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Social Choice and Welfare

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01-03-2014 | Original Paper

Singularity and Arrow’s paradox

Author: Wu-Hsiung U. Huang

Published in: Social Choice and Welfare | Issue 3/2014

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Abstract

In this thesis we establish a completion theory of Arrow’s work, extending it from quasi-form to absolute form and generalizing it to a degree theorem of various forms. It is proved that Arrow’s independence of irrelevant alternatives (AI ) is inconsistent with some forms of Pareto condition, e.g. with the strong Pareto condition. Based on these observations, we try to resolve Arrow’s paradox by introducing the “extent principle” and set up a weak Arrow’s framework to show the consistency of the weak AI , the weak Pareto condition, anonymity, no decisive minority group and other widely-accepted rationality principles. Singularity is the core concept of the paper.

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A binary relation $\succsim $ on $X$ is complete if $ \forall x , y \in X $, either $ x \succsim y $ or $ y \succsim x $ , or both. It is transitive if $\forall x , y , z \in X,\,x \succsim y$ and $y \succsim z \Rightarrow x \succsim z $. A binary relation $\succsim $ on $X$ which is complete and transitive is called a preference on $X$.

$x\precsim y$ will be used as equivalent to $y\succsim x$. “$\succ $” is the asymmetric component of “$\succsim $”, i.e. $x\succ y$ iff $x\succsim y$ but not $y\succsim x$. When completeness is given, we see that $x\succ y$ iff $y\succsim x$ does not hold. Also, the indifference sign “$\thicksim $” is defined by: $x\thicksim y$ iff $x\succsim y$ and $y\succsim x$.

The definition of an “absolute dictator” is stronger than a “dictator” in Arrow’s framework. The latter will be called “quasi dictator” in this paper, as it only requires

$$\begin{aligned} x \succ y \text { in } p_k \Rightarrow x \succ y \text { in } F(\mathbf{p}) \end{aligned}$$

for any pair $x,y$ in $X$. We will discuss this in detail in Sect. 3.

Arrow’s theorem says that QPC and AI imply the quasi-dictatorship QD (see footnote 3). Both of the assumption QPC and the conclusion QD are weaker than Corollary 1. In Corollary 3, we will obtain Arrow’s theorem as a corollary of Theorem 3, the quasi-degree theorem.

Later we will introduce the “Extent Principle” to modify Arrow’s independence.

The corresponding statements for the quasi-form and the weak form are not true. This is a more sophisticated part of this work. See Remark 4 and the counterexample (Example 10).

An ordered pair $(x,y)$ is different from $(y,x)$, but a pair $\{ x,y\}$ is simply a set and hence $\{ x,y\}=\{ y,x\}$.

A ranking of $X$ with respect to $ (\sim ,\succ ) $ in $ B$ means a partition of $X$ into equivalent classes $\{ X_\alpha ; \alpha \in I\}$, in which $x \sim y$ in $ B$ iff $x$ and $y$ belong to a same equivalent class, and $\{ X_\alpha \}$ is linearly ordered, according to “$\succ $ in $ B$”.

For precision about $(\approx , \gg )$ see Definition 13 in Sect.4.

Since $\mathbf{p},\varvec{q}$ and $x,y$ are arbitrary, the definition automatically implies

$$\begin{aligned} x \prec y \text { in } F(\mathbf{p}) \Rightarrow x \precsim y \text { in } F(\mathbf{q}). \end{aligned}$$

By calling $\mu $ a permutation, we mean it is a bijective map.

Anonymity involves renaming the individuals behind the preferences within a profile and equality of alternatives (i.e. neutrality) involves renaming the alternatives.

A topological preference $p$ on $X$ is a complete and transitive binary relation on $X$ such that its graph $ G \equiv \{ (x , y) ; x \succsim y \text { in } p\}$ is a closed set in $X \times X$, or equivalently, “$ x_n \succsim y_n$ in $p$, when $x_n$ tends to $x$ and $y_n$ tends to $y$” implies “$x \succsim y$ in $p$”.

The author provides a complete study for the global topological preferences in a previous paper (Huang 2009).

Here “localness” means the restriction to a pair $\{x,y\}$ of alternatives. Hence “local decisiveness” is weaker than “decisiveness”.

“Outcome rationality” is also called “voting system rationality” by Tao. It was defined earlier by Kalai (2002) to mean that the outcome of a social choice function is “rational” in the sense that the social preferences are all linear orders. We will use hereafter the term “outcome rationality” in place of “voting system rationality” to avoid confusion.

Kalai defines a social choice function by a map: $R=F(R_1, \ldots , R_N)$, where each $R_i$ is a (linear) order relation on alternatives and $R$ is an asymmetric relation. It is slightly different from the social welfare function considered in this thesis. A social choice function $F$ is called “symmetric” if $F$ is neutral and invariant under some transitive group of permutations on $\{1, 2,\ldots , N\}$. Note that neutrality implies almost Pareto condition and symmetry implies non-dictatorship.

In order to apply Kalai’s result to the social welfare functions considered in this thesis, we can adjust a social choice function $F$ by attaching the transitivity map $\sigma $ (see Definition 13) so that the composition $\sigma \circ F$ has images in our preference space $P$. Now it is clear that an outcome of $F$ is rational iff that of $\sigma \circ F$ is non-singular.

As a static model, the alternative voting is a likeable design, but it is still far from perfect.

That is, it contains no singularity.

Precisely, it means that there exists a diffeomorphism $\alpha : W\equiv D^{m-n}\times V\rightarrow M$, where $D^{m-n}$ is an ($m-n$)-dimensional open disk, $V$ a neighborhood of $y$ in $N,\,\alpha (0,y)=x$ and

$$\begin{aligned} \alpha (D\times \{y'\})=f^{-1}(y')\cap \alpha (W), \; \forall y' \in V. \end{aligned}$$

In this sense, we call it “evenly distributed”.

In fact, we should extend the definition of singular points to include the boundary of $\Omega $. For precise definition, see Def. 2.4.2 of (Huang 2009).

Arrow KJ (1951) Social choice and individual values. Wiley, New York

Baigent N (1987) Preference proximity and anonymous social choice. Q J Econ 102(1):161–169CrossRef

Chichilnisky G (1982) Social aggregation rules and continuity. Q J Econ 97(2):337–352CrossRef

Crypton D (1985) Is democracy mathematically unsound? Sci Dig 93:56–84

Huang WH (2004) Is proximity preservation rational in social choice theory? Soc Choice Welf 23(3):315–332CrossRef

Huang WH (2009) Is a continuous rational social aggregation impossible on continuum spaces? Soc Choice Welf 32(4):635–686CrossRef

Mas-Colell A, Whinston MD, Green JR (1995) Microeconomic theory. Oxford University Press, Oxford

Saari D (2008) Disposing dictators, demystifying voting paradoxes: social choice analysis. Cambridge University Press, New YorkCrossRef

Sen A (1970) Collective choice and social welfare, chapter 6. Holden-Day, San Francisco

Sen A (1995) Rationality and social choice. Am Econ Rev 85(1):1–24

Title: Singularity and Arrow’s paradox
Author: Wu-Hsiung U. Huang
Publication date: 01-03-2014
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 3/2014
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-013-0750-2

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