Singularity and Arrow’s paradox

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.

Appendix: Singularity from the viewpoint of mathematical structuralism

The notion of singularity is defined relative to the system under consideration, or more precisely, it is subject to the structure of the given theory. It appears where the regular behavior of the system breaks down. For example, in fluid dynamics a vortex with zero velocity is a singularity. In linear algebra, a square matrix is singular when the determinant is zero. Also, consider for instance a presidential election for social choice theory, where singularity occurs when two candidates obtain an equal number of votes. Clearly, in order to define singularity, we have to define what is “regularity” relative to the system in the first place. A rigorous treatment of singularity based on mathematical structuralism is provided as follows.

(I) Given a geometric space $(X, \gamma )$, where $X$ is a set and $\gamma$ is a given geometric structure of $X$. For example, $X = \mathbf R ^2$, the plane, and $\gamma$ is the canonical differentiable structure $\gamma _d$. Another example is when $X$ is a topological space with its topological structure $\gamma _t$. Consider a differentiable manifold ($\mathbf R ^2$ is its special case), with a differentiable structure $\gamma _d$. Here the differentiable structure means $C^k$ with some $k \ge 1$. Let $f$ be a differentiable function: $M^m \rightarrow N^n$, where $M^m$ and $N^n$ are differentiable manifolds of dimension $m$ and $n$ respectively, assuming $m \ge n$. Given $x \in M$, if the differential $df$ at $x$ mapping the tangent space $T_x M$ of $M$ at $x$ to the tangent space $T_y N$ of $N$ at $y = f(x)$ is an onto map, i.e.

$$\begin{aligned} df : T_x M \xrightarrow {\text {onto}} T_y N, \end{aligned}$$

we call $x$ a regular point of $f$, or $f$ is regular (or nonsingular) at $x$. Otherwise, it is a singular point of $f$. We see that if $x$ is a regular point of $f$ and $y=f(x)$, then “locally” (i.e. in a small neighborhood of $x$ in $M$) the preimage set $f^{-1}(y)$ must be a differentiable submanifold of $M$ with dimension $m-n$, where $y=f(x) \in N^n$. And furthermore, by the continuity of the derivatives of $f$, the differential $df$ is an onto map not only at $x$, but also at any point in a neighborhood of $x$, so that a foliation of $(m-n)$-dimensional submanifolds $\{ f^{-1} (y'); y' \in$ a neighborhood of $y$ in $N\}$ are “evenly distributed”²¹ in a neighborhood of $x$. See them in the dot circle in Fig. 4a. Notice that if $x$ is a singular point of $f$, the behavior of the preimages $f^{-1} (y')$ may be not well-behaved like the regular case and is usually out of control. We therefore call it singular. Given $y \in N$, if $f$ is regular at any $\overline{x} \in f^{-1}(y)$, we say $y$ is a regular value of $f$, otherwise a singular value. Furthermore, when $f$ is regular at any point $x \in M$, we say that $f$ is a regular map from $M$ to $N$. See the corresponding level curves (i.e. preimage sets) of $f$ in Fig. 4a for this case. By the singular set of $f$, we mean the set of all singular points of $f$.

Fig. 4

a $p$ is regular, relative to differentiable structure $\gamma _d$. b $p$ is singular at $B$, but regular at $C$, relative to $\gamma _d$, where $\Lambda$ is the line of cusp points. c $p$ is singular at $O$, but regular at $D$, relative to topological manifold structure $\gamma _M$. d Relative to topological structure $\gamma _t$, the dark area $\varOmega$ is an indifference set of $p$ on which every point is singular. However, $p$ is regular elsewhere

As a special case, we consider a differentiable function: $f: X\rightarrow \mathbf R$, where $X$ is an open set of $\mathbf R ^2$. Note that at any $x = (x_1,x_2) \in X,\,f$ is regular if and only if the gradient $\nabla f \ne 0$ at $x$, where $\nabla f = ( \frac{\partial f}{\partial x_1} , \frac{\partial f}{\partial x_2})$. In fact, $df$ is onto at $x$ iff $\nabla f \ne 0$ at $x$. It characterizes the regularity of $f$ at $x$. When $f$ is regular at $x$, the level sets $f^{-1}(y),\,y \in \mathbf R$, around $x$ are all differentiable curves “locally” and they are “evenly distributed” in a neighborhood of $x$. This is the case we often consider in economics where $f$ is a differentiable utility function on a domain $X$ in $\mathbf R ^2$. The level curves are the so-called indifference curves with respect to $f$.

(II) In case we are given only differentiable preference $p$ on $X$, rather than a differentiable utility function $f: X\rightarrow \mathbf R$, we say a preference $p$ is regular at a point $x \in X$, if the indifference sets are differentiable curves “evenly distributed” in a neighborhood of $x$. In particular, the indifference set containing $x$ is a differentiable curve $\varGamma$. Thus we can consider the unit normal vector $u(\mathbf{x})$ of $\varGamma$ pointing towards the side of better preference. In this sense, Chichilnisky defines a differentiable preference by a differentiable unit vector field $u$ on its regular portion of $X$ and a zero vector at its singular points (Chichilnisky 1982). The figures(Fig. 4a, 4b) illustrate the above concepts, when the differentiable structure $\gamma _d$ is considered.

(III) Based on the above analysis, we remark that given a preference $p$ on $X,\,p$ is regular at a point $x$ of $X$, if and only if the indifference sets around $x$ are differentiable and evenly distributed in a neighborhood of $x$. Conceptually, it requires that the indifference sets around $x$ have the “best” feature which the differentiable structure $\gamma _d$ inherits. In other words, the differentiable structure $\gamma _d$ can detect whether the geometry of indifference sets is “good”. Differentiable curves are “good” relative to the differentiable structure $\gamma _d$. Otherwise, they are “bad”. If “good”, the points are regular. If “bad”, the points are singular. We extend this criterion to topological manifolds. Given a topological manifold $(X, \gamma _M)$ where $\gamma _M$ now is a topological manifold structure. Although the preference in Fig. 4b is singular at a cusp point, such as $B$ on the line $\Lambda$, relative to the differentiable structure $\gamma _d$, it is regular for the topological structure $\gamma _M$. In fact, $\gamma _M$ can not detect the difference of a cusp point from a differentiable point. In other words, “smoothness” is not a notion that a topological manifold structure $\gamma _M$ can detect. Both the indifference curves around $B$ and $C$ are “equally good” in the sense of $\gamma _M$. However, $\gamma _M$ can distinguish the saddle point $O$ in Fig. 4c from the rest of points in $X-\{ O\}$. The latter are all regular with respect to $\gamma _M$. More precisely, at a point $D \in X-\{ O\}$, the indifference set containing $D$ is a manifold-like curve, but the indifference set containing $O$ is a set of two cross curves which is not manifold-like, i.e. not a topological submanifold of $X$. Hence, relative to $\gamma _M,\,D$ is a “good” point but $O$ is a “bad” one. Applying the same argument, we also see that in the category of topological manifolds, the preference defined by Fig. 4d is singular on the dark area $\varOmega$ in $X$ and regular elsewhere, relative to the topological manifold structure $\gamma _M$. We summarize the previous observations as the following. Given a preference $p$ on a geometric space $(X,\gamma )$, there are two criteria for an alternative $\mathbf{x}_0$ in $X$ to be regular:

• (C1) The indifference set containing $\mathbf{x}_0$ is a “good” set relative to $\gamma$.

• (C2) The indifference sets in a neighborhood of $\mathbf{x}_0$ are all “good” relative to $\gamma$ and evenly distributed.

If (C1) and (C2) are not both satisfied, $\mathbf{x}_0$ is singular. Remark that (C2) implies (C1).

(IV) For a differentiable manifold $(X, \gamma _{d})$ and a differentiable utility function f on X, we have noticed that f is defined singular at x, if  $\nabla f = 0$ at x, i.e. if x is a critical point of f. In classical terminology, a critical point is also called a “stationary” point, which means that f is “infinitesimally constant” at the point. Now given a topological space where $\gamma _t$ is a topological structure of X, it is natural to define that a utility function f on X is singular at x, if f is “constant around x”, i.e. the level set of f contains an open neighborhood of x. Correspondingly, we define that a preference p on X is singular at x if the indifference set of p contains an open neighborhood of x. Thus the point in $\varOmega$ of Fig. 4d is singular²², while the points $A, B, C, D, E, O$ of Fig. 4d are now regular relative to $\gamma _t$. Although $\gamma _t$ can detect, for instance, the difference of indifference set at O from those at A, B, C, D, E, but $\gamma _t$ can not evaluate which one of A, B, C, D, E, O is “better” than another-yet $\gamma _M$ can. In fact, A,B,C,D,E and O are equally “good” in $(X,\gamma _t )$. They are all regular relative to $\gamma _t$, satisfying (C1) and (C2) in a modified sense.

(V) Given a finite set $X$, the geometric structure $\gamma _0$ attached to $X$ is just discreteness and cardinality. A subset $S$ of $X$ has its cardinal number $|S|$. For example, a set $S$ containing three alternatives is different from a singleton set $S_0$ in the sense of $\gamma _0$. When preferences on $X$ based on $\gamma _0$ are concerned, especially when we consider social preferences, the “best” case now is that all the alternatives of $X$ are “linearly ordered”, so that they are lined up one after another along increasing preference. This kind of preference $p_0$ is a case where (C2) is satisfied and hence regarded regular relative to the finite discrete structure in transitivity preference theory. Otherwise, a preference on $X$ is called singular. This is why Definition 11 is adopted. For non-transitivity preference theory, a cycle is called a presingularity, because by applying the transitivity map $\sigma$, it becomes a singularity. This is analogous to the concept stated in the first few paragraphs of Sect. 6 for the continuum case that Chichilnisky’s local setting is transformed into our global setting (Huang 2009). Remark that in the discrete case, a singularity which is an indifference set with more than one alternative is now analogous to a singularity for the continuum case, which is a dark area $\varOmega$ as shown in Fig. 4d with non-empty interior.