Continuity and Completeness

McCarthy, D., Mikkola, K., Continuity and completeness of strongly independent preorders. Mathematical Social Sciences 93 (2018): 141–145. PDF | Abstract | Discussion

Expected utility theory has three main axioms: completeness, continuity, and independence. Completeness is dubious in many normative settings, so what happens when completeness is dropped? We show that there is a surprising difficulty.

Independence and continuity axioms

Let $X$ be a convex set, that is, $\alpha x + (1-\alpha) y \in X$ for any $x, y \in X$, $\alpha \in (0,1)$. An example is the set of probability measures on a finite set of outcomes. Let $\succsim$ be a preorder (a reflexive, transitive binary relation) on $X$.

Consider the strongest and most natural version of independence, strong independence.

(SI) For $x,y,z\in X$ and $\alpha\in(0,1)$, $x\succsim y \iff \alpha x+(1-\alpha)z\succsim \alpha y+(1-\alpha)z$.

Continuity axioms also come in different forms. The two most common are Archimedean and mixture continuity.

(Ar) For $x$, $y$, $z\in X$, if $x \succ y \succ z$, then $(1-\alpha)x+\alpha z \succ y$ for some $\alpha \in (0,1)$.
(MC) For $x$, $y$, $z \in X$, if $(1 – \alpha) x+\alpha y \succ z \text{ for all } \alpha \in (0,1]$, then $x \succsim z$.

If $\seX$ can be incomplete, it is natural to consider a slight strengthening of Ar.

(Ar$^{+}$) For $x,y,z\in X$, if $x \succ y$, then $(1-\alpha)x+\alpha z\succ y$ for some $\alpha \in (0,1)$.
Ar and Ar$^+$ express similar ideas. Suppose $x \sX y$, and $z$ is a third alternative. Then Ar says that $z$ cannot be so much worse than $x$ that mixing $x$ with any positive chance of $z$, no matter how small, would disturb the preference $x \succ y$. Ar$^+$ extends this by replacing “worse than” with “worse than or incomparable with”.

Main result

Theorem 1. For an SI preorder $\seX$ on a convex set $X$,
1. Ar and MC essentially rule out incompleteness.
2. Ar$^+$ and MC formally rule out incompleteness.

Stated more precisely, the first claim is that Ar and MC imply that comparability is an equivalence relation. But this fits poorly with intuitions about comparability.

To use an example of Joseph Raz, suppose that a legal career and a musical career are incomparable. Then if comparability is an equivalence relation, it cannot be that both are better than a career cleaning toilets.

To gain some intuition about why the theorem is true, it is worth working through the following example, adapted from Aumann (1962).

Example 1. Let $X = \RRR^2$. Define the commonly used weak and strong Pareto preorders as follows.
\begin{align*}
x \succsim_{\text{wp}} y &\iff x = y \text{ or } x_i > y_i \text{ for } i=1,2. \\
x \succsim_{\text{sp}} y &\iff x_i \geq y_i \text{ for } i=1,2.
\end{align*}
Then $\succsim_{\text{wp}}$ satisfies Ar and Ar$^+$, but not MC. Conversely, $\succsim_{\text{sp}}$ satisfies MC but not Ar or Ar$^+$.

Discussion

Here are a couple of reasons why the theorem is relevant for philosophers.

First, it raises a puzzle about what we should think about continuity axioms, and about the reliability of our intuitions about continuity. There are many discussions of incomplete strongly independent preorders in the economics literature. Both Mixture Continuity and variations on the Archimedean axioms feature heavily. So it seems that both styles of axioms are regarded as (a) plausible, and (b) comparably plausible. But in the presence of incompleteness, they say essentially opposite things. If one is right, the other is wrong. What should we make of this?

Second, moral philosophers often assume incompleteness, then present informal continuity arguments, whether they are explicitly stated that way or not. Since continuity behaves quite subtly and counterintuitively in the presence of incompleteness, I think they should be more careful.

Theorem 1 extends results of Aumann (1962), Dubra (2011), and Schmeidler (1971).

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