Econometrica, Vol. 88, No. 1 (January, 2020), 203–205
SAVAGE’S P3 IS REDUNDANT
LORENZ HARTMANN
Department of Quantitative Finance, University of Freiburg
Savage (1954) provided the first axiomatic characterization of expected utility without relying on any given probabilities or utilities. It is the most famous preference axiomatization existing. This note shows that Savage’s axiom P3 is implied by the other
axioms, which reveals its redundancy. It is remarkable that this was not noticed before
as Savage’s axiomatization has been studied and taught by hundreds of researchers for
more than six decades.
KEYWORDS: Savage axioms, subjective expected utility.
LET S BE a state space and X a set of consequences, both nonempty. Subsets of S are called
events. 2S denotes the set of all events.1 Acts are mappings from S to X. The set of acts is
denoted by F . With the usual abuse of notation, X also denotes the set of constant acts.
A decision maker’s preferences over acts are modeled by a binary relation over F . The
asymmetric and symmetric parts of are denoted by and ∼, respectively. For f g ∈ F
and A ∈ 2S , fA g denotes the act resulting in f on A and g on Ac . A preference relation
has a subjective expected utility representation if there exists a utility function u : X → R
and a probability measure P on (S 2S ) such that, for all f g ∈ F ,
f g ⇐⇒
u f (s) dP(s) ≥ u g(s) dP(s)
S
S
A probability measure P on (S 2S ) is nonatomic if, for every A ∈ 2S and r ∈ [0 1], there
exists some B ⊆ A such that P(B) = rP(A). Following Savage (1954), only finite additivity
of P is assumed: for A B ∈ 2S , we have A ∩ B = ∅ =⇒ P(A ∪ B) = P(A) + P(B).
Savage (1954) introduced seven axioms.
P1. is complete and transitive, that is, is a weak order.
P2. For f f g g ∈ F and E ∈ 2S ,
fE g fE g
⇐⇒
fE g f E g For f g ∈ F and A ∈ 2S , f A g means that fA h gA h holds for all h ∈ F . Axioms P1
and P2 ensure that these conditional preferences are well-defined and are weak orders. An
event A ∈ 2S is null if f ∼A g for all f g ∈ F . Otherwise A is nonnull.
P3. For all nonnull E ∈ 2S , f ∈ F and x y ∈ X,
xy
⇐⇒
xE f yE f
Lorenz Hartmann: lorenz.hartmann@vwl.uni-freiburg.de
I thank two anonymous referees for very insightful comments. I thank Dieter Balkenborg, Jürgen Eichberger, Simon Grant, David Kelsey, Yves Le Yaouanq, Yang Liu, Eva Lütkebohmert-Holtz, Massimo Marinacci, Lorenzo Stanca, Qizhi Wang, and in particular, Jean Baccelli and Itzhak Gilboa for comments and
valuable discussions. My biggest thank-you goes to Peter Wakker for numerous vital comments and a correspondence which prepared the ground for the insights of this note.
1
Savage (1954, p. 42) pointed out that all his results remain valid if all acts are measurable with respect to a
σ-algebra of events. This is also the case for the results of this note.
© 2020 The Econometric Society
https://doi.org/10.3982/ECTA17428
204
LORENZ HARTMANN
P4. For A B ∈ 2S and x y z w ∈ X with x y and z w,
xA y xB y
⇐⇒
zA w zB w
P5. There exist f g ∈ F such that f g.
P6. For f g h ∈ F with f g, there exists a partition of S, {A1 An }, such that for
every i ≤ n,
h Ai f g
and
f hAi g
P7. For f g ∈ F and A ∈ 2S ,
1 f A g(s) ∀s ∈ A
=⇒
f A g
∀s ∈ A
=⇒
g A f
2 g(s) A f
The following theorem shows that P3 is implied by the other axioms. Axioms P2 and P7
are primarily responsible for this. Axioms P1 and P4 are also needed.2
THEOREM 1: If satisfies P1 P2 P4, and P7, then satisfies P3.
The theorem directly implies the redundancy of P3 in Savage’s characterization of subjective expected utility preferences.3,4
COROLLARY 1: Let be a preference relation on F . The following are equivalent:
1. satisfies P1 P2 P4 P5 P6, and P7.
2. There exists a nonconstant bounded function u : X → R and a nonatomic finitely additive probability measure P on (S 2S ) such that, for all f g ∈ F ,
u f (s) dP(s) ≥ u f (s) dP(s)
f g ⇐⇒
S
S
Furthermore, P is unique and u is unique up to positive affine transformations.
The boundedness of utility was shown in Fishburn (1970, p. 206).
Two lemmas pave the way for the proof of the theorem. The first lemma states that,
under P1, P2, and P7, the ordinal ranking over constant acts is unaffected by conditioning
on events.
LEMMA 1: Assume that satisfies P1 P2, and P7. If, for z w ∈ X, we have z w, then
z E w for all E ∈ 2S .
PROOF: Consider an arbitrary E ∈ 2S . For z w ∈ X, assume z w. Then z w(s) for
all s ∈ S. Of course also z z(s) for all s ∈ S. This implies z (wE z)(s) for all s ∈ S.
Consider in P7.1 the case A = S to conclude z wE z. P2 implies z E w.
Q.E.D.
2
From P1, only completeness is needed. Transitivity is not.
Savage (1954) showed that when only simple (finite-valued) acts are considered, P1 to P6 are necessary
and sufficient for representation. He then used P7 to extend this result to arbitrary acts. It must be noted that
in the representation without P7, P3 is not redundant.
4
Fishburn (1970) considered a weaker version of P7. One can show that Fishburn’s P7 is equivalent to
Savage’s P7 under P1, P2, and P6. Hence P3 is still redundant if we use Fishburn’s version of Savage’s axioms.
I thank an anonymous referee for making me aware of this.
3
SAVAGE’S P3 IS REDUNDANT
205
The second lemma states that, under P1, P2, and P7, for every nonnull event, there exist
constant acts amongst which, conditional on the event, strict preference holds.
LEMMA 2: Assume that satisfies P1 P2, and P7. For every nonnull event E ∈ 2S , there
exist z w ∈ X such that z E w. We then also have z w.
PROOF: Consider E ∈ 2S nonnull. There exist h1 h2 ∈ F such that h1 E h2 . Completeness and the contrapositive of P7.2 imply the existence of a state s ∈ E such that
h1 E h2 (s). For this s, completeness and the contrapositive of P7.1 imply the existence
of a state s ∈ E such that h1 (s ) E h2 (s). Choose z = h1 (s ) and w = h2 (s) to obtain
z E w. Assume for contradiction that z w. Completeness implies w z. Lemma 1
Q.E.D.
implies w E z, a contradiction. Therefore, we have z w.
The theorem can now be proved.
PROOF OF THEOREM 1: Assume that satisfies P1, P2, P4, and P7. Consider f ∈ F ,
x y ∈ X, and a nonnull event E ∈ 2S . Due to completeness, it suffices to show x y =⇒
xE f yE f and x y =⇒ xE f yE f . Assume x y. Lemma 1 implies x E y. P2 implies
xE f yE f . Assume x y. Since E is nonnull, it follows from Lemma 2 that there exist
z w ∈ X such that z E w and z w. From the former, P2 implies zE w w. In P4,
consider the case A = E and B = ∅ to conclude xE y y. P2 implies xE f yE f . Q.E.D.
REFERENCES
FISHBURN, P. C. (1970): Utility Theory for Decision Making. New York: John Wiley and Sons. [204]
SAVAGE, L. (1954): The Foundations of Statistics. New York: John Wiley and Sons. [203,204]
Co-editor Bart Lipman handled this manuscript.
Manuscript received 26 June, 2019; final version accepted 16 August, 2019; available online 27 August, 2019.