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Keywords:

  • subjective belief;
  • imprecise information;
  • ambiguity;
  • Steiner point
  • D81

Abstract

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Setting
  5. 3 Basic axioms
  6. 4 Inter-informational axioms and characterization
  7. 5 Comparative ambiguity aversion
  8. 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

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Setting
  5. 3 Basic axioms
  6. 4 Inter-informational axioms and characterization
  7. 5 Comparative ambiguity aversion
  8. 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.

Motivated by the Ellsberg paradox (1961), decision models accommodating ambiguity aversion are established by many studies such as Gilboa and Schmeidler (1989), Epstein (1999), Ghirardato and Marinacci (2002).

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

  • image

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 Ep of state-contingent utility inline image 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 inline image, 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 inline image, where inline image 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.

2 Setting

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Setting
  5. 3 Basic axioms
  6. 4 Inter-informational axioms and characterization
  7. 5 Comparative ambiguity aversion
  8. References

Let inline image be a finite set of states of the world. Let X be a compact metric space of pure outcomes. We consider lotteries as outcomes, and let inline image be the set of Borel probability measures over X, which is again a compact metric space with respect to the Prokhorov metric. A lottery-act is a mapping from states into lottery outcomes, and let inline image be the domain of lottery-acts (Anscombe and Aumann 1963), which is endowed with the product topology. Note that inline image is viewed as a subset of inline image consisting of constant mappings. The set inline image is a mixture space in which mixture of measures is defined as follows: given inline image and inline image, the mixture inline image is defined by

  • image

for all Borel subsets B of X. Thus, the set of lottery-acts inline image is a mixture space in which mixture of acts is defined as follows: given inline image and inline image, the mixture inline image is defined by

  • image

for each inline image.

Let inline image be the set of probability measures over inline image, which is the inline image dimensional unit simplex in inline image. Let inline image be the set of compact convex subsets of inline image, which is endowed with the Hausdorff metric. An element of inline image is called a probability–possibility set. When inline image is given, the decision maker knows that the true probability law lies in P, but she does not know anything about which one in it is true or more likely to be true. Also for each probability measure inline image, the singleton probability–possibility set consisting of it only is denoted by inline image.

Given a probability–possibility set inline image, let inline image denote the preference relation defined over inline image. For inline image and inline image, the ranking inline image states that the decision maker weakly prefers action f to action g under information P. Let inline image be the family of preference relations index by probability–possibility sets.

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Setting
  5. 3 Basic axioms
  6. 4 Inter-informational axioms and characterization
  7. 5 Comparative ambiguity aversion
  8. References

The first four axioms are taken from Gilboa and Schmeidler (1989), which are applied to each fixed probability–possibility set.2

Axiom 1 (Order) For allinline image,inline imageis complete, transitive and continuous.

Axiom 2 (Monotonicity) For allinline imageandinline image, ifinline imagefor allinline image, theninline image.

Axiom 3 (Certainty Independence) For allinline image, inline image, inline imageandinline image, ifinline imagetheninline image.

Axiom 4 (Ambiguity Aversion) For allinline image, inline imageandinline image, ifinline imagetheninline image.

The following is an immediate consequence of applying the Gilboa–Schmeidler result to each P.

Lemma 1 A family of preferences inline imagesatisfies Axioms 1–4 if and only if there exist a family of mixture-linear continuous real-valued functionsinline imageoverinline imageand a mappinginline imagesuch that for eachinline image, the rankinginline image is represented in the form

  • image

Moreover, inline imageis unique; and ifinline imagerepresents the same family of preferences there exist numbersinline imagesuch thatAP >0andvP=APuP+BPfor eachinline image.

In the next section we introduce axioms which deal with variable information. Before this we impose a minimal consistency condition across probability–possibility sets, that outcome (risk) preference is independent of information.

Axiom 5 (Outcome Preference) For all inline imageandinline image, inline imageif and only ifinline image.

Proposition 1 A family of preferences inline imagesatisfies Axioms 1–5 if and only if there exist a mixture-linear continuous real-valued functionuoverinline imageand a mappinginline imagesuch that for eachinline image, the rankinginline imageis represented in the form

  • image

Moreover,inline imageis unique anduis unique up to positive affine transformations.

Proof:  Necessity of the axiom is obvious. To see its sufficiency, notice that inline image over inline image is identical across P and falls in the expected utility theory. Hence the von Neumann–Morgenstern index uP is cardinally equivalent across P, and without loss of generality it is set to be u which is independent of P. inline image

4 Inter-informational axioms and characterization

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Setting
  5. 3 Basic axioms
  6. 4 Inter-informational axioms and characterization
  7. 5 Comparative ambiguity aversion
  8. References

Now we introduce axioms which deal with variable information. First, we introduce an axiom which states that when information is precise it is used as subjective belief as it is. Given inline image and inline image, let inline image be the lottery induced by p and f.

Axiom 6 (Reduction under Precise Information) For allinline imageandinline image, inline image.

Lemma 2 Assume thatinline imagesatisfies Axioms 1–5. Letinline imagebe the pair representinginline image. Then,inline imagesatisfies Axioms 6 if and only if

  • image

for allinline image.

Proof:  Necessity of the axiom is straightforward. To prove sufficiency, assume without loss of generality that any random variable inline image is generated by some f in the form inline image. Then we have inline image. For each inline image, by taking inline image and inline image for all inline image, we have inline image. This is true only when inline image. inline image

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.

Axiom 7 (Dominance) For allinline imageandinline image, ifinline imagefor allinline imagetheninline image.

Lemma 3 Assume thatinline imagesatisfies Axioms 1–5. Letinline imagebe the pair representinginline image. Ifinline imagesatisfies Axioms 7 then

  • image

for allinline image.

Proof:  Suppose inline image. Then a separating hyperplane argument yields that there is inline image such that

  • image

Then we have inline image for all inline image.

Let inline image and inline image be such that inline image and u(l)=0. Then we have inline image for all inline image but inline image, a contradiction. inline image

Converse of Lemma 3 is not true unless Axiom 6 is assumed. For example, let inline image, and consider that a possibility set P is divided to two subsets P1 and P2. Let inline image for all inline image and inline image for all inline image and inline image. Then inline image. Let f=(0, 1, 1) and g=(0.9, 0.9, 0.9). Then we have inline image despite that inline image for all inline image.

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

Lemma 4 Assume thatinline imagesatisfies Axioms 1–5. Letinline imagebe the pair representinginline image. Then,inline imagesatisfies Axioms 6 and 7 if and only if

  • image

for allinline image.

Proof:  Sufficiency of the axioms follows from the preceding two lemmata. To show necessity, assume inline image for all P. It is already known that Axiom 6 is necessary. To show necessity of Axiom 7, it is without loss of generality to work on the space of payoff vectors. Because of the certainty independence property, it suffices to consider inline image with inline image. Then the set inline image is a closed convex cone pointed to the origin, where its dual cone is inline image. Suppose inline image, then by the property of dual cone there is inline image such that inline image for all inline image with inline image. In particular, this implies inline image. Since inline image, we have inline image. inline image

Remark 1 Reduction under Precise Information and Dominance appear to be related, but they are logically independent of each other. To see that Dominance does not imply Reduction under Precise Information, consider the representation withinline image, whereinline imageis a fixed probability measure andinline image, which satisfies Axioms 1–5 and 7 but not 6. Since Reduction under Precise Information involves only precise information, it cannot imply Dominance.

Next axiom states that preference is preserved under mixtures of probability–possibility sets. Given inline image and inline image, let inline image be the mixture of P and Q with proportion inline image vs. inline image.

Axiom 8 (Combination) For allinline image, inline image, inline image andinline image,inline image and inline image imply inline image; and inline image and inline image imply inline image.

To understand, provided that the rankings inline image and inline image hold, consider that the decision maker faces information P with probability inline image and Q with probability inline image. Now consider choice between ‘receiving f whichever she faces P or Q’ and ‘receiving l if she faces P and m if she faces Q.’ Given the preceding two rankings, the former is naturally preferred to the latter. Because the latter is nothing but receiving l with probability inline image and m with probability inline image, it is natural to conclude that she prefers f to inline image under inline image. An underlying assumption here is that the decision maker is indifferent to the order of resolution of uncertainty, so that she identifies information ‘P is true with probability inline image and Q is true with probability inline image’ with the mixture of probability–possibility sets inline image, as well as the standard timing indifference assumption that she identifies ‘receiving l with probability inline image and m with probability inline image’ with the mixture of lotteries inline image.

Lemma 5 Assume thatinline imagesatisfies Axioms 1–5. Letinline imagebe the pair representinginline image. Then,inline imagesatisfies Axioms 8 if and only if

  • image

for allinline imageandinline image.

Proof: Necessity of the axiom: Suppose inline image and inline image. Under Axioms 1–5, this implies inline image and inline image. Then we have

  • image

which implies inline image.

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

  • image

One may take inline image, inline image without loss of generality so that inline image, inline image and inline image. Then we have inline image, inline image but inline image, which contradicts to the axiom. Hence inline image. Similar proof shows inline image. inline image

Next axiom states that preference is invariant under transformations of probability measures such that preference under precise information is unchanged. Let inline image be the uniform distribution, and let inline image be the expected value of random variable inline image according to the uniform distribution.

Definition 1 Aninline imagematrixTis said to be a unitary transformation if

  • 1 . 
    it is a doubly stochastic matrix, that is,inline image, inline imageandinline imagefor allinline image.
  • 2 . 
    for allinline imageandinline imagewithinline image, inline imageimpliesinline image.

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. Letinline imagedenote the set of all the unitary transformations.

The following lemma provides a simple characterization of unitary transformation.

Lemma 6 Assume Axioms 1–6. Then aninline imagedoubly stochastic matrixTis a unitary transformation if and only if there existsinline imagesuch thatinline image, whereIis the identity matrix andEis the matrix with all the entries being 1.

Proof: Sufficiency of the condition: Suppose T satisfies inline image for some inline image. Let inline image and inline image. Then we have

  • image

Necessity of the condition: Suppose that inline image and inline image always imply inline image. This implies that

  • image

Let inline image, inline image and inline image where inline image are distinct. It is immediate that inline image and pt (xy)=0. Then we have inline image, where inline image is the inline image-th row of Tt and inline image is the inline image-th column of T, respectively. Since inline image is arbitrary, all the off diagonal entries of TtT are the same. Therefore, all the diagonal entries of TtT are the same. Since TtT is also a doubly stochastic matrix, we have inline image with inline image. If inline image, inline image and inline image imply inline image, a contradiction to the assumption. Hence inline image. inline image

Given inline image and inline image, let inline image be the image of P by T. Given inline image and inline image, let inline image be the image of f by T, which is defined by

  • image

for each inline image.

Now we state the invariance axiom.

Axiom 9 (Invariance to Unitary Transformations) For allinline image, inline imagewithinline image, andinline image, inline imageimpliesinline image.

Lemma 7 Assume thatinline imagesatisfies Axioms 1–5. Letinline imagebe the pair representinginline image. Then,inline imagesatisfies Axioms 9 if and only if

  • image

for allinline imageandinline image.

Proof: Necessity of the axiom: Suppose inline image and inline image. Let inline image and inline image. Then inline image. Since

  • image

we have inline image.

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

  • image

where x is taken so that inline image without loss of generality.

Let inline image be such that x=Ty. Then inline image. Then we have

  • image

for some inline image.

Let inline image and inline image be such that inline image and 0=u(l). Then we have inline image and inline image but inline image, which contradicts to the axiom. Hence inline image. Similar proof shows inline image. inline image

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

Axiom 10 (Information Continuity) For allinline imageand all sequenceinline image, ifinline imagefor allnandinline imageconverges toPin Hausdorff metric, theninline image.

Lemma 8 Assume thatinline imagesatisfies Axioms 1–5. Letinline imagebe the pair representinginline image. Then,inline imagesatisfies Axioms 10 if and only ifinline imageis continuous.

Proof:  Necessity of the axiom follows from the fact that the inline image is continuous over inline image when inline image is continuous. To show sufficiency of the axiom, suppose there is a sequence of probability–possibility sets inline image which converges to P in the Hausdorff metric but inline image does not converge to inline image in the Hausdorff metric. Because inline image is compact, we may assume that inline image is convergent with out loss of generality. Let inline image, then inline image. There is inline image such that

  • image

Then for all sufficiently large n we have inline image. Let inline image and inline image be such that inline image and 0=u(l) then we obtain a contradiction. inline image

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.

Definition 2 Letinline imagebe theinline imagedimensional unit sphere orthogonal toe. Letinline imagebe the uniform distribution overinline image. Then, Steiner point of compact convex setinline image, denoteds(P), is defined by

  • image

Now we state the main theorem.

Theorem 1 A family of preferencesinline imagesatisfies Axioms 1–10 if and only if there exist a mixture-linear continuous real-valued functionuoverinline imageand a numberinline imagesuch that for eachinline image, the rankinginline imageis represented in the form

  • image

whereinline imagehas the form

  • image

Moreover,inline imageis unique anduis unique up to positive affine transformations.

Proof:  The above sequence of lemmata shows that inline image satisfies Axioms 1–10 if and only if there exist a mixture-linear continuous real-valued function u over inline image and a number inline image such that for each inline image, the ranking inline image is represented in the form

  • image

where inline image satisfies

  • 1
    inline image for all inline image;
  • 2
    inline image for all inline image and inline image;
  • 3
    inline image for all inline image and inline image;
  • 4
    continuity with respect to the Hausdorff metric.

From the proof of Theorem 6 in Gajdos, Hayashi, Tallon, and Vergnaud (2008), this is the case if and only if there exists a unique inline image such that inline image. inline image

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 inline image being given by
    • image
    where inline image 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 inline image. Consider a representation with inline image being given by
    • image
    with inline image 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 inline image being given by
    • image
    with inline image, where inline image is a mapping which satisfies inline image for all inline image, 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 inline image being given by
    • image
    where inline image denotes generalized Steiner point which is defined for a non-atomic Borel probability measure inline image over inline image, not necessarily uniform, in the form
    • image
    The family of preferences represented by such class satisfies all the axioms but Invariance to Unitary Transformations.
  • Information Continuity : 
    Consider a representation with inline image being given by
    • image
    with inline image, where inline image is a mapping which satisfies inline image for all inline image, 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

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Setting
  5. 3 Basic axioms
  6. 4 Inter-informational axioms and characterization
  7. 5 Comparative ambiguity aversion
  8. 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.

Definition 3 inline imageis more ambiguity averse thaninline imageif for allinline image, inline imageandinline image,

  • image

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.

Proposition 2 Assume thatinline imageandinline imagesatisfy Axioms 1–5 and letinline imageandinline imagebe their representations respectively. Theninline imageis more ambiguity averse thaninline imageif and only if there exist numbersA, BwithA>0such thatu1=A u2+Bandinline imagefor allinline image.

Proof:  It follows from applying the result by Ghirardato and Marinacci (2002) to each inline image.inline image

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

Proposition 3 Assume that inline imageandinline imagesatisfy Axioms 1–10 and letinline imageandinline imagebe their representations respectively. Theninline imageis more ambiguity averse thaninline imageif and only if there exist numbersA, BwithA>0such thatu1=A u2+Bandinline image.

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

Footnotes
  • 1

    For an overview of the literature, see the recent survey by Giraud and Tallon. An earlier paper by Jaffray (1989) adopts capacity as the description of imprecise information, and establishes a model of preference over capacities over outcomes.

  • 2

    Gilboa and Schmeidler (1989) assumed that the set of outcomes consist of simple lotteries over pure outcomes, but the current extension can be done based on the result by Grandmont (1972) which establishes the expected utility theory with a compact metric space being the set of pure outcomes.

  • 3

    One can easily show independence of Ambiguity Aversion, though, by replacing min by max.

  • 4

    Consider for example a sequence of triangles converging to a segment. Center of gravity of each triangle divides its midlines by one versus two, but it does not converge to the midpoint of the segment in the limit.

References

  1. Top of page
  2. Abstract
  3. 1 Introduction
  4. 2 Setting
  5. 3 Basic axioms
  6. 4 Inter-informational axioms and characterization
  7. 5 Comparative ambiguity aversion
  8. References
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