A simple mean–dispersion model of ambiguity attitudes

Journal of Mathematical Economics 58 (2015) 25–31

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A simple mean–dispersion model of ambiguity attitudes Mark A. Schneider ∗ , Manuel A. Nunez Department of Operations and Information Management, University of Connecticut, 2100 Hillside Road Unit 1041, Storrs, CT 06269-1041, USA

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Article history: Received 7 October 2014 Received in revised form 5 February 2015 Accepted 9 March 2015 Available online 23 March 2015 Keywords: Ambiguity aversion Translation invariance Dispersion Uncertainty Probabilistic sophistication

abstract Several characterizations of ambiguity aversion decompose preferences into the expected utility of an act and an adjustment factor, an ambiguity index, or a dispersion function. In each of these cases, the adjustment factor has very little structure imposed on it, and thus these models provide little guidance as to which function to use from the infinite class of possible alternatives. In this paper, we provide a simple axiomatic characterization of mean–dispersion preferences which uniquely determines a subjective probability distribution over a set of possible priors and which uniquely identifies the dispersion function. We provide an algorithm for determining this subjective probability distribution and the coefficient in the dispersion function from experimental data. We also demonstrate that the model accommodates ambiguity aversion in the Ellsberg paradox. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The subjective expected utility (SEU) model (Anscombe and Aumann, 1963; Savage, 1954) is the foundational approach to decision making in economics, game theory, and other disciplines throughout the social sciences. Despite its generality and mathematical elegance, it has long been criticized on descriptive grounds, primarily by simple experiments in which the axioms of the theory are systematically violated. Perhaps the most important limitation of SEU is that it requires a decision maker to have neutral attitudes toward ambiguity, which conflicts with the widespread observation that many people are ambiguity-averse (Ellsberg, 1961). For instance, in one of Ellsberg’s classic examples, a decision maker is given a choice between two urns, and is told that each urn contains 100 balls, where 50 balls are red and 50 are black in Urn 1, but Urn 2 contains red and black balls in an unspecified proportion. The decision maker is asked to choose between winning $100 if a red ball is drawn from Urn 1, and winning $100 if a red ball is drawn from Urn 2. In such cases, the unambiguous urn (Urn 1) is frequently chosen. When given a choice between the same bets if a black ball is drawn, the decision maker again selects Urn 1. This preference for known risks over unknown risks is called ambiguity aversion. Ellsberg observed that such a preference pattern is not compatible with SEU, and thus implies that the decision maker does not have a unique subjective probability distribution over the number of red balls in Urn 2.

∗

Corresponding author. Tel.: +1 860 486 6485; fax: +1 860 486 4839. E-mail address: [email protected] (M.A. Schneider).

http://dx.doi.org/10.1016/j.jmateco.2015.03.002 0304-4068/© 2015 Elsevier B.V. All rights reserved.

Recent years have seen a plethora of studies aimed at modeling ambiguity aversion. One popular approach to modeling attitudes toward ambiguity is to decompose preferences into the expected utility of an act and an ambiguity index (Maccheroni et al., 2006b), or an adjustment factor (Siniscalchi, 2009), or a dispersion function (Grant and Polak, 2013). The most general of these specifications is the class of mean–dispersion preferences, axiomatized in the Anscombe–Aumann framework by Grant and Polak (2013). In particular, Grant and Polak characterize preferences which can be represented as: V (f ) = µ(f , π ) − ρ(d(f , π )), where µ(f , π ) is the mean utility of the act f with respect to a vector probability distribution π across all states of nature; and d(f , π ) is the vector of deviations from the mean given, that is, ds = U (f (s)) − µ(f , π ), where U (f (s)) is the expected utility of f in state s. The function ρ(·) is a measure of (aversion to) dispersion. The class of mean–dispersion preferences is quite large and includes leading theories such as the multiple priors model (Gilboa and Schmeidler, 1989), Choquet expected utility (Schmeidler, 1989), the variational representation of preferences (Fishburn, 1994; Maccheroni et al., 2006a), and vector expected utility (Siniscalchi, 2009) as special cases. While the generality of a representation theorem is very desirable, Grant and Polak (2013) comment that their main theorem is ‘‘too general to be very useful’’ (p. 1367). In particular, the dispersion function in the Grant–Polak representation, like the ambiguity index for variational preferences and the adjustment factor in vector expected utility has very little structure imposed on

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M.A. Schneider, M.A. Nunez / Journal of Mathematical Economics 58 (2015) 25–31

it. In addition, the main representation theorem by Grant and Polak (2013) does not uniquely identify the probability distribution, π . Grant and Polak remark, ‘‘Typically, we will be interested in mean–dispersion preferences that at least partially tie down the admissible probabilities and that put more structure on the dispersion function’’ (p. 1367). Working in a generalization of the Anscombe–Aumann framework, we provide a simpler axiomatic characterization of mean–dispersion preferences, which uniquely determines a subjective probability distribution over a set of possible priors, and which uniquely selects the dispersion function from an infinite class of possible alternatives. We show that the model accommodates attitudes which depend on the aversion to ambiguity in the Ellsberg paradox. In addition, we demonstrate how both the subjective probability distribution and the coefficient in the dispersion function can be uniquely identified from experimental data. 2. Objective and subjective lotteries In our analysis, we generalize the subjective expected utility (SEU) theory of Anscombe and Aumann (1963) by allowing a decision maker to exhibit optimism or pessimism toward desirable states. We set our analysis in a variant of the framework of Anscombe and Aumann (1963), in which the objects of choice are objective lotteries and subjective lotteries (Anscombe and Aumann refer to these objects as roulette lotteries and horse lotteries, respectively). Let X be a finite nonempty set of potential consequences, representing the payoff space. To avoid trivial scenarios, we assume that X contains at least two elements. An objective lottery, p : X → [0, 1], is a probability distribution over  outcomes; that is, p(x) ∈ [0, 1] for all x ∈ X , and x∈X p(x) = 1. We denote by ∆(X ) the set of objective lotteries and assume that it is a mixture space. As usual, a von Neumann–Morgenstern utility function is an application U : ∆(X ) → R defined as U (p) :=



p(x)u(x),

(1)

x∈X

where u : X → R is a utility function on the outcomes set. While the assumption that X is finite is common in models of choice under uncertainty, we note that our results continue to hold even if this assumption is further relaxed. In particular, X can be countably infinite, and the results can be extended to this case by using a standard inductive argument (see Kreps, 1988). Let S = {s1 , . . . , sn } be a finite set representing all possible states of nature. We define a subjective lottery or act f as any mapping f : S → ∆(X ). Hence, according to f , each state of nature determines a particular roulette lottery (objective lottery) to be played. We will use both f (s) and fs to denote the value of f at  state s. Notice that fs (x) ∈ [0, 1] for all s ∈ S and x ∈ X , and x∈X fs (x) = 1 for all s ∈ S. We denote the set of subjective lotteries (acts) by F and assume that it is a mixture space. We call F the set of acts over states. The set of probability vectors on S is denoted by ∆(S ). Given a utility function u : X → R and π ∈ ∆(S ), we denote by µ(f , π ) the mean utility of the subjective lottery f in F . That is,

µ(f , π) :=

 s∈S

πs U (f (s)) =



πs fs (x)u(x).

for all f and g exactly one of f ≻ g , g ≻ f , or f ∼ g holds; and % is a complete and transitive relation (Kreps, 1988). We denote by F c the set of constant acts, that is, f ∈ F c represents a subjective lottery that yields the same objective lottery in each state of nature: f (s) = p for all s ∈ S, where p ∈ ∆(X ) is an objective lottery. In this case, and when the context is clear, we also let p ∈ ∆(X ) denote the corresponding constant subjective lottery. Accordingly, we can naturally extend the preference relation from F c to ∆(X ) by letting p ≻ q, for p, q ∈ ∆(X ), whenever the constant act yielding lottery p for all states is strictly preferred to the constant act yielding lottery q for all states. Furthermore, a degenerate objective lottery that yields outcome x ∈ X with probability 1 is, once again abusing notation, denoted by x. Hence, we denote x ≻ y for x, y ∈ X when outcome x is preferred to outcome y. Our framework is a generalization of Anscombe and Aumann (1963) in which there are two qualitatively different types of subjective lotteries. In one type of subjective lottery (an act over states), there is uncertainty over both the identity of the true state and the outcome which will obtain, for a given state. An SEU maximizer forms a unique prior over this state space. In another type of subjective lottery (an act over priors), there is also uncertainty over the subjective probability that a given prior assigned to states is the ‘right one’. For instance, in an Ellsberg urn, an act over states reflects uncertainty over both the composition of the urn and the color which will be drawn, given the urn’s composition. An act over priors additionally reflects the decision maker’s uncertainty surrounding her subjective probability distribution of the urn’s composition. In this sense, our framework is similar to the setup in Klibanoff et al. (2005), but it is simpler as they work in a richer variant of the setup of Savage (1954). An illustrative example of acts over priors and a more detailed analysis of Ellsberg’s two-color paradox will be provided in Section 4. To model ambiguity, we assume that the decision maker does not have enough information to determine a single probability vector π ∈ ∆(S ) to assess the expected utility of the acts, and instead, she has a finite set of candidate priors Π ⊂ ∆(S ) that are likely to represent the actual probability vector over states. We refer to the probability vectors in Π by using index set M := {1, . . . , m}, where |Π | = m. There is another set of subjective lotteries consisting of all mappings fˆ : M → ∆(X ). The interpretation of fˆ is that nature chooses a prior distribution, say πj ∈ Π , and then, based on that distribution, the decision maker receives a randomized payoff according to the objective lottery fˆ (j). A similar notion features prominently in the Bayesian games of Harsanyi (1967–1968), in which nature selects a probability distribution over the players’ types. We denote the set of such subjective lotteries based on prior distributions by Fˆ and call it the set of acts over priors. We assume ˆ that Fˆ is a mixture space, and that there is a binary relation ≻ over Fˆ . Notice that when m = 1, all of the probability is placed on a single prior, and the framework reduces to the classical setup of Anscombe and Aumann (1963). Note also that even though we have assumed that Π is a finite set, this is without loss of generality since it can be any set for which we are able to find a probability measure (like in Klibanoff et al., 2005); our finiteness assumption concerning Π is just to simplify the presentation and the mathematical expressions. ˆ on the set Fˆ of The following axioms concern the relation ≻ acts over priors:

s∈S x∈X

We denote by ‘‘≻’’ ⊂ F × F a binary relation over F . The relation ≻ is called a preference relation if it is asymmetric and negatively transitive, and in that case, we say that f is preferred to g if f ≻ g. In addition, we say that a subjective lottery f is weakly preferred to another subjective lottery g, denoted as f % g, if g ̸≻ f . Moreover, we say that f is indifferent to g, denoted as f ∼ g, if f ̸≻ g and g ̸≻ f . Observe that if ≻ is a preference relation, then

ˆ on Fˆ is a preference relation. Axiom 1 (Preference). ≻ ˆ gˆ ≻ ˆ hˆ implies that Axiom 2 (Continuity). For every fˆ , gˆ , hˆ ∈ Fˆ , fˆ ≻ ˆ ˆ gˆ ≻ ˆ β fˆ +(1−β)h. there exist α, β ∈ (0, 1) such that α fˆ +(1−α)hˆ ≻ ˆ gˆ in Fˆ implies α fˆ + (1 − α)hˆ ≻ ˆ α gˆ + Axiom 3 (Independence). fˆ ≻ (1 − α)hˆ for all hˆ ∈ Fˆ and α ∈ (0, 1].

M.A. Schneider, M.A. Nunez / Journal of Mathematical Economics 58 (2015) 25–31

ˆ gˆ . Axiom 4 (Nontriviality). There exist fˆ , gˆ in Fˆ such that fˆ ≻

Table 1 Examples of risk-sensitive objective lotteries.

ˆ gˆ (j) for all Axiom 5 (Monotonicity). For every fˆ , gˆ ∈ Fˆ , fˆ (j) ≻ ˆ gˆ . j ∈ M implies fˆ ≻ From the classic Anscombe–Aumann Theorem (Gilboa, 2009; Kreps, 1988), we obtain the following result.

ρ



0.0 0.2 0.5 1.0

νj hˆ j (x)u(x),

(3)

In light of Proposition 1 and as in Klibanoff et al. (2005), notice that we can alternatively replace Axioms 1–5, together with Proposition 1, by the following axiom: Axiom (Subjective Expected Utility on Second-Order Acts). There exist a unique up to a positive affine transformation nonconstant function u : X → R and a unique probability distribution ν on M such that

ˆ gˆ if and only if Vˆ (fˆ ) > Vˆ (ˆg ), fˆ ≻ where Vˆ is the map from Fˆ to R defined as

νj hˆ j (x)u(x),

j∈M x∈X

for all hˆ ∈ Fˆ . 3. Risk-sensitive acts and mean–dispersion across states The decision maker characterized in Section 2 has SEU preferences for acts over priors, which, in the presence of Axiom 6 (introduced in this section), implies mean–dispersion preferences for acts over states (i.e., Anscombe–Aumann acts). In particular, Axiom 6 connects preferences over the two domains of subjective lotteries, while accounting for the decision maker’s degree of optimism or pessimism toward desirable states. For an arbitrary probability distribution π ∈ ∆(S ), we motivate Axiom 6 by introducing the new concept of a risk-sensitive objective lottery, derived from a subjective lottery f ∈ F . To do so, we partition the state set S into two sets S π (f ) and S π (f ) as follows

 S π (f ) :=

x1

x2

x1

x2

0.38 0.34 0.32 0.30 0.29 0.26 0.23

0.62 0.66 0.68 0.70 0.71 0.74 0.77

0.50 0.45 0.42 0.40 0.38 0.35 0.30

0.50 0.55 0.58 0.60 0.62 0.65 0.70

0.58 0.54 0.52 0.50 0.49 0.46 0.43

0.42 0.46 0.48 0.50 0.51 0.54 0.57

Definition 1. For a scalar ρ ∈ [−1, 1] and vector π ∈ ∆(S ), the risk-sensitive objective lottery ℓρπ (f ) ∈ ∆(X ) derived from f ∈ F is defined as s∈S π (f )

for all hˆ ∈ Fˆ . Moreover, ν is unique and u is unique up to a positive affine transformation in this representation.



(3/4, 1/4)

x2

   πs f (s) ℓρπ (f ) := 1 − ρ aπ (f )

j∈M x∈X

Vˆ (hˆ ) :=

(1/2, 1/2)

x1

(2)

where Vˆ is the map from Fˆ to R defined as Vˆ (hˆ ) :=

(π1 , π2 ) (1/4, 3/4)

−1.0 −0.5 −0.2

Proposition 1. Axioms 1 through 5 are necessary and sufficient for there to exist a nonconstant function u : X → R and a probability distribution ν on M such that

ˆ gˆ if and only if Vˆ (fˆ ) > Vˆ (ˆg ), fˆ ≻

27

s′ ∈ S : f (s′ ) %

 

πs f (s) ,

s∈S

and S π (f ) := S \S π (f ). In other words, we partition the state set S into a set of ‘‘desirable’’ states S π (f ), i.e., states that, under act f , have an associated objective lottery that the decision maker would prefer, or remain indifferent, to the objective lottery obtained by averaging with respect to π all of the objective lotteries defined by the act; and ‘‘undesirable’’ states S π (f ), i.e., the remaining states in S.  Next, we define aπ (f ) := s∈S π (f ) πs , and aπ (f ) := 1 − aπ (f ). The probability aπ (f ) corresponds to the likelihood of choosing an arbitrary desirable state according to distribution π and subjective lottery f ; and aπ (f ) is the complementary probability of aπ (f ).



+ (1 + ρ aπ (f ))

πs f (s).

(4)

s∈S π (f )

When ρ is in (0, 1], the decision maker is averse to the risk that the true state is undesirable and places less weight on the probability of a desirable state relative to the likelihood that an undesirable state obtains. When ρ is [−1, 0), the transformation works in the opposite direction. Thus, ρ may be viewed as reflecting a decision maker’s degree of optimism (if ρ is in [−1, 0)) or pessimism (if ρ is in (0, 1]) toward the likelihood that the true state is desirable. Alternatively, ρ may be interpreted as a perturbation in how a decision maker perceives the risk associated with each act. This may reflect ambiguity in the mind of the decision maker as to how the act is perceived, or it may be of interest to the theorist whether paradoxes such as those by Ellsberg still arise if there is a small perturbation in how an act is perceived. In this sense, we could view a risk-sensitive act over priors as a ‘perturbed act’, in which case Axioms 1–5 imply that a decision maker has SEU preferences over perturbed acts. We show in Section 4 that, under an informationally symmetric prior, the Ellsberg paradox is resolved for any ρ > 0. Thus even a tiny perturbation (any arbitrarily small ρ > 0) in how an act is perceived naturally resolves the paradoxes posed by Ellsberg (1961). To check that ℓρπ (f ) ∈ ∆(X ), notice that



   ℓρπ (f )x = 1 − ρ a πs fs (x) + (1 + ρ a) π s f s ( x)

x∈X

x∈X s∈S

= 1 − ρa 

 s∈S

πs + (1 + ρ a)

x∈X s∈S



πs

s∈S

  = 1 − ρ a a + ( 1 + ρ a) a = 1 , where we have omitted the dependence on f and π to simplify the expressions. To illustrate the concept of risk-sensitive lotteries, consider a scenario where there is an urn containing 100 balls, where each ball is either black or white. There are two states s1 and s2 . In state s1 the urn contains 60 black balls and 40 white balls. In state s2 the urn contains 20 black balls and 80 white balls. A ball is randomly drawn from the urn and the decision maker receives a payoff x1 with utility u(x1 ) = 1 if the ball is black and receives a payoff x2 with utility u(x2 ) = 0 if the ball is white. Table 1 shows the results corresponding to Π consisting of three different probability vectors (π1 , π2 ) = (1/2, 1/2), (π1 , π2 ) = (1/4, 3/4), and (π1 , π2 ) = (3/4, 1/4), respectively; and ρ taking values in {−1, −0.5, −0.2, 0, 0.2, 0.5, 1}. Letting f

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M.A. Schneider, M.A. Nunez / Journal of Mathematical Economics 58 (2015) 25–31

represent the act corresponding to this experiment, we obtain µ(f , π) = 0.6π1 + 0.2π2 . Moreover, assuming that the decision maker would prefer an objective lottery with higher probability of drawing a black ball, we have S π (f ) = {s1 } and S π (f ) = {s2 } for each π ∈ Π , and so aπ (f ) = π1 and aπ (f ) = π2 for all π ∈ Π . As expected, the resulting risk-sensitive objective lotteries ℓρπ (f ) assign (proportionally to ρ ) larger probabilities to the least favorable outcome x2 as ρ increases from −1 to 1 in each of the scenarios corresponding to the three distributions (π1 , π2 ). This reflects the increasing concern about risk of a risk-sensitive decision maker as ρ increases. The increase is more prominent in the first scenario ((π1 , π2 ) = (1/4, 3/4)) where the least desirable state s2 is more likely than the most desirable state s1 . Notice that if f is a constant subjective lottery, that is, there is an objective lottery p ∈ ∆(X ) such that f (s) = p for all s ∈ S, then S π (f ) = S and S π (f ) = ∅. Therefore, aπ (f ) = 1, aπ (f ) = 0, and ℓρπ (f ) = p. The risk-sensitive lotteries derived from an act f ∈ F by considering ρ ∈ [−1, 1], and all the probability vectors π ∈ Π , generate an act fˆρ in Fˆ defined as follows. Definition 2. Let f ∈ F , ρ ∈ [−1, 1], and π (j) ∈ ∆(S ) be the jth probability vector in Π . Then, the act fˆρ ∈ Fˆ defined as fˆρ (j) := ℓρπ (j) (f ),

(5)

for all j ∈ M, is called the risk-sensitive act over priors derived from f.

Using risk-sensitive objective lotteries over the space of acts over states, we now derive a mean–dispersion measure across states. This measure will represent the decision maker’s preferences among subjective lotteries weighted according to her attitude toward uncertainty. To do so, we first define the mean absolute semideviation of the utility of f ∈ F across states as



r (f , π ) :=

Notice that from the well-known result





|βk | =

r (f , π ) =

1 2 s∈S

10

40

5

+

10 1 10

ρ

1 2 1 2

− +

3

ρ



40  ,  3 40

ρ

πs |µ(f , π ) − U (f (s))| .

Definition 3. A mean–dispersion representation across states is a given tuple (u, ν, Π , ρ, V ), where u : X → R is a utility function; ν is a probability vector over M; Π is a set of probability vectors over S; ρ ∈ [−1, 1]; and preferences over subjective lotteries are given by



for all f ∈ F .

5 3

ρ

βk ,

k,βk <0

Using this definition, we introduce the concept of a mean–dispersion representation across states.

3 3 − ρ  10 40 fˆρ =  7 3 + ρ

1



for any real vector (β1 , . . . , βn ), it follows that r (f , π ) can be also expressed as

V (f ) :=

−

βk −

k,βk >0

k

For example, going back to our scenario from Table 1, fˆρ can be expressed as a function of ρ , as follows: 2

πs [µ(f , π ) − U (f (s))] .

s∈S ,U (f (s))<µ(f ,π )

 µ(f , π (j) ) − ρ r (f , π (j) ) νj ,

(6)

j∈M

Now, combining our result from Proposition 1 and Axiom 6, we obtain our main result from this section:

where each column corresponds to an objective lottery for a given prior and the top row is the probability of drawing a black ball in each case. A risk-sensitive act over priors assigns a risk-sensitive objective lottery to each prior over states. Using this definition, we state our final axiom, connecting the two spaces F and Fˆ :

Proposition 2. Axioms 1 through 6 hold if and only if there exists ρ ∈ [−1, 1] such that preferences admit a mean–dispersion representation across states (u, ν, Π , ρ, V ) such that

Axiom 6 (Consistency with Preferences Over Second-Order Acts). There exists ρ ∈ [−1, 1] such that for every f , g ∈ F ,

Moreover, probability vector ν is unique and the utility function u in the mean–dispersion representation is nonconstant and unique up to a positive linear transformation.

ˆ gˆρ . f ≻ g if and only if fˆρ ≻

ˆ gˆ if and only if Vˆ (fˆ ) > Vˆ (ˆg ), and fˆ ≻

(7)

f ≻ g if and only if V (f ) > V (g ).

(8)

Proof. See Appendix. Axiom 6 says that a preference for f over g implies that the decision maker has accounted for the risk that the true state is undesirable in each act, according to each prior over states, and in accordance with her own degree of optimism or pessimism toward desirable states. Observe that if f ∈ F c is a constant act that yields lottery p ∈ ∆(X ) for all states in S, then fˆρ is the constant act yielding the same lottery p for all priors j ∈ M. Combining this observation with ˆ q, Axiom 6, and once again abusing notation, it follows p ≻ q iff p ≻ for all p, q ∈ ∆(X ). Therefore, we obtain the following result:

ˆ are equivalent Lemma 1. Axioms 1 and 6 imply that ≻ and ≻ preference relations on ∆(X ). In light of Lemma 1, from now on we will use either notation, ˆ q, to mean the same: that the decision maker prefers p ≻ q or p ≻ objective lottery p over objective lottery q on either the set of acts over states or the set of acts over priors, respectively.



By accounting for the dispersion of expected utility in each state from the mean utility across all states, Proposition 2 explains ambiguity-averse behavior, consistent with Ellsberg’s paradox. We provide a more detailed analysis in Section 4. We conclude this section by presenting a few properties of our mean–dispersion representation. To V as V (f ) = do so, we re-write (j) W (f ) − ρ R(f ), where W (f ) := j∈M µ(f , π )νj , and R(f ) :=



j∈M

r (f , π (j) )νj , for all f ∈ F . Notice that R(f ) ≥ 0 for all f .

Lemma 2. Let f , g ∈ F , p ∈ ∆(X ), x ∈ X , and α ∈ [0, 1]. The following properties hold true: 1. 2. 3. 4.

V (α f V (α f V (α f V (α f

+ (1 − α)g ) ≥ α V (f ) + (1 − α)V (g ) for ρ ≥ 0; + (1 − α)g ) ≤ α V (f ) + (1 − α)V (g ) for ρ ≤ 0; + (1 − α)p) = α V (f ) + (1 − α)U (p); + (1 − α)x) = α V (f ) + (1 − α)u(x).

M.A. Schneider, M.A. Nunez / Journal of Mathematical Economics 58 (2015) 25–31

Aligned with Theorem 2 from Grant and Polak (2013), Lemma 2 implies that V is a concave (convex) function on the mixture space F for ρ ≥ 0 (ρ ≤ 0), V satisfies the constant absolute uncertainty aversion property, and the certainty betweenness property (Grant and Polak, 2013). Following Siniscalchi (2009) and Grant and Polak (2013), we say that two acts f and f are complementary if for any two states s, s′ ∈ S we have 12 f (s) + 21 f (s) ∼ 12 f (s′ ) + 21 f (s′ ). In this case, we

refer to (f , f ) as a complementary pair. For a complementary pair we have R(f ) = R(f ), and so, V also satisfies the complementary independence property (Grant and Polak, 2013). When Π consists of a single vector, say Π = {π}, representation (8) might have a vector expected utility (VEU) interpretation. Let us define

 π ζi (s) := i πi − 1

if s ̸= i, if s = i,

for all i ∈ S. Then, we have Eπ [ζi ] = 0 and Eπ [ζi U (f )] = πi (µ(f , π ) − U (f (i))) for f ∈ F . Moreover, let A : R|S | → R be defined as A(Φ ) := −

ρ 2 s∈S

|Φs |.

Then, we have V (f ) = Eπ [U (f )] + A (Eπ [ζi U (f )])i∈S ,





where the random variables ζi are adjustment factors and A is linear homogeneous (A(λΦ ) = λA(Φ ) for all λ > 0), A(0) = 0, and A(−Φ ) = A(Φ ). This functional form of V satisfies all the conditions for a VEU representation (see Siniscalchi, 2009) except for monotonicity, that is, V can be seen as a VEU representation depending on whether the relation ≻ satisfies the monotonicity axiom: Axiom (Monotonicity). f (s) % g (s) for all s ∈ S implies f % g. Since there exist values of ρ for which the corresponding representation V cannot satisfy the monotonicity axiom, it follows that Axiom 6 and the monotonicity axiom cannot be consistent for certain values of ρ . On the other hand, notice that the adjustment factors covariances with respect to U (f ) (i.e., Covπ [ζi , U (f )] = Eπ [ζi U (f )]) are proportional to the distances from the U (f (s)) values to their expected value µ(f , π ). This shows a natural consistency between our approach and a VEU approach as the decision maker is adjusting her assessment over the baseline expected-utility evaluation µ(f , π ) according to the dispersion of the U (f (s)) values. On the other hand, our approach may be viewed as less demanding than a VEU approach since it only requires standard assumptions (Axˆ and one connecting axiom between ioms 1–5) from the relation ≻ ˆ (Axiom 6). In addition, our approach prothe two relations ≻ and ≻ vides a specific parametrization of the dispersion measure which may be more convenient in applications. Finally, for certain values of ρ , the existence of representation V implies that ≻ would satisfy the required axioms for a multiple prior representation as described in Gilboa and Schmeidler (1989). In other words, for some values of ρ there exists a compact set C ⊂ ∆(S ) such that V (f ) = min µ(f , π ), π ∈C

which provides an alternative characterization for a multiple prior representation in some cases. The required values of ρ are those for which the monotonicity axiom (see above) and the uncertainty aversion axiom hold.

29

4. The Ellsberg paradox The reader may find it readily apparent that our dispersion measure provides a resolution to the paradoxes of Ellsberg (1961). For completeness, we include a formal demonstration in this section. Consider Ellsberg’s two-color paradox which involves a choice between an objective and a subjective lottery. The objective lottery is based on drawing a ball from an urn (Urn 1) containing only 50 red balls and 50 black balls. The subjective lottery is based on drawing a ball from an urn (Urn 2) containing 100 balls, where each ball may be either red or black, but the true proportion of each color is unknown. The decision maker is confronted with the following pair of binary choices: 1. Choose between (a) and (b): (a) Receive $100 if a red ball is drawn from Urn 1, or (b) Receive $100 if a red ball is drawn from Urn 2; 2. Choose between (a) and (b): (a) Receive $100 if a black ball is drawn from Urn 1, or (b) Receive $100 if a black ball is drawn from Urn 2. In Ellsberg’s experiments (Ellsberg, 1961), many people reported that they would strictly prefer 1(a) over 1(b), indicating a strong preference for betting on an outcome with a known probability than on an outcome with an unknown probability. Similarly, the same people reported that they would strictly prefer 2(a) over 2(b). Since bets 1(a) and 2(a) involve known probabilities, the preference for these bets is termed ambiguity aversion. As observed by Gilboa (2009), such preferences would have to reflect the belief that it is more likely that a red ball will be drawn from Urn 1 than from Urn 2 and that it is more likely that a black ball will be drawn from Urn 1 than from Urn 2, which is impossible because the probabilities of the two colors have to add up to 1 in each urn. To formalize this intuition, we first normalize the utility function such that u($100) = 1 and u($0) = 0. The state space is S := {0, . . . , 100}, where s ∈ S represents the number of red balls in an urn. Note that s ∈ S fully characterizes the state of the experiment, since the number of black balls is 100 − s. We associate acts f1 , f2 , f3 , and f4 with bets 1(a), 1(b), 2(a), and 2(b), respectively, as follows: f1 (s) := (1 : 1/2; 0 : 1/2) , f2 (s) := (1 : s/100; 0 : 1 − s/100) , f3 (s) := (0 : 1/2; 1 : 1/2) , f4 (s) := (0 : s/100; 1 : 1 − s/100) , for all s ∈ S; where the notation (u1 : p1 ; u2 : p2 ) represents the utility u1 on a red ball outcome with probability p1 and the utility on a black ball outcome with probability p2 , for each of the four acts. Let π be an arbitrary vector probability distribution in ∆(S ) and define Aπ :=

100  k=0

kπk ,

Bπ :=

100 

|Aπ − k| πk .

k =0

It is easy to see that µ(f1 , π ) = µ(f3 , π ) = 1/2, µ(f2 , π ) = Aπ /100, and µ(f4 , π ) = 1 − Aπ /100. Hence, if a person has preferences f1 ≻ f2 and f3 ≻ f4 , and a SEU representation holds for a given π , we must simultaneously have µ(f1 , π ) > µ(f2 , π ) and µ(f3 , π ) > µ(f4 , π ), which lead to the contradictory inequalities 50 > Aπ and 50 < Aπ , respectively. On the other hand, we can see how Ellsberg’s paradox is resolved by the model in Proposition 2 by noticing that r (f1 , π ) = r (f3 , π ) = 0, and r (f2 , π ) = r (f4 , π ) = Bπ /200. Therefore, V (f1 ) > V (f2 ) and V (f3 ) > V (f4 ) will hold when |100 − 2Aπ | < ρ Bπ for all π ∈ Π , which can be satisfied for an infinite number of values of π and ρ > 0. For example, if the decision maker believes that all states have the same likelihood, that is, πk = 1/101 for all

30

M.A. Schneider, M.A. Nunez / Journal of Mathematical Economics 58 (2015) 25–31

k, then Aπ = 50 and the inequality is satisfied for all ρ > 0. Thus, under informational symmetry, a decision maker chooses A and C, and Ellsberg’s 2-color paradox is resolved for any ρ > 0. Through a similar analysis it can be shown that the same condition resolves Ellsberg’s 3-color paradox.

ρ = (W (g ) − α) /R(g ). Therefore, to determine ρ , we need to find a value of α that makes the decision maker indifferent between the acts gˆρ and α fˆρ + (1 − α)hˆ ρ , which can be computed by asking the decision maker to choose between acts gρ and αk fˆρ + (1 − αk )hˆ ρ for a sequence of values αk ∈ (0, 1) determined according to a

5. Determination of the mean–dispersion function

traditional binary (also known as half-interval) search (Luce and Raiffa, 1989 have a description of such a binary search).

We discuss how we can experimentally infer ν and ρ in the mean–dispersion representation (6) from a decision maker’s preferences across acts. Throughout this section we assume that the decision maker satisfies Axioms 1–6. V as V (f ) = Also, we re-write (j) W (f ) − ρ R(f ), where W (f ) := j∈M µ(f , π )νj , and R(f ) :=

r (f , π (j) )νj , for all f ∈ F . Notice that R(f ) ≥ 0 for all f . Given a set of objective lotteries ∆(X ) and a preference relation defined on this set, we assume that we can determine a von Neumann–Morgenstern (vN–M) utility function U like in (1), consistent with the decision maker’s preferences, as long as the decision maker satisfies the classic von Neumann–Morgenstern axioms. This is a well-known result for which there exists an extensive literature (see Kreps, 1988 or Luce and Raiffa, 1989 for a discussion on this subject and further references). Notice that from Lemma 1, any decision maker satisfying Axioms 1–6 will also satisfy the vN–M axioms on ∆(X ). Therefore, we assume that U (and so u(x) for all x ∈ X ) in (6) has already been determined. To obtain ν , notice that from Propositions 7.4 and 7.5 from   ˆ Kreps (1988), Vˆ can be written as Vˆ (hˆ ) = j∈M x∈X hj (x)uj (x),



j∈M

for all hˆ ∈ Fˆ , where each uj is a utility function on X satisfying uj (x) = aj u(x) + bj , ∀x ∈ X , for some aj , bj ∈ R, aj  ≥ 0. Moreover, because of Axiom 4 not all aj are zero and νj = aj / i∈M ai . Hence, we can obtain vector ν by determining aj for each j ∈ M. We do this by telling the decision maker to assume that the states in S follow a distribution given by prior π (j) , for a fixed j ∈ M. Knowing this, the decision maker will re-scale her preferences on ∆(X ) (but still obey the vN–M axioms) consistently with preference relation ˆ . As indicated before, using the resulting preferences on ∆(X ), ≻ we can determine uj (x) for all x. Let x and x denote the most and least preferred outcomes in X , respectively. Hence, x ≻ x and u(x) > u(x). By normalizing function u, we can assume without loss of generality that u(x) = 1 and u(x) = 0. It is easy to see that aj = uj (x) − uj (x) and bj = uj (x). Finally, to determine ρ , without loss of generality we assume that there exist a prior π (j) and a state s ∈ S such that νj > 0 and (j)

0 < πs < 1 in (6). Notice that if this is not the case, that is, for all (j) positive νj there does not exist a state s with 0 < πs < 1, then we would be dealing with a decision problem that does not require the framework of the candidate set of prior distributions. Also, it is not difficult to show that for an arbitrary act f , u(x) ≥ V (f ) ≥ u(x) for all ρ ∈ [−1, 1]; and that if R(f ) > 0, then V (f ) is strictly decreasing as a function of ρ in [−1, 1]. Now, denote by f the (degenerate) constant act that awards the same outcome x for all the states in S and by h the (degenerate) constant act that awards the same outcome x for all the states in S. Then, we have V (f ) = u(x), V (h) = u(x), and f ≻ h. Define act g such that gs is the degenerate objective lottery that always yields outcome x and gs′ is the degenerate objective lottery that always yields outcome x for alls′ ̸= s. Then r (g , π (j) ) = (j)

= πs(j) 1 − πs(j) > 0, and hence, R(g ) > 0. It follows that V (f ) > V (g ) > V (h), which implies ˆ gˆρ ≻ ˆ hˆ ρ . From Axiom 2, it follows that there f ≻ g ≻ h and fˆρ ≻ exists α ∈ (0, 1) such that gˆρ ∼ α fˆρ + (1 − α)hˆ ρ . By using u(x) − u(x) πs





(j)

1 − πs

again the mean–dispersion representation, this entails that there exists α ∈ (0, 1) such that the V (g ) = α V (f ) + (1 − α)V (h) = α u(x) + (1 − α)u(x) = α . Since V (g ) = W (g ) − ρ R(g ), we obtain

6. Conclusion We have provided a simple axiomatic approach to mean– dispersion preferences which uniquely selects a subjective probability distribution over a given set of candidate priors and yields an absolute semideviation dispersion measure from the myriad dispersion measures possible. Our approach is thus complementary to the mean–dispersion preferences of Grant and Polak (2013). While the Grant–Polak approach has greater generality, it does not uniquely identify either the dispersion function or subjective probability distribution. We demonstrated that our model accommodates ambiguity aversion in the Ellsberg paradoxes. We subsequently provided an algorithm for uniquely eliciting the model parameters (e.g., the subjective probability distribution over priors and the dispersion function coefficient) from experimental data. Acknowledgment We would like to thank the anonymous reviewer who provided significant improvements and insights to our original manuscript; it really helped us to enhance our research. Appendix. Proof of main result Proof of Proposition 2. We first prove that Axioms 1 through 6 are sufficient, so let us assume that those axioms hold. Statement (7) follows directly from Proposition 1. From Axiom 6, there exists ρ ∈ [−1, 1] such that f ≻ g if and only if ˆ gρ , for all f , g ∈ F . Thus, from Proposition 1, f ≻ g iff Vˆ (fˆρ ) > fˆρ ≻ˆ

Vˆ (ˆgρ ). For any f ∈ F and π ∈ ∆(S ), we have



  ℓρπ (f )(x)u(x) = 1 − ρ a πs fs (x)u(x)

x∈X

x∈X s∈S

+ ( 1 + ρ a)



πs fs (x)u(x)

x∈X s∈S

= µ(f , π ) − ρ a



πs fs (x)u(x) + ρ a



s∈S x∈X

= µ(f , π ) − ρ a



πs fs (x)u(x)

s∈S x∈X

πs fs (x)u(x)

s∈S x∈X

  +ρ 1 − a πs fs (x)u(x) s∈S x∈X





= µ(f , π ) − ρ aµ(f , π ) −



πs fs (x)u(x)

s∈S x∈X

 = µ(f , π ) − ρ

 

πs µ(f , π ) −

s∈S



 = µ(f , π ) − ρ

 s∈S

πs fs (x)u(x)

s∈S x∈X

πs µ(f , π ) −

  x∈X

fs (x)u(x) .

M.A. Schneider, M.A. Nunez / Journal of Mathematical Economics 58 (2015) 25–31

V (f ) > V (g ). On the other hand, as we showed above, we have V (f ) = Vˆ (fˆρ ) and V (g ) = Vˆ (ˆgρ ). Thus, Vˆ (fˆρ ) > Vˆ (ˆgρ ), and so,

We claim that

 



πs µ(f , π ) −

s∈S



fs (x)u(x)

= r (f , π ).

(A.1)

x∈X

If this is true, then we obtain that f ≻ g if and only if V (f ) =



 µ(f , π (j) ) − ρ r (f , π (j) ) νj = Vˆ (fˆρ ) > Vˆ (ˆgρ )

j∈M

=



 µ(g , π (j) ) − ρ r (g , π (j) ) νj = V (g ).

j∈M

Therefore, ≻ admits a mean–dispersion representation across states and Axioms 1 through 6 are sufficient. To prove  that (A.1) holds, notice that s′ ∈ S (f ) if and only if f (s′ ) ≺ s∈S x∈X πs f (s). Using Lemma 1, it follows

 Vˆ (f (s )) < Vˆ ′

  s∈S

<

 s∈S x∈X

πs f (s) ⇒



fs′ (x)u(x)

x∈X

πs fs (x)u(x) ⇒

31



fs′ (x)u(x) < µ(f , π ).

x∈X

′ Thus, we obtain that s′ ∈ S (f ) if and only if x∈X fs (x)u(x) < µ(f , π ), and (A.1) follows. In the second part of the proof, we show that Axioms 1 through 6 are necessary. First, notice that Proposition 1 and statement (7) imply that Axioms 1–5 hold, and so, we only need to show that Axiom 6 also holds. Assume that there exists ρ ∈ [−1, 1] such that (8) holds. Let f , g be acts in F such that f ≻ g. Then, we have



fˆρ ≻ gˆρ . Similarly, if f ∼ g, then fˆρ ∼ gˆρ . Therefore, Axiom 6 holds and the result follows.  References Anscombe, F.J., Aumann, R.J., 1963. A definition of subjective probability. Ann. Math. Statist. 34 (1), 199–205. Ellsberg, D., 1961. Risk, ambiguity, and the Savage axioms. Quart. J. Econ. 75 (4), 643–669. Fishburn, P., 1994. A variational model of preference under uncertainty. J. Risk Uncertain. 8 (2), 127–152. Gilboa, I., 2009. Theory of Decision Under Uncertainty. Cambridge University Press, New York, NY. Gilboa, I., Schmeidler, D., 1989. Maxmin expected utility with non-unique prior. J. Math. Econom. 18 (2), 141–153. Grant, S., Polak, B., 2013. Mean-dispersion preferences and constant absolute uncertainty aversion. J. Econom. Theory 148, 1361–1398. Harsanyi, J., 1967–1968. Games of incomplete information played by Bayesian players. Parts I–III. Manag. Sci. 14, 159–182. 320–334, 486–502. Klibanoff, P., Marinacci, M., Mukerji, S., 2005. A smooth model of decision making under ambiguity. Econometrica 73 (6), 1849–1892. Kreps, D.M., 1988. Notes on the Theory of Choice. Westview Press, Inc., Boulder, CO. Luce, D., Raiffa, H., 1989. Games and Decisions: Introduction and Critical Survey. Dover Publications, Inc.. Maccheroni, F., Marinacci, M., Rustichini, A., 2006a. Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica 74 (6), 1447–1498. Maccheroni, F., Marinacci, M., Rustichini, A., 2006b. Dynamic variational preferences. J. Econom. Theory 128 (1), 4–44. Savage, L.J., 1954. The Foundations of Statistics. John Wiley & Sons, Inc., New York, NY. Schmeidler, D., 1989. Subjective probability and expected utility without additivity. Econometrica 57 (3), 571–587. Siniscalchi, M., 2009. Vector expected utility and attitudes toward variation. Econometrica 77 (3), 801–855.