### Abstract

- Top of page
- Abstract
- 1 Introduction
- 2 Setting
- 3 Basic axioms
- 4 Inter-informational axioms and characterization
- 5 Comparative ambiguity aversion
- References

This paper axiomatically studies a family of preference orderings indexed by variable, objective and generally imprecise information. It characterizes functional relationships between objective information and subjective beliefs. In particular, it establishes the contraction model due to Gajdos, Hayashi, Tallon, and Vergnaud (2008) without relying on identifying direct preference over information.

### 1 Introduction

- Top of page
- Abstract
- 1 Introduction
- 2 Setting
- 3 Basic axioms
- 4 Inter-informational axioms and characterization
- 5 Comparative ambiguity aversion
- References

The classical subjective expected utility (SEU) theory (Savage 1954; Anscombe and Aumann 1963) identifies the set of axioms on preference ranking over actions that characterizes the decision criterion in which the decision maker holds some subjective belief over possible states of the world, and evaluate actions in the expected utility form based on her belief.

The SEU theory puts no restriction on what to believe, however. It only establishes the existence of *some* subjective belief which could be any, and does not explain how it is related to objective information. It leaves information to be implicit and fixed, and thus remains silent about how subjective belief is related to objective information.

Since the models of ambiguity aversion are built on the same domain as adopted in the SEU theory, they also leave information to be implicit and fixed. Thus they cannot distinguish if the observed ambiguity aversion is due to objective degree of imprecision of information or to the decision maker’s subjective interpretation of such imprecise information.

As the leading case, consider the multiple-priors model (Gilboa and Schmeidler 1989) in which an act *f*, mapping from states into outcomes, is evaluated in the form

in which *u* is the von Neumann–Morgenstern index over outcomes and *C* is the set of probability measures over states, and the decision maker takes expectation *E*_{p} of state-contingent utility with the probability measure *p* being the worst-case scenario in *C*. There are two existing interpretations of set *C*. One is that the decision maker with larger *C* exhibits larger ambiguity aversion, in the sense that she is more pessimistic and takes a wider range of worst-case scenarios. The other is that larger *C* describes that the decision maker is facing larger (non-probabilistic) uncertainty.

However, the set *C* here is obtained as a *part of the representation* of preference over acts, and it is a composite of subjective attitude toward uncertainty and objective property of information that is not explicit in the model. In other words, it is not clear if larger *C* means the decision maker being more pessimistic or less confident, which is a subjective nature, or means information being more imprecise, which is an objective nature.

Recent theoretical works attempt to relate subjective beliefs to objective information, and separate ambiguity aversion into the above noted subjective and objective factors, by incorporating objective but imprecise information as a variable. They assume that information comes in the form of a set of *objectively possible* probability laws, while the decision maker does not know anything about which one in the set is true or more likely to be true. Let us call it a *probability–possibility set*.

Among them, recent papers by Olszewski (2007), Ahn (2008), Stinchcombe (2003), Gajdos, Tallon, and Vergnaud (2004), Gajdos, Hayashi, Tallon, and Vergnaud (2008) (henceforth GHTV) and Giraud (2005) explicitly include imprecise information as a part of *objects of choice*.^{1} They explicitly look at preference over imprecise information and give axiomatic characterization of decision models which allow aversion to imprecision of information. In particular, Gajdos, Hayashi, Tallon, and Vergnaud (2008), the most related paper here, consider preference over pairs of probability–possibility sets and acts. Let *P*, *Q* denote probability–possibility sets and *f*, *g* denote acts, random variables which map states into outcomes. They consider preference in the form , which means the decision maker prefers taking action *f* under information *P* to taking *g* under *Q*.

Although explicit attitude toward imprecise information is of significant interest, identifying preference/choice over information as well as actions requires a richer set of observations than identifying preference just over actions. This is actually too much if we are just interested in identifying a functional relationship between imprecise information and subjective beliefs, and ambiguity attitude as relevant to choice over acts. In this paper we consider a primitive such that a smaller set of observations suffices for its identification. We consider a family of preference rankings over acts which is indexed by probability–possibility sets. It is given in the form , where means the decision maker prefers action *f* to action *g* under information *P*. Our setting is rather closer to the one by Damiano (2006), which considers a family of preference rankings over acts which is indexed by cores of convex capacities over states.

In this setting we provide an axiomatic analysis of functional relationship between objective information and subjective (and possibly ambiguous) beliefs. In particular, we demonstrate that the contraction model by Gajdos, Hayashi, Tallon, and Vergnaud (2008) is established without relying on identifying direct preference over information.

### 4 Inter-informational axioms and characterization

- Top of page
- Abstract
- 1 Introduction
- 2 Setting
- 3 Basic axioms
- 4 Inter-informational axioms and characterization
- 5 Comparative ambiguity aversion
- References

It is also natural to assume that if every possible probability law in a probability–possibility set supports one action over another, so does the imprecise information as a whole.

However, conjunction of Axiom 6 and 7 leads to the following characterization.

**Definition 1** *An**matrix**T**is said to be a unitary transformation if*

*The Birkhoff–von Neumann theorem states that any doubly stochastic matrix is obtained as a convex combination of permutation matrices, hence unitary transformation is viewed as a stochastic generalization of permutation. The second condition states that unitary transformation does not change rankings between acts when information is precise. Let**denote the set of all the unitary transformations.*

The following lemma provides a simple characterization of unitary transformation.

Now we state the invariance axiom.

*Sufficiency of the axiom*: Suppose . Then a separating hyperplane argument yields that there is such that

where *x* is taken so that without loss of generality.

Let be such that *x*=*Ty*. Then . Then we have

for some .

The last axiom is a technical one, but it excludes certain class of selection mappings. For example, center of gravity is not continuous.

The main result states that the decision maker’s preference given each probability–possibility set is represented in the maximin expected utility form with subjective set of priors, where the subjective set is obtained by shrinking the probability–possibility set toward its ‘center’ at a constant rate. The notion of center characterized here is Steiner point. See Schneider (1993) for more detailed properties of Steiner point.

Now we state the main theorem.

**Independence of the axioms**

Here we discuss independence of the axioms. Because our variable information argument is built on the family of preferences represented in the maximin form with risk preference being independent of information, we take the first five axioms, Order, Monotonicity, Certainty Independence, Ambiguity Aversion and Outcome Preference as ‘ground axioms,’ and discuss independence of the second five axioms which deal with variable information.^{3}

*Reduction under Precise Information* : Consider a representation with

being given by

where

is the vector of uniform distribution. The family of preferences represented by such class satisfies all the axioms but Reduction under Precise Information.

*Dominance* : For simplicity of illustration, restrict attention to probability–possibility sets which are sufficiently away from the boundary of

. Consider a representation with

being given by

with

is sufficiently close to 1. This is rather inflating probability–possibility sets rather than shrinking. Then the family of preferences represented by such class satisfies all the axioms but Dominance.

*Combination* : Consider a representation with

being given by

with

, where

is a mapping which satisfies

for all

, commutes with unitary transformations and satisfies continuity with respect to the Hausdorff metric, but does not satisfy mixture linearity. Center of gravity commutes with unitary transformation and violates mixture linearity, but it also violates continuity when the dimension of sets changes.

^{4} However, it is continuous within the space of compact convex sets with the same dimension.

*Invariance to Unitary Transformations* : Consider a representation with

being given by

where

denotes

*generalized Steiner point* which is defined for a non-atomic Borel probability measure

over

, not necessarily uniform, in the form

The family of preferences represented by such class satisfies all the axioms but Invariance to Unitary Transformations.

*Information Continuity* : Consider a representation with

being given by

with

, where

is a mapping which satisfies

for all

, satisfies mixture linearity and commutes with unitary transformations, but does not satisfy continuity with respect to the Hausdorff metric. Such an example of mapping is found in Schneider (

Schneider 1993, page 170). The family of preferences represented by such class satisfies all the axioms but Information Continuity.

### 5 Comparative ambiguity aversion

- Top of page
- Abstract
- 1 Introduction
- 2 Setting
- 3 Basic axioms
- 4 Inter-informational axioms and characterization
- 5 Comparative ambiguity aversion
- References

In the existing literature, ambiguity aversion is defined for preference over acts, where information is taken to be implicit and fixed (see for example Epstein 1999; Ghirardato and Marinacci 2002). Hence one cannot distinguish if any observed ambiguity aversion is due to objective degree of imprecision of information or to the decision maker’s subjective interpretation of such imprecision. Here we separate the two, and identify the purely subjective part of ambiguity aversion. Thus, two decision makers facing the same information can exhibit different degrees of ambiguity aversion which are revealed from choices over acts.

The following result states that a more ambiguity averse decision maker holds a larger set of beliefs than a less ambiguity averse one does, whenever they face *the same* information.

The result below shows that in the contraction model ambiguity aversion is described by one parameter.

Proof: In the contraction model, the assertion of the previous proposition holds if and only if we have .