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Theory and Decision

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01.10.2013

Social choice, the strong Pareto principle, and conditional decisiveness

verfasst von: Susumu Cato

Erschienen in: Theory and Decision | Ausgabe 4/2013

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Abstract

This paper examines social choice theory with the strong Pareto principle. The notion of conditional decisiveness is introduced to clarify the underlying power structure behind strongly Paretian aggregation rules satisfying binary independence. We discuss the various degrees of social rationality: transitivity, semi-transitivity, the interval-order property, quasi-transitivity, and acyclicity.

Vorheriger Artikel Maximin play in completely mixed strategic games

Nächster Artikel Characterizing referenda with quorums via strategy-proofness

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Cato (2011) provides sufficient conditions for the equivalence between the weak Pareto and strong Pareto principles.

An exception is the maximin criterion. Sen (1970) proposes a lexicographic refinement of the maximin criterion.

Arrow’s theorem does not hold in the case of an infinite population. See Fishburn (1970).

Campbell and Kelly (2002) provide a survey of Arrovian social choice theory.

The basic properties of filters and ultrafilters are explained by Aliprantis and Border (2006) and Willard (1970).

Another expression is as follows: there exists a permutation \(\rho :\mathcal{N } \rightarrow \mathcal{N }\) such that for all \(i\), if \((x,y)\in P(R_i)\) and there exists no \(j\) such that \(\rho (j)<\rho (i)\) and \((y,x)\in P(R_j)\), then \((x,y)\in P(W(\mathbf{R}))\).

Similarly, the structure of \(\mathcal{N }\) is unspecified in later theorems (Theorems 2– 5), and subsequent discussions are only sketches of the finite case.

Some might consider our definition of the serial oligarchy to be weak. An alternative definition is as follows: there exists a partition \((B_1,\dots ,B_K)\) such that for all \(k\), if \((x,y)\in P(\bigcap _{i \in B_k}R_i)\) and \((x,y)\in \bigcap _{i \in C_{k}}I(R_i)\), then \((x,y)\in P(W(\mathbf{R}))\). This definition is natural, but it is not equivalent to the filter structure of the conditionally decisive sets.

Similar to the serial oligarchy, the serial collegium provides only a partial description of the collective process; there are many SDFs with the same structure of the serial collegium.

The proof of the theorem is as follows. Condition (i) immediately follows from strong Pareto. If condition (ii) is violated, \((x,y) \in P(W(\mathbf{R}))\) and \((y,x) \in P(W(\mathbf{R}))\) whenever \((x,y) \in \bigcap _{i \in M \cap M^{\prime }} I(R_i),\,(x,y) \in \bigcap _{i \in M - M^{\prime }} P(R_i)\), and \((y,x) \in \bigcap _{i \in M^{\prime } - M} P(R_i)\).

An exception is Theorem 5. In the theorem, binary independence is not required to be satisfied, but acyclicity of social preference has a critical role for the consequence.

In the rest of this section, we assume that both \(\mathcal{N }\) and \(X\) are finite.

The above-mentioned rule generates Suzumura-consistent social preferences. A binary relation is Suzumura-consistent if and only if, for all \(K \in \mathbb{N }\) and for all \(x^0, \dots , x^K,\,(x^{k-1},x^k) \in R \ \forall k \in \{1,\dots , K\} \Rightarrow (x^K,x^0) \notin P(R)\). The concept is proposed by Suzumura (1976); Bossert and Suzumura (2008)) characterize a class of CCRs generating Suzumura-consistent social preferences. See Bossert and Suzumura (2010) for a comprehensive argument.

They also imposed the axiom of stationarity on CCRs.

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Titel: Social choice, the strong Pareto principle, and conditional decisiveness
verfasst von: Susumu Cato
Publikationsdatum: 01.10.2013
Verlag: Springer US
Erschienen in: Theory and Decision / Ausgabe 4/2013
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-013-9352-9

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