fairness and expectation
TRANSCRIPT
Atti del Congresso Logica e filosofia della scienza: problemi e prospettive. Lucca, 7-10 gennaio 1993. EDIZIONI ETS, PISA (lTAL Y) © 1994
FAIRNESS AND EXPECTATION
Alberto Murat
Depanment olPhilosophy University 01 Pisa, Italy
1. Introduction
The theory of personal probability as a fair betting quotient goes back to Ramsey
[1926/90] and de Finetti [1931]. Both claimed that this theory presupposes the mIe of
maximization of expected gain (supposedly linear in utility). Ramsey [p. 79] writes that the
theory «is based throughout on the idea of mathematical expectation.» De Finetti uses almost
the same words [p. 304] when he writes that the theory «is in substance based on the notion
of mathematical expectation.» This claim has never, as far as I know, been questioned, so
that it is quite genernlly taken for granted.
On the other hand, there are go od reasons for considering too strong the axioms ofprefer
ence that lead to the linear utility functions. In particular, Allais [1953, 1987/90] convinc
ingly asserts that a violation of Savage's sure-thing principle [1954/72] is implied by very
reasonable preferences. If Ramsey's and de Finetti's thesis were correct, any weakening of
the sure-thing principle would imply loss of the Dutch Book argument along with loss of
any characterisation of personal probability as a fair betting quotient 1.
t I would like to thank Clark Glymour, Richard C. Jeffrey, Brian Skynns, Patrick Suppes and espe
cia!ly Teddy Seidenfeld for helpful discussions on the topic investigated in this paper. I arn a!so in
debted to Ken Gemes and Elena Russo for their stylistic suggestions. Responsibility for errors and
opinions rests however entirely with me.
1 In recent years severa! theories have been proposed that imply weakening of Savage's axiom sy
stem while retaining additive personal probabilites as derivable from preferences. Unfortunately,
none of these theories seems 10 be able to save the characterization of personal probability as a fair
456 A.MURA
In this paper I cOOllenge de Finetti's and Rarnsey's thesis. I propose a new nonlinear the
ory such tOOt the Dutch Book argument can be formulated within il. This nonlinear theory
provides a new formula for attaching numerical desirability values to gambles, which turns
out to coincide with mathematical expectation in special cases.
I confine my analysis to gambles, as in the originaI approach of Rarnsey and de Finetti. In
fact, the consideration of acts OOving an infinite number of possibie outcomes, aIthough im
portant under other respects, would involve a lot of technicalities, without adding anything
essential to the topic under consideration.
2. Two Kinds of Fairness
Let ;r. be a finite Boolean algebra of events with n atoms Eh... ,En (whose order is fixed
once for ali in advance). Suppose we OOve a currency !inear in utility. A gamble can be con
sidered a situation in which, for every atom Ei a person X receives or pays a certain amount
of money Xi, if Ei obtains. From a mathematical point of view, a gamble may be represented
by a vector (Xh"',x,,) in Rn
• A \inear utility function can be defined as a Iinear mapping from
f:Rn ~ R satisfying the so called weak dominance principi e, namely:
l. given two vectors (Xl> ••• ,x,,) and (Yh ... ..Yn), if for every i (l S; i S; n) Xi S; Yi then f(Xl, ... ,xn) S; f(Yb ... ..Yn).
Given a constant vector (x, ... ,x), the value f(x, ... ,x) is equal to kx + c for some real posi
tive constant k and some real constant c. Since the choice of values for k arid c is arbitrary,
we can suppose without Ioss of generality k l and c O, so that the following relation
holds:
2. f(x, ... ,x) = x.
A bet on an event E given an event H is a special kind of gamble where gain equals a
constant a (O S; a) for ali atoms that imply E (ì H, where gain equals a constant b S; O(b < a)
for ali atoms entailing --, E (ì H and where gain is Ofor every atom that entails ......JI. The ratio
-bla is called the odds ratio ofthe bet (provided a> O), while the ratio -bICa-b) is called the
betting quotient. See, for example, Gilboa (1985), Fishburn and La Valle (1987), Machina and
Schmeidler (1990). For an excellent introduction to nonlinear decision theories see Fishburn (1988).
457 FAIRNESS ANO EXPECTATION
betting quotient of the bet. If H is the necessary event, the bet E given H is said to be an ab
so Iute bet on E.
The key idea in betting theory is the notion ofjairness. There are two distinct concepts of
fairness that we need to distinguish. In a fmt sense, you can consider as fair any bet on an
event E whenever you find yourself indifferent between accepting that bet and abstaining
from betting. In a second sense, you can consider as fair a bet on an event E at odds ratio O if
you are indifferent between accepting that bet and betting against E with the odds ratio 110.
In technical terms, a bet vector x is said to be fair in the first sense when f(x) = O, and it is
said to be fair in the second sense when f(x) f(- x).
There is no reason to distinguish these two different kinds of fairness as far as it is pre
supposed that the values of gambles are determined by mathematical expectation, since in
that case the two definitions turn out to be equivalent. So the mie of maximizing expected
utility (cali it RMEU) implies that you must be indifferent between betting and abstaining
from betting whenever you do not consider betting on E at certain odds preferable to betting
against E at the same odds.
There are at least two reasons for rejecting this consequence ofRMEU. The first reason is
that you hardly ever have so definite a degree of belief that it may be expressed by a single
real number. In most situations an interval better represents your credal state. In such a case
the minimum odds on which you are prepared to bet on an event E do not coincide with the
maximum odds on which you are prepared to bet against E. As a result, there are odds on
which you are neither prepared to bet on E nor prepared to bet against E. In such a case, ab
staining from betting would be for you preferable to both betting on E and to betting agrunst
E. A generalization ofbet theory along these lines was outlined by Smith [196]] and carried
out in detail by Levi [1974, 1980, p. 151ff., 1986, p. 122ff.] and Seidenfeld, Schervish, and
Kadane [1990]. In this generalized theory, a credal state is represented by a c1ass ofprobabil
ity functions, each function in the class satisfYing the usuallaws ofprobability.
The second reason for rejecting the received identification of the two ideas of fairness is
that it mles out any preference between two betting situations when both are fair in the sec
ond sense. It may be that you prefer to bet at odds 9: l on an event E to which you assign
probability 0.9 than to bet at even odds on an event E' to which you assign probability Y2.
This may happen even if you have no doubts about the probability values of E and E'. The
458 A.MURA
two situations differ in the degree of uncertainty about the truth-value of E and E'. I see no
reason to condemn as irrational a preference depending on this factor (at least in one-stage
situations). Indeed, preference for a less uncertain bet is a fonn of risk aversion that cannot
be explained in tenns of marginaI utility of money, since we are supposing that the payoffs
are linear in utility. Accommodating this kind of preference, in the presence of welI defined
subjective probabilities and utilities, requires some weak:ening of Savage's sure-thing prin
ciple [1954/72, pp. 21-26]. The theory that I introduce in this paper is an attempt in that di
rection.
3. Fairness and Linearity
Although both Ramsey and de Finetti independently claimed that the theory of fair bets
(calI this theory TFB) is based on mathematical expectation, no strict proof of this claim has
been given by them. What they actualIy showed is that if we presuppose that the bettor's
system ofpreferences is in accordance with RMEU then, under this assumption, TFB may be
developed. This can be summarized by saying that RMEU is a sufficient condition for the
characterisation of subjective probability as a betting quotient. We can presume that they be
lieved that it is also a necessary one, meaning that this characterisation is impossible when
ever RMEU is violated.
It has to be recognised that the premises of the celebrated Dutch Book argument, stated
by both authors and proved by de Finetti, strongly suggest the inseparability of TFB and
RMEU. Let us see why. Both Ramsey and de Finetti presented their Dutch Book argument
in a fonn that is based on the folIowing premises:
(1) Any linear combination of fair gambles is in turn a fair gamble;
(2) For any event E there exists one and only one betting quotient such that any bet on E
with that betting quotient is a fair gamble;
(3) A gamble whose gains are negative in every event is not fair.
We can consider conditions (1 )-(3) as an axiomatization of the idea of faimess. What de
Finetti actually proved is that for every set S of bets on events E1, ...,E whose betting quon
tients do not satisfy probability laws, there exists a linear combination of bets in S whose to
tal balance is negative in every event. By conditions (1) and (3) at least one of the elements
459 FAIRNESS AND EXPEcrATION
in S is not fair. On the other band, by condition (2), a set S' of fair bets on El'....En exists.
Since, by conditions (l) and (3), no linear combination of elements of S' implies a total bal
ance negative in every case, the betting quotients of the bets in S satisfY the axioms of finite
probability. Hence for every El' ... .E a single set of fair betting quotients exists and it satisn
fies the axioms of finite probability. But condition (1) also implies that the set of fair gam
bles is c10sed under linear combination and indeed constitutes a \inear vector space. Now, it
is very natural to justifY condition (I) by saying that the set offair gambles must be a linear
vector space because it is the kemel of a \inear mapping f:R" ~ R. In that case the first
meaning of faimess seems to be primary and the second merely a derived one. This suggests
that the theory of fair bets can hardly be maintained if we tak:e a non\inear mapping
f: R" ~ R as representing the values ofgambles.
In spite ofthe reasonableness ofthese considemtions, I shall show that this conclusion is
incorrect. To begin with, I observe that an alternative version of the Dutch Book argument,
using only the second notion of faimess, can be provided. To avoid confusion, I shall reserve
the word 'fair' for gambles that are fair in the sense that an indifference exists between each
of them and the status quo, while I shall cali 'balanced' those gambles that are fair in the
other sense. Clearly, ali conditions (1)-(3) remain tme if we substitute the term 'fair' with
the term 'balanced', so that we have the following axioms for the idea ofbalanced gamble:
(I ') Any Iinear combination ofbalanced gambles is in turn a balanced gamble;
(2') For any event E there exists one and only one betting quotient such that any bet
on E with that betting quotient is a balanced gamble;
(3') A gamble whose gains are negative in every event is not balanced.
Since axioms (I ')-(3') are parallel to axioms (\)-(3), a pamllel Dutch Book argument
can be carried out using the idea of balanced gambI e, thus without presupposing that ali bal
anced gambles are ranked with the status quo. In this new form, the argument proves that for
every E1, ... ,En a single set of balanced betting quotients exists, one that satisfies the axioms
offinite probability.
The preceding analysis opens the door to a new way to merge the Iinear vector space of
balanced gambles in a nonlinear structure. Sepamting the idea of balanced gamble from the
idea of faimess, we have no need to require that the vector space of balanced gambI es would
be the kemel of a mapping f:R" ~ R. It is sufficient that this vector space is kept as the set
460 A.MURA
of gambles whose value does not change when the payoffs are multiplied by -1. In such a
case there would be no basis for the alleged dependence of subjective probabiIity on the idea
of mathematical expectation. In what foIlows I shall prove that this is the case.
4. Nonlinear Expectation
In order to introduce a new parameter for risk attitude, I shall consider two extreme cases:
maximum risk aversion and maximum risk attraction, so that the generai case can be
achieved as a !inear combination of these two extreme cases; Let f:Rn ~ R a !inear utiIity
function. Consider the following transformation of f:
f.(y) f«min(f(y),yl)" .. ,min(f(y),y,,)
Clearlyadopting formula (A) in pIace of expectation to calculate desirability values presup
poses a strong risk aversion. My proposal is to take (A) as representing the utiIity function
with maximum risk aversion. Formula (A) has an optimistic counterpart in the foIIowing
formula:
(B) r(y) f«max(f(y),yl),···,max(f(y),y,,)
So (B) may be taken to represent the utiIity function of a decision-maker with maximum
risk attraction. It is possible to reach a continuum of degrees of risk attitudes that goes from
the maximum risk aversion function f. to the maximum risk attraction function r consider
ing every convex Iinear combination off. and r in the folIowing way:
U/I(x) =/3f.(x)+ {(x)(C) /3+1
2 Rere I aro supposing that for every constant vector x=(x, ...,x), f(x)=x holds. This is a simpIifying as
sumption that implies no loss of generality. But relaxing this assumption requires a generalization of
formula (A), namely:
(AG) f.(y) = f(min(f(y),f(Yl'···'Yl»,···,min(f(y),f(y", ... ,y,,)))
as well as (in a similar manner) of subsequent formulas (B),(C),(A'),(B'), and (C').
-
w)
FAIRNESS AND EXPECTATION 461
lt is quite intuitive that when 13 I, (C) yields again the mathematical expectation. This is
proved by the following result:
Proposition 1.
'<:Ix (f.(x) + r(x) = 2f(x»
Proof.
By definition:
(j) f.(x) = f(min(xl,f(x», ... ,min(xn,f(x»)
and
By substitution we bave:
f.(x) + r{x) = f(min(x.,f(x», ... ,min{xn,f(x») + f(max{xl,f(x», ... ,max{xn,f(x)))
Since f is an homomorfism:
f(min(x.,f(x», ... ,min(xn,f(x») + f(max(xl>f(x», ... ,max(xn,f(x))) =
f«min(xl,f(x», ... ,min(xn,f(x))) + (max(x.,f(x», ... ,max(xn,f(x»)))
For every i (I S; i ~ n) it holds:
ifmin(xj,f(x» Xi then max(x;,f(x» = f(x)
and
ifmin(xj,f(x» = f(x) then max(xi,f(x)) = Xi'
Therefore:
f«min(xt,f(x», ... ,min(xn,f(x))) + (max(xd(x»•...• max(xn,f(x»»
462 A.MURA
f(x+(f(x), ... ,f(x» f(x) + f(x) = 2f(x) •
By Proposition l, u](x) f(x), so tbat (C) can be considered as a genuine generalization
of expectation. When ~ > 1 we bave risk aversion and when ~ < l risk attraction. So, as re
quired, my theory is able to introduce only one parameter, representing the generai attitude
towards risk.
To show that this theory satisfies other requisites as well, like stochastic dominance, it is
necessary to generalize our approach, so that different probabilities distributions may be
compared. This is reached if we define a gamble as a pair of vectors having the same dimen
sion. Simply, for any natura! n (n > O), given a vector x (x], ... ,xn)' and a vector
p = (p], ... ,Pn) ofprobability values such tbat Lp; = l and O:::; Pi for every i (let #Rn the sel of
such vectors), I consider as a gamble the pair (x,p). I cali the number f(x,p) = LXiP; the lin
ear utility of (x,p). The ~-continuum of utility nonlinear function may be defined as foIlows:
f3f.(x,p)+ {(x,p)(C') ( )uf3 x,p = 13+1
where f. and r are respectively defined as follows:
(A') f.(Y,p) f«min(f(y,p)J']), ... ,min(f(Y,P)J'n)
(B') r(y,p) f«max(f(y,p)J']), ... ,max(f(Y,P)J'n)
The following result proves that the class of functions u~ fulfils aH requirement to char
acterize probability as a fair betting quotient and to assure one stage coherence in the sense
ofDI.:
Theorem 1.
Every u~fuTlction satisfies the following conditions:
1. ifx belongs to Rn
and x = (x, ... ,x), then for every p belonging to #Rn, u~(x,p) = x;
2. for every n (1:::; TI < ro), every x (xl, ... ,xn)' Y (r1, ... J'n)' and p (p belonging to
#Rn), iffor every i (l:::; i:::; n) Xi :::; y;, then u~(x,p):::; u~(y,p)(weak dominance);
-
463 FAIRNESS AND EXPECTAnON
3. for every n (l ~ n < co), every x = (xl, .. ·,xn)' q = (qj, ... ,qn)' and p = (Pt, ... ,Pn) (p and
q belonging to #R8), if for every j (l ~ j < n) Pl+",+Pj ~ qt+...+qj and xj+l ~Xj' then
up(x,p) ~ up(x,q) (first order stochastic dominance);
4. (linearity in utility)
(a) for every rea! À > O, ÀUp(x.p) = Up(Àx,p)
(b) for every constant vector y "" (y, ... ,y), ulI(x+y,P) up(x,p) + Y
5. up{x,p) = ulI(-x,P) iff f(x,p) O
Proof.
L ifx1= ...= xn' up(x,p) f(x,p), so that condition l is immediate!y satisfied .•
2. ifx = (xt,. .. ,xn) and y (Yl, ... ,yn) are such that for every i (l~ i ~ n) Xi ~ Yi' then:
f(x,p) ~ f(y,p)·
Therefore, for every i (l ~ i ~ n) :
min(xi,f(x,p» ~ min(yj,f(y,p»
and
It follows, by the fact that f satisfies the dominance condition, that
f.(x,p) ~ f.(y,p)
and
!*(x,p) ~ !*(y,p)
from which soon it follows:
up(x,p) ~ ulI(y,p).•
3. From the fact that f satisfies first order stochastic dominance, it follows that:
f(x,p) ~ f(x,q).
Now, for somej,r (l ~j ~ r < n):
464 A.MURA
f.(x,p) = f«f(x,p), ... ,f(x,p),xj, ... ,xn),p)
and
f.(x,q) = f«(f(x,q), ... .f(x'q),xr' ... 'xn),q)
Since j:::; rand f(x,q) :::;xj +s for some s (O:::; s:::; n- j+ 1)
f«f(x,p), ... .f(x,p),xj' ... ,xn).q) :::; f.(x,q)
Now,
f.(x,p) = f«(f(x,p), ... ,f(x,p),xj,· .. ,xn)'P) :::; f«f(x,p), ... ,f(x,p),xj, ... ,xn),q)
It follows (by transitivity):
f.(x,p):::; f.(x,q)
By analogous (omitted) reasoning it can be proved:
r(x,p) :::; r(x,q),
so that:
u~(x,p) :::; u~(x,q).•
4. (a) Since À > O:
f.(Àx,p)
Analogously (by substituting min with max):
À!*(x,p) !*(Àx,p)
and therefore:
ÀU~(x,p) u~(Àx,p).•
~---------------------------------------------------------------------
465 FAIRNESS ANO EXPECTA110N
(b) Since f(x+y) = f(x) + y,
f .(x+y) f(min((x;+y),(f(x)+y», ... ,min«xn+y),(f(x)+y»
f(min(x;,f(x» + y, ... ,min(xn,f(x» + y)
f(min(xj,f(x», ... ,min(xn,f(x» + y)
f(min(xj,f(x», ... ,min(xn,f(x») + f(y)
f.(x) + Y
Analogously:
r(x + y) r(x) + y.
It follows immediately:
5. Suppose f(x,p) O. Then:
f.(x,p) = LXiPi x/so
and
f.(-x,p) LXiPi' Xi> o
It holds:
f(x,p) = LXiPi LXjPi + LXjPi f.(x,p) + f.(-x,p) O. xiS'O xj>O
Hence:
f.(x,p) = f.(-x,p).
Analogously:
r(x,p) r(-x,p).
466 A.MURA
It follows:
Uf3(x,p) = uf3(-x,p) •
Suppose now that f(x,p) * O. In that case: ~>iPi * - LXiPi so that: x, o: o x/>o
(1) f.(x,p) * f.(-x,p).
Analogously:
(2) j*(x,p) * j*(-x,p)
Furthennore, since
LXiP; - LX;P; =LX,P, - L -xip; Xl~O xj>o xi>o X(so
it holds:
(3) f.(x,p) - f.(-x,p) = j*(x,p) f*(-x,p)
Reasoning ad absurdum, suppose IIp(x,p) = u~(-x,p), so that uf3(x,p) - u~(-x,p) O. It follows:
(Pf.(x,p) + !*(x,p» - (Pf.(-x,p) + f+(-x,p» = O.
Therefore:
(4) P(f.(x,p) - f.(-x,p» + (f(x,p) - !*C-x,p» o.
Now ifp 00, (4) contradicts (I); ifp 0(4) contradicts (2) and ifO < P < 00, (4) is
inconsistent with (1), (2), and (3) taken together. Thus we conclude:
u~(x,p) = u~(-x,p) •
..,...----------------------------------
FAIRNESS AND EXPECT A TION 467
5. From Preference to Utility and Probability
The Ramsey-von Neumann-Savage approach to utility is based on a set ofaxioms about
preferences and on a representation theorem that shows that whenever those axioms are sat
isfied by a decision-maker X, a utility function exists unique up a positive linear affine
transformation that preserves X's preferences. The problem arises whether a similar theorem
can be proved with respect to our theory. From our point of view it is important to derive
both probabìlìties and utìlìties from preferences, as it is in the theory ofRamsey, Savage and
Jeffrey. Nevertheless, as a first step I shall try to use the simpler approach of von Neumann
that presupposes external probabìlìties.
In the von Neumann approach, a procedure exists allowing the attachment of numerical
utiIity values to outcomes. Von Neumann's basic idea is that, given a finite set of outcomes
(xl, ... ,x,,), ordered according to the preference relation, we can attach aibitrary values to XI
and x" and determine the other values, finding for every Xi (l < i < n) that probability value p
such that x is indifferent between the gamble 'Xl with probability p, x" with probability l-p'
and Xi for certainty. The existence of this probability value is assured by a specific axiom.
Then the attached value to Xi is given by the formula of mathematical expectation
xlP +x,,(l-p). Clearly this procedure is entirely based on the mathematical expectation. So
we cannot use it in our theory to determine utìlìty values of outcomes, simply because
xlP + x,,(l-p) does not fit in geneml our formula (C'). Using (C') instead of mathematical
expectation would lead to a formula containing the parameter 13 and it would not be working
unti! we had an independent way to determine the value of 13. There is no simple way to do
that. So we have to invent an alternative way, not dependent on a 13 value.
Since (C') is a genemlization ofthe formula ofmathematical expectation, the method we
are looking for should be an alternative method valid also with respect to the linear theory.
My proposal consists in tak:ing the idea of a balanced gamble as a basis, since the linear
structure of balanced gambles is perfectly preserved in our theory. So the first problem we
have to solve is to define the concept of a balanced gamble in terms ofpreferences.
Given a certain outcome 0, the idea of a balanced gamble requires the existence of an in
verse outcome, that is, it requires the idea of an outcome -O that has the same distance in
utility from the point O as 0, but is preferred to the point O if and only if the point O is pre
ferred to O. Clearly this idea requires, in turn, the idea ofa point O with an intrinsic meaning.
468 A.MURA
I bave said before tbat a natural way to give this meaning is to identifY the point O with the
utility ofthe status quo. Unfortunately. there is no idea of status quo to borrow from the von
Neumann's or Savage's theory. So we bave to find a new approach, one tbat consistently
allows definition of these concepts. I shall not try here to find a rigorous axiornatic founda
tion of my theory, leaving this technical task to the future. I sball try, instead, to show in an
informaI way wbat to do in order to pass from preferences to utilities according to the generai
formula (C') instead oftbat ofthe mathematical expectation.
My basic idea is to define preferences among subject state transitions in the following
sense. I suppose that there is certain finite set A {SI"",Sn} of n states and that in a certain
instant the decision maker X can be in one (and only one) of them. I suppose that X may
move from any state Si to any other state S;. This transition is represented here simply with
the ordered pair (Si'S;), I assume that when X is not indifferent between an event and its ne
gation, the state of X changes when the tmth-value of this event becomes known to X. We
can assume that the ultimate reason for any preference lies in a causai connection between
the objects of preferences and subjective state transitions.
Suppose now that X is put in the following situation: X wìll move from the present state
to any one ofthe possible states, and X attributes to every possible state the same probability
of being selected. It is well known that uniform personal probabìlìty distributions can be de
fined without reference to a numerical scale, simply in terms of indifference attitudes, under
certain existential assumptions. This is tme in the framework of the !inear theory, but re
mains tme also in the present theory. Suppose in fact that we have Boolean algebra B with n
ethically neutral atoms3 whose truth value is unknown. Suppose further that to every state Si
is associated an atom ai' such that X will transit to the state Si iff ai obtains. Therefore we
bave a one '"Co one mapping between the set of states and the set of atoms of B. Suppose that
X is completely indifferent to the possible mappings to be used. This defines the idea of
equiprobability of the atoms in terms of preference dispositions. Suppose also that before X
carne to be informed about the atom that actually obtains, she is offered the opportunity to
acquire the right to transit from one state to another state. Clearly X will be able to use the
3 In the present theory we can define ethica1 neutrality in the following way: an event E is said to be
ethically neutral for X iff for every state S ifX is in S the sole knowledge by X of the truth value of E
leaves X in the state S.
469 FAIRNESS AND EXPECT A nON
right to tmnsit from state Sto state T only if the state S obtains. In these circumstances we
can conceive that X has a definite attitude of preference between any pair of tmnsitions.
Actually these tmnsitions are conditional upon an uncertain event (namely the initial state of
.K). But since this event has the same probabiIity for every tmnsition, that fact does not dis
tort the preferences between tmnsitions in themselves, due to the principle of wea.k domi
nance. We can suppose tbat the preferences ofX concerning these transitions define a weak
order relation on the set LhA. This preference relation defines also a relation of preference
among states, since we can say that the state S is not preferred to the state T iff the tmnsition
(S,1) is not preferred to the tmnsition (T,S).
A distinguishing feature of the present approach is tbat it allows the definition of the no
tion of status quo. In fact, the status quo can be represented as the transition (S,S) from a
state S to itself. l suppose that for every S and T, X is indifferent between (S,S) and (T,1).
This is completely natural in the circumstances I bave previously defined. As usual, we can
consider as values the equivalence c1asses with respect to the relation of indifference, so that
the status quo can be considered as the equivalence c1ass of any tmnsition (S,S).
Coming to the notion ofbalanced gamble, we have to define, given a transition (SI,S2) the
idea of the inverse transition -{SI,S2) of (SI,S2)' l simply set -{SI,S2) (S2,SI)' I sball as
sume that if X does not prefer (Sj'S2) to (TI,T]), X also does not prefer (T],T1) to (S2,Sj)'
From this assumption it follows immediately that if X is indifferent between (SI,S2) and
(TI,T]), X is also indifferent between (S2,SI) and (T2,Tj). This consequence allows the defi
nition of inverse value in terms the idea of inverse transition. For, given a value a, the in
verse of a, say -a is the equivalence c1ass to which ali the inverses of the elements of a be
long.
Once we bave defined the idea of inverse transition, we are able to define the idea of a
balanced gamble as well. Suppose that in the situation defined before bets will be offered to
X involving rights to tmnsitions, so that, for example, given an ethically neutral event A, the
(right to) transition (SpTj) is offered ifA happens and the (right to) transition (S2,T2) is of
fered if -.A happens. l sball say that this gamble is balanced iff there is an indifference be
tween it and the gamble (TI,Sj) ifA, (T2,S2) if -,A. I sball assume tbat for every equivalence
class Wand every ethically neutral event E there exists one and only one equivalence class V
470 A.MURA
whose e!ements are such that the bet consisting of the prospect of an element of W if E and
an e!ement of V if ......E is balanced.
Now we are able to assign to every transition a definite numerical va!ue provided we bave
events with any possible probability, as in the von Neumann approach. Suppose, in fact, tbat
we bave a set oftransitions (ShTI) ... (Sk,TIr) in order of weak preference, such tbat (Sk,TIr) is
strictly preferred to (SI,TI). The procedure to attach a numerical value to every transition
(Si,Ti) (l < i < k) consists ofthe following steps:
81. Determine for every transition 'Twhether it is strictly preferred, indifferent or strict!y
not preferred to the status quo '10. S2. Give arbitrary value 1 to a transition s" strict!y preferred to the status quo and the
value -I to its inverse -s".
S3. For every transition 'T strictly preferred to the status quo, find tbat probability P'T at
which the prospect 'Twith probability P'1' -s" with probability I-P'1' is ba!anced.
84. For every transition 'Tstrictly not preferred to the status quo, find that probability P'Tat
which the prospect 'Twith probability P'1' and s" with probability l-p'1' is baianced.
85. Assign:
(a) to every transition ranked with the status quo the value O';
(b) to every transition 'T strictly preferred to the status quo the numerica! value:
(c) to every transition 'Tstrictly not preferred to the status quo the numerical value:
The problem to determine the value of the pararneter pthat appears in our theory arises
now. This is a very easy task once we bave reached the numerical scale of utilities. Suppose
4 This is actua1ly a simplifying convention that implies no loss of generality. You can assign to the
status quo areai value k whatsoever, but in such a case the inverse of a ìransition of value z, would
receive the value 2k-z.
....
471FAIRNESS AND EXPECf A TION
in fact that the preferences of X fit the formula (C'). Then p may be determined simply by
determining the utility value V of the bet: "I at probability Y:z, -I at probability W'. Since V,
according to (C'), is confined in the c10sed interval [:""Y:, Y:z] we OOve:
1-2V if V>-Y:zf3 =2V+I
In the case V = -Y:z we put ~ = 00. This formula is easily derived from (C').
So far I OOve shown that my generalized theory allows determination of utility from pref
erences, provided we have external probabilities at hand. The next step consists of finding a
procedure to derive probabilities as well as utilities from preferences. In Savage's theory, the
inference from preferences to probabilities is ultimately based on the fact the relation of
comparative probability (A is more probable than B) can be defined in the preference lan
guage. We can borrow this idea from Savage's theory. Clearly, within our approach, it is per
fectly possible to characterize the idea of de Finetti-Savage qualitative (also called compara
tive) probability. The relation 'A is not less probable than B', denoted with A ;;:: B can indeed
be defmed in the following way: given a transition 'Tsuch tOOt the status quo '10 is not pre
ferred to it and two events E and E', we say that E is not less probable than E' iffthe gamble
"'Tif E', '10 if -,E'" is not preferred to the gamble "'T if E, '10 if -,E". From tbis definition
we can obtain, in the usual way, the derived relations of 'A is not more probable than B' (A :s;
B), 'A is strictly more probable than B' (A > B) and 'A is strictly less probable than B' (A <
B). Moreover, we can define the ideas of probability O in the following way. An event E is
said to be ofprobability O ifffor every transition 'Tstrictly preferred to the status quo '10, the
gamble 'TifE '10 if..,E is ranked together wÌth '10. The relation 'not less prohable' so defined, satisfies the first three of the folIowing four
axioms of comparative probability proposed by de Finetti [1931, p. 321], namely:
l. Given two events E', E", eitherE' ;;:: EH or E" <::: E' (comparability).
2. If T is the necessary event, and E a factual event not of probability O, holds:
T <::: E >..,T . 5 (nontriviality).
5 In the originai paper of de Finetti [1931 p. 321] the cJause that E is not an event of probability O is
not present and instead of the inequality T ~ E > ...,T the inequality T> E > ...,T appears. There is su
rely a misprint in the de Finetti paper, because de Finetti's fonnulation implies that every factual
472 A. MURA
3. IfE' <! E, and E <! EH, then E' <! EH (transitivity).
4. IfEl and E2 and E are such that El and E2 both are logically incompatible with E, then
El <!E2 ,iffE,UE <! E2 UE (additivity).
Proo[ ~
l. Immediate trom the fact that the weak preference relation satisfies the comparability
condition.•
2. Let 'Tbe a transition strictly preferred to the status quo '1Q. By definition, the gamble
"'Tif T, '1Q if ....,1"', is ranked together with "'TifE, 'Tif ....,E" and the gamble "'Tif
....,T, '1Q if1'" is ranked with "'1Q if E, '1Q if ....,E". Consider the gamble "'Tif E, '1Q if
...., EH. By weak dominance, it is strictly preferred to '1Q and it is not preferred to 'T. It
follows: T <! E > ....,T.•
3. Immediately deduced trom the transitivity ofthe weak preference relation .•
It seems that the proof of 4 cannot be carried out without new principles. In Savage's
book the proof is omitted. A proof using the sure-thing principle (which is not valid in our
theory) in an essential manner turns out to be indeed very easy. In any case, 4 is clearly sat
isfied in our theory based on (C'), because it is a consequence of the stochastic dominance
principle. So I consider 4 as an axiom.
It is weIl known that the conditions 1-4 are not sufficient in generai to the ensure the ex
istence of a unique finitely additive probability function P such that P(A) <! P(B) iff A <! B.
New conditions are necessary. So far no sufficient set of necessary conditions for existence
and uniqueness is known in a generic finite Boolean algebra .!il, unless we suppose that no
atom of Jll has probability 0.6 This assumption is in generai rejected by Bayesian subjectiv
event has probability strictly greater than O, a constraint rejected by de Finetti even in the same pa
per. We notice that the set ofaxioms 1-4 are sligthly stronger than Savage's axioms for qualitative
probability [1954 p. 32J, because our system implies that a factual event has the same probability of
a contradìction iff it is of probability O. Thls consequence is not deducible from Savage's definition
ofqualitative probability, although it is a consequence of its whole theory of preference.
6 For more details see the excellent artiele of Fishburn [1986J and the comments on it by Suppes
[1986] and Seidenfeld [1986J.
473
I I
~
FAIRNESS AND EXPECT A TION
ists.7 I think that this rebuttal is correct because the assumption is extraneous to the meaning
ofprobability. Therefore, the appeal to an infinite algebra seems to be unavoidable.
Given the conditions 1-4, to ensure the existence of a unique agreeing probability func
tion, the way generally followed consists of adding structural constraints. Several alternatives
are possible. The common basic idea goes back to de Finetti [1931 pp. 322-3]. Intuitively
speaking, it consists in assuming that for every event B (no matter how improbable it is), the
logical space is partitionable into events ali ofwhich are less probable than B. De Finetti re
quired partitioning into equiprobable events, but less constraining alternatives are available.
Savage proposed a postulate (P6) [1954 p. 39] in a form that turns out to be stronger than re
quired by our theory, which is concemed only with gambles. But Savage formulated also a
weaker form P6' ofP6 that is sufficient for our purposes. It goes as follows:
5. lfB < C, there exists a partition of the algebra S of events, the union of each element
ofwhich with Bis less probable than C.
Savage claimed, without proof, the existence of a single agreeing probability function de
fined on the algebra S of events satisfying 5. A proof ofthis result, valid for any Boolean al
gebra whatsoever, has been reached by Wakker [1981]. In any event, 1 shall adopt 5, so that,
in the present theory as well, the existence of a unique probability function induced by the
set of preferences is guaranteed.
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....