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Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension

Decisions under risk and partial knowledge modelling uncertainty and risk aversion

Giulianella Colettia Davide Petturitia,b Barbara Vantaggib

a University of Perugia b “La Sapienza” University of Rome

9th Int. Symp. on Imprecise Probability: Theories and Applications July 20-24, 2015 Pescara, Italy

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension

Classical decision theory under risk

• L = {L = (XL, PL)}, a set of lotteries on X = {x1, . . . , xn},

where PL is a probability distribution with support XL ⊆ X

• , a preference relation on L

von Neumann-Morgenstern’s axioms are equivalent to the existence of a linear function U : L → R (unique up to positive linear transformations) representing , i.e., for L, L′ ∈ L L L′ ⇔ U(L) ≤ U(L′)

⇒ If L contains the degenerate lotteries L0 = {δx : x ∈ X}

then there exists u : X → R s.t. for L ∈ L U(L) = EPL(u) =

• x∈X

u(x)PL(x)

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension

Problems

• The DM cannot consider just a finite set of lotteries L
• The DM has to provide comparisons between “certainty

equivalents” and “risky prospects”

• It is not possible to consider imprecise probabilities

Imprecise information

In order to deal with imprecise information some axiomatizations, that generalizes von Neumann-Morgenstern theory, have been provided (see e.g. Jaffray (1989), Gaidos et al. (2004)). Actually, these axiomatizations are not structure free.

Aim

The aim is to provide a rational criterion where the DM expresses just few preferences.

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension

Partial preferences on an arbitrary set L

Strengthened preference relation

Given an arbitrary set of lotteries L, we consider a pair of consistent relations (, ≺) where none of or ≺ is assumed to be complete and ≺= ∅:

• L L′ stands for “L is not preferred to L′”;
• L ≺ L′ stands for “L′ is preferred to L”.

Representation

We search for a function U : L → R representing (, ≺), i.e., for every L, L′ ∈ L L L′ ⇒ U(L) ≤ U(L′) and L ≺ L′ ⇒ U(L) < U(L′).

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension

A paradigmatic example

U1 L2 ≺ L1, L4 ≺ L3

P1 = {Pθ} with θ ∈

• 0, 2

3

• for no θ there exists u : {0, 100} → R

{w} {b} {r} Pθ

1 3

θ

2 3 − θ

s.t. EPθ(u(L)) represents ≺ w b r L1 100e 0e 0e L2 0e 0e 100e L3 0e 100e 100e L4 100e 100e 0e Problem: Probability is not suit- able to measure uncertainty in situations as those described above: non-additive uncertainty measures come to the fore

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension

Generalized lotteries

Definition [Jaffray 1987]

A generalized lottery, or g-lottery for short, on a finite set XL is a pair L = (℘(XL), BelL) where BelL is a belief function on ℘(XL), i.e.: (i) BelL(∅) = 0 and BelL(XL) = 1; (ii) BelL n

i=1 Ai

• ≥
• ∅=I⊆{1,...,n}

(−1)|I|+1BelL

• i∈I Ai
• for every Ai ∈ ℘(XL).

⇒ A g-lottery could be equivalently defined as L = (℘(XL), mL), where mL is

the basic assignment associated to BelL defined for every A ∈ ℘(XL) as mL(A) =

• B⊆A

(−1)|A\B|BelL(B)

⇒ mL is a function from ℘(XL) to [0, 1] s.t. mL(∅) = 0 and

• A∈℘(XL)

mL(A) = 1

⇒ Probability measures on ℘(XL) are particular belief functions

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension

Generalized convex lotteries

Definition

A gc-lottery on a finite set XL is a pair L = (℘(XL), ϕL) where ϕL is a convex capacity on ℘(XL), i.e. ϕ (A ∪ B) ≥ ϕ(A) + ϕ(B) − ϕ(A ∩ B). (1)

⇒ A gc-lottery could be equivalently defined as L = (℘(XL), mL), where

m : ℘(X) → R is the basic assignment associated to ϕL.

⇒ For every A ∈ ℘(X) with |A| ≥ 2 and every {xi, xj} ⊆ A, it satisfies

• {xi ,xj }⊆B⊆A m(B) ≥ 0

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension

Operations on gc-lotteries ⇒ Given L = {L = (℘(XL), ϕL)}, if X = ∪{XL : L ∈ L} is finite,

then all gc-lotteries can be rewritten on X: they reduce to convex capacities on ℘(X)

Convex combination of gc-lotteries

For L1, . . . , Lt ∈ L and k = (k1, . . . , kt) with ki ≥ 0 (i = 1, . . . , t) and t

i=1 ki = 1, the convex combination of L1, . . . , Lt according to k is the

gc-lottery on X k(L1, . . . , Lt) =

• A

t

i=1 kimLi (A)

• for every A ∈ ℘(X) \ {∅}.

(2)

Set of degenerate gc-lotteries

L∗

0 = {δB : B ∈ ℘(X) \ {∅}}, where mδB (B) = 1 for B ∈ ℘(X) \ {∅}

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension

Jaffray’s linear representation for g-lotteries

If L is closed under convex combinations of g-lotteries, von Neumann-Morgenstern axioms are equivalent to the existence of a linear function LU : L → R (unique up to p.l.t.) representing , i.e., for L, G ∈ L L G ⇔ LU(L) ≤ LU(G)

⇒ If L contains also the degenerate g-lotteries, then there exists

v : ℘(X) → R s.t. for L ∈ L LU(L) =

• B∈℘(X)

v(B)mL(B)

⇒ The semantic interpretation of “utility” function v on ℘(X) is not

clear, moreover, it requires to specify a number of parameter in the order

• f 2card X

⇒ A possible alternative is to search for a function u : X → R and to

use the Choquet expected utility functional [Schmeidler 1989] defined for L ∈ L as CEU(L) = C

• u dBelL

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension

Ordered set of prizes

Consider

• L, set of gc-lotteries
• X = {XL : L ∈ L} = {x1, . . . , xn} totally ordered as x1 < . . . < xn

Aggregated basic assignment

The aggregated basic assignment of L ∈ L is defined for every xi ∈ X as ML(xi) =

• xi ∈B⊆Ei

mL(B), where Ei = {xi, . . . , xn} for i = 1, . . . , n.

⇒ ML is a “pessimistic” probability dis-

tribution on X induced by ϕL

⇒ If u : X → R is strictly increasing

c

• u dϕL = n

i=1 u(xi)ML(xi)

Assumpion on L

(AO) L0 = {δ{x} : x ∈ X} ⊆ L and ∀x, x′ ∈ X, x ≤ x′ ⇔ δ{x} δ{x′}

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension

Generalized Choquet rationality condition (g-CR)

Definition

A strengthened preference relation (, ≺) on an arbitrary set L of gc-lotteries is said to be generalized Choquet rational if it satisfies: (g-CR) For all h ∈ N and Li, L′

i ∈ L with Li L′ i (i = 1, . . . , h), if

k(ML1, . . . , MLh) = k(ML′

1, . . . , ML′ h)

with k = (k1, . . . , kh), ki > 0 (i = 1, . . . , h) and h

i=1 ki =

1, then it can be Li ≺ L′

i for no i = 1, . . . , h.

⇒ (g-CR) involves aggregated basic assignments: convex combinations are in

the usual sense, i.e., among probability distributions

⇒ If is complete and L is convex, (g-CR) implies von

Neumann-Morgenstern axioms and (*) for every L, L′ ∈ L, ML = ML′ ⇒ L ∼ L′

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension

CEU representation theorem

Theorem

Let L be a finite set of gc-lotteries, X = {XL : L ∈ L} = {x1, . . . , xn} and let ≤∗ be a total preorder

• n X. For a strengthened preference relation (, ≺) on L

satisfying (A0) the following statements are equivalent: (i) (, ≺) is Choquet rational (i.e., it satisfies (gc-CR)); (ii) there exists a strictly increasing function u : X → R, whose CEU functional defined, for every L ∈ F CEUF(L) = C

• uF dϕL =

n

• i=1

uF(xi)ML(xi) represents (, ≺).

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension

Risk aversion in the case of money payoffs

Suppose X = {x1, . . . , xn} ⊂ R and ≤∗≡≤ with x1 < . . . < xn: ML is a probability distribution on X for every L ∈ L.

Assumptions (A1) and (A1*)

Let ki = (ki, 1 − ki) be with ki =

xi+1−xi xi+1−xi−1 (i = 2, . . . , n − 1), define

L1 =

• ki
• δ{xi−1}, δ{xi+1}
• : i = 2, . . . , n − 1
• (A1) L1 ⊆ L and ki
• δ{xi−1}, δ{xi+1}
• ≺ δ{xi} or

ki

• δ{xi−1}, δ{xi+1}
• ∼ δ{xi}.

(A1*) L1 ⊆ L and ki

• δ{xi−1}, δ{xi+1}
• ≺ δ{xi}.

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension

Risk aversion in the case of money payoffs

Proposition [Risk aversion in case of money payoffs]

Assume (, ≺) satisfies (A0) and (gc-CR) and let u be a utility whose CEU represents (, ≺). The following statements hold: (i) if (A1) holds then u extends to a strictly increasing concave function v ∈ C 0([x1, xn]); (ii) if (A1*) holds then u extends to a strictly increasing strictly concave function w ∈ C 2([x1, xn]).

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension

Second order stochastic dominance:

Proposition [S.O.S.D. in case of money payoffs]

Assume (A0) and (A1) are satisfied. If (, ≺) satisfies (gc-CR) then for every complete preference relation ′ on L extending (, ≺) and satisfying (gc-CR) the following condition holds for every L1, L2 ∈ L: (S2) if x

−∞ FL1(t)dt ≤

x

−∞ FL2(t)dt for every x ∈ R, it

cannot be L1 ≺′ L2. where FL(x) =

xi≤x ML(xi)

for every x ∈ R and L ∈ L.

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension

Extension of a Choquet rational relation

Theorem [Extension]

Let X = {x1, . . . , xn} be a finite set with a total prorder ≤∗, L and L′ finite sets of gc-lotteries on X, with L ⊆ L′, and (, ≺) a strengthened preference relation on L satisfying (A0). Then if (, ≺) satisfies condition (gc-CR) there exists a family {γ : γ ∈ Γ} of complete relations on L′ satisfying (gc-CR) which extend (, ≺). Moreover, denoting with ≺γ and ∼γ, respectively, the strict and symmetric parts of γ, for γ ∈ Γ, condition (gc-CR) singles out the relations ≺⋆= {≺γ : γ ∈ Γ} and ∼⋆= {∼γ : γ ∈ Γ}.

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension

Extension of a Choquet rational relation

Extension

The extension of (, ≺) on a new pair of gc-lotteries (F, G) can be computed solving at most three linear systems

S≺⋆ :

• A′w > 0

Bw ≥ 0 S≻⋆ :

• A′′w > 0

Bw ≥ 0 S∼⋆ : Aw > 0 B′w ≥ 0

Lotteries Generalized lotteries Rationality conditions for gc-lotteries Extension

References

• Coletti, G., Petturiti, D., Vantaggi, B.: Rationality principles

for preferences on belief functions. Accepted in Kybernetika.

• Gajdos, T., Tallon, J.M., Vergnaud, J.C.: Decision making

with imprecise probabilistic information, J. of Math. Ec., 40(6), 647-681 (2004).

• Gilboa, I., Schmeidler, D.: Maxmin expected utility with

non-unique prior. J. of Math. Econ., 18(2), 141–153 (1989).

• Jaffray, J.-Y.: Linear utility theory for belief functions. Op.
• Res. Let., 8 (2), 107–112 (1989).