We translate preferences over lotteries into preferences over “shifts in probabilities”. We define the independence axiom for such preferences. Next we construct and expected utility function to represent the preferences and present the corresponding additive geometry.expected utility theory, cardinality

David Cantalá

English

Research Papers in Economics

PREFERENCES FOR SHIFTS IN PROBABILITIES 
AND EXPECTED UTILITY THEORY* 
David Cantalá 
El Colegio de Mexico, A. C. 
Resumen: Las preferencias definidas sobre loterías inducen preferencias sobre 
cambios en probabilidades. Definimos el axioma de independencia 
para preferencias definidas sobre cambios en probabilidades. Al usar 
esta versión del axioma de independencia construimos una función de 
utilidad esperada. Finalmente, desarrollamos la geometría aditiva cor-
respondiente. 
Abstract: We translate preferences over lotteries into preferences over "shifts in 
probabilities". We define the independence axiom for such preferences. 
Next we construct and expected utility function to represent the pref-
erences and present the corresponding additive geometry. 
Clasificación JEL: D81 
Palabras clave: teoría de la utilidad esperada, cardinalidad, expected utility the-
ory, cardinality 
Fecha de recepción: 29 V 2006 Fecha de aceptación: 11 XII 2006 
* I wish to thank Simon Grant, Peter Hammond and John Weymark for their 
comments. 
Estudios Económicos, vol. 22, núm. 1, enero-junio 2007, páginas 99-109 100 ESTUDIOS ECONÓMICOS 
1. Motivation 
Consider the following situation. Initially, an agent faces an even 
lottery: with probability \ she wins 5 pesos and with probability \ 
she wins 10 pesos. Consider a shift in probability of \ from outcome 
"wining 5 pesos" to outcome "wining 6 pesos": 
Outcomes 
5 
6 
7 
8 
9 
10 
Lottery 
l 
2 
0 
0 
0 
Shift in probability 
l 
"3 
+ 3-
2
n
d
 Lottery 
0 
i 
2 
r 
0 
0 
Which shift in probability from the second lottery might lead 
to a third lottery indifferent to the first one. For some preferences, 
an answer might be a shift of \ from outcome "wining 10 pesos" to 
outcome "wining 8 pesos": 
Outcomes 
5 
6 
7 
8 
9 
10 
Lottery 
l 
6 
1 
3 
0 
0 
Shift in probability 
+ 4 
1 
"4 
2
n
d
 Lottery 
l 
6 
1 
3 
0 
0 
1 
0  0 
Thus, from the initial lottery, the first shift in probability is com-
pensated, thus comparable with the second shift in probability, al-
though 5, 6, 8 nor 10 are indifferent outcomes. Actually, even if the 
answer should be "none", the question makes sense. 
Any preferences over lotteries induces preferences over shifts in 
probabilities. We present a complementary approach to expected 
utility theory
1 by presenting a characterization of expected utility 
1 References on the subject are, for instance, Hammond (1998), Kreps (1988), 
and Wakker (1988). PREFERENCES FOR SHIFTS 101 
where the independence axiom deals with preferences defined over 
shifts in probabilities. 
The relevance of our approach is three-fold. First it makes it 
clear that the cardinal nature of expected utility is only a behavioral 
characteristic rooted in the independence axiom and has no normative 
appeal. Second, our proof is entirely constructive, which provides 
insight into the way the independence axiom leads to expected utility. 
Third, we deal strictly with preferences defined over simple lotteries. 
Moreover, applying Thales's Theorem, we relate the fact that 
indifference "curves" are straight lines with weighted additive geom-
etry in the Marshak-Machina triangle, thus providing a geometric 
interpretation of expected utility. 
2. Preferences Over Shifts in Probabilities 
Let C = {ao„} be the set of finite outcomes and L a lottery 
defined over C. Given a lottery L which assigns to event a a proba-
bility larger than a and to event b a probability smaller than 1 - a, 
a e [0,1], L(aA
b
a) is the lottery where the probability assigned to 
outcome a has been lowered by an amount of a and the probabil-
ity assigned to outcome b has increased by an amount of a. For 
instance, if C = {a,b,c,d,e), L = (.1; .4; .3;0; .2) and a = .1, then 
L(aA
b
a) = (0; .5; .3;0; .2). The concatenation of the operation is de-
noted by L(aA
b
a + ... + a'A*,). 
The space of preferences over simple lotteries is denoted by C, 
a preference relation in C is given by >. Given a lottery L, let pa 
(Pb) be the probability of a (b) in the lottery L, and DL = {(a, a, b) € 
[0,1]*C*<? | a < mm{pa,l-pb}}. Consider a preference relation > in 
C, we define the induced preference relation over shifts in probabilities 
>
A as follows: for any L and (a, a, b)mDL 
[aA*>
A/?Af for L] if and only if [L(aA
b
a)>L{l3A
d
c)]. 
If the binary relation > is complete and transitive, so is >
A. To 
see it, think of the contrapositive statement. 
These are heavy notations. They emphasize the richness of pref-
erences over lotteries and how restrictive the independence axiom is. 
It say that the ordering of shifts in probabilities is independent of its 
weight and the original lottery. 102 ESTUDIOS ECONÓMICOS 
DEFINITION 1. The preference relation >
A satisfies the independence 
axiom if, for all L, (a, a, b) and (/?, c, d) in VL 
aA
b
a>
ApA
d
c for L if and only if (7a)A* >
A(7/?)Af 
for all V and 7 > 0 where 7a and j/3 belong to DL and DL>. 
The equivalence holds with strict preferences or indifference. 
Consider the concatenation such that 
L(aA
b
a + f3Ai)~ L 
It tells us in which proportion shifts in probabilities from a to 
b and from c to d keep the agent indifferent. By independence this 
marginal rate of substitution, |, is constant (independent of the orig-
inal lntterv and of the size of the shift) Quoting- Macfiina (19871 
Hammond (19981 discusses how ratios of utility differences are equal 
to the marrinal rates of substitution between probability shifts which 
are constant when indifference curves are linear. Although it is some-
how indirect to caDture a DroDertv about Dreferences bv dealing with 
the utility representation it is an illuminating way to explain the 
cardinal nature of expected utility. Thus, we know that indifference 
curves are indeed straight lines In our focus in contrast the constant 
marginal rate of substitution between probability shifts is a restric¬
tion put on the preferences over shifts in probabilities. It is clearly 
a choice behavior pattern, thus, our approach agrees with Weymark 
(2005) in the sense that cardinality has no normative appeal. 
We also need a continuity condition to state our theorem. 
DEFINITION 2. The preference relation > is continuous if for any L, 
L' and final outcomes a and b 
{a E [0,1] : L{aA
b
a)>L'} and {a € [0,1] : L'>L{aA
b
a)} 
are closed. 
THEOREM 3. Continuous preferences > in C are represented by an 
expected utility function if and only if induced preferences >
A satisfy 
the independence axiom. Moreover, functions obtained from positive 
affine transformations of the utility function also represent these pref-
erences. PREFERENCES FOR SHIFTS 103 
The proof of the theorem is entirely constructive, which allows to 
us exhibit, both geometrically and analytically, how the independence 
axiom leads to expected utility.
2 
Figure 1 
Additive shifts in probability 
a3 
indifference line 
PROOF. The complete proof is in the appendix. There, we adapt usual 
arguments to prove that: 
1. There exists a worst lottery, L, wich is a degenerated lottery 
and assigns probability 1 to event, say, ai, 
2. There exists a preferred lottery, L, wich is degenerated and 
assigns probability 1 to event, say, an, 
3. For all lotteries L, there exists a lottery L(aLA%) indifferent 
to £, and the higher aL is, the greater the preference for L. 
Thus, assigning to all lotteries L in £ the utility uL = aL is 
always feasible and allows us to represent preferences. 
Then uai = 0, u0„ = 1 and for all degenerated lotteries L which 
assign probability 1 to a,, let uai = on where ~ L. 
2 We do not need to prove that the function which represents the preferences 
is linear and rely on the result which states that a utility function is linear if and 
only if it has an expected utility form. 104 ESTUDIOS ECONÓMICOS 
Moreover, since L(aiA^) ~ L, the shift in probability from 
L = (0,...,1,...,0) to L(aiA^) - (1 - on,0,a<) indicates that 
the constant rate of substitution between probability shifts which 
keeps the agent indifferent with respect to L, is the following: for 
any lottery, if the probability assigned to 0i (i / l,n) is lowered by 
Pi, to compensate the agent the probability assigned to an has to 
be increased by Pi * ai and the probability assigned to ax has to be 
increased by Pi * (1 - ai). 
Figure 1 displays the geometry of the analysis in the Marshak-
Machina triangle when n = 3, where L(a2A^) is denoted by L'. The 
indifference line of the left hand triangle exhibits the constant rate 
of substitution. By construction a2a = 1, and we choose c such that 
a2c = P2 (= Lh in the right hand triangle), L'f = a2 and L'b = l-a2. 
By Thales' Theorem: 
— = — (in triangle a2, a, L'), 
a2a a2V
 y 5 '
 ; 
^L = 1L (in triangle a2, L', f), 
a2L' L'f
 y 5 ' 
= — (in triangle a2, b, L'). 
a2L' bL<
 y & ' ' 
Thus eg = p2«2 and de = p2(l - a2). 
The right hand triangle exhibits the additive nature of the shifts 
in probabilities. It is the geometric interpretation of the expected 
utility form of the function which represents the preferences, since 
LL" =P3+ p2a2 = P3u3 + P2u2 + pmL
3 
We now see analytically how these constant shifts in probability 
is performed by the independence axiom. Moreover, it is performed 
additively for a2, an-i, which explains the expected utility form 
of the utility function. 
We consider non degenerated-lotteries L and compute aL. All 
lotteries L = (Pl, ...,pn) E C can be expressed as the concatenation 
L(p2A
a
a\ + ... +pnA£) = L. 
Since by the definition of a2, 1A^ ~ a2A% for L, independence 
implies 
3 We do not need to deal with induced preferences to develop the geometric 
interpretation. PREFERENCES FOR SHIFTS 105 
L(p2A£ + ... +p„A£) ~ L(p2a2A£ + p3A£ + - + P»A£). 
Repeating the argument for i = 3, n - 1 leads to 
£(P2«2A£ +p3A£...+p„A£) 
~ £(p2aaA£ + p3a3A£ + ... + p„A^) 
L = L(p2A£ + ... + p„A£) ~ L.((P2a2 + ... + P»)A£). 
Since an = 1, the probability distribution of L({P2a2 + ... + 
Pn)A£) is 
n n 
(l-g(Pia0,0,...l0>g(Pia0). 
Remember that wai = ax = 0 and wQ„ = an = 1. Hence the 
utility associated with L is Ml = a/, = £^=2^«*) = £"=i(Pi«i); 
namely 
n 
UL = y^(pi«z). 
¿=1 
This utility function has the expected utility form. Obviously all 
positive affme transformations also represent the preferences. B 
3. Concluding Remarks 
Now, the set of final outcomes C is a continuous interval in SR. A 
lottery L is a probability distribution over C. The space of compound 
lotteries is L
C. Suppose that L has the same probability distribution 
as TTILI + 7r2L2 + ... + TTILI, a compound lottery over simple lotteries 
Li, L2, ... and Lx in C°. Then, L(cvA£
2) is the lottery where the 
weight given to Li has been lowered by a < TT and the weight given 
to L2 has been increased by a. 
Given a lottery L, let Pl (p2) be the probability of Lx (L2) in the 
lottery L, and 
DL = {(a, Li,L2) e [0,1] * C * C \ a < min {Pl, 1 - p2}} . 106 ESTUDIOS ECONÓMICOS 
For any (a,LuL2) in DL, we define the induced preference relation 
>
A as follows 
[aA^>
A/?A^ for L] if and only if [L(aA^)>L(/?A^)]. 
The independence axiom is rephrased. 
DEFINITION 4. The preference relation >
A satisfies the independence 
axiom if for all L and (a,L1;L2) in DL 
aA£>
A/?A% for L if and only if (7a)A£>
A(7/J)A£ 
for all V and 7 where 7a and 7(0 are in DL and DL>. 
The independence axiom states that 
«A^>
AaA^ for L if and only if (7a)A^>
A(7a)A^ 
for all V and 7 well defined. When 7a = 1 and L = V, the reader has 
recognized the usual (strong) independence axiom defined on prefer-
ences over lotteries. Now, if these preferences are also continuous, 
they are represented by an expected utility function. Finally, when 
preferences are represented by an expected utility function, it is easy 
to check, as in the first part of the proof of our theorem, that the in-
dependence axiom for induced preferences (definition 4) holds. Thus, 
our approach is consistent with the approach of Von Neuman and 
Morgenstern (1944) when dealing with compound lotteries. 
References 
Hammond P. J. (1998). Objective Expected Utility: A Consequential Per-
spective, in S. Barbera, P. J. Hammond, and Ch. Seidl (Eds.)Handbook of 
Utility Theory, vol. 1, chap. 5, Kluwer Academic Publishers. 
Kreps D. M. (1988). Notes on the Theory of Choice, Boulder, Westview. 
Machina M. J. (1987). Choice Under Uncertainty: Problems Solved and Un-
solved, Journal of Economic Perspectives, I, 121-154. 
Von Neumann J. and O. Morgenstern (1944). Theory of Games and Economic 
Behavior, 3rd. edition, 1953, Princeton University Press. PREFERENCES FOR SHIFTS 107 
Wakker P. P. (1988). Additive Representation of Preferences, Kluwer Academic 
Publisher. 
Weymark J. A. (2005). Measurement Theory and the Foundation of Utilitarian-
ism, Social Choice and Welfare, vol. 25, no. 2-3, 527-555. 
Appendix 
PROOF. (=>) Suppose that all preferences > in C are represented by an 
expected utility function «(.). Thus, for all a< in C, there exists a fi-
nite real number u* such that u(L) = £"=1piWi for all L = (pi,...,p„). 
Without loss of generality, since all functions reached from affine 
transformation of the utility function also represent preferences, au 
an are positive real numbers. Thus, when 0 < a < p, < 1 and 
0 < 0 < Pj < 1, 
u(L(aA£»)) = pi«i + ... + (pi - a)ui + ... + (pm + a)um + .... + Pnun, 
and 
u{L(f3A^)) =Pi«i + ..- + (Pj -P)Uj + ... + (pk + f3)uk + .... +P„un. 
Suppose aA%>>*0A% for L, i.e., «(L(aA^™)) > u(L((3A^)), 
namely 
w(L(aA£"))-u(L(/?A°
fe)) >0« -aui+aum+pUj-i3uk > 0, 
<^ 7(-aM; + aum + - 0uk) > 0 forall, 
7 > 0 independently of the original lottery. 
Thus, whenever L'(7aA£") and Z/(7/3A^) are well defined, 
7aA£™ >
A
7/3A£
f
c
 for all V. Hence, induced preferences satisfy the 
independence axiom. Continuity is direct since expected utility is 
continuous in probabilities. 
(«=) STEP 1. One of the degenerated lotteries is the worst among all 
lotteries, another one is the best one. 108 ESTUDIOS ECONÓMICOS 
Since > is a preference relation over C, degenerated lotteries can 
be ordered from worst to best. Let L be the worst (or one of the 
worst) degenerated lottery which assigns probability 1 to, say, ax and 
L be the best (or one of the best) degenerated lottery which assigns 
probability 1 to, say, an. 
All degenerated lotteries L>L where ai is assigned probability 1 
can be reached as L = L(IA^); thus 
1A«>
A1A£ (1) 
All lotteries L = (pi, ...,pn) in C can be reached from a sequence 
of lotteries L1=L, L2=L1(p2Al'1), Ln=Ln^(pnA^) = L and by 
(1) and independence, at all iterations i, Li+i>Li; thus L >L by 
transitivity of the preferences. Hence L is the worst (or one of the 
worst) lottery in C. _ 
A similar argument establishes that L is the best (or one of the 
best) lottery in C. 
STEP 2. // 0A
a
a{ y
A 0A%i for some V and 0, then for all a, a' e 
[0,Pi], aAl{ y a'A
a
a{ if and only if a > a' for all L. 
Without loss of generality, let a > a', i.e., 
L{aA%) = L
1^ - a')A%) where L
1 = L{a'A%). 
Suppose 0A
aJ y
A 0A%. for some V and 0, by independence for 
any L we have 
L(aA%) = L\(a - a')A%) >~ L\{a - a')A%) = L(a'A
a
a{) 
i.e., aA% y
A a'A% 
With a < a', the argument proves the contrapositive statement. 
STEP 3. For all L eC there exists a E [0,1] such that I(aA^) ~ L. 
If L ~ L then a = 0 and if L ~ L, a = 1. 
We consider now that L y L y L. Suppose there is no such a 
for L. By Step 2, we know that the strictly larger a is, the strictly PREFERENCES FOR SHIFTS 109 
better i(oAJ) is. Moreover L = L(1A^) y L y L = L(0Af>), 
thus:
 1 
=» either there is one a' € [0,1] such that L(aA%») -< L for all 
0 < a < a' and L(aA»«) >- L for all 1 > a > a'; continuity would 
not hold in this case since, {a G [0,1] : L(aA
b
a)>L} =]<*', 1] is open* 
or there is one a' G [0,1] such thai I(aA°") -< L for all 0 < 
a < a' and i(aA^) >- L for all 1 > a > a
}; in this case con-
tinuity would not hold since, {a G [0,1] : L >L(aA*)} = [0, a'[ is 
open. H 

Additive Representation of Preferences,

Choice Under Uncertainty: Problems Solved and Unsolved,

Measurement Theory and the Foundation of Utilitarianism,

Notes on the Theory of Choice,

Objective Expected Utility: A Consequential Perspective,

Preferences for Shifts in Probabilities and Expected Utility Theory

http://codex.colmex.mx:8991/exlibris/aleph/a18_1/apache_media/JAJPLN386DF3DYXPKGXARL1N8YUU3Q.pdf

Preferences for Shifts in Probabilities and Expected Utility Theory

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics