A “threesentence proof” of Hansson’s theorem
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Abstract
We provide a new proof of Hansson’s theorem: every preorder has a complete preorder extending it. The proof boils down to showing that the lexicographic order extends the Pareto order.
Keywords
Ordering extension theorem Lexicographic order Pareto order PreferencesJEL Classification
C65 D011 Introduction
We thus argue that abstract extension theorems of orders can be embodied in this simple principle.“Every preorder is essentially a Pareto order and the lexicographic order extends the Pareto order. Since the lexicographic order is complete every preorder has an ordering extension”.
The note is organized as follows. Section 2 contains notation and preliminaries. In Sect. 3, we present our proof of Hansson’s theorem. Section 4 concludes. We gather some standard results in set theory in Appendix A.
2 Notation and preliminaries

Reflexive If \(x \succsim x\) for all \(x \in X\).

Transitive If \(x \succsim y\), \(y \succsim z\) implies \(x \succsim z\) for all \(x,y,x \in X\).

Complete If \(x \succsim y\) or \(y \succsim x\) for all \(x,y \in X\).

Antisymmetric If \(x \succsim y\) and \(y \succsim x\) implies \(x=y\) for all \(x,y \in X\).
If \(\succsim \) is an order we define a relation \(\succ \) on X by \(x \succ y\) if and only if \(x \succsim y\) and not \(y \succsim x\). We also define an order \(\sim \) on X by \(x \sim y\) if and only if \(x \succsim y\) and \(y \succsim x\). A complete preorder \(\succsim ^{\prime }\) on X extends (or is an ordering extension of) a preorder \(\succsim \) on X if for all \(x,y \in X\): \(x \succsim y\) implies that \(x \succsim ^{\prime } y\) and \(x \succ y\) implies that \(x \succ ^{\prime } y\).
Given a set I and for each \(i \in I\) a complete preorder \(\succsim _i\) on X the Pareto order \(\ge \) on X is defined by for all \(x,y \in X\): \(x \ge y\) if and only if \(x \succsim _i y\) for all \(i \in I\). We will sometimes refer to \(\succsim _i\) as a coordinate relation.
Let I be a set with linear order \(\le \) on I and a collection of complete preorders \(\succsim _i\) on X for all \(i \in I\). Define a lexicographic relation \(\ge _{L}\) on X by \(x \ge _{L} y\) if and only if \(x \succsim _i y\) for all \(i \in I\), or there is a \(j \in I\) such that \(x \succ _j y\) and \(x \sim _i y \) for all \(i \in I\) with \(i < j\). It is a standard fact in basic set theory that if \(\le \) is a well order then \(\ge _{L}\) is a complete preorder on X. For example, Mandler (2015) alludes to this result, see also Ciesielski (1997). For completeness a proof is presented in Lemma A.1 in the Appendix A.
3 The proof
We now give our proof of Hansson’s theorem. A first crucial ingredient in our proof is a corollary to a result in Evren and Ok (2011). It shows that every preorder essentially is a Pareto order. The result has the same universal character as a representation result by Chipman (1960), which shows that every complete preorder essentially is a lexicographic order:
Lemma 3.1
For every preorder \(\succsim \) there is a Pareto order \(\ge \) with coordinate relations \(\succsim _i\) for all \(i \in I\) such that \(x \succsim y\) if and only if \(x \ge y\) for all \(x,y \in X\).
Proof
Let \(I=X\) and for each \(x \in X\) let \(u_x(y):={\mathbf{1}}_{\{z \in X  z \succsim x\}}(y)\) for all \(y \in X\) (where \({\mathbf{1}}_A\) denotes the indicator function of a set A). Define a relation \(\succsim _x\) by \(z \succsim _x y\) if and only if \(u_x(z) \ge u_x(y)\). The result follows by Evren and Ok (2011, Proposition 1). \(\square \)
Another important observation in our proof is that the lexicographic order extends the Pareto order. This observation is recorded as Lemma 3.2.
Lemma 3.2
Let \(\ge \) be a Pareto order on X with coordinate relations \(\succsim _i\). Then the lexicographic order \(\ge _{L}\) on X with coordinate relations \(\succsim _i\) for all \(i \in I\) extends \(\ge \).
Proof
Let \(x,y \in X\) with \(x \ge y\). If \(y >_{L} x\) then there is an \(i \in I\) such that \(y \succ _i x\), contradicting that \(x \succsim _i y\) for all \(i \in I\). Let \(x,y \in X\) with \(x >y\). Then \(x \succ _j y\) for some \(j \in I\) and \(x \succsim _i y\) for all \(i \in I\). If \(y \ge _{L} x\) then either \(x \sim _i y\) for all \(i \in I\) or \(y \succ _i x\) for some \(i \in I\), a contradiction. \(\square \)
We are now ready for our proof of Hansson’s theorem:
Theorem 3.3
Let \(\succsim \) be a preorder order on X. Then there exists a complete preorder \(\succsim ^{\prime }\) on X extending \(\succsim \).
Proof
By Lemma 3.1 there is a Pareto relation \(\ge \) with coordinate relations \(\succsim _i\) such that \(x \ge y\) if and only if \(x \succsim y\) for all \(x,y \in X\). Well order I by \(\le \) and let \(\ge _L\) be the lexicographic relation with coordinate relations \(\succsim _i\) for all \(i \in I\). Then \(\ge _L\) is a complete preorder by Lemma A.1 and Lemma 3.2 implies that \(\ge _{L}\) extends \(\succsim \). \(\square \)
4 Concluding remarks
 (a)
The theorem follows from intuitively plausible and simple principles. Once it is understood that the lexicographic order extends the Pareto order, the rest of the proof follows smoothly.
 (b)
The use of the well ordering theorem is transparent. We only had to use the well ordering theorem once in the second sentence of Theorem 3.3. Hence if I is some set that is known to be well ordered (like the set of natural numbers or a finite set), we see that the proof of Hansson’s theorem follows without invoking the well ordering Theorem 3.3.
References
 Chipman, J.S.: The foundations of utility. Econometrica 28(2), 193–224 (1960)CrossRefGoogle Scholar
 Ciesielski, K.: Set Theory for the Working Mathematician, vol. 39. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
 Evren, Ö., Ok, E.A.: On the multiutility representation of preference relations. J. Math. Econ. 47(4), 554–563 (2011)CrossRefGoogle Scholar
 Hansson, B.: Choice structures and preference relations. Synthese 18(4), 443–458 (1968)CrossRefGoogle Scholar
 Mandler, M.: The lexicographic method in preference theory. Available at SSRN 2192127 (2015)Google Scholar
 Petri, H., Voorneveld, M.: Characterizing lexicographic preferences. J. Math. Econ. 63, 54–61 (2016)CrossRefGoogle Scholar
 Szpilrajn, E.: Sur l’extension de l’ordre partiel. Fund. Math. 1(16), 386–389 (1930)CrossRefGoogle Scholar
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