- Email: [email protected]
On the uses of the monotonicity and independence axioms in models of ambiguity aversion
Mathematical Social Sciences 59 (2010) 326–329
Contents lists available at ScienceDirect
Mathematical Social Sciences journal homepage: www.elsevier.com/locate/econbase
On the uses of the monotonicity and independence axioms in models of ambiguity aversionI Leandro Nascimento a,∗ , Gil Riella b a
New York University, 10012 New York, NY, United States
b
Universidade de Brasília, Brazil
article
info
Article history: Received 21 August 2009 Received in revised form 25 January 2010 Accepted 27 January 2010 Available online 2 February 2010
abstract This paper suggests an alternative axiomatization of two canonical models of ambiguity aversion. Instead of relaxing the independence axiom to accommodate uncertainty aversion, we impose independence on constant acts only. Maxmin and variational preferences are characterized by different strengthenings of the monotonicity axiom. © 2010 Elsevier B.V. All rights reserved.
JEL classification: D81 Keywords: Monotonicity Independence Ambiguity
1. Introduction The canonical models of ambiguity aversion of Gilboa and Schmeidler (1989) and Maccheroni et al. (2006) accommodate Schmeidler’s (1989) uncertainty aversion postulate by imposing weaker versions of the independence axiom.1 The main motivation behind the weakenings of the independence axiom is its incompatibility with the type of behavior suggested by Ellsberg (1961) in the presence of a subjective uncertainty. Maxmin preferences are characterized by the ‘‘certainty independence’’ axiom, while variational preferences require a weaker version, the ‘‘weak certainty independence’’ axiom. The interpretation of certainty independence is that constant acts are not good for hedging in the presence of uncertainty. Maccheroni et al. (2006) motivate their weak certainty independence axiom by suggesting that mixing two nonconstant acts at a time with the same constant act and using the same weights may induce different orderings depending on the weights. If the purpose of the model is to explain the ‘‘Ellsbergian choices’’, then it is reasonable to impose the independence axiom on risky prospects. That is, if we do not try to account for the Allais paradox, the independence axiom does not seem too problematic when imposed only on constant acts. Therefore, if one agrees that the independence axiom holds in the context of objective
I We thank Efe Ok and an anonymous reviewer for helpful suggestions.
∗
Corresponding author. E-mail addresses: [email protected] (L. Nascimento), [email protected] (G. Riella).
1 Chateauneuf (1991) independently axiomatized maxmin preferences. 0165-4896/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2010.01.003
uncertainty, the natural course of action would be to impose independence on constant acts only. In fact, the axiomatizations of Gilboa and Schmeidler (1989) and Maccheroni et al. (2006) imply independence on constant acts. The purpose of this paper is to provide an alternative (and equivalent) axiomatization of those models of ambiguity aversion by imposing independence on constant acts only. We show how to strengthen the monotonicity axiom so as to obtain these models. Variational preferences require the weakest strengthening of the monotonicity axiom, whereas the subjective expected utility (SEU) model of Anscombe and Aumann (1963) requires a very strong version of monotonicity. The monotonicity axiom for the maximin preferences lies in between. In the next section we describe the setup and the basic axioms. In Section 3 we present the strengthenings of the monotonicity axiom, and show how to obtain the canonical models of ambiguity aversion. While Section 4 concludes the paper with a discussion on how the Ellsberg paradox relates to our novel axioms and an additional remark, the Appendix contains the proofs of our main results. 2. Setup and basic axioms The state space is denoted by S. Let Σ be an algebra of subsets of S, and denote by ∆(S ) the set of all finitely additive probability measures on S. The set of consequences X is a nontrivial convex subset of a vector space. An act is a Σ -measurable function f : S → X such that |f (S )| < ∞. The domain of preferences is F , the set of all such acts. We denote by Fc the set of constant acts, i.e., Fc =
L. Nascimento, G. Riella / Mathematical Social Sciences 59 (2010) 326–329
{f ∈ F : ∃x ∈ X s.t. f (S ) = {x}}, and identify it with X . We represent by xf any constant act that is indifferent to f . Given the weight α ∈ [0, 1], the pointwise mixture of the acts f and g is denoted by α f + (1 − α)g. As a last piece of notation, if ∅ 6= Γ ⊆ R, then we denote by B0 (Σ , Γ ) the set of Σ -measurable finite step functions ξ : S → R satisfying ξ (S ) ⊆ Γ , and endow it with the sup norm. The binary relation <⊆ F × F satisfies the following set of axioms. Axiom A1 (Weak Order). < is complete and transitive. Axiom A2 (Constant Continuity). For all f ∈ F , and x, y ∈ X , the sets {α ∈ [0, 1] : α x + (1 − α)y < f } and {α ∈ [0, 1] : f < α x + (1 − α)y} are closed. Axiom A3 (Risk Independence). For all x, y, z ∈ X , and α ∈ (0, 1], x < y iff α x + (1 − α)z < α y + (1 − α)z. Axiom A4 (Constant Monotonicity). For all f ∈ F , and x ∈ X , if x < f (s) for all s ∈ S, then x < f , and if f (s) < x for all s ∈ S, then f < x. Axiom A5 (Non-triviality). 6= ∅.2 Axioms A1, A3 and A5 are quite common. The Constant Continuity axiom is a weaker form of the more standard version in which the constant acts x and y are replaced by the general elements of F . The reason why we use that particular version is that, for the models of ambiguity aversion considered below, the strengthenings of the monotonicity axiom already impose a substantial continuity on the representation. Axiom A4 is not standard as well. Note that the usual monotonicity axiom is the following: Axiom A4∗ (Monotonicity). For all f , g ∈ F , if f (s) < g (s) for all s ∈ S, then f < g. Given Axioms A1 and A2, Constant Monotonicity is the weakest form of A4∗ to ensure the existence of a certainty equivalent xf ∈ X for each act f ∈ F . In fact, given any act f ∈ F , let x, y ∈ X be such that x < f (s) < y for all s ∈ S. Using Axioms A1, A2 and A4 one can show that the sets {α ∈ [0, 1] : α x + (1 − α)y < f } and {α ∈ [0, 1] : f < α x + (1 − α)y} are nonempty, closed, and must have a nonempty intersection. Axiom A4∗ will be a consequence of our axioms after we add the strong versions of monotonicity that characterize each class of preferences. Now observe that, when A4∗ holds and < |X admits a utility representation by u, then the utility function U on F , as defined by U (f ) = u(xf ), represents <. The next lemma is wellknown, and its proof follows from the observation just made plus standard arguments, which are omitted. Lemma 1. < satisfies A1–A3, A4∗ , and A5 iff there exists a nonconstant and affine function u : X → R, and I : B0 (Σ , u(X )) → R such that I is increasing, I |u(X ) is the identity map, 0 ∈ int(u(X )) 6= ∅, and, for all f , g ∈ F , f < g iff I (u(f )) ≥ I (u(g )). 3. Additional axioms and representations 3.1. Axioms for variational preferences The binary relation < admits a variational preference represenR tation when the functional I satisfies I (ξ ) = minµ∈∆(S ) [ ξ dµ + c (µ)], for some lower semi-continuous, convex and grounded cost function c : ∆(S ) → R+ ∪{+∞}.3 In order to obtain this representation, we first add the following stronger monotonicity condition: Axiom A6 (Strong Monotonicity 1). For all f , g ∈ F , x, y ∈ X , and α ∈ [0, 1], if α f (s) + (1 − α)x < α g (s) + (1 − α)y for all s ∈ S, x < f , and g < y, then α x + (1 − α)x < α y + (1 − α)y.
2 stands for the asymmetric part of <. 3 The cost function c is grounded if inf
µ∈∆(S )
c (µ) = 0.
327
What makes A6 stronger than A4∗ is that, in the presence of Axioms A1, A2 and A4, it not only implies A4∗ , but Axiom A6 also yields monotonicity when one replaces, for example, the nonconstant acts f and g by their certainty equivalents in those mixtures of acts. If we see the (objective) mixture of an act with a constant act as a two-stage lottery, Strong Monotonicity 1 has the following interpretation. We are told that the decision maker prefers the two-stage lottery α f (s)+(1 −α)x over α g (s)+(1 −α)y for each possible state s. The standard monotonicity axiom would tell us that the decision maker prefers the two-stage lottery that gives f with probability α and x with probability 1 − α to the twostage lottery that gives g with probability α and y with probability 1 − α . Ex-ante he is ignorant about the realization of the state. But he knows that the risky prospect x is preferred to f , and that the act g is preferred to y, and therefore he preserves the ranking when we replace f by x, and g by y at the same time. Technically, Axiom A6 can be interpreted as an ordinal characterization of a niveloid. Dolecki and Greco (1995) define a niveloid as a functional J : [−∞, +∞]S → [−∞, +∞] that is isotone (ξ ≥ ζ implies J (ξ ) ≥ J (ζ )) and vertically invariant (a ∈ R implies J (ξ + a) = J (ξ ) + a). An alternative characterization of a niveloid given by their Corollary 1.3 is J (ξ ) − J (ζ ) ≤ sups∈S [ξ (s) − ζ (s)], under the convention that (−∞) + (+∞) = (+∞) + (−∞) = (+∞) − (+∞) = −∞. In our setup, one says that b I : B0 (Σ , R) → R is a niveloid if b I is isotone and vertically invariant. The main role of Axiom A6 is to ensure I, as obtained in Lemma 1, satisfies I (ξ ) − I (ζ ) ≤ sups∈S [ξ (s) − ζ (s)], so it admits a (least) niveloidal extension to B0 (Σ , R). The final axiom is Schmeidler’s (1989) uncertainty aversion postulate: Axiom A7 (Uncertainty Aversion). For all f , g ∈ F , and α ∈ [0, 1], if f ∼ g, then α f + (1 − α)g < f . The next theorem characterizes the variational preferences under our axioms. Its content is certainly not new, but we believe that part of the proof is. Our approach is to use our axioms to derive the basic properties of the functional I, and then use the more technical results of Maccheroni et al. (2006) to complete the proof. Theorem 1. The binary relation < admits a variational preference representation iff it satisfies A1–A7. 3.2. Axioms for multiple priors and SEU The binary relation < admits a multiple priors preference representation if it admits a variational preferences representation in which there exists a nonempty, closed and convex set M ⊆ ∆(S ) such that c (µ) = 0 if µ R ∈ M, and c (µ) = +∞ otherwise. That is, when I (ξ ) = minµ∈M ξ dµ. The next axiom is a strengthening of Axiom A6. Axiom A60 (Strong Monotonicity 2). For all f , g ∈ F , x, y ∈ X , and α1 , α2 ∈ [0, 1], if α1 f (s) + (1 − α1 )x < α2 g (s) + (1 − α2 )y for all s ∈ S, x < f , and g < y, then α1 x + (1 − α1 )x < α2 y + (1 − α2 )y. Strong Monotonicity 2 has a similar interpretation when compared to Axiom A6. The difference is that now the decision maker does not care about the weights being used in the mixtures of acts.4 Theorem 2. The binary relation < admits a multiple priors representation iff it satisfies A1–A5, A60 , and A7. Now we turn to the last strengthening of the monotonicity axiom. We say that < admits a SEU representation when it admits a multiple priors representation with M = {µ}, for some µ ∈ ∆(S ).
4 The technical reason is that the maxmin preferences are characterized by a functional in the induced domain of util acts that is also positively homogenous (besides being vertically invariant as in the variational preferences).
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L. Nascimento, G. Riella / Mathematical Social Sciences 59 (2010) 326–329
Axiom A600 (Strong Monotonicity 3). For all f , g , h, i ∈ F , and α1 , α2 ∈ [0, 1], if α1 f (s) + (1 − α1 )h(s) < α2 g (s) + (1 − α2 )i(s) for all s ∈ S, x < f , z 1 < h, g < y, and i < z 2 , then α1 x + (1 − α1 )z 1 < α2 y + (1 − α2 )z 2 . Theorem 3. The binary relation < admits a SEU representation iff it satisfies A1–A5, and A600 . If we had required Axiom A600 to hold only if the acts f and h are comonotonic, and g and i are comonotonic, then we would have obtained Schmeidler’s (1989) representation with a convex capacity after adding the uncertainty aversion Axiom A7. In fact, by employing an argument similar to that in the proof of Theorem 3, one can show that, for any two comonotonic acts f and g, and α ∈ [0, 1], we have I (α u(f ) + (1 − α)u(g )) = α I (u(f )) + (1 − α)I (u(g )). Therefore, when f , g , h ∈ F are pairwise comonotonic, if I (u(f )) > I (u(g )), then α I (u(f )) + (1 − α)I (u(h)) = I (α u(f ) + (1 −α)u(h)) > I (α u(g )+(1 −α)u(h)) = α I (u(g ))+(1 −α)I (u(h)). Now the representation follows from the same arguments used in the proof of the main theorem of Schmeidler (1989). 4. Discussion 4.1. Ellsberg paradox: Example The weakenings of the independence axiom that have been proposed in the literature are appealing for at least two reasons. They not only have a simple interpretation as we mentioned in the introduction, but also help to accommodate the uncertainty aversion postulate and build a model of decision making that can account for the Ellsberg paradox. The alternative axiomatization proposed in this paper has an interpretation, as we argued above, albeit not as simple as the weakenings of the independence axiom. A natural question is how the Ellsbergian choices and the strengthenings of the monotonicity axiom relate. Here we present an alternative perspective on the incompatibility between the Ellsberg-type behavior and the SEU model. It allows us to see Axioms A6 and A60 as weakenings of Axiom A600 , which characterizes SEU, and understand why they are necessary to explain that type of behavior. Consider the following version of the Ellsberg paradox. An urn contains 90 balls, and 30 balls are red (R). The remaining 60 balls are either black (B) or yellow (Y ), but the proportion is unknown. The state space is S = {R, B, Y }. Four alternative bets (monetary acts) are available to the decision maker. They are vectors where the first coordinate represents the prize (in $) in case a red ball is drawn, and the second and third coordinates correspond to prizes associated to black and yellow balls being drawn, respectively. In other words, a bet is a function f : S → [0, 100], where the minimum prize is zero dollars, and the maximum prize is one hundred dollars. We consider: " # " # 100 0 0 , fR = fB = 100 , 0 0 100 0 , 100
" fRY =
#
0 100 . 100
" and fBY =
#
The framework in this example is one with monetary acts. The risk independence Axiom A3 allows us to take convex combinations of monetary prizes. The following rankings induce typical Ellsbergian choices: fR $32 =: xfR xfB := $18 fB and fRY ≺ xfRY := $52 ≺ $64 =: xfBY ≺ fBY .5 Observe that the
rankings are incompatible with the Strong Monotonicity 3 axiom: for all s ∈ S, 1
1
1
1
fRY (s) , 2 2 2 2 but 1 1 1 1 xf + xfBY = $16 + $32 > $9 + $26 = xfB + xfRY . 2 R 2 2 2 fR (s) +
fBY (s) = $50 =
fB (s) +
This violates Axiom A600 (because it would require 12 xfR + 21 xfBY ≤
+ 12 xfRY ). One interpretation of Axiom A600 is that, provided one has information about the pointwise comparison of the mixture of two pairs of acts, one can make an inference, for example, about the mixture of the certainty equivalents associated with those two pairs. This is a very strong requirement in the presence of subjective uncertainty if the decision maker is ambiguity averse. In the Ellsberg example, the acts fR and fBY are much more attractive to the decision maker because probabilities are completely specified. The agent will be willing to pay more money for the acts fR and fBY than he is willing to sell fB and fRY on average. In order to accommodate the Ellsbergian choices one needs to find weaker versions of Axiom A600 . The maxmin preferences force the mixture to be of a pair of acts containing at least one constant act. The weights of the two pairs of acts may differ, though. Variational preferences place the same sort of constraints on the pairs of acts, but require the weight to be the same when mixing each pair. They are both compatible with the Ellsbergian choices in our example. 1 x 2 fB
4.2. Remark In general, Strong Monotonicity 1 alone need not imply weak certainty independence. But one can show that, when Axioms A1–A4 and A6 hold, the implication α f + (1 − α)x < α g + (1 − α)x ⇒ α f + (1 − α)y < α g + (1 − α)y is true (i.e., weak certainty independence also holds). In fact, take any f , g ∈ F , x, y ∈ Fc , α ∈ [0, 1], and assume α f + (1 − α)x < α g + (1 − α)x. 1 If we define γ := 2−α , then γ [α f (s) + (1 − α)y] + (1 − γ )x = γ [α f (s)+(1 −α)x]+(1 −γ )y, and γ [α g (s)+(1 −α)y]+(1 −γ )x = γ [α g (s) + (1 − α)x] + (1 − γ )y for all s ∈ S. Hence, using the existence of certainty equivalents plus Axioms A3 and A6 we obtain γ xα f +(1−α)y + (1 − γ )x ∼ γ xα f +(1−α)x + (1 − γ )y < γ xαg +(1−α)x + (1 − γ )y ∼ γ xαg +(1−α)y + (1 − γ )x, so that α f + (1 − α)y < α g + (1 − α)y. Appendix. Proofs We only show the ‘‘if’’ part of the theorems. A.1. Proof of Theorem 1 Because A6 implies A4∗ when A1–A4 are satisfied, from Lemma 1 we obtain the representation I ◦ u : F → R. Take any two acts f , g ∈ F . Using the fact |f (S )|, |g (S )| < ∞, there exists s∗ ∈ S such that u(g (s∗ )) − u(f (s∗ )) = sups∈S [u(g (s)) − u(f (s))]. Define x := g (s∗ ) and y := f (s∗ ), so that 21 f (s) + 12 x < 21 g (s) + 12 y for all s ∈ S. By Axiom A6 we obtain 12 xf + 12 x < 12 xg + 12 y, which implies I (u(g )) − I (u(f )) ≤ sups∈S [u(g (s)) − u(f (s))]. Therefore, for all ξ , ζ ∈ B0 (Σ , u(X )), I (ξ ) − I (ζ ) ≤ sups∈S [ξ (s) − ζ (s)]. The representation now follows from the same arguments in Maccheroni et al. (2006, pp. 1480–1481). A.2. Proof of Theorem 2
5 The numbers follow from maxmin preferences with set of priors co{(1/3, 1/6, 1/2), (1/3, 1/2, 1/6)}, where the symbol co stands for the convex hull.
Because A60 implies A4∗ when A1–A4 are satisfied, from Lemma 1 we obtain the representation I ◦ u : F → R. It remains to
L. Nascimento, G. Riella / Mathematical Social Sciences 59 (2010) 326–329
show that I is positively homogeneous and vertically invariant. It sure is vertically invariant because A60 implies A6. Assume w.l.o.g. that [−1, 1] ⊆ u(X ). Now take λ > 0, and ξ ∈ B0 (Σ , u(X )) such that λξ ∈ B0 (Σ , u(X )), and let f , g ∈ F be such that u(f ) = ξ and u(g ) = λξ , and x0 ∈ X satisfy u(x0 ) = 0. Therefore, for all s ∈ S, 1 λ 1 λ ξ (s) + 1+λ u(x0 ) = 1+λ u(x0 ) + 1+λ λξ (s), which translates into 1+λ λ 1 f (s) + 1+λ x0 1+λ 0
λ 1 x + 1+λ g (s) for all s ∈ S. As a consequence 1+λ 0 λ 1 λ 1 I (ξ ) + u(x0 ) = 1+λ u(x0 ) + 1+λ I (λξ ), 1+λ 1+λ
329
using A600 , for h = α f +(1−α)g, we have α f (s)+(1−α)g (s) ∼ h(s) for all s ∈ S, and therefore α xf + (1 − α)xg ∼ h, which implies α I (ξ ) + (1 − α)I (ζ ) = I (αξ + (1 − α)ζ ). Therefore I is affine, and we obtain the representation using well-known arguments. References
∼
of A6 we obtain and hence I (λξ ) = λI (ξ ). The representation now follows from Theorem 1 above and Lemmas 3.4 and 3.5 of Gilboa and Schmeidler (1989). A.3. Proof of Theorem 3
We need to show that I, as defined in Lemma 1, satisfies I (αξ + (1 − α)ζ ) = α I (ξ ) + (1 − α)I (ζ ) for all ξ , ζ ∈ B0 (Σ , u(X )), α ∈ [0, 1]. Take f , g ∈ F such that u(f ) = ξ , and u(g ) = ζ . Hence,
Anscombe, F., Aumann, R., 1963. A definition of subjective probability. The Annals of Mathematical Statistics 34, 199–205. Chateauneuf, A., 1991. On the use of capacities in modeling uncertainty aversion and risk aversion. Journal of Mathematical Economics 20, 343–369. Dolecki, S., Greco, G., 1995. Niveloids. Topological Methods in Nonlinear Analysis 5, 1–22. Ellsberg, D., 1961. Risk, ambiguity, and the Savage axioms. The Quarterly Journal of Economics 75, 643–669. Gilboa, I., Schmeidler, D., 1989. Maxmin expected utility with non-unique prior. Journal of Mathematical Economics 18, 141–153. Maccheroni, F., Marinacci, M., Rustichini, A., 2006. Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica 74, 1447–1498. Schmeidler, D., 1989. Subjective probability and expected utility without additivity. Econometrica 57, 571–587.
Contents lists available at ScienceDirect
Mathematical Social Sciences journal homepage: www.elsevier.com/locate/econbase
On the uses of the monotonicity and independence axioms in models of ambiguity aversionI Leandro Nascimento a,∗ , Gil Riella b a
New York University, 10012 New York, NY, United States
b
Universidade de Brasília, Brazil
article
info
Article history: Received 21 August 2009 Received in revised form 25 January 2010 Accepted 27 January 2010 Available online 2 February 2010
abstract This paper suggests an alternative axiomatization of two canonical models of ambiguity aversion. Instead of relaxing the independence axiom to accommodate uncertainty aversion, we impose independence on constant acts only. Maxmin and variational preferences are characterized by different strengthenings of the monotonicity axiom. © 2010 Elsevier B.V. All rights reserved.
JEL classification: D81 Keywords: Monotonicity Independence Ambiguity
1. Introduction The canonical models of ambiguity aversion of Gilboa and Schmeidler (1989) and Maccheroni et al. (2006) accommodate Schmeidler’s (1989) uncertainty aversion postulate by imposing weaker versions of the independence axiom.1 The main motivation behind the weakenings of the independence axiom is its incompatibility with the type of behavior suggested by Ellsberg (1961) in the presence of a subjective uncertainty. Maxmin preferences are characterized by the ‘‘certainty independence’’ axiom, while variational preferences require a weaker version, the ‘‘weak certainty independence’’ axiom. The interpretation of certainty independence is that constant acts are not good for hedging in the presence of uncertainty. Maccheroni et al. (2006) motivate their weak certainty independence axiom by suggesting that mixing two nonconstant acts at a time with the same constant act and using the same weights may induce different orderings depending on the weights. If the purpose of the model is to explain the ‘‘Ellsbergian choices’’, then it is reasonable to impose the independence axiom on risky prospects. That is, if we do not try to account for the Allais paradox, the independence axiom does not seem too problematic when imposed only on constant acts. Therefore, if one agrees that the independence axiom holds in the context of objective
I We thank Efe Ok and an anonymous reviewer for helpful suggestions.
∗
Corresponding author. E-mail addresses: [email protected] (L. Nascimento), [email protected] (G. Riella).
1 Chateauneuf (1991) independently axiomatized maxmin preferences. 0165-4896/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2010.01.003
uncertainty, the natural course of action would be to impose independence on constant acts only. In fact, the axiomatizations of Gilboa and Schmeidler (1989) and Maccheroni et al. (2006) imply independence on constant acts. The purpose of this paper is to provide an alternative (and equivalent) axiomatization of those models of ambiguity aversion by imposing independence on constant acts only. We show how to strengthen the monotonicity axiom so as to obtain these models. Variational preferences require the weakest strengthening of the monotonicity axiom, whereas the subjective expected utility (SEU) model of Anscombe and Aumann (1963) requires a very strong version of monotonicity. The monotonicity axiom for the maximin preferences lies in between. In the next section we describe the setup and the basic axioms. In Section 3 we present the strengthenings of the monotonicity axiom, and show how to obtain the canonical models of ambiguity aversion. While Section 4 concludes the paper with a discussion on how the Ellsberg paradox relates to our novel axioms and an additional remark, the Appendix contains the proofs of our main results. 2. Setup and basic axioms The state space is denoted by S. Let Σ be an algebra of subsets of S, and denote by ∆(S ) the set of all finitely additive probability measures on S. The set of consequences X is a nontrivial convex subset of a vector space. An act is a Σ -measurable function f : S → X such that |f (S )| < ∞. The domain of preferences is F , the set of all such acts. We denote by Fc the set of constant acts, i.e., Fc =
L. Nascimento, G. Riella / Mathematical Social Sciences 59 (2010) 326–329
{f ∈ F : ∃x ∈ X s.t. f (S ) = {x}}, and identify it with X . We represent by xf any constant act that is indifferent to f . Given the weight α ∈ [0, 1], the pointwise mixture of the acts f and g is denoted by α f + (1 − α)g. As a last piece of notation, if ∅ 6= Γ ⊆ R, then we denote by B0 (Σ , Γ ) the set of Σ -measurable finite step functions ξ : S → R satisfying ξ (S ) ⊆ Γ , and endow it with the sup norm. The binary relation <⊆ F × F satisfies the following set of axioms. Axiom A1 (Weak Order). < is complete and transitive. Axiom A2 (Constant Continuity). For all f ∈ F , and x, y ∈ X , the sets {α ∈ [0, 1] : α x + (1 − α)y < f } and {α ∈ [0, 1] : f < α x + (1 − α)y} are closed. Axiom A3 (Risk Independence). For all x, y, z ∈ X , and α ∈ (0, 1], x < y iff α x + (1 − α)z < α y + (1 − α)z. Axiom A4 (Constant Monotonicity). For all f ∈ F , and x ∈ X , if x < f (s) for all s ∈ S, then x < f , and if f (s) < x for all s ∈ S, then f < x. Axiom A5 (Non-triviality). 6= ∅.2 Axioms A1, A3 and A5 are quite common. The Constant Continuity axiom is a weaker form of the more standard version in which the constant acts x and y are replaced by the general elements of F . The reason why we use that particular version is that, for the models of ambiguity aversion considered below, the strengthenings of the monotonicity axiom already impose a substantial continuity on the representation. Axiom A4 is not standard as well. Note that the usual monotonicity axiom is the following: Axiom A4∗ (Monotonicity). For all f , g ∈ F , if f (s) < g (s) for all s ∈ S, then f < g. Given Axioms A1 and A2, Constant Monotonicity is the weakest form of A4∗ to ensure the existence of a certainty equivalent xf ∈ X for each act f ∈ F . In fact, given any act f ∈ F , let x, y ∈ X be such that x < f (s) < y for all s ∈ S. Using Axioms A1, A2 and A4 one can show that the sets {α ∈ [0, 1] : α x + (1 − α)y < f } and {α ∈ [0, 1] : f < α x + (1 − α)y} are nonempty, closed, and must have a nonempty intersection. Axiom A4∗ will be a consequence of our axioms after we add the strong versions of monotonicity that characterize each class of preferences. Now observe that, when A4∗ holds and < |X admits a utility representation by u, then the utility function U on F , as defined by U (f ) = u(xf ), represents <. The next lemma is wellknown, and its proof follows from the observation just made plus standard arguments, which are omitted. Lemma 1. < satisfies A1–A3, A4∗ , and A5 iff there exists a nonconstant and affine function u : X → R, and I : B0 (Σ , u(X )) → R such that I is increasing, I |u(X ) is the identity map, 0 ∈ int(u(X )) 6= ∅, and, for all f , g ∈ F , f < g iff I (u(f )) ≥ I (u(g )). 3. Additional axioms and representations 3.1. Axioms for variational preferences The binary relation < admits a variational preference represenR tation when the functional I satisfies I (ξ ) = minµ∈∆(S ) [ ξ dµ + c (µ)], for some lower semi-continuous, convex and grounded cost function c : ∆(S ) → R+ ∪{+∞}.3 In order to obtain this representation, we first add the following stronger monotonicity condition: Axiom A6 (Strong Monotonicity 1). For all f , g ∈ F , x, y ∈ X , and α ∈ [0, 1], if α f (s) + (1 − α)x < α g (s) + (1 − α)y for all s ∈ S, x < f , and g < y, then α x + (1 − α)x < α y + (1 − α)y.
2 stands for the asymmetric part of <. 3 The cost function c is grounded if inf
µ∈∆(S )
c (µ) = 0.
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What makes A6 stronger than A4∗ is that, in the presence of Axioms A1, A2 and A4, it not only implies A4∗ , but Axiom A6 also yields monotonicity when one replaces, for example, the nonconstant acts f and g by their certainty equivalents in those mixtures of acts. If we see the (objective) mixture of an act with a constant act as a two-stage lottery, Strong Monotonicity 1 has the following interpretation. We are told that the decision maker prefers the two-stage lottery α f (s)+(1 −α)x over α g (s)+(1 −α)y for each possible state s. The standard monotonicity axiom would tell us that the decision maker prefers the two-stage lottery that gives f with probability α and x with probability 1 − α to the twostage lottery that gives g with probability α and y with probability 1 − α . Ex-ante he is ignorant about the realization of the state. But he knows that the risky prospect x is preferred to f , and that the act g is preferred to y, and therefore he preserves the ranking when we replace f by x, and g by y at the same time. Technically, Axiom A6 can be interpreted as an ordinal characterization of a niveloid. Dolecki and Greco (1995) define a niveloid as a functional J : [−∞, +∞]S → [−∞, +∞] that is isotone (ξ ≥ ζ implies J (ξ ) ≥ J (ζ )) and vertically invariant (a ∈ R implies J (ξ + a) = J (ξ ) + a). An alternative characterization of a niveloid given by their Corollary 1.3 is J (ξ ) − J (ζ ) ≤ sups∈S [ξ (s) − ζ (s)], under the convention that (−∞) + (+∞) = (+∞) + (−∞) = (+∞) − (+∞) = −∞. In our setup, one says that b I : B0 (Σ , R) → R is a niveloid if b I is isotone and vertically invariant. The main role of Axiom A6 is to ensure I, as obtained in Lemma 1, satisfies I (ξ ) − I (ζ ) ≤ sups∈S [ξ (s) − ζ (s)], so it admits a (least) niveloidal extension to B0 (Σ , R). The final axiom is Schmeidler’s (1989) uncertainty aversion postulate: Axiom A7 (Uncertainty Aversion). For all f , g ∈ F , and α ∈ [0, 1], if f ∼ g, then α f + (1 − α)g < f . The next theorem characterizes the variational preferences under our axioms. Its content is certainly not new, but we believe that part of the proof is. Our approach is to use our axioms to derive the basic properties of the functional I, and then use the more technical results of Maccheroni et al. (2006) to complete the proof. Theorem 1. The binary relation < admits a variational preference representation iff it satisfies A1–A7. 3.2. Axioms for multiple priors and SEU The binary relation < admits a multiple priors preference representation if it admits a variational preferences representation in which there exists a nonempty, closed and convex set M ⊆ ∆(S ) such that c (µ) = 0 if µ R ∈ M, and c (µ) = +∞ otherwise. That is, when I (ξ ) = minµ∈M ξ dµ. The next axiom is a strengthening of Axiom A6. Axiom A60 (Strong Monotonicity 2). For all f , g ∈ F , x, y ∈ X , and α1 , α2 ∈ [0, 1], if α1 f (s) + (1 − α1 )x < α2 g (s) + (1 − α2 )y for all s ∈ S, x < f , and g < y, then α1 x + (1 − α1 )x < α2 y + (1 − α2 )y. Strong Monotonicity 2 has a similar interpretation when compared to Axiom A6. The difference is that now the decision maker does not care about the weights being used in the mixtures of acts.4 Theorem 2. The binary relation < admits a multiple priors representation iff it satisfies A1–A5, A60 , and A7. Now we turn to the last strengthening of the monotonicity axiom. We say that < admits a SEU representation when it admits a multiple priors representation with M = {µ}, for some µ ∈ ∆(S ).
4 The technical reason is that the maxmin preferences are characterized by a functional in the induced domain of util acts that is also positively homogenous (besides being vertically invariant as in the variational preferences).
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Axiom A600 (Strong Monotonicity 3). For all f , g , h, i ∈ F , and α1 , α2 ∈ [0, 1], if α1 f (s) + (1 − α1 )h(s) < α2 g (s) + (1 − α2 )i(s) for all s ∈ S, x < f , z 1 < h, g < y, and i < z 2 , then α1 x + (1 − α1 )z 1 < α2 y + (1 − α2 )z 2 . Theorem 3. The binary relation < admits a SEU representation iff it satisfies A1–A5, and A600 . If we had required Axiom A600 to hold only if the acts f and h are comonotonic, and g and i are comonotonic, then we would have obtained Schmeidler’s (1989) representation with a convex capacity after adding the uncertainty aversion Axiom A7. In fact, by employing an argument similar to that in the proof of Theorem 3, one can show that, for any two comonotonic acts f and g, and α ∈ [0, 1], we have I (α u(f ) + (1 − α)u(g )) = α I (u(f )) + (1 − α)I (u(g )). Therefore, when f , g , h ∈ F are pairwise comonotonic, if I (u(f )) > I (u(g )), then α I (u(f )) + (1 − α)I (u(h)) = I (α u(f ) + (1 −α)u(h)) > I (α u(g )+(1 −α)u(h)) = α I (u(g ))+(1 −α)I (u(h)). Now the representation follows from the same arguments used in the proof of the main theorem of Schmeidler (1989). 4. Discussion 4.1. Ellsberg paradox: Example The weakenings of the independence axiom that have been proposed in the literature are appealing for at least two reasons. They not only have a simple interpretation as we mentioned in the introduction, but also help to accommodate the uncertainty aversion postulate and build a model of decision making that can account for the Ellsberg paradox. The alternative axiomatization proposed in this paper has an interpretation, as we argued above, albeit not as simple as the weakenings of the independence axiom. A natural question is how the Ellsbergian choices and the strengthenings of the monotonicity axiom relate. Here we present an alternative perspective on the incompatibility between the Ellsberg-type behavior and the SEU model. It allows us to see Axioms A6 and A60 as weakenings of Axiom A600 , which characterizes SEU, and understand why they are necessary to explain that type of behavior. Consider the following version of the Ellsberg paradox. An urn contains 90 balls, and 30 balls are red (R). The remaining 60 balls are either black (B) or yellow (Y ), but the proportion is unknown. The state space is S = {R, B, Y }. Four alternative bets (monetary acts) are available to the decision maker. They are vectors where the first coordinate represents the prize (in $) in case a red ball is drawn, and the second and third coordinates correspond to prizes associated to black and yellow balls being drawn, respectively. In other words, a bet is a function f : S → [0, 100], where the minimum prize is zero dollars, and the maximum prize is one hundred dollars. We consider: " # " # 100 0 0 , fR = fB = 100 , 0 0 100 0 , 100
" fRY =
#
0 100 . 100
" and fBY =
#
The framework in this example is one with monetary acts. The risk independence Axiom A3 allows us to take convex combinations of monetary prizes. The following rankings induce typical Ellsbergian choices: fR $32 =: xfR xfB := $18 fB and fRY ≺ xfRY := $52 ≺ $64 =: xfBY ≺ fBY .5 Observe that the
rankings are incompatible with the Strong Monotonicity 3 axiom: for all s ∈ S, 1
1
1
1
fRY (s) , 2 2 2 2 but 1 1 1 1 xf + xfBY = $16 + $32 > $9 + $26 = xfB + xfRY . 2 R 2 2 2 fR (s) +
fBY (s) = $50 =
fB (s) +
This violates Axiom A600 (because it would require 12 xfR + 21 xfBY ≤
+ 12 xfRY ). One interpretation of Axiom A600 is that, provided one has information about the pointwise comparison of the mixture of two pairs of acts, one can make an inference, for example, about the mixture of the certainty equivalents associated with those two pairs. This is a very strong requirement in the presence of subjective uncertainty if the decision maker is ambiguity averse. In the Ellsberg example, the acts fR and fBY are much more attractive to the decision maker because probabilities are completely specified. The agent will be willing to pay more money for the acts fR and fBY than he is willing to sell fB and fRY on average. In order to accommodate the Ellsbergian choices one needs to find weaker versions of Axiom A600 . The maxmin preferences force the mixture to be of a pair of acts containing at least one constant act. The weights of the two pairs of acts may differ, though. Variational preferences place the same sort of constraints on the pairs of acts, but require the weight to be the same when mixing each pair. They are both compatible with the Ellsbergian choices in our example. 1 x 2 fB
4.2. Remark In general, Strong Monotonicity 1 alone need not imply weak certainty independence. But one can show that, when Axioms A1–A4 and A6 hold, the implication α f + (1 − α)x < α g + (1 − α)x ⇒ α f + (1 − α)y < α g + (1 − α)y is true (i.e., weak certainty independence also holds). In fact, take any f , g ∈ F , x, y ∈ Fc , α ∈ [0, 1], and assume α f + (1 − α)x < α g + (1 − α)x. 1 If we define γ := 2−α , then γ [α f (s) + (1 − α)y] + (1 − γ )x = γ [α f (s)+(1 −α)x]+(1 −γ )y, and γ [α g (s)+(1 −α)y]+(1 −γ )x = γ [α g (s) + (1 − α)x] + (1 − γ )y for all s ∈ S. Hence, using the existence of certainty equivalents plus Axioms A3 and A6 we obtain γ xα f +(1−α)y + (1 − γ )x ∼ γ xα f +(1−α)x + (1 − γ )y < γ xαg +(1−α)x + (1 − γ )y ∼ γ xαg +(1−α)y + (1 − γ )x, so that α f + (1 − α)y < α g + (1 − α)y. Appendix. Proofs We only show the ‘‘if’’ part of the theorems. A.1. Proof of Theorem 1 Because A6 implies A4∗ when A1–A4 are satisfied, from Lemma 1 we obtain the representation I ◦ u : F → R. Take any two acts f , g ∈ F . Using the fact |f (S )|, |g (S )| < ∞, there exists s∗ ∈ S such that u(g (s∗ )) − u(f (s∗ )) = sups∈S [u(g (s)) − u(f (s))]. Define x := g (s∗ ) and y := f (s∗ ), so that 21 f (s) + 12 x < 21 g (s) + 12 y for all s ∈ S. By Axiom A6 we obtain 12 xf + 12 x < 12 xg + 12 y, which implies I (u(g )) − I (u(f )) ≤ sups∈S [u(g (s)) − u(f (s))]. Therefore, for all ξ , ζ ∈ B0 (Σ , u(X )), I (ξ ) − I (ζ ) ≤ sups∈S [ξ (s) − ζ (s)]. The representation now follows from the same arguments in Maccheroni et al. (2006, pp. 1480–1481). A.2. Proof of Theorem 2
5 The numbers follow from maxmin preferences with set of priors co{(1/3, 1/6, 1/2), (1/3, 1/2, 1/6)}, where the symbol co stands for the convex hull.
Because A60 implies A4∗ when A1–A4 are satisfied, from Lemma 1 we obtain the representation I ◦ u : F → R. It remains to
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show that I is positively homogeneous and vertically invariant. It sure is vertically invariant because A60 implies A6. Assume w.l.o.g. that [−1, 1] ⊆ u(X ). Now take λ > 0, and ξ ∈ B0 (Σ , u(X )) such that λξ ∈ B0 (Σ , u(X )), and let f , g ∈ F be such that u(f ) = ξ and u(g ) = λξ , and x0 ∈ X satisfy u(x0 ) = 0. Therefore, for all s ∈ S, 1 λ 1 λ ξ (s) + 1+λ u(x0 ) = 1+λ u(x0 ) + 1+λ λξ (s), which translates into 1+λ λ 1 f (s) + 1+λ x0 1+λ 0
λ 1 x + 1+λ g (s) for all s ∈ S. As a consequence 1+λ 0 λ 1 λ 1 I (ξ ) + u(x0 ) = 1+λ u(x0 ) + 1+λ I (λξ ), 1+λ 1+λ
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using A600 , for h = α f +(1−α)g, we have α f (s)+(1−α)g (s) ∼ h(s) for all s ∈ S, and therefore α xf + (1 − α)xg ∼ h, which implies α I (ξ ) + (1 − α)I (ζ ) = I (αξ + (1 − α)ζ ). Therefore I is affine, and we obtain the representation using well-known arguments. References
∼
of A6 we obtain and hence I (λξ ) = λI (ξ ). The representation now follows from Theorem 1 above and Lemmas 3.4 and 3.5 of Gilboa and Schmeidler (1989). A.3. Proof of Theorem 3
We need to show that I, as defined in Lemma 1, satisfies I (αξ + (1 − α)ζ ) = α I (ξ ) + (1 − α)I (ζ ) for all ξ , ζ ∈ B0 (Σ , u(X )), α ∈ [0, 1]. Take f , g ∈ F such that u(f ) = ξ , and u(g ) = ζ . Hence,
Anscombe, F., Aumann, R., 1963. A definition of subjective probability. The Annals of Mathematical Statistics 34, 199–205. Chateauneuf, A., 1991. On the use of capacities in modeling uncertainty aversion and risk aversion. Journal of Mathematical Economics 20, 343–369. Dolecki, S., Greco, G., 1995. Niveloids. Topological Methods in Nonlinear Analysis 5, 1–22. Ellsberg, D., 1961. Risk, ambiguity, and the Savage axioms. The Quarterly Journal of Economics 75, 643–669. Gilboa, I., Schmeidler, D., 1989. Maxmin expected utility with non-unique prior. Journal of Mathematical Economics 18, 141–153. Maccheroni, F., Marinacci, M., Rustichini, A., 2006. Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica 74, 1447–1498. Schmeidler, D., 1989. Subjective probability and expected utility without additivity. Econometrica 57, 571–587.