Theoretical Economics Letters, 2013, 3, 17-21
http://dx.doi.org/10.4236/tel.2013.35A2004 Published Online September 2013 (http://www.scirp.org/journal/tel)
Action-Independent Subjective Expected
Utility without States of the Wo rl d
Andreas Duus Pape
Binghamton University Economics Department, Binghamton, USA
Email: apape@binghamton.edu
Received August 2, 2013; revised September 2, 2013; accepted September 10, 2013
Copyright © 2013 Andreas Duus Pape. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
This paper develops an axiomatic theory of decision-making under uncertainty that has no state-space. The choice set-
ting follows Karni [1,2]: a set of effects (outcomes), a set of actions which induce these effects, and a set of real-valued
bets over effects. In Karni’s representation, a preference over action/bet pairs yields utility, which is action-dependent.
In our representation, utility is action-independent. This is achieved by augmenting Karni’s choice set with lotteries
over actions. Identification is achieved similarly to Anscombe-Aumann [3], in which there are objective “roulette” lot-
teries over subjective “horse race” lotteries.
Keywords: Subjective Expected Utility; States of the World; Action Independence
1. Introduction
This paper develops an axiomatic theory of decision-
making under uncertainty that has no state-space. The
resulting representation has action-independent prob-
abilities.
Karni [1,2] proposes a decision theory without refer-
ence to states of the world. The primitives are effects
(outcomes); actions, which induce these effects; and
real-valued bets over which effect obtains. A preference
relation over action/bet pairs yields a utility representa-
tion with subjective probability distributions over effects,
one for each action. Karni’s “main motivation” is to re-
place the state-space; he argues that “the relevant state-
space is often unintuitive and too complex to be com-
patible with decision makers’ perception of choice prob-
lems [1]”. Our representation, like Karni, has actions,
effects, bets, and no state-space.1
In Karni’s representation, utility cannot be compared
across actions; in our representation, it can. Establishing
an action-independent representation in the Karni setting
is the main motivation for our work. (Karni discusses
action-independent preferences in his framework, but
they do not emerge as a necessary consequence of a set
of axioms.) Action-independence is similar to state-inde-
pendence. As Karni points out [1], when his representa-
tion exhibits action-independence, “the probabilities ...
represent the decision maker’s beliefs in the sense of
Ramsey [5]”. Since our axioms deliver Karni’s represen-
tation with action-independence, this applies to our rep-
resentation (see Corollary 1).
The choice set is larger in our representation than that
in Karni. Under Karni, the choice set is bets paired with
actions, while in our representation, the choice set is bets
paired with objective lotteries over actions. Ancom-
be-Aumann [3] augmented the Savage [6] choice set in a
similar way. They added a layer of objective “roulette”
bets on subjective “horse race” bets. Like Anscombe-
Aumann, our representation requires objective lotteries
(“roulette wheel” lotteries) to define the subjective prob-
ability distributions, which in the Anscombe-Aumann
case are called “horse races”, and here are the distribu-
tions over outcomes that result from actions.
Accordingly, like Anscombe-Aumann, the “essential
device” of our representation “is to apply von Neumann-
Morgenstern’s [7] utility theory twice over [3]”. The
proof here uses a similar process. We establish a utility
representation over a subset of all actions: the “determi-
nistic actions”, which are actions that are known by the
decision maker to achieve particular outcomes with cer-
tainty. Then we use the vNM axioms to extend this rep-
resentation over the whole choice set.
1Gilboa and Schmeidler [4] also develop a representation without a
state-space. It differs from ours as it does from Karni; they take utility
as given instead of deriving a utility function over bets and outcomes.Unlike Anscombe-Aumann, who used this technique
C
opyright © 2013 SciRes. TEL
A. D. PAPE
18
to provide a representation, which was the same as the
Savage representation, we provide for a representation
which differs from the Karni representation in an impor-
tant way: the application of this device in the Karni set-
ting delivers an action-independent utility representation,
which Karni’s original framework does not deliver.
The remainder of our paper presents a set of axioms, a
theorem with a corollary, and a proof. The theorem pro-
vides an equivalence between the axioms and an action-
independent representation without a state-space. The
corollary states that our axioms imply an action-inde-
pendent version of Karni, when attention is restricted to
the Karni choice set.
2. Notation and Framework
, with typical element
, is a finite set of effects.
is the set of all bets . , with typical ele-
ment , is the possibly infinite set of actions. For any
set , define
:b
a
S
S

,b
as the set all of simple lotteries
over . is a binary relation defined on the choice
set , where , with typical element
, .
S




b,
a
will be understood to
be the probability that the action occurs under lottery
and
a
a

will be understood to be the (finite) sub-
set of upon which a puts positive probability. For
notational simplicity, we will also interpret
a
to
mean the degenerate lottery over actions, which yields
action with certainty. We define a mixture of ele-
ments of element-wise:
a
 
 
,1 ,
1,1
bb
bb
 
 



 
where
is some objective probability, and where
1

b and are also defined
element-wise. Definition of “null”: an effect θ is null
given the action a if


1b


,,ab ab
for all bets ,bb
such that
 
bb

b
and

b

for all
. The definitions of , , , and “null given
action a” follow Karni [1].
 
A deterministic action a
is an action which yields
the effect
with certainty, in the sense that: given ac-
tion a
, all
are null and
is not null. We also
assume that there is no more than one deterministic ac-
tion for each outcome. Let
be the set of all deter-
ministic actions. We suppose that , i.e. for all 
,a
 . This assumption is critical for our repre-
sentation; it is similar to Karni’s assumption A0 in that it
generates sufficient richness of the choice set.
An example: , = {Stay in Room;
Walk without Umbrella, Walk with Umbrella; Take a
Shower}, and = {Stay in Room; Take a Shower}
under the assumption that Stay in Room determines the
effect Dry and that Take a Shower determines the effect
Wet.
Dry, Wet
3. Axioms
Axioms 1 through 3 are standard von Neuman-Morgen-
stern axioms over objects in . We will refer to them
collectively as “the vNM axioms”.2
Axiom 1. (Preference Relation.) on is asy-
metric and negatively transitive.
Axiom 2. (Independence.) Suppose ,

,b
,b
,
,b
 
. Then
 
,,b


b if and only if
 
,1 ,
,1,for any 0,1
bb
bb
 
 


  
 
Axiom 3. (Continuity.) Suppose ,

,b
,b
,
,b
 
, where

,,bb
 
 



,b. Then
there exist β, γ (0, 1) such that

,1 ,
,,1
bb
bb
 
 
 

,b

 
Axiom 4. (Bet monotonicity.) Let such that
,bb
>bb
for some
and

bb

for
all ,


. Then, for all such that

>0a
,
bb,,

.
Axiom 4 considers two bets, and b, which are
identical but for one effect,
b
. On effect
, bet
yields a higher payoff than . Then it considers lotter-
ies over deterministic actions which yield effect
b
b
with
positive objective probability. Each of those lotteries,
paired with bet , must be preferred to the same lottery
paired with bet
b
b
. The logic is, the only difference be-
tween those lottery/bet pairs is what bet payoff is re-
ceived when effect
occurs, and in that case, the agent
should prefer more wealth to less. This axiom, then,
codifies that more wealth is better. Importantly, this
axiom also delivers that association of effects with bets
which pay off when that effect occurs. That is, it assures
that, when the decision maker is considering the choice
object
,ab
, the payoff given by bet under some
other effect
b
is irrelevant.
Axiom 5. Suppose Then there exists .a
such that
,, abb b

This axiom encodes the idea that an action is of inter-
est to the decision maker only through the bouquet of
effects it delivers by requiring that each action is in es-
sence identical to some lottery over deterministic actions.
In our example with
Dry, Wet , it would require
that action Walk with Umbrella be indifferent for all bets
to some lottery over Stay in Room and Take a Shower.
From this perspective, this axiom can be thought of as
a lottery-reduction axiom: that the decision maker is only
2This formulation of the vNM axioms follows Kreps [8], p. 43-44, with
the corresponding vNM theorem for an arbitrary prize space and sim-
p
le lotteries given by Theorem 5.15, p. 58.
Copyright © 2013 SciRes. TEL
A. D. PAPE 19
ultimately concerned about the outcomes and not the
means by which those outcomes are achieved.
The lottery
which corresponds to action a is
unique for each action (shown below, Section 4, step
three). Given this fact, the weight puts on determi-
nistic action
a
a
will be interpreted as the probability the
agent believes the action induces effect a
.
This interpretation might raise an objection from some
readers, which we address here. Consider a lottery over
deterministic actions, which yields outcome
which
probability p and outcome
with probability 1p
:
i.e. the lottery
1pap a
 . Now suppose the action
a also yields the same outcomes
and
with the
same subjective probabilities p and : In the identi-
fication of the representation below, Axiom 5 is used in
such a way that, it must be the case that
1p
1apa
pa
, else the representation yields the
incorrect subjective probabilities.

The reader may object to this interpretation for the
following reason: it could be the case that the lottery is
strictly preferred to action a. Why? This would be the
case, it could be argued, if action a is associated with
some cost that the lottery over deterministic actions is not
subject to. Therefore, the argument goes, interpreting this
axiom in this way, and using it to specify subjective
probabilities, is equivalent to assuming that all actions
are costless (or, more generally, that they have the same
cost.)
The conclusion of this objection, however, is unwar-
ranted, which can be seen when one considers the set of
outcomes . is an arbitrary set. Suppose that the
elements of are final levels of wealth (excluding
wealth from the bets), so that = R. Seen from this
perspective, it is possible to imagine two actions that
have different costs: Consider, as before, an action a
which yields the outcome
 
with subjective probability
p and the outcome
with subjective probability
:
Now consider action which yields the outcomes
with subjective probability p and outcome
with subjective probability 1, for some c >
0: It could be said that action is identical to action
; but for a cost c. This examples reveals that this
frame- work does allow for actions with different costs;
those costs must be expressed, however, in terms of final
outcomes. It does not allow for costs that are not observ-
able to the modeler.
a
c
c
a
p
a
4. A Representation Theorem
As per the standard definition, we say repre-
sents if
.
Then:
,Ub

,b


,>Ub
 
,, ,U bbb

 
Theorem 1. The following two statements are equiva-
lent:
1) satisfies Axioms 1-5;
2) There exists :u
 unique up to a positive
affine transformation and unique
π: such
that
,Ub
represents , where


 



,,π
a
Uba uba

 


.
Corollary 1. If satisfies Axioms 1 - 5, then on

,,
Kab ab, can be represented by:
 


,,π
Karni a
Uabfub a
 




for some :u
 unique up to a positive affine
transformation, and unique , and for all

π:
a
, a
f
is the identity function.
Corollary 1 is written in this way to emphasize that
,
Karni
Uab is the representation given in Karni, save
that the functions fa in Karni may vary across actions.
Hence our representation implies an action-independent
version of Karni’s representation [1].
Proof. The corollary is an immediate implication of
theorem 1 and requires no proof. The proof that the rep-
resentation implies the axioms (i.e. statement 2 implies
statement 1) is straightforward and is omitted. What fol-
lows is the proof that the axioms imply the representation.
The proof proceeds in four steps.
First, we establish a von Neumann-Morgenstern rep-
resentation on lotteries over deterministic actions paired
with bets.
Consider the preferences on the set
. The vNM axioms and the vNM theorem
imply that there exists
ˆ
u, unique up to a positive,
affine transformation, such that represents
on
,b
ˆ
, where

ˆ
 
ˆ,,bau
ab

 . Now
define

,b
ˆ
ua,ub
. This is well-defined because
we assume that every outcome has exactly one corre-
sponding deterministic action and vice versa. Then we
can equivalently define
ˆ
as:


,,ba
u


ˆ
b
.
Second, we establish that does not depend on
all values in the vector b; instead, it only depends on the
value of
,ub
b
. Then, we introduce a representation
which takes advantage of this fact.
Suppose ,bb
such that
 
bb
. Let
be the lottery in
which puts all weight on action
. Note that
,b
,b

because bets b and
b
differ only on events that are null given action
.


ˆˆ
,,, ,
,,
bb b
ubub



b




Define
,ur
, r
as:

,,
r
urub

where br is a bet that yields r for every effect. Define
Copyright © 2013 SciRes. TEL
A. D. PAPE
20

as:




,,baub


th
.
Given that only the
element of b matters, that
represents on implies also repre-
sents on .

ˆ

ˆ
Third: By Axiom 5, each action is indifferent to at
least one lottery over deterministic actions. Here we es-
tablish that each action is indifferent to no more than one
such lottery.
Let and
a
, such that
is indif-
ferent to a for all bets, in the sense of Axiom 5. Suppose
toward a contradiction that there exists
,
such that
and
is indifferent to a for all bets
as in the sense of Axiom 5. Then,
is indifferent to
for all bets as in the sense of Axiom 5.
, and that they are distributions, implies that,
for at least one
 
,>
 
 . For notational
simplicity suppose that
has index 1. Then interpret
and
as -dimensional vectors and express
them as:





 


T
2
T
10,,,
11
1, 0,,01, 0,,0


  
  


 




 
T






 
T
2
T
10,,,
11
1, 0,, 0


  





Since
for all bets, then, by the independence
axiom, the following two lotteries,
and
, are also
indifferent for all bets:





 
 
T
2
T
10,, ,
111
1,0, ,0
1

 
 
 






T
2
0,,,
11


 


Now consider some bet b and some bet b
, where
bb
 for all ,



, but

>bb
.
By independence (axiom 2),
,b,b

since the
action a
occurs with probability zero under
. But
,b

,b
by bet monotonicity (Axiom 4). This
implies that
and
are not indifferent for at least
one bet of b; b0: Contradiction.
Fourth, we interpret the unique corresponding lottery
over deterministic actions as a subjective probability dis-
tribution, and apply vNM to the entire choice set .
For each a, let a
be the unique deterministic action
lottery
a such that a
is indifferent to a
for all bets in the sense of Axiom 5. Define the family of
probability distributions as: π


πa
aa

.
Consider some
. By the vNM axioms and
the vNM theorem,


,,
avaba

Ub
represents on , where
,vab
is unique up to a
positive affine transformation. Now consider an arbitrary
degenerate lottery a and bet b.
,b vab,Ua and
a
,,πb

abu
. Since a and b were
chosen arbitrarily, this implies can be affine-
transformed into such that
,va
b
v
,,πvabu b

a
for all ,.ab

Therefore the following function represents on :


 



,,π
a
Ubauba

 


where the are unique and u is unique up to a positive
affine transformation.
π
5. Conclusion
This paper develops an axiomatic theory of decision-
making under uncertainty that has no state-space. The
resulting representation has action-independent prob-
abilities. The primitives are effects (outcomes); actions,
which induce these effects; and real-valued bets over
which effect obtains. This representation is most closely
related to Karni [1,2], which also proposes a decision
theory without reference to states of the world. In Karni’s
representation, unlike ours, however, the probability dis-
tributions are action dependent, which results in prob-
abilities that can not be compared across actions. In our
representation, the probability distributions are action
independent, so they can be compared across actions and
therefore are probabilities in the sense of Ramsey [5].
REFERENCES
[1] E. Karni, “Subjective Expected Utility Theory without
States of the World,” Journal of Mathematical Economics,
Vol. 42, No. 3, 2006, pp. 325-342.
http://dx.doi.org/10.1016/j.jmateco.2005.08.007
[2] E. Karni, “A Theory of Bayesian Decision Making with
Action-Dependent Subjective Probabilities,” Economic
Theory, Vol. 48, No. 1, 2011, pp. 125-146.
[3] F. Anscombe and R. Aumann, “A Definition of Subjec-
tive Probability,” The Annals of Mathematical Statistics,
Vol. 34, No. 1, 1963, pp. 199-205.
[4] I. Gilboa and D. Schmeidler, “Subjective Distributions,”
Theory and Decision, Vol. 56, No. 4, 2004, pp. 345-357.
Copyright © 2013 SciRes. TEL
A. D. PAPE
Copyright © 2013 SciRes. TEL
21
http://dx.doi.org/10.1007/s11238-004-2596-7
[5] F. P. Ramsey, “The Foundations of Mathematics and
Other Logical Essays: Truth and Probability,” Routledge
and Kegan Paul, London, 1931, pp. 156- 198.
[6] L. J. Savage, “The Foundations of Statistics,” Wiley, Ho-
boken, 1954.
[7] J. von Neumann and O. Morgenstern, “Theory of Games
and Economic Behavior,” Princeton University Press,
Princeton, 1944.
[8] D. M. Kreps, “Notes on the Theory of Choice. Under-
ground Classics in Economics,” Westview Press, Inc.,
Fredrick A. Praeger, 1988.