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.
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Notes
Svensson (1980) proves a possibility result by modifying the definition of continuity.
An ordering is a transitive and complete binary relation. Transitivity requires that for all \({\mathbf{x}},\mathbf{y},{\mathbf{z}} \in X^\infty \), if \({\mathbf{x}} R_t{\mathbf{y}}\) and \({\mathbf{y}} R_t{\mathbf{z}}\), then \({\mathbf{x}} R_t{\mathbf{z}}\); completeness requires that, for all \({\mathbf{x}},{\mathbf{y}}\in X^\infty \), \({\mathbf{x}} R_t{\mathbf{y}}\) or \({\mathbf{y}} R_t{\mathbf{x}}\). Our definition of completeness implies reflexivity, which requires that, for all \({\mathbf{x}}\in X^\infty \), \({\mathbf{x}} R_t{\mathbf{x}}\). Cato (2016) discusses implications of completeness and transitivity. There is another way of defining completeness, see Bossert and Suzumura (2010).
See Willard (1970) for basic results on filters and ultrafilters.
When we consider utility representations, we assume that there exist the upper bound \({\overline{u}} \in {\mathbb {R}}\) and the lower bound \({\underline{u}} \in {\mathbb {R}}\) such that \({\overline{u}}>u_t(x_t)>{\underline{u}}\) for all \(t \in {\mathbb {N}}\) and all \(x \in X\). Another point is that we implicitly consider a mapping that assigns a utility function \(u_t\) to each preference \(\succsim _t\) (or \(R_t\)). That is, there is a function v such that \(u_t(x_t)=v(x_t;\succsim _t)\).
Indeed, multi-profile stationarity imposes some priority only on the first generation and then even the second generation cannot have a decisive power.
Packel (1980, p. 226, Theorem 4) provides a possibility result in his framework, which is different from ours. He introduces cardinal comparability of individual utilities, while we deal with ordinal utilities. Also, he drops Arrow’s axioms in his possibility result, while we impose these axioms in our possibility result.
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I thank an anonymous reviewer for valuable comments and suggestions. This work was financially supported by JSPS KAKENHI (18K01501).
Appendix
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
Let \({\mathbf {R}} \in {\mathcal {R}}_S^\infty \) be such that
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
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
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
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
Let \({\mathbf {R}} \in {\mathcal {R}}_S^\infty \) be such that
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
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:
By repeating the same argument, we have
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
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
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 \({\mathbf{x}}, {\mathbf{y}} \in X^\infty \),
Now, take the following streams:
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
Since \(b_1=c_1\) and \(b_2=c_2\), quasi-multi-profile stationarity implies that
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
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 \({\mathbf{x}}, {\mathbf{y}} \in X^\infty \),
Now, take the following streams:
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
Note that \(b_1=c_1\) and \(b_2=c_2\). By quasi-multi-profile stationarity, we have
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
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
By Lemma 3, \(\{3,5,7,\ldots \} \notin {\mathcal {T}}(f)\). From (U4) of an ultrafilter, it follows that
From (2), (3), and (U3) of an ultrafilter, it follows that
\(\square \)
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
By Lemma 4,
By (4), (5), and (U3) of an ultrafilter, it follows that
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 \({\mathbf{x}}, {\mathbf{y}} \in X^\infty \),
Now, take the following streams:
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
Since \(b_t=c_t\) for all \(t \le \tau \), \(\tau \)-quasi-multi-profile stationarity implies that
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 \(\{\tau +2,\tau +4,\tau +6,\ldots \} \in {\mathcal {T}}(f)\),
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 \({\mathbf{x}}, {\mathbf{y}} \in X^\infty \),
Now, take the following streams:
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:
Note that \(b_t=c_t\) for all \(t \le \tau \). By \(\tau \)-quasi-multi-profile stationarity, we have
Then we have \({\mathbf{a}} P(f({\mathbf {R}}_{\ge 2})){\mathbf{b}}\). However, \(\{\tau +1,\tau +3,\tau +5,\ldots \} \in {\mathcal {T}}(f)\),
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
By Lemma 6, \(\{\tau +1,\tau +3,\tau +5,\ldots \} \notin {\mathcal {T}}(f)\). From (U4) of an ultrafilter, it follows that
By (6), (7), and (U3) of an ultrafilter, it follows that
\(\square \)
Proof of Theorem 4
Suppose that \(\{ t \} \notin {\mathcal {T}}(f)\) for all \(t \le \tau \). From (U4) of an ultrafilter, it follows that
By Lemma 7,
From (8), (9), and (U3) of an ultrafilter, it follows that
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 \({\mathbf{x}}, {\mathbf{y}} \in X^\infty \),
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:
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
Note that
By (U2), we have
From (U3), it follows that
\(\square \)
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, \({\mathbf{x}}_{\ge 2}I(f({\mathbf{R}}_{\ge 2})) {\mathbf{y}}_{\ge 2}\). \(\square \)
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Cato, S. Quasi-stationary social welfare functions. Theory Decis 89, 85–106 (2020). https://doi.org/10.1007/s11238-020-09746-4
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DOI: https://doi.org/10.1007/s11238-020-09746-4