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Group Decision and Negotiation

Erschienen in:

01.05.2014

Menu Dependence and Group Decision Making

verfasst von: Susumu Cato

Erschienen in: Group Decision and Negotiation | Ausgabe 3/2014

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Abstract

This paper is concerned with the problem of group decision making. We introduce the notion of a collective system rule. A collective system rule maps each preference profile to a group-preference system, which is a collection of social preferences on the subsets of the alternatives. By formulating the Arrovian conditions, we show the Arrow-type impossibility theorems. We also discuss how our approach is related to the standard group decision-making process.

Vorheriger Artikel Exploring Individual Trust Factors in Computer Mediated Group Collaboration: A Case Study Approach

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Sen (1970), Suzumura (1983), Campbell and Kelly (2002), and Gaertner (2009) provide surveys on the extensions of Arrow’s theorem. See Sen (1979); Sen (1995) for a concise proof of Arrow’s theorem. See also Cato (2010).

For criticism of the full rationality of administrative behavior, see Simon (1957, 1979, 1997).

Tyson (2008) also discusses the relationship between a nested preference system and Simon’s satisficing criterion.

That is, the two binary relations $I(R)$ and $P(R)$ are defined by $(x,y) \in I(R) \Leftrightarrow [(x,y) \in R \text{ and}\, (y,x)\,{\in }\,R]$ and $(x,y) {\in } P(R) {\Leftrightarrow } [(x,y) {\in } R \text{ and} (y,x) \notin R]$, respectively.

A binary relation $R$ on $X$ is complete if and only if $\forall x,y \in X,\,(x,y) \in R$ or $(y,x) \in R$; a binary relation $R$ on $X$ is transitive if and only if $\forall x,y,z \in X,\,[(x,y \in R \text{ and}\,(y,z) \in R] \Rightarrow (x,z) \in R$; a binary relation $R$ on $X$ is reflexive if and only if $\forall x \in X,\,(x,x) \in R$.

A binary relation is asymmetric if and only if $\forall x,y \in X,\,(x,y) \in P^*_A \Rightarrow (y,x) \notin P^*_A$; a binary relation is negatively transitive if and only if $\forall x,y,z \in X,\,[(x,y) \notin P^*_A \text{ and}\,(y,z) \notin P^*_A \Rightarrow (x,z) \notin P^*_A$.

A direct proof is as follows: suppose that $C$ is generated by a nested system $\langle R_A \rangle _{A \in \mathcal X }$ of weak orders. Let $x \in C(A \cup C(B))$. Then, $(x,y) \in R_{A \cup B}$ for al $y \in A \cup C(B)$. If $y \in C(B)$, then $(y,z) \in R_B$ for all $z \in B$. Since $R_B \subseteq R_{A \cup B},\,(y,z) \in R_{A \cup B}$ for all $z \in B$. By the transitivity of $R_{A \cup B},\,(x,z) \in R_{A \cup B}$ for all $z \in B$. Thus, it follows that $(x,y) \in R_{A \cup B}$ for all $y \in A \cup B$. We have $x \in C(A \cup B)$.

In Sen’s definition of menu-independence, preferences in the system are allowed to be intransitive. (Sen (1997), Theorem 3.3) shows that $C$ is menu-independent if and only if it satisfies contraction consistency and expansion consistency, which is defined as follows: for a collection $\mathcal A \subseteq \mathcal X $ of subsets of $X$,

$$\begin{aligned} x \in \bigcap _{A \in \mathcal A }C(A) \Rightarrow x \in C\left( \bigcup _{A \in \mathcal A }A\right) \!. \end{aligned}$$

In the standard framework of social choice, Gibbard (1969), Guha (1972), Mas-Colell and Sonnenschein (1972) independently show that there exists a vetoer for every Arrovian rule that generates quasi-transitive social preferences.

Sen (1993) reformulates axiom I and shows an impossibility result with no consistency axiom.

According to Theorem 1, SEC and CC require group preferences to be menu-independent. Under Sen’s (1977) impossibility result, PWP and I are equivalent to WP and BI, respectively.

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Cato S (2010) Brief proofs of Arrovian impossibility theorems. Soc Choice Welf 35:267–284CrossRef

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Titel: Menu Dependence and Group Decision Making
verfasst von: Susumu Cato
Publikationsdatum: 01.05.2014
Verlag: Springer Netherlands
Erschienen in: Group Decision and Negotiation / Ausgabe 3/2014
Print ISSN: 0926-2644
Elektronische ISSN: 1572-9907
DOI: https://doi.org/10.1007/s10726-013-9343-9

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