Extending an order on a set to the power set: Some remarks on Kannai and Peleg's approach
type: Item
instance of: scholarly article
title: Extending an order on a set to the power set: Some remarks on Kannai and Peleg's approach
zbMATH Open document ID: 0548.90002
DOI: 10.1016/0022-0531(84)90083-8
author: Prasanta K. Pattanaik
author: Salvador Barberà
published in: Journal of Economic Theory
publication date: 1984-01-01T00:00:00Z
review text: Given a reflexive linear order R on a set \(\Omega\), and hence given its asymptotic part P, a weak order \(\gtrsim\) on the power set \(2^{\Omega}\) is called an extension of P, if for all x,\(y\in\Omega \) for which xPy, it follows that \(\{\) \(x\} \succ\{y\}\). Kannai and Peleg proposed two appealing axioms for such extensions, (GP) and (M), and established their impossibility theorem that there is no extension of a reflexive linear order P on \(\Omega\) to \(2^{\Omega}\backslash\{\emptyset\}\) which satisfies both (GP) and (M) and is also a weak ordering, if {\#}\(\Omega \geq 6\) [see \textit{Y. Kannai} and \textit{B. Peleg}, ibid. 32, 172-175 (1984; Zbl 0533.90005)]. The authors commented on the Kannai-Peleg result by establishing two theorems. Their axioms are: (B) For all distinct x,\(y\in\Omega \), if xRy, then \(\{\) \(x\} \succ\{x,y\}\) and \(\{\) x,y)\(\succ\{y\}.\) (K) Let \(A,B\in 2^{\Omega}\backslash\{\Phi\}\). If [for all \(x\in A\) and for all \(y\in B\), xRy] and [for some \(x\in A\) and some \(y\in B\), xRy and not yRx], then \(A\succ B.\) (M') If \(A,B,C\in 2^{\Omega}\backslash\{\emptyset\}\), \(A\cap B=A\cap C=\emptyset\), and \(B\succ C\), then \(A\cup B\succ A\cup C.\) Axiom (K) is weaker than (GP) and axiom (B) is weaker than (K), while axiom (M') is stronger than M. The theorems are: (1) If \(\Omega\) is finite there exists a weak order over \(2^{\Omega}\backslash\{\emptyset\}\) satisfying (K) and (M). (2) If {\#}\(\Omega \geq 3\) there exists no binary relation \(\gtrsim\) on \(2^{\Omega}\backslash\{\emptyset\}\) which satisfies (B) and (M').
Mathematics Subject Classification ID: 91B08
Mathematics Subject Classification ID: 91B14
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 3874910
zbMATH Keywords: extension of a linear order to the power set
zbMATH Keywords: Gärdenfors principle
zbMATH Keywords: monotonicity property
zbMATH Keywords: reflexive linear order
zbMATH Keywords: weak order
reviewed by: Tatsuro Ichiishi
MaRDI profile type: Publication
full work available at URL: 0022-0531(84)90083-8
OpenAlex ID: W1983226570
cites work: The Manipulation of Social Choice Mechanisms that Do Not Leave "Too Much" to Chance
cites work: Comment on the Kannai-Peleg impossibility theorem for extending orders
cites work: Manipulation of social choice functions
cites work: A note on the extension of an order on a set to the power set
cites work: Strategy-Proofness and Social Choice Functions without Singlevaluedness
cites work: Q5639713
cites work: Q3252269
cites work: Plausibility orderings and social choice
cites work: Q4196192
description: scientific article; zbMATH DE number 3874910