Sketch of Richter’s Proof on Revealed Preference
Chiaki Hara
Institute of Economic Research, Kyoto University
April 20, 2010
Abstract
We provide a sketch of Richter’s (1966) proof on revealed preference in general choice
spaces based directly on the Axiom of Choice.
1
Setup and Theorem
Let X be a non-empty set. Let (B, C) be a choice structure defined on X, that is, B is
a non-empty set of non-empty subsets of X and C is a mapping from B to the set of all
non-empty subsets of X such that C(B) ⊆ B for every B ∈ B.
Let R0 be the revealed-to-be-at-least-as-preferable-as relation of (B, C), that is, for all
x ∈ X and y ∈ X, xR0 y if and only if there exists a B ∈ B such that {x, y} ⊆ B and
x ∈ C(B). Denote by R1 the transitive closure of R0 . We say that (B, C) satisfies the Strong
Axiom of Revealed Preference if x ∈ C(B) whenever x ∈ X, y ∈ X, xR1 y, B ∈ B, {x, y} ⊆ B,
and y ∈ C(B).
Theorem 1 (Richter (1966)) The choice structure (B, C) satisfies SARP if and only if
there exists a complete and transitive binary relation R such that
C(B) = {x ∈ B | xRy for every y ∈ B}
(1)
for every B ∈ B.
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Sketch of the proof
The If part (the existence of a complete and transitive binary relation implies that the choice
structure C defined by (1) satisfies SARP) is easy to prove. In the following, we prove the
Only-If part.
Let D = {(x, x) ∈ X × X | x ∈ X}. Let R2 = R1 ∪ D, that is, we let, for all x ∈ X and
y ∈ X, xR2 y if and only if xR1 y or x = y. Then R2 is a reflexive and transitive relation.
Denote by R2i the indifference part of R2 and by R1s the strict part of R2 . Then R2i is an
equivalence relation (a reflexive, symmetric, and transitive binary relation). Let X = X/R2i ,
1
the equivalence class on X generated by R2i . Define a binary relation P0 on X by letting,
for all Y ∈ X and Z ∈ X , Y P0 Z if and only if yR2s z for all y ∈ Y and z ∈ Z. Then, for all
Y ∈ X and Z ∈ X , Y P0 Z if and only if yR2s z for some y ∈ Y and z ∈ Z. Since R1s = R2s ,
P0 is transitive and irreflexive.
Denote by P the set of all transitive and irreflexive binary relations on X that includes
P0 , endowed with the set-theoretic inclusion ⊇. Let Q be a totally ordered subset of P (that
S
is, if P ∈ Q and Q ∈ Q, then either P ⊇ Q or Q ⊇ P). Then the binary relation P∈Q P
is transitive and irreflexive, and includes P0 . Hence it is an upper bound of Q with respect
to ⊇. By Zorn’s Lemma, there exists a maximal element, denoted by P1 , of P with respect
to ⊇.
We claim that P1 is total, that is, for all Y ∈ X and Z ∈ X , if Y 6= Z, then either
Y P1 Z or ZP1 Y . Indeed, if not, then there are Y ∈ X and Z ∈ X with Y 6= Z such
that neither Y P1 Z nor ZP1 Y . Denote by P2 the transitive closure of P1 ∪ {(Y, Z)}. To
show that it irreflexive, suppose that there is a W ∈ X such that W P2 W . Then there are
Y1 , Y2 , . . . , YN such that Y1 = W , YN = Y , and Yn P1 Yn+1 for every n ≤ N − 1, and there are
Z1 , Z2 , . . . , ZM such that Z1 = Z, ZM = W , and Zm P1 Zm+1 for every m ≤ M − 1. Since
P1 is transitive, W P1 Y , ZP1 W , and, hence, ZP1 Y . This is a contradiction. Hence P2 is
irreflexive, and, of course, transitive. But since P2 ) P1 , this contradicts the maximality of
P1 . We can therefore conclude that P1 is total.
Let R = P1 ∪ D, where D = {(Y, Y ) | Y ∈ X }. Then R is complete and transtive.
Define a binary relation R on X by letting, for all y ∈ X and z ∈ X, yRz if and only if Y RZ,
where y ∈ Y ∈ X and z ∈ Z ∈ X . Then R is complete and transtive.
It now suffices to prove that R satisfies (1). Suppose first that x ∈ C(B) and y ∈ B. Then
xR0 y. Hence xR1 y. Hence xR2 y. Let x ∈ W ∈ X and y ∈ Y ∈ X . Then W RY .1 Thus
xRy.
It remains to prove that for all x ∈ C(B) and y ∈ B \ C(B), xRs y. It is this stage of
proof where SARP is used. Indeed, let x ∈ B and y ∈ B \ C(B). A first application of SARP
implies that xR0s y.2 A second application of SARP implies that xR1s y. Let x ∈ W ∈ X and
y ∈ Y ∈ X . Then W P0 Y . Thus W P1 Y . Since P1 is irreflexive, W R s Y . Thus xRs y. The
proof is thus completed.
References
[1] Marcell K. Richter, 1966, Revealed preference theory, Econometrica 34, 635–645.
1
2
To be exact, consider the case of xR2i y and that of xR2s y separately.
In fact, WARP suffices to establish this claim.
2