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Theory and Decision

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11-02-2020

Quasi-stationary social welfare functions

Author: Susumu Cato

Published in: Theory and Decision | Issue 1/2020

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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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See Arrow (1951, 1963, 2012).

See Lauwers and Van Liedekerke (1995), Noguchi (2011), Cato (2013), and Takayama and Yokotani (2017).

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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Title: Quasi-stationary social welfare functions
Author: Susumu Cato
Publication date: 11-02-2020
Publisher: Springer US
Published in: Theory and Decision / Issue 1/2020
Print ISSN: 0040-5833
Electronic ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-020-09746-4

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