The variance is a classic example of a measure of dispersion. The purpose of such measures is to describe how dispersed is a random variable relative to some reference point (the mean in the case of the variance). We now present a general deÞnition
17In the second line we used the equality M = {u (f) + t : f ∈ G, t ∈ R}. Since u (X) is unbounded,
of measure of dispersion, that suggests a natural way to generalize mean-variance preferences.
Let B (Σ) be the set of all bounded real-valued Σ-measurable functions deÞned on S. Given a base probability q ∈ ∆σ(Σ), consider a vertically invariant reference functional τq : B (Σ)→ R, like the mean functional τq(ϕ) =R
ϕdq for each ϕ ∈ B (Σ).
A measure of dispersion Hq : B (Σ)→ R about τq is a convex functional satisfying the following conditions, discussed in Bickel and Lehmann [4] and [5]:
(i) Hq(0) = 0 and Hq(ϕ + k) = Hq(ϕ) for each ϕ ∈ B (Σ) and each constant function k ∈ B (Σ);
(ii) Hq(ϕ1) ≥ Hq(ϕ2) if |ϕ1− τq(ϕ1)| stochastically dominates |ϕ2− τq(ϕ2)|, that is, if F|ϕ1−τq(ϕ1)|(t)≤ F|ϕ2−τq(ϕ2)|(t) for each t ≥ 0.19
For example, important classes of measures of dispersion about the mean ϕ ≡R ϕdq where γ is any positive constant and µ is any probability distribution on (0, 1).
Variance Varq(ϕ) and standard deviation SDq(ϕ) are the special cases of, respec-tively, Hqv and Hqσ when γ = 2 and µ is the uniform distribution. SpeciÞcally,
A preference% on the set of all acts F on S is a dispersion preference if there is a dispersion measure Hq : B (Σ) → R such that, for all f, g ∈ F, is a dispersion preference, which for γ = 2 reduces to the mean-variance preference we
studied before.
19F|ϕ−τq(ϕ)| : R+ → [0, 1] is the cumulative distribution of |ϕ − τq(ϕ)| ∈ B (Σ) w.r.t. the base probability q; i.e., F|ϕ−τq(ϕ)|(t) = q ({s ∈ S : |ϕ (s) − τq(ϕ)| ≤ t}) for all t ≥ 0.
While dispersion measures describe the dispersion of a random variable about some reference point, variability measures try to describe the “intrinsic” dispersion of a random variable. For example, a classic variability measure is Gini’s mean difference:
G (ϕ) = Z
S×S
(ϕ (s)− ϕ (s0))2d (q⊗ q) .
In the same way dispersion measures induce dispersion preferences, variability measures as well can be used to introduce variability preferences. For brevity, here we just give a simple example involving Gini’s mean difference.
Example 35 Suppose S = {s1, s2}, and Σ = 2S. Given a base probability q ∈ ∆ (Σ), deÞne a preference%v on F as follows: for all f, g ∈ F,
f %v g ⇔ Z
u (f ) dq−θ
2(u (f (s1))− u (f (s2)))≥ Z
u (g) dq−θ
2(u (g (s1))− u (g (s2))) , where θ > 0 and u : X → R is an affine function. The preference %v may not be monotone, unless its domain is suitably restricted. Set
M (θ) =
½
ϕ∈ R2 : ϕ2 −1− q1
θ ≤ ϕ1 ≤ q1
θ + ϕ2
¾ .
It is easy to check that%vis a continuous variational preference on Fθ ={f : u (f) ∈ M (θ)}, with index of ambiguity aversion c? given by:
c?(p) =−(p1− q1)2 2θ . Hence,
f %v g ⇔ min
p∈∆σ(Σ,q)
µZ
u (f ) dp + 1
2θ(p1− q1)2
¶
≥ min
p∈∆σ(Σ,q)
µZ
u (g) dp + 1
2θ (p1− q1)2
¶
for all f, g ∈ F. N
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APPLIED MATHEMATICS WORKING PAPER SERIES
1. Luigi Montrucchio and Fabio Privileggi, “On Fragility of Bubbles in Equilibrium Asset Pricing Models of Lucas-Type,” Journal of Economic Theory 101, 158-188, 2001 (ICER WP 2001/5).
2. Massimo Marinacci, “Probabilistic Sophistication and Multiple Priors,”
Econometrica 70, 755-764, 2002 (ICER WP 2001/8).
3. Massimo Marinacci and Luigi Montrucchio, “Subcalculus for Set Functions and Cores of TU Games,” Journal of Mathematical Economics 39, 1-25, 2003 (ICER WP 2001/9).
4. Juan Dubra, Fabio Maccheroni, and Efe Ok, “Expected Utility Theory without the Completeness Axiom,” Journal of Economic Theory, forthcoming (ICER WP 2001/11).
5. Adriana Castaldo and Massimo Marinacci, “Random Correspondences as Bundles of Random Variables,” April 2001 (ICER WP 2001/12).
6. Paolo Ghirardato, Fabio Maccheroni, Massimo Marinacci, and Marciano Siniscalchi, “A Subjective Spin on Roulette Wheels,” Econometrica, forthcoming (ICER WP 2001/17).
7. Domenico Menicucci, “Optimal Two-Object Auctions with Synergies,” July 2001 (ICER WP 2001/18).
8. Paolo Ghirardato and Massimo Marinacci, “Risk, Ambiguity, and the Separation of Tastes and Beliefs,” Mathematics of Operations Research 26, 864-890, 2001 (ICER WP 2001/21).
9. Andrea Roncoroni, “Change of Numeraire for Affine Arbitrage Pricing Models Driven By Multifactor Market Point Processes,” September 2001 (ICER WP 2001/22).
10. Maitreesh Ghatak, Massimo Morelli, and Tomas Sjoström, “Credit Rationing, Wealth Inequality, and Allocation of Talent”, September 2001 (ICER WP 2001/23).
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15. Steven Haberman and Elena Vigna, “Optimal Investment Strategies and Risk Measures in Defined Contribution Pension Schemes,” Insurance: Mathematics and Economics 31, 35-69, 2002 (ICER WP 2002/10).
16. Enrico Diecidue and Fabio Maccheroni, “Coherence without Additivity,” Journal of Mathematical Psychology, forthcoming (ICER WP 2002/11).
17. Paolo Ghirardato, Fabio Maccheroni, and Massimo Marinacci, “Ambiguity from the Differential Viewpoint,” April 2002 (ICER WP 2002/17).
18. Massimo Marinacci and Luigi Montrucchio, “A Characterization of the Core of Convex Games through Gateaux Derivatives,” Journal of Economic Theory, forthcoming (ICER WP 2002/18).
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24. Claudio Mattalia, “Existence of Solutions and Asset Pricing Bubbles in General Equilibrium Models”, January 2003 (ICER WP 2003/2).
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34. Bruno Bassan, Olivier Gossner, Marco Scarsini, and Shmuel Zamir, “Positive Value of Information in Games”, International Journal of Game Theory, forthcoming (ICER WP 2003/26).
35. Marco Dall’Aglio and Marco Scarsini, “Zonoids, Linear Dependence, and Size-Biased Distributions on the Simplex”, Advances in Applied Probability forthcoming (ICER WP 2003/27).
36. Taizhong Hu, Alfred Muller, and Marco Scarsini, “Some Counterexamples in Positive Dependence”, Journal of Statistical Planning and Inference, forthcoming (ICER WP 2003/28).
37. Massimo Marinacci, Fabio Maccheroni, Alain Chateauneuf, and Jean-Marc Tallon, “Monotone Continuous Multiple Priors” August 2003 (ICER WP 2003/30).
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