{\mathbf{x}})$.

(iii) From (i) and (ii), we have the following:

$$\begin{aligned}{}[T \text{ is } \text{ decisive } \text{ over } ({\mathbf{x}},{\mathbf{y}})] \Rightarrow [T \text{ is } \text{ decisive } \text{ over } ({\mathbf{y}},{\mathbf{z}})]. \end{aligned}$$

By repeating the same argument, we have

$$\begin{aligned}{}[T \text{ is } \text{ decisive } \text{ over } ({\mathbf{y}},{\mathbf{z}})] \Rightarrow [T \text{ is } \text{ decisive } \text{ over } ({\mathbf{z}},{\mathbf{w}})]. \end{aligned}$$

Then T is decisive over all distinct pairs $({\mathbf{w}},{\mathbf{z}})$. The proof is complete. $\square$

Proof of Theorem 2

(U1) and (U2) directly follow from weak Pareto and the definition of decisiveness. Thus, it suffices to show (U3) and (U4).

(U3) Suppose that $T,T' \in {\mathcal {T}}(f)$. Choose any triple ${\mathbf{x}},{\mathbf{y}},{\mathbf{z}} \in X^\infty$ such that $x_t,y_t,z_t$ are distinct for all $t \in {\mathbb {N}}$. Since there exist at least four per-period outcomes, this pair exists. Let ${\mathbf {R}} \in {\mathcal {R}}_S^\infty$ be such that

$$\begin{aligned} {\mathbf{x}} P(R_t){\mathbf{y}}&\text{ for } \text{ all } t \in T {\setminus } T',\\ {\mathbf{x}} P(R_t) {\mathbf{y}} P(R_t) {\mathbf{z}}&\text{ for } \text{ all } t \in T \cap T',\\ {\mathbf{y}}P(R_t) {\mathbf{z}}&\text{ for } \text{ all } t \in T' {\setminus } T. \end{aligned}$$

Since $T \in {\mathcal {T}}(f)$ and ${\mathbf{x}} P(R_t){\mathbf{y}}$ for all $t \in T$, we have ${\mathbf{x}} P(f({\mathbf {R}})) {\mathbf{y}}$. Since $T' \in \Omega (F)$ and ${\mathbf{y}}P(R_t) {\mathbf{z}}$ for all $t \in T'$, we have ${\mathbf{y}} P(f({\mathbf {R}})) {\mathbf{z}}$. By the transitivity of $f({\mathbf {R}})$, ${\mathbf{x}} P(f({\mathbf {R}})) {\mathbf{z}}$. Since the preferences of individuals outside of $T \cap T'$ over ${\mathbf{x}}$ and ${\mathbf{z}}$ are not specified, independence of irrelevant alternatives implies that $T \cap T'$ is decisive over $({\mathbf{x}},{\mathbf{z}})$. By Lemma 1, $T \cap T' \in {\mathcal {T}}(f)$.

(U4) Suppose that $T \notin {\mathcal {T}}(f)$. Then, there exists a profile ${\mathbf {R}} \in {\mathcal {R}}_S^{\infty }$ and ${\mathbf{x}},{\mathbf{y}} \in X^\infty$ such that ${\mathbf{x}} P(R_t){\mathbf{y}}$ for all $t \in T$ and ${\mathbf{y}} f({\mathbf {R}}) {\mathbf{x}}$. Let ${\mathbf {R}}' \in {\mathcal {R}}_S^{\infty }$ be such that

$$\begin{aligned}&{\mathbf {R}}|_{\{ {\mathbf{x}}, {\mathbf{y}}\}} = {\mathbf {R}}'|_{\{ {\mathbf{x}}, {\mathbf{y}}\}},\\&{\mathbf{x}}P(R_t') {\mathbf{z}} \text{ for } \text{ all } t \in {\mathbb {N}}. \end{aligned}$$

Independence of irrelevant alternatives implies that ${\mathbf{y}} f({\mathbf {R}}') {\mathbf{x}}$, and weak Pareto implies that ${\mathbf{x}} P(f({\mathbf {R}}')) {\mathbf{z}}$. By the transitivity of $f({\mathbf {R}}')$, we have ${\mathbf{y}} P(f({\mathbf {R}}')) {\mathbf{z}}$. Since the preferences of individuals outside of ${\mathbb {N}} {\setminus } T$ over ${\mathbf{y}}$ and ${\mathbf{z}}$ are not specified, independence of irrelevant alternatives implies that $T \subseteq {\mathbb {N}}$ is decisive over $({\mathbf{y}},{\mathbf{z}})$, By Lemma 1, ${\mathbb {N}} {\setminus } T \in \mathcal{T}(f)$. $\square$

1.2 Proof of Theorem 3

To prove Theorem 3, we first establish two lemmas, which might be insightful in themselves.

Lemma 2

If a social welfare functionfsatisfies selfish domain and quasi-multi-profile stationarity, then$\{4,6,8,\ldots \} \notin {\mathcal {T}}(f)$.

Proof

Let f be a social welfare function satisfying selfish domain and quasi-multi-profile stationarity. Also, let x and y be two distinct elements of X, and $\succsim$ be an ordering on X such that $x \succ y$. By selfish domain, we can define a profile ${\mathbf {R}} \in {\mathcal {R}}_S^\infty$ by letting, for all $t \in N$ and for all ${\mathbf{x}}, {\mathbf{y}} \in X^\infty$,

$$\begin{aligned} {\mathbf{x}} R_t {\mathbf{y}} \Leftrightarrow x_t \succsim y_t. \end{aligned}$$

Now, take the following streams:

$$\begin{aligned}&{\mathbf{a}}=(y,x,y,x,y,x,\ldots ),\\&{\mathbf{b}}=(y,y,x,y,x,y,x,\ldots )=(y, {\mathbf {a}}),\\&{\mathbf{c}}=(y,y,y,x,y,x,y,x,\ldots )=(y, {\mathbf {b}}). \end{aligned}$$

Suppose that $\{4,6,8,\ldots \} \in {\mathcal {T}}(f)$. Since $\{t \in {\mathbb {N}}| {\mathbf{c }} P(R_t) {\mathbf{b}}\} = \{4,6,8,\ldots \}$, we have

$$\begin{aligned} {\mathbf{c}} P(f({\mathbf {R}})){\mathbf{b}}. \end{aligned}$$

Since $b_1=c_1$ and $b_2=c_2$, quasi-multi-profile stationarity implies that

$$\begin{aligned} {\mathbf{b}} f({\mathbf {R}}_{\ge 2}){\mathbf{a}} \Leftrightarrow {\mathbf{c}} f({\mathbf {R}}){\mathbf{b}}. \end{aligned}$$

Then ${\mathbf{c}} P(f({\mathbf {R}})){\mathbf{b}} \Leftrightarrow {\mathbf{b}} P(f({\mathbf {R}}_{\ge 2})){\mathbf{a}}$. We obtain ${\mathbf{b}} P(f({\mathbf {R}}_{\ge 2})){\mathbf{a}}$. However, since $\{2, 4,6,8,\ldots \} \in {\mathcal {T}}(f)$, it follows that

$$\begin{aligned} {\mathbf{a}} P(f({\mathbf {R}}_{\ge 2})){\mathbf{b}}. \end{aligned}$$

This is a contradiction. $\square$

Lemma 3

If a social welfare functionfsatisfies selfish domain and quasi-multi-profile stationarity, then$\{3,5,7,\ldots \} \notin {\mathcal {T}}(f)$.

Proof

Let f be a social welfare function satisfying selfish domain and quasi-multi-profile stationarity. Also, let x and y be two distinct elements of X and $\succsim$ be an ordering on X such that $x \succ y$. By selfish domain, we can consider a profile ${\mathbf {R}} \in {\mathcal {R}}^\infty$ by letting, for all $t \in N$ and for all ${\mathbf{x}}, {\mathbf{y}} \in X^\infty$,

$$\begin{aligned} {\mathbf{x}} R_t {\mathbf{y}} \Leftrightarrow x_t \succsim y_t. \end{aligned}$$

Now, take the following streams:

$$\begin{aligned}&{\mathbf{a}}=(y,x,y,x,y,x,\ldots ),\\&{\mathbf{b}}=(y,y,x,y,x,y,x,\ldots )=(y, {\mathbf {a}}),\\&{\mathbf{c}}=(y,y,y,x,y,x,y,x,\ldots )=(y, {\mathbf {b}}). \end{aligned}$$

Suppose that $\{3,5,7,\ldots \} \in {\mathcal {T}}(f)$. Since $\{t \in {\mathbb {N}}| {\mathbf{b }} P(R_t) {\mathbf{c}}\} = \{3,5,7,\ldots \}$, we have

$$\begin{aligned} {\mathbf{b}} P(f({\mathbf {R}})){\mathbf{c}}. \end{aligned}$$

Note that $b_1=c_1$ and $b_2=c_2$. By quasi-multi-profile stationarity, we have

$$\begin{aligned} {\mathbf{a}} P(f({\mathbf {R}}_{\ge 2})){\mathbf{b}} \Leftrightarrow {\mathbf{b}} P(f({\mathbf {R}})){\mathbf{c}}. \end{aligned}$$

Then we have ${\mathbf{a}} P(f({\mathbf {R}}_{\ge 2})){\mathbf{b}}$. However, since $\{3, 5,7,9,\ldots \} \in {\mathcal {T}}(f)$, it follows that

$$\begin{aligned} {\mathbf{b}} P(f({\mathbf {R}}_{\ge 2})){\mathbf{a}}. \end{aligned}$$

This is a contradiction. $\square$

Theorem 2 and Lemmas 2 and 3 together imply the following result.

Lemma 4

Suppose that$|X| \ge 4$. If a social welfare functionfsatisfies selfish domain, weak Pareto, independence of irrelevant alternatives and quasi-multi-profile stationarity, then$\{1,2\} \in {\mathcal {T}}(f)$.

Proof

Let f be a social welfare function satisfying selfish domain, weak Pareto, independence of irrelevant alternatives and quasi-multi-profile stationarity. By Theorem 2, the family ${\mathcal {T}}(f)$ of decisive sets forms an ultrafilter on ${\mathbb {N}}$.

By Lemma 2, $\{4,6,8,\ldots \} \notin {\mathcal {T}}(f)$. From (U4) of an ultrafilter, it follows that

$$\begin{aligned} {\mathbb {N}} {\setminus } \{4,6,8,\ldots \} \in {\mathcal {T}}(f). \end{aligned}$$

(2)

By Lemma 3, $\{3,5,7,\ldots \} \notin {\mathcal {T}}(f)$. From (U4) of an ultrafilter, it follows that

$$\begin{aligned} {\mathbb {N}} {\setminus } \{3,5,7,\ldots \} \in {\mathcal {T}}(f). \end{aligned}$$

(3)

From (2), (3), and (U3) of an ultrafilter, it follows that

$$\begin{aligned} ({\mathbb {N}} {\setminus } \{4,6,8,\ldots \}) \cap ({\mathbb {N}} {\setminus } \{3,5,7,\ldots \})=\{1,2\} \in {\mathcal {T}}(f). \end{aligned}$$

$\square$

Proof of Theorem 3

Assume that $\{ 1 \} \notin {\mathcal {T}}(f)$ and $\{ 2 \} \notin {\mathcal {T}}(f)$. From (U4) of an ultrafilter, it follows that

$$\begin{aligned} {\mathbb {N}} {\setminus } \{1\} \in {\mathcal {T}}(f) \text{ and } {\mathbb {N}} {\setminus } \{2\} \in {\mathcal {T}}(f). \end{aligned}$$

(4)

By Lemma 4,

$$\begin{aligned} \{1,2\} \in {\mathcal {T}}(f) . \end{aligned}$$

(5)

By (4), (5), and (U3) of an ultrafilter, it follows that

$$\begin{aligned} ({\mathbb {N}} {\setminus } \{1\}) \cap ({\mathbb {N}} {\setminus } \{ 2 \}) \cap \{1,2\} \in {\mathcal {T}}(f). \end{aligned}$$

This contradicts (U1) of an ultrafilter. $\square$

1.3 Proof of Theorem 4

Lemma 5

If a social welfare functionfsatisfies selfish domain and$\tau$-quasi-multi-profile stationarity, then$\{\tau +2,\tau +4,\tau +6,\ldots \} \notin {\mathcal {T}}(f)$.

Proof

Let f be a social welfare function satisfying selfish domain and T-quasi-multi-profile stationarity. Also, let x and y be two distinct elements of X and $\succsim$ be an ordering on X such that $x \succ y$. With selfish domain, we can consider a profile ${\mathbf {R}} \in {\mathcal {R}}^\infty$ by letting, for all $t \in N$ and for all ${\mathbf{x}}, {\mathbf{y}} \in X^\infty$,

$$\begin{aligned} {\mathbf{x}} R_t {\mathbf{y}} \Leftrightarrow x_t \succsim y_t. \end{aligned}$$

Now, take the following streams:

$$\begin{aligned}&{\mathbf{a}}=(({\mathbf{y}}_\mathrm{{con}})_{\le \tau -1},x,y,x,y,x,\ldots ),\\&{\mathbf{b}}=(({\mathbf{y}}_\mathrm{{con}})_{\le \tau },x,y,x,y,x,\ldots )=(y, {\mathbf {a}}),\\&{\mathbf{c}}=(({\mathbf{y}}_\mathrm{{con}})_{\le \tau },y,x,y,x,y,x,\ldots )=(y, {\mathbf {b}}). \end{aligned}$$

Suppose that $\{\tau +2,\tau +4,\tau +6,\ldots \} \in {\mathcal {T}}(f)$. Since $\{t \in {\mathbb {N}}| {\mathbf{c }} P(R_t) {\mathbf{b}}\} = \{\tau +2,\tau +4,\tau +6,\ldots \}$, we have

$$\begin{aligned} {\mathbf{c}} P(f({\mathbf {R}})){\mathbf{b}}. \end{aligned}$$

Since $b_t=c_t$ for all $t \le \tau$, $\tau$-quasi-multi-profile stationarity implies that

$$\begin{aligned} {\mathbf{b}} f({\mathbf {R}}_{\ge 2}){\mathbf{a}} \Leftrightarrow {\mathbf{c}} f({\mathbf {R}}){\mathbf{b}}. \end{aligned}$$

Then ${\mathbf{c}} P(f({\mathbf {R}})){\mathbf{b}} \Leftrightarrow {\mathbf{b}} P(f({\mathbf {R}}_{\ge 2})){\mathbf{a}}$. We obtain ${\mathbf{b}} P(f({\mathbf {R}}_{\ge 2})){\mathbf{a}}$. However, since $\{\tau +2,\tau +4,\tau +6,\ldots \} \in {\mathcal {T}}(f)$,

$$\begin{aligned} {\mathbf{a}} P(f({\mathbf {R}}_{\ge 2})){\mathbf{b}}. \end{aligned}$$

This is a contradiction. $\square$

Lemma 6

If a social welfare functionfsatisfies selfish domain and$\tau$-quasi-multi-profile stationarity, then$\{\tau +1,\tau +3,\tau +5,\ldots \} \notin {\mathcal {T}}(f)$.

Proof

Let f be a social welfare function satisfying selfish domain and $\tau$-quasi-multi-profile stationarity. Also, let x and y be two distinct elements of X and $\succsim$ be an ordering on X such that $x \succ y$. With selfish domain, we can consider a profile ${\mathbf {R}} \in {\mathcal {R}}^\infty$ by letting, for all $t \in N$ and for all ${\mathbf{x}}, {\mathbf{y}} \in X^\infty$,

$$\begin{aligned} {\mathbf{x}} R_t {\mathbf{y}} \Leftrightarrow x_t \succsim y_t. \end{aligned}$$

Now, take the following streams:

$$\begin{aligned}&{\mathbf{a}}=(({\mathbf{y}}_\mathrm{{con}})_{\le \tau -1},x,y,x,y,x,\ldots ),\\&{\mathbf{b}}=(({\mathbf{y}}_\mathrm{{con}})_{\le \tau },x,y,x,y,x,\ldots )=(y, {\mathbf {a}}),\\&{\mathbf{c}}=(({\mathbf{y}}_\mathrm{{con}})_{\le \tau },y,x,y,x,y,x,\ldots )=(y, {\mathbf {b}}). \end{aligned}$$

Suppose that $\{\tau +1,\tau +3,\tau +5,\ldots \} \in {\mathcal {T}}(f)$. Since $\{t \in {\mathbb {N}}| {\mathbf{b }} P(R_t) {\mathbf{c}}\} = \{\tau +1,\tau +3,\tau +5,\ldots \}$, we have the following:

$$\begin{aligned} {\mathbf{b}} P(f({\mathbf {R}})){\mathbf{c}}. \end{aligned}$$

Note that $b_t=c_t$ for all $t \le \tau$. By $\tau$-quasi-multi-profile stationarity, we have

$$\begin{aligned} {\mathbf{a}} P(f({\mathbf {R}}_{\ge 2})){\mathbf{b}} \Leftrightarrow {\mathbf{b}} P(f({\mathbf {R}})){\mathbf{c}}. \end{aligned}$$

Then we have ${\mathbf{a}} P(f({\mathbf {R}}_{\ge 2})){\mathbf{b}}$. However, $\{\tau +1,\tau +3,\tau +5,\ldots \} \in {\mathcal {T}}(f)$,

$$\begin{aligned} {\mathbf{b}} P(f({\mathbf {R}}_{\ge 2})){\mathbf{a}}. \end{aligned}$$

This is a contradiction. $\square$

Lemma 7

If a social welfare functionfsatisfies selfish domain, weak Pareto, independence of irrelevant alternatives, and$\tau$-quasi-multi-profile stationarity, then$\{1,2,\ldots , \tau \} \in {\mathcal {T}}(f)$.

Proof

Let f be a social welfare function satisfying selfish domain, weak Pareto, independence of irrelevant alternatives, and quasi-multi-profile stationarity. By Theorem 2, the family ${\mathcal {T}}(f)$ of decisive sets forms an ultrafilter on ${\mathbb {N}}$.

By Lemma 5, $\{T+2,T+4,T+6,\ldots \} \notin {\mathcal {T}}(f)$. From (U4) of an ultrafilter, it follows that

$$\begin{aligned} {\mathbb {N}} {\setminus } \{\tau +2,\tau +4,\tau +6,\ldots \} \in {\mathcal {T}}(f). \end{aligned}$$

(6)

By Lemma 6, $\{\tau +1,\tau +3,\tau +5,\ldots \} \notin {\mathcal {T}}(f)$. From (U4) of an ultrafilter, it follows that

$$\begin{aligned} {\mathbb {N}} {\setminus } \{\tau +1,\tau +3,\tau +5,\ldots \} \in {\mathcal {T}}(f). \end{aligned}$$

(7)

By (6), (7), and (U3) of an ultrafilter, it follows that

$$\begin{aligned} ({\mathbb {N}} {\setminus } \{\tau +2,\tau +4,\tau +6,\ldots \}) \cap ({\mathbb {N}} {\setminus } \{\tau +1,\tau +3,\tau +5,\ldots \})=\{1,2,\ldots ,\tau \} \in {\mathcal {T}}(f). \end{aligned}$$

$\square$

Proof of Theorem 4

Suppose that $\{ t \} \notin {\mathcal {T}}(f)$ for all $t \le \tau$. From (U4) of an ultrafilter, it follows that

$$\begin{aligned} {\mathbb {N}} {\setminus } \{t\} \in {\mathcal {T}}(f) \text{ for } \text{ all } t \le \tau . \end{aligned}$$

(8)

By Lemma 7,

$$\begin{aligned} \{1,2,\ldots ,\tau \} \in {\mathcal {T}}(f) . \end{aligned}$$

(9)

From (8), (9), and (U3) of an ultrafilter, it follows that

$$\begin{aligned} \{1,2,\ldots , \tau \} \cap \bigcap _{t\le T}{\mathbb {N}} {\setminus } \{t\} \in {\mathcal {T}}(f), \end{aligned}$$

contradicting (U1) of an ultrafilter. $\square$

1.4 Proof of Theorem 5

We can prove this theorem by following the same step as for Theorem 4.

Lemma 8

If a social welfare functionfsatisfies selfish domain and generalized multi-profile stationarity, then$\{\tau +2,\tau +4,\tau +6,\ldots \} \notin {\mathcal {T}}(f)$for some$\tau \in {\mathbb {N}}$.

Proof

Let f be a social welfare function satisfying selfish domain and generalized multi-profile stationarity. Also, let x and y be two distinct elements of X and $\succsim$ be an ordering on X such that $x \succ y$. With selfish domain, we can consider a profile ${\mathbf {R}} \in {\mathcal {R}}^\infty$ by letting, for all $t \in N$ and for all ${\mathbf{x}}, {\mathbf{y}} \in X^\infty$,

$$\begin{aligned} {\mathbf{x}} R_t {\mathbf{y}} \Leftrightarrow x_t \succsim y_t. \end{aligned}$$

By generalized multi-profile stationarity, there exists some $T \in {\mathbb {N}}$ such that for all ${\mathbf{x}}, {\mathbf{y}} \in X^\infty$, if $x_t = y_t$ for all $t \in \{1,2,\ldots ,\tau \}$, then ${\mathbf{x }} f({\mathbf {R}}) {\mathbf{y}} \Leftrightarrow {\mathbf{x}}_{\ge 2} f({\mathbf {R}}_{\ge 2}) {\mathbf{y}}_{\ge 2}$. Now, take the following streams:

$$\begin{aligned}&{\mathbf{a}}=(({\mathbf{y}}_\mathrm{{con}})_{\le \tau -1},x,y,x,y,x,\ldots ),\\&{\mathbf{b}}=(({\mathbf{y}}_\mathrm{{con}})_{\le \tau },x,y,x,y,x,\ldots )=(y, {\mathbf {a}}),\\&{\mathbf{c}}=(({\mathbf{y}}_\mathrm{{con}})_{\le \tau },y,x,y,x,y,x,\ldots )=(y, {\mathbf {b}}). \end{aligned}$$

The rest of the proof is the same as in Lemma 5. $\square$

The following result can be proven in a similar way to the proof of Lemma 8.

Lemma 9

If a social welfare functionfsatisfies selfish domain and$\tau$-quasi-multi-profile stationarity, then$\{\tau +1,\tau +3,\tau +5,\ldots \} \notin {\mathcal {T}}(f)$for some$\tau \in {\mathbb {N}}$.

Note that $\tau$ in Lemma 9 may be different from that in Lemma 8.

Lemma 10

If a social welfare functionfsatisfies selfish domain, weak Pareto, independence of irrelevant alternatives, and generalized multi-profile stationarity, then$\{1,2,\ldots ,\tau \} \in {\mathcal {T}}(f)$for some$\tau \in {\mathbb {N}}$.

Proof

Let f be a social welfare function satisfying selfish domain, weak Pareto, independence of irrelevant alternatives, and generalized multi-profile stationarity. Then the family ${\mathcal {T}}(f)$ of decisive sets forms an ultrafilter on ${\mathbb {N}}$.

By Lemma 8, $\{\tau +2,\tau +4,\tau +6,\ldots \} \notin {\mathcal {T}}(f)$ for some $\tau \in {\mathbb {N}}$. Then it follows that ${\mathbb {N}} {\setminus } \{\tau +2,\tau +4,\tau +6,\ldots \} \in {\mathcal {T}}(f)$. By Lemma 9, $\{\tau '+1,\tau '+3,\tau '+5,\ldots \} \notin {\mathcal {T}}(f)$ for some $\tau ' \in {\mathbb {N}}$. Thus, it follows that ${\mathbb {N}} {\setminus } \{\tau '+1,\tau '+3,\tau '+5,\ldots \} \in {\mathcal {T}}(f)$. Let

$$\begin{aligned} \tau ^* = \max \{ \tau ,\tau '\}. \end{aligned}$$

Note that

$$\begin{aligned} {\mathbb {N}} {\setminus } \{\tau +2,\tau +4,\tau +6,\ldots \} \subseteq {\mathbb {N}} {\setminus } \{\tau ^*+2,\tau ^*+4,\tau ^*+6,\ldots \};\\ {\mathbb {N}} {\setminus } \{\tau '+1,\tau '+3,\tau '+5,\ldots \} \subseteq {\mathbb {N}} {\setminus } \{\tau ^*+1,\tau ^*+3,\tau ^*+5,\ldots \}. \end{aligned}$$

By (U2), we have

$$\begin{aligned} {\mathbb {N}} {\setminus } \{\tau ^*+2,\tau ^*+4,\tau ^*+6,\ldots \} \in {\mathcal {T}}(f);\\ {\mathbb {N}} {\setminus } \{\tau ^*+1,\tau ^*+3,\tau ^*+5,\ldots \} \in {\mathcal {T}}(f). \end{aligned}$$

From (U3), it follows that

$$\begin{aligned} ({\mathbb {N}} {\setminus } \{\tau ^*+2,\tau ^*+4,\tau ^*+6,\ldots \}) \cap ({\mathbb {N}} {\setminus } \{\tau ^*+1,\tau ^*+3,\tau ^*+5,\ldots \})=\{1,2,\ldots ,\tau ^* \} \in {\mathcal {T}}(f). \end{aligned}$$

$\square$

The rest of the proof is the same as for Theorem 4.

1.5 Proof of Theorem 6

Proof

It suffices to show that $f^*_{{\mathcal {U}}}$ satisfies Pareto indifference and cofinite-multi-profile stationarity. To show Pareto indifference, suppose that ${\mathbf{x}}I(R_t) {\mathbf{y}}$ for all $t \in N$. Since $N \in {\mathcal {U}}$, $\{ t \in {\mathbb {N}}: {\mathbf{x}}R_t {\mathbf{y}} \} \in {\mathcal {U}}$ and $\{ t \in {\mathbb {N}}: {\mathbf{y}}R_t {\mathbf{x}} \} \in {\mathcal {U}}$. Then we have ${\mathbf{x}}I(f({\mathbf{R}})) {\mathbf{y}}$.

Now, we show that cofinite-multi-profile stationarity is satisfied. Suppose that there exists T such that $|T^\mathrm{{c}}| <\infty$, $1 \in T$, and $x_t = y_t$ for all $t \in T$. Since each individual is selfish, ${\mathbf{x}}I(R_t) {\mathbf{y}}$ for all $t \in T$. We can show that T is an element of ${\mathcal {U}}$. If T is not, then $T^\mathrm{{c}}$ is an element because ${\mathcal {U}}$ is an ultrafilter. However, $T^\mathrm{{c}}$ is finite. Then a singleton must be an element of ${\mathcal {U}}$. For any ${\mathbf{R}}\in {\mathcal {R}}^{\infty }_S$, $\{ t \in {\mathbb {N}}: {\mathbf{x}}I(R_t) {\mathbf{y}} \} \in {\mathcal {U}}$ and thus, $\mathbf{x}I(f({\mathbf{R}})) {\mathbf{y}}$.

Let us consider the ranking between ${\mathbf{x}}_{\ge 2}$ and $\mathbf{y}_{\ge 2}$ under ${\mathbf{R}}_{\ge 2}$. Let ${\mathbf{R}}'={\mathbf{R}}_{\ge 2}$, ${\mathbf{x}}'={\mathbf{x}}_{\ge 2}$ and ${\mathbf{y}}'={\mathbf{y}}_{\ge 2}$. Because of the existence of T, there exists S such that $|S^\mathrm{{c}}|=|T^\mathrm{{c}}| <\infty$ and $x'_t = y'_t$ for all $t \in S$. Because of our domain restriction, ${\mathbf{x}}'I(R'_t) {\mathbf{y}}'$ for all $t \in S$. We have ${\mathbf{x}}' I(f({\mathbf{R}}')) {\mathbf{y}}'$ and thus, ${\mathbf{x}}_{\ge 2}I(f({\mathbf{R}}_{\ge 2})) {\mathbf{y}}_{\ge 2}$. $\square$