ELSEVIER
Journal of Mathematical
Economics
24 (1995) 371-381
How complicated are betweenness preferences?
Zvi Safra a, Uzi Segal b~c,*
aFaculty of Management, Tel Aviv University, Tel Aviv 69978, Israel
h Department of Economics, University of Toronto, 150 St. George Street, Toronto,
Ontario, M5S IA, Canada
’ Division of the Humanities and Social Sciences, California Institute of Technology,
Pasadena, CA 91125, USA
Submitted July 1993; accepted January
1994
Abstract
In this paper we suggest a complexity-based
taxonomy of some families of preferences
over lotteries. We define the complexity of a given family to be of order n if the question
whether or not a given preference belongs to this family can be determined by comparing
lotteries who differ on only n outcomes at most. We first show that the complexities of the
expected utility family, the weighted utility family and the betweenness family are all of
order 3. We then show that the complexity of singletons is always equal to 3. Finally, we
provide examples of families of preferences with arbitrary orders of complexities. The
paper also provides a better understanding of the large body of experimental results in the
theory of decision making under risk.
Keywords: Non-expected
utility; Complexity;
Experiments
JEL classification: D81; C90
1. Introduction
In recent years there has been a growing interest in non-expected utility theories
of decision making under risk, mainly because they explain a wide range of
* Corresponding
author.
0304-4068/95/$09.50
0 1995 Elsevier Science B.V. All rights reserved
SSDI 0304-4068(94)00693-8
372
Z. Safia, U. Segal/ Journal of Mathematical Economics 24 (1995) 371-381
behavioral patterns and experimental results that are inconsistent with expected
utility theory.
Several families of transitive non-expected utility preferences have been identified and axiomatized. They include the betweenness family (Chew, 1983, 1989;
Fishburn, 1983; Dekel, 19861, the rank dependent expected utility family (Quiggin, 1982; Yaari, 1987; Segal, 1993), the weighted utility family (Chew, 1983),
the quadratic utility family (Machina, 1982; Chew, Epstein and Segal, 1991,
19941, and the disappointment aversion family (Gul, 1991).
One of the purposes of this paper is to suggest a complexity-based
taxonomy of
some of these families. The idea is that a family of preferences is of high
complexity level if complicated lotteries are needed in order to determine whether
a given preference relation belongs to this family or not. Complicated lotteries are
lotteries with many prizes. More specifically, we define the complexity of a given
family to be of order IZ if the question whether or not a given preference belongs
to this family can be determined by comparing lotteries who differ on only n
outcomes at most. (The formal definition appears in Section 2.)
In this paper we concentrate on the complexity levels of preferences that are
sub-families
of the betweenness family: the expected utility and the weighted
utility relations, together with the betweenness family itself and prove that all
these families are of complexities of order 3. We then consider families which are
singletons (not necessarily of betweenness preferences) and show that all these
families have complexities of order 3, independently of the preference relation that
belong to such a singleton. Finally, we provide examples for families of preference
relations with arbitrary orders of complexity. These families are all subsets of the
expected utility family.
Another purpose of this paper is to provide a better understanding of the large
body of experimental results in the theory of decision making under risk. Most of
the experiments are restricted to three-prizes lotteries (see for example, MacCrimmon and Larsson (1979). Even the famous Allais paradox and the common ratio
effect experiments deal with lotteries of three prizes only. Clearly, to reject the
conjecture that a given preference relation belongs to, say, the expected utility
family it is sufficient to find one such experiment in which the choices (restricted
to three-prizes lotteries) contradict the known properties of this family. However,
in order to determine whether a given preference relation belongs to the expected
utility family, many more choices should be investigated. Nevertheless, it may
well happen that we do not have to make all possible comparisons to conclude
(assuming continuity, monotonicity
and transitivity) that the preference relation
belongs to the expected utility family. A natural question is how many, and what
choices should be examined? A partial answer to this question is given by our
results, since they imply that, for some sub-families of the betweenness family, all
that is needed is to compare lotteries which differ in three outcomes at most.
A stronger result holds if one is interested in a unique preference relation (and
not in a family of preference relations). We show that, for al preference relations,
2. safra, U. Segal / Journal of Mathematical
373
Economics 24 (I 99.5) 371-381
the information given comparing pairs of lotteries which differ in the probabilities
of three outcomes is always sufficient. Note that this result does not depend on the
preference relation being a member of the betweenness family.
2. Definitions
Let L be the set of lotteries over the compact interval [a, bl E R. Finite
lotteries are denoted (x,, p,; . . . ; x,, p,), where Cp, = 1 and pi > 0, i = 1, . . , n.
Let L, = 1(x,, p1; f . . ; x,, p,)} be the set of lotteries with n different outcomes at
most. For y=(y ,,..., y,)~[rt,
blm, x1,..., x,~[a, 61 and q=(q,,...,q,,,)
such that Cq, < 1 and q,, . . . , q, a 0, let A"- '(y,q){x,, . . ., x,,} be the set of
lotteries of the form (yl, q,;...;y,,
q,; xl, p,;...;
xnP1, pn_,; x,, 1 -c?==,qi
- c:,:p,).
For XEL,
let Fx be the cumulative distribution function of X, given by
F,(x)=Pr(X<x).
For X, YEL and (YELO, 11, let c-uX+(l-a)Y
be the
lottery with the distribution function aFx + (1 - aIF,.
It follows immediately
that for X=(x1,
pl;. ..; x,, p,> and Y= (y,, 4,;. . .; y,, q,), ax + (1 - a)Y
= (x 1, UpI;. . . ; x,,
ffpn; y,, (1 - “14,;.
. .; y,,
(1 - dq,).
be a preference relation over L, with strict preference + and indifference N. We assume throughout that 2 is complete, transitive, continuous, and
monotonic with respect to first-order stochastic dominance. ’ The functional
V : I_.+ R represents the preference relation 2 if for every X, YE L, X 2 Y if
and only if V(X) > V(Y). Let L’ c L. An indifference set through X in L’ is the
set of all lotteries in L’ that are indifferent to X. We say that k satisfies
betweenness on L’ if for every X, YE L’ such that X 2 Y and for every (YE [O, 11,
X 2 aX + (1 - (Y)Y 2 Y. By continuity, this condition is equivalent to the following requirement:
For all X and Y such that X N Y and for all (YE LO, 11,
X N aX + (1 - (Y)Y N Y. For a discussion of betweenness preferences, see Chew
(1983) and Dekel (1986).
The expected utility functional, given by
Let
2
V(X)
=/4x) dFx(x)
satisfies the betweenness
functional, given by
V(X)
=
/ 4x)
condition.
(1)
So does Chew’s
(1983)
weighted
utility
dF&)
(2)
/ W(X) d&(x)
We say that the functional
’ The preference
[t/z, F,(z) Q F,(z)
at Eq. (2) is proper weighted if it is not expected utility.
relation
2 is monotonic with respect to first-order
and 3z such that F,(z) < F,(z)] * X + Y.
stochastic
dominance
if
374
Z. Safra, U. Segal/ Journal of Mathematical Economics 24 (1995) 371-381
3. Betweenness-supporting
preferences
As mentioned in the Introduction, we are interested in the question how much
information is needed before we can determine that a given preference relation
belongs to a certain family, or that it satisfies a certain set of axioms. It turns out
that, with respect to betweenness functionals, it is insufficient to require that the
preference relation is of one of the above discussed forms on all lotteries with no
more than n possible outcomes. This claim is formalized in the following theorem.
Theorem 1. Let the preference relation 2 satisfy property LY on L,, CY=
betweenness, expected utility, weighted utility. Then 2 does not have to satisjj
property ff on L.
2 * such that 2 * satisfies
Proof. Let V” represent the preference relation
property (Y on L. Let X” be the lottery such that Fx* is the uniform distribution
on [a, b]. Let 11. II be the L, norm. Define
L~={XEL:IIF~-F,*(I
<(b-n)/4n}.
Note that L, f~ L*, = 0. Denote z, = L \L*,.
For X E Lz, define the sets X, = {YE L : X weakly dominates Y by first-order
stochastic dominance} and X, = (YE L : Y weakly dominates X by first-order
stochastic dominance}. Obviously, z,, X,, and X, are closed sets. The set L is
compact (recall that the segment [a, b I is compact), hence so are the sets z,,c X,
and z, fl X,, for all X E L*,. For such X, define V,*(X) = max{V*(Y) : YE L, I?
X,} and V:(X)
= min{V*(Y): YES, nX,}. By construction,
V,*(X) < V*(X)
< v;(x).
Define a functional V on L such that for Y EL,,, V(Y) = V*(Y). On L*,, we
want V to satisfy the following conditions: It should be monotonic with respect to
first-order stochastic dominance, should not satisfy betweenness,
and for every
X E L*,, V:(X)
G V(X) G VU*(X). Clearly, such functions exist. A preference
relation
2 represented by such a functional
is monotonic,
does not satisfy
betweenness
(hence also weighted utility and expected utility), but satisfies
property (Y on L,.
0
Note that the proof holds for the disappointment
aversion family as well, since
this family of preferences is a subset of the betweenness class.
Theorem 1 shows that the behavior of a preference relation on the sets L, does
not uniquely determine its overall behavior. We therefore have to consider sets of
the form A”- ‘(y, q){x,, . . . , xn).
Definition 1. We say that the complexity of a certain family of preference relations
9 is of order n if n is the smallest integer for which the following statement
holds:
Let 2 be a preference relation. If on every A"- '(y, qxx,,
preference relation 2 agrees with a member of ST, then 2 ELM.
. . . , xn} the
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Z. Safia, U. Segal/ Journal of Mathematical Economics 24 (I 995) 371-381
Let
0
9-a be the set of betweenness
l ST, be the set of expected
l FW
preference
relations.
utility preference
relations.
be the set of proper weighted utility preference
relations.
Clearly, if the preference relation 2 satisfies the betweenness axiom, then it
satisfies it on all the sets A”- ‘(y, q){x,, . . . , x,). Also, if it satisfies the independence axiom (and therefore, belongs to the set F,), then it can be represented by
an expected utility functional on each of these sets. The same result for the
weighted utility functionals follows from the following lemma.
Lemma 1. Consider the continuous and monotonic preference relation
A”- ‘(y, q){x,, . . , x,,}. The following two conditions are equiualent.
2
on
1. The order 2 can be represented by a proper weighted utilityfunctional.
2. All the indifference sets of 2 are hyperplanes intersecting at one n - 3 plane.
Proof. (1) * (2): Denote u(xi) = ui and w(xi) = w,. It follows from Eq. (2) that
indifference sets are planes of the form
CPi’i
==/3;
I I
thatis, cpi[ui-pw,]=O
Let p # p’. The vector (pl, . . . , p,) solves the set of equations
CPi[‘i-Pwil
cp,[ui-p’wJ
=O
=o
if and only if it solves the set of equations
cpiL+ = 0
cpiwi = 0.
This solution is independent of p and p’.
(2) * (1): Suppose that all the hyperplanes containing the indifference curves
of 2 intersect at the (n - 3)-dimensional
plane K, spanned by the points
x’, . . .) xnp2, where xi = (xi,. . ., XL_,>, i = 1,. . . , n - 2. The hyperplane defined by these n - 2 points and by p* = ( pT , . . . , p,*_ 1> E A contains the indifference curve through p*. It is given by
n-2
Pn-I= CPjPj+@n-I
j=l
376
Z. Safra, U. Segal/Journal
of Mathematical Economics 24 (1995) 371-381
the vector p = ( /3,, . . . , p,_ 1>solves the set of linear equations
In particular,
n-2
xi
n-1
=
Cp,x;+p,_,
i=l,...,n-2
j=l
I
I
P,*-1=
n-2
c
PiPI” +&-I
j=l
Using Cramer’s rule we obtain that @_;, (say) is the ratio between two linear
functions of p:, . . . , p,*_ 1. The value of p,, _ 2 can be used as a representation
function of 2 , hence the lemma.
•I
Remark. Although it is probably well known, we could not find a proof (or a
formal statement) of this lemma in the literature.
Theorem 2. The complexity of the betweenness family of preference
order 3.
Proof. We prove the Theorem
relations is of
by induction. Let n > 3. Suppose that for every
and x =x1,. . . , xnr x1 < . . . <x,, the order 2
satisfies
the betweenness
hypothesis.
Let
A” =
on
A’-‘( y, q)(x)
A”( y, qh,,
. . . , x,+ t}. We want to show that the order 2 on A” satisfies
betweenness.
) E Int(A”).
Consider the two hyperplanes
Hi =
Let p* =(pT,...,pn*+l
p,
+
1):
p,
=
p:},
i
=
1,
2.
By
the
induction
hypothesis,
the
order
on
H,
I,.“,
{(P
and Hz satisfies betsveenness. Let Gi be the indifference set through p* in Hi,
i = 1, 2. Note that all points in G, are indifferent to all points in G,. By
betweenness, G, and G, are (n - 2)-dimensional
planes. By monotonicity, G, +
G,. Otherwise, it follows that for every ( pr , pz, p3, . . . , pn+ ,)- p* and for
every E>O, there is no S>O such that (p;“+~,
pz, ~~-a,...,
p,,+,)-p*.
Let [* >0 such that there exist (~7, p2, pz +(-l)‘f*,
p4,...,
pn+,)~G1
i
=
1,
2.
Let
H,,
be
the (n and (p,, p;, P: +(--1)‘5*,
p4,...,pn+,)~G2,
1)-dimensional
hyperplane
spanned
by G, and G,. Let (p,, p2, pg +
5, P4r.. .7 p,~+,)EH12suchthat1~lE(0,5*).LetH,={pEA:p,=p~+~}.
Again by the induction hypothesis, the order on H, satisfies betweenness. The sets
H, n G, and H3 n G, are (n - 3)-dimensional
planes. Denote the affine set
spanned by these two sets by H( J ). Its dimension is at least n - 2. Since the
order on H3 satisfies betweenness, the set H(c) is (a subset) of an indifference
set in H,, all its members being indifferent to p*. It also follows that its
dimension
is n - 2. By construction,
it is a subset of H,, n Hz. Since the
dimension of this last set is n - 2, it follows that H( 5) = H,, n H3. It thus
follows that in a neighborhood around p*, p* is indifferent to all points in HI2
such that their third co-ordinate is different from p:. By continuity and mono-
y=y,,...,y,,
4=41,..*,qm,
Z. Safia, U. Segal /Journal
of Mathematical Economics 24 (1995) 371-381
377
tonicity it follows that in a neighborhood of p*, the indifference set through p* is
planar.
Next we show that the indifference sets are hyperplanes. Let p E Int( A) with
the planar portion I of the indifference set through p around it. Let H be the
hyperplane
containing
I. If all points in H are indifferent
to p, then by
monotonicity
H contains all the indifference set through p. Suppose that there
exists p’EH such that p’*p.
Let (Y=max{cr* :Vcz~[0, a*], ap’+(l-a)p
-p}. The maximum exists by continuity. By the definitions of I and H, this
maximum is positive. Let p = Zp’ + (1 - Z)p. There exists a neighborhood of j5
where the indifference set through 3 is planar. Since there is E > 0 such that the
chord [(G - l
)p’ + (1 - Z + E)P, Cup’+ (1 - Cr)p] is in this indifference
set,
there exists E’ > 0 such that the chord [ (Yp’ + (1 - Z>p, ((;u + E’)P’ + (1 - (Y~‘)p] is in this indifference set, in contradiction with the definition of U. It thus
follows that all points in H are indifferent to p and the proof of the induction is
complete.
In particular we obtain that if X and Y belong to I>,, (and have the same
possible outcomes) such that X N Y, then for every cy E [O, 11, aX + (1 - cr)Y N
X. If X and Y do not belong to L, for any n, then it follows by continuity that
X N Y implies, for all CYE [0, 11, that crX + (1 - cr)Y N X.
0
Theorem 3. The complexity of the sets FE and .Fw is of order 3.
Proof. The proof is based on the geometric
preference relations.
properties
of the two families
of
Expected utility. Suppose that for every y, q, and x = (x,, x2, x,), the order 2
on A’(y, q)(x) satisfies the expected utility hypothesis. By Theorem 2, the order
satisfies betweenness globally. If it does not satisfy expected utility, then it follows
by continuity that there are three lotteries X, Y, 2 EL, such that X N Y but +X
+ $2 + $Y + $Z. (Otherwise, the preference relation 2 satisfies the independence axiom and can therefore be represented by an expected utility functional.)
Assume, without loss of generality, that X, Y, and Z have the same possible
outcomes. It thus follows that in the (n - l)-dimensional
simplex A{ x1,. . . , x,J
there are two non-parallel
indifference
hyperplanes,
say Cj’= 1 a/pi = /3’,
j = 1, 2 where for some i and k, CY~/CY~
# $/LX;.
Suppose, without loss of
generality,
that i = 1 and k = 2. Consider
the simplex
A2(x,, . . . , x,,
indifference sets,
p:,...,
P,T)IX,F x2, x3). This simplex contains two non-parallel
given by CT= 1 CY,‘p,= p’ - c:=,
a/p:,
j = 1, 2, a contradiction
with the
assumption of the theorem.
Weighted Utility. Suppose that for every y, q, and x = (x,, x2, x,), the order 2
on A2(y, q){ x ] can be represented by a weighted utility functional. As before, it
follows that the order satisfies betweenness globally. If it cannot be represented by
378
2. Safra U. Segal/ Journal
of Mathematical Economics 24 (1995) 371-381
a weighted utility functional, then it follows by continuity that there are x1,. . . , x,
such that three indifference hyperplanes in A”- ‘{x1,. . . , x$ do not have a
common intersection. (Otherwise, Chew’s (1983) axioms are satisfied for all finite
lotteries and by continuity, for all lotteries). Let Hj = {(p,, . . . , p,> : Za/pi = /3j},
j = 1, 2, 3 be three such indifference hyperplanes. Let p* = (pT,. . . , p,* > E H’
and consider the sets Ki -{(PI, pt ,..., pi*-1, pi, pF+l,... ,p,*-,)), i=2,...,n
- 1. We can assume, without loss of generality, that the three hyperplanes H’,
H2, and H3 are close enough to each other so that they are all intersected by
By the assumption of the Theorem, on each of these sets the order
&,...,K,_1.
can be represented by a proper weighted utility functional. Therefore, indifference
sets in Ki intersect at a point. Denote it by qi. Note that qi = (qI, p:, . . . , P;_~,
qi7 P?+19***, p,*_ 1>. Clearly, q2,. . . , q”-’ E H’ n Hz f7 H3 := H’23. Moreover,
since H’, H 2, and H3 are hyperplanes, their intersection is a plane. The
dimension of the hyperplanes is n - 2. Therefore, to prove the claim of the
Theorem, it is sufficient to show that H’23 is of dimension n - 3. This follows
readily from the fact that the n - 3 points q3 - q2,. . . , q”- ’ - q2 are linearly
independent. For this, note that for all j Z i, the i-th co-ordinate of q’ is p*. On
the other hand, since qi is on the continuation of an indifference set through p* in
Ki, qi #p;.
•I
4. Singletons
Suppose that the set of preferences 9 consists of one relation only, 2 *. We
show in this section that the complexity of the singleton F is of order 3. This
result is equivalent to the following: If two preference relations 2 1 and 2 2
agree on all the sets A’(y, q){x,, x2, x3), then they are the same.
Theorem 4. Let 9 = { 2 *}. Then the complexity of F is of order 3.
Assume that 2 1 is different from 2 2 even though they agree on all
two-dimensional (conditional) simplices. By continuity there exist two lotteries X
and Y such that X + lY, Y > 2X, both X and Y have the same n + 1 outcomes
(x1,..., x,-r such that xi <x,,+~ ) and both are interior points of the corresponding n-dimensional simplex. We show that this is in contradiction with the
assumption that 2 1 and 2 2 agree on two-dimensional (conditional) simplices.
We use an induction argument and, with no confusion, use the above notations for
the general induction step.
Hence, assume that 2 1 and 2 2 agree on all (n - l)-dimensional (conditional)
simplices (n 3 3) and, by way of negation, there exist two lotteries X and Y as
before. Identify X with the vector ( pl, p2, . . . , p, + 1> and Y with
1. Without loss of generality we may assume that pi b qj for
(417
q27...,qn+1
j=l,
2. Now consider the set of lotteries fi={(pl,y2,
r3,...,rn+I):rjER,
j
=3,...,
n + 11. Clearly, X and G (and Y and IV> belong to an (n - l)Proof.
Z. Safra, U. Segal/ Journal
of Mathematical Economics 24 (1995) 371-381
379
dimensional subsimplex on which, by assumption,
t I and t 2 are identical. By
monotonicity,
the lottery W, = (p,, q2, p3,. . . , p,, + 1 + (p2 - q2)) satisfies W,
2,Xandthelottery
W,=(p,,q,,q,,...,q,+,-(p,--q,))satisfiesYkLWW,.
Hence, for both i, W, 2 i X and Y 2 i W, which, since the simplex and W are
connected sets, imply the existence of a lottery W in L? such that W N 1X, i = 1,
2. However, W > ,Y and Y > ,W is in contradiction with the induction assumption.
0
5. Special cases
In this section we consider some special cases of the functionals we already
discussed. With the aid of these special cases we demonstrate the existence of
classes of preference relations with arbitrary orders of complexity.
Definition 2. The preference relation satisfies constant relative risk aversion if
(x1, p,;...;x,,
“((“Xl,
p,)
t(y1,
41;...;ym,
p1;...; Ax,, p,) t
(AY,,
4m)
q,;...;*y,,
4,)
for all A > 0.
Lemma 2. The complexity of the expected
relative risk aversion is 4.
utility preferences
satisfying
constant
Proof. It is well known that an expected utility preference relation satisfies
constant relative risk aversion if and only if its utility function is either of the form
U(X) = ux”l + b or a In x + b, a > 0. We now show that if 2 is expected utility
with the convex utility function U(X), then on all the sets A’(y, q){x,, x2, x3} it
agrees with an expected utility relation with constant relative risk aversion. To see
this, observe that if 2 is expected utility, then one indifference curve determines
the whole preference relation. Suppose that x1 <x2 < xj and let p* such that
u(x,) =p* u(x,> + (1 -p*)u(x,).
We are looking for a number (Y that solves the
equation x2* = p* xl* + (1 - p* )x;. That is,
4x3) - 4x2) =p* _
4 9) - 4x1)
xy-xz
XT-X;
(3)
Observe that for (Y= 1 the left-hand side is greater than the right-hand side, while
as (Y--) m, the right-hand side approaches 1 and is therefore larger than the
expression on the left-hand side of the last equation. Moreover, the right-hand side
of Eq. (3) is increasing in (Y. To see this, note that for & > (Y, the function x’ is
less concave then x”I. Taking an affine transformation of x6 such that U-X,?= x,?
380
Z. Safra, U. Segal/ Journal of Mathematical Economics 24 (1995) 371-381
for j = 1, 3 now implies that the right-hand side of Eq. (3) is larger with 15 than
with cr. By continuity, there exists a unique number CYthat solves Eq. (3).
Next we show that the complexity of this set of functionals is 4. Suppose that
the preference relation 2 agrees with expected utility with constant relative risk
aversion on all the sets A3(y, q){x,, . . . , x4}. By Theorem 3 it follows that t is
expected utility with a utility function u. Let x1 < . . . <x4 and let p* such that
u(xJ =p*u(x,) + (1 -p,*)u(x,),
i = 2, 3. Jet f, be the function
x”I for (Y> 0
and In x for (Y= 0. The monotonicity
of the right-hand side of Eq. (3) with
respect to (Y implies that there exist unique oj, i = 2, 3, such that
fcJx4) -f&J
p* = f&J -f&J
However, since on A3(y, q){x,, . . . , x4} the preference relation agrees with
constant relative risk aversion, it follows that CQ = a3 = CY*. By changing the
value of x3 we thus obtain that the utility function u coincides with the function
fa*. 0
Suppose now that a = 0 and b = 1 and consider the sets of utility functions
given by
2Yn=
iu(x)
=a,+x+
i=o,2
,a..,
1
. II.
1
1
[yy- 1, >+7
~UjX’x+
i=2
1.
It.
FZn
1
9
CC
1
Let S, be the set of the expected utility preferences with a utility function out of
Z&. Similarly to the proof of Lemma 2, it follows that the complexity of the set of
relations F,, is of order n + 2. Note, however, that fl:=,Zn
= {u(x) = e’}. By
Theorem 4, the complexity of the set of the expected utility preferences with the
utility functions u(x) = ex is of order 3.
The last example demonstrates a discontinuity in the definition of complexity.
This discontinuity should not be considered a flaw in our definition since, even in
other situations, there seems to be nothing that relates continuity with complexity.
For example, (simple) smooth functions
are always the limit of (complex)
everywhere non-differentiable
functions.
6. Conclusions
This paper offers a natural definition of complexity of preference relations over
lotteries. We used it to check the complexity of some betweenness functionals, but
the definition has nothing to do with linearity of indifference curves. It can be
Z. Safia, U. Segal/Journal
of Mathematical
Economics 24 (199.5) 371-381
381
applied to quadratic or to the rank-dependent
functionals.
We showed that
expected utility, weighted utility, and general betweenness preferences are relatively simple (according to our definition). On the other hand, adding further
restrictions, for example, constant relative risk aversion, may complicate a simple
preference relation.
Acknowledgments
We gratefully acknowledge the financial support of the Social Sciences and
Humanities Research Council of Canada and of the Israeli Institute for Business
Research.
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