On the Formal Properties of Weighted Averaging as a Method of Aggregation
Author(s): Carl Wagner
Source: Synthese, Vol. 62, No. 1, Consensus (Jan., 1985), pp. 97-108
Published by: Springer
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CARL
ON THE FORMAL
AVERAGING
WAGNER
PROPERTIES
AS A METHOD
OF WEIGHTED
OF AGGREGATION
I first encountered
Keith
work on consensus
Lehrer's
the
during
summer of 1977,l and was immediately
of
intrigued by the possibility
a formal account of his model
of rational group decision
developing
I adopted as a model
for this enterprise
the axiomatic method
making.
was
not
to discover
of social choice
and
that the
theory
surprised
was as complex
of how to aggregate
and prob
probabilities
question
lematic as the question
of how to aggregate
and
utilities.
preferences
The former issue has only recently received
the kind of attention which
at the latter for several decades,
has been directed
and is finally
as
as
to
to
be
attested
debated,
by the essays in
beginning
vigorously
this volume. I am grateful to Barry Loe wer for organizing
this forum on
in Science
Rational Consensus
critical essays follow.
1.
and Society. My
DAVIS
replies
to the preceding
BAIRD
I agree with Baird that averages may conceal differences
in standard
as well as in other distributional
a
set
deviations
of
of
numerical
aspects
to
In certain decisionmaking
estimates.
situations one is well advised
a
com
to
to
attention
such
differences
and
eschew
pay
premature
In other situations
in the form of an average.
it is not so clear
promise
that distributional
information
is useful. Suppose, for example,
that I am
a
to
to
decide
whether
and
trying
undergo
particular surgical procedure
consult a group of experts on the probability
that the operation will
or not to undergo
that I must decide whether
succeed. Assume
this
on
their
based
advice.
that
rule
circumstances
operation
solely
(Imagine
out delaying
in order to give these experts time for further
the decision
research, and that there is no more qualified group to whom I can turn
for advice.) Suppose
that they report to me a consensual
of
probability
success equal to 0.6, based on a consensual weighted
of
their
average
to use this number
initial estimates.
It seems to me that I ought
in
the expected
calculating
tive of the distributional
Synthese 62 (1985) 97-108.
?
1985
by D.
Reidel
Publishing
the operation,
irrespec
utility of undergoing
set of estimates.
of
initial
their
aspects
0039-7857/85/0621-0097
Company
$01.20
CARL
98
WAGNER
in which
Baird may reply that the scenarios
(a) they all initially agree
that 0.6 is the probability
of success, (b) their initial opinions are widely
are concentrated
scattered
about 0.6, and (c) their initial opinions
in
on either side of 0.6 represent
two "camps"
sorts of
very different
I agree that one can make
situations.
about the
interesting
conjectures
state of the art involved
in making
in question,
the prediction
depend
on
seems
to me that such
of
scenarios
but
it
which
these
obtains,
ing
are irrelevant
to my decision.
Perhaps Baird would
speculations
reply
that scenario
the knowledge
(b) obtained
prior to the group's
me
consensus
to
to
should
incline
and that I should
caution
coming
in
calculations
of
the
initial
smallest
my
employ
expected
utility
in question. Even
if the individual respon
estimate of the probability
received
the lowest consensual weight of any of
sible for that estimate
in the group? This is not caution, but pessimism,
the surgeons
and is no
more warranted
than embracing
the largest
initial estimate
of the
that
in question.
probability
that scenario
(c) obtained
suppose
prior to the group's
Similarly,
consensus.
to
I adopt
On
what
should
the prior
grounds
coming
to that of the other? Remember
that
estimate of one camp in preference
of judging the relative expertise
I hive no independent means
of the
two
groups.
in any case, the distributional
relates
accord more prominence
to
information which Baird wishes
to the initial set of estimates
of the
are
in question. The crucial issue is not how these estimates
probability
if
the
final
consensual
estimate
is.
but
what,
distributed,
any,
point
to
our
to
the
in
forces
individuals
method
grant weight
Nothing
if not on the basis
opinions of others. How should Imake my decision,
consensus
of the experts?
of an uncoerced
If my scheme of utilities
is such that a consensual
estimated
prob
success
to undergo
to
in
the
of
0.6
results
my choosing
ability
equal
room
I
enter
of
the
in
somewhat
course,
may,
operating
operation,
on the prior distribution
of
different
states, depending
psychological
in question.
Imay choose to be encouraged
estimates of the probability
the probability
initially estimated
by the fact that one or more surgeons
success
to be plagued
to be 0.8 or allow myself
by apprehension
to be
this probability
about the fact that some of them initially estimated
case it might
be best for me not to be apprised of this
0.4 (in which
con
since anxiety can have physiological
distributional
information,
are
to
distractions
But
such
emotional
endemic
decision
sequences).
of
WEIGHTED
AVERAGING
99
Indeed, the normative model which endorses decisionmaking
making.
on the basis of expected utilities may be viewed
in part as an attempt to
of such considerations.
counteract
influence
the retrograde
Baird,
to bringing
them to the fore.
albeit unwittingly,
may be contributing
the case in
Baird also notes that in, say, a 2-person decision problem,
same
small weight and the case
which each person assigns the other the
same
in which each person assigns the other the
substantial weight both
vector
is that our
(2,2)- His conclusion
weight
yield the consensual
to
low
from
fails
differentiate
mutual
mutual
method
regard
high
of our elementary
the applicability
regard. He is of course assuming
so that the weights
in question are granted across a hierarchy of
model,
seems
from this perspective,
his example
evaluative
skills. Viewed
come
at
not
to
than
I
first
less disturbing
glance. May
considerably
accord another's opinion the same weight as my own by different paths?
even grudging,
acts of concession may, after
sequence of measured,
an
same
as
of
in
the
result
immediate
all, cumulate
appreciation
of
view.
another's point
In both of the foregoing
the issue between
Baird and
examples
a
we have
that
is whether
differences
make
ourselves
ignored
seem to me to
which he wishes to emphasize
difference. The differences
theorists.
be data for psychologists
rather than decision
A
2.
ROBERT
LADDAGA
AND
BARRY
LOEWER
n have assigned subjective
i= 1,...,
that individuals
prob
Supposing
a sequence
to
of
exhaustive
abilities
pairwis^
contradictory,
Pi(Sj)
=
we have proposed
their
l,...,fe,
s7, y
aggregating
propositions
a
a
of
form
into
consensual
rule
the
p by
assignment
assignments
single
n
(2.1)
The
p(Sj)= ??=1 WiPiiSj).
function
propositions
(2.2)
p is extended
by the natural
u = sh v
to disjunctions
and obvious
rule
v sjr of atomic
p(u)=?k=i p(sjk\
and consensual
conditional
be computed
by the rule
probabilities
for disjunctions
u and
t are to
100
(2.3)
CARL
p(u\t)
=
WAGNER
p(uAt)/p(t).
to one
occur
to try alternatively
to calculate
it might
the
as
a
consensual
conditional
of
the
average
probability
weighted
p(u \ t)
individual conditional
and Loe wer note
probabilities
Pi(u\ t). Laddaga
are
unconditional
that the weights
vv?,used to aggregate
probabilities,
not suitable for this task since, in general,
Now
(2.4)
p(u\t)?t
i= \
WiP?u\t)
as Laddaga
and
This is not a proof of some grave defect in our method,
seem to think, but just an indication
of the fact that the
Loewer
of conditional
involves subtler considerations
probabilities
aggregation
of unconditional
than the aggregation
The correct way to
probabilities.
as
is
well
known
conditional
1968,
(Raiffa,
aggregate
probabilities,
8, ?11), is given by the weighted
average
Chapter
(2.5)
?7=1 ?iPi(u\t\
where
(2.6)
?j=wiPi(t)/i
wiPi(t).
It is easy to check that (2.5) yields the same result as (2.3), a fact which
and
Levi (1980) puts to crucial use in his theorem on conditionalization
convex sets of probability
distributions.
in
that under aggregation
It is obvious
by arithmetic
averaging,
to the atomic propositions
dividuals may assign probabilities
...,
sk
su
u and i, of these
that some pair of disjunctions,
in such a way
for each of their assignments,
turn out to be independent
propositions
I have argued
for the consensual
but not independent
assignments.
is a bad thing
at length (Lehrer and Wagner,
that
this
elsewhere
1983)
in the probability
is of epistemic
significance
only if independence
at
in
and
that
where
the initial acts
assessment
hand,
problems
problem
a
are directed
to
at assigning
of
of assessment
sequence
probabilities
an
formal
is
after-the-fact
atomic propositions,
curiosity.
independence
r
that you observe
A further example may be in order. Suppose
successes
r' successes
in n' (different)
in n trials and I observe
trials,
WEIGHTED
101
AVERAGING
the data generating
process agreed by both of us to be a Bernoulli
on
our
data I would
select as the density
process. Based
respective
function for the parameter
p of this process
=
r- 1)!,
l)!(rc
(n
xT'^/irl)!xrl(l
?(n, r)
with
0<jc<1,
and you would
select ?(n\
consensual
r'). The appropriate
density
of
function here is not a weighted
r'), but
average
?(n, r) and ?(n\
that we
rather j3(n + n\ r+ r'). I cite this example both to emphasize
as a method
have not endorsed weighted
of aggregating
averaging
to
show
that
and
also
the
functions
perfectly
unexceptionable
density
is not
function
consensual
?(n + n\r+r')
independence
density
1 success in 2 trials, we
For if we each separately observe
preserving.
on [0,1], as
select j3(2,1), the uniform distribution
separately
of
For
the
uniform
the
the density
function
p.
density,
propositions
p e [1/4, 3/4] and p e [5/8, 7/8] are independent. This is not the case for
?
=
and Loewer
the consensual
6x(l
x). Are Laddaga
density ?(4, 2)
one
to
that
this
should
be
disturbed
failure to
argue
prepared
by
preserve
independence?
are of course other kinds of assessment
where
There
problems
is
Such
epistemically
problems
significant.
typically
independence
for exam
involve a sequence of independent
trials. Suppose,
repeated
to the 4 possible
ple, that you and I are asked to assign probabilities
outcomes
of flipping a (possibly biased) coin twice. I have information
will
each
that the coin showed heads 3 times in 10 flips and you have information
that coin showed heads 8 times in 20 (different) flips. Based on my
of HH, HT,
the probabilities
TH, and
sample alone I would estimate
as
and
and
based
TT, respectively,
(0.7)(0.3),
(0.3)2, (0.3)(0.7),
(0.7)2
on your sample alone, you would estimate
these probabilities,
respec
and (0.6)2. In this situation
it
(0.6)(0.4),
tively, as (0.4)2, (0.4X0.6),
our
to take a weighted
of
of
the
would be ridiculous
estimates
average
a consensual
of these events
in order to produce
dis
probabilities
and Loewer
think that our
reason, Laddaga
to this sort of thoughtless
of any
averaging
cross
we
our
that
It
distributions
What
should
doesn't.)
path.
probability
our
to
is
combine
the
consensual
do, clearly,
samples,
producing
11/30 of heads on any toss, and then invoke independence
probability
on each toss (on which we agree)
to produce
of the outcomes
the
tribution.
method
(For some
us
commits
consensual assignments (11/30)2, (ll/30)(19/30),
(19/30)(ll/30),
and
102
CARL
WAGNER
one hardly notices it in such a simple case, weighted
(19/30)2. Although
is
used
here, not to combine our estimates of the probabilities
averaging
of the events HH, HT,
TH, and TT, but to combine our estimates of
outcomes
of the possible
the probabilities
{H, T} on any toss. For
our
our estimates
to
is
of
combining
samples
combining
equivalent
to the sample sizes
with weights
these basic probabilities,
proportional
=
+
on which we based these estimates:
11/30 and
(l/3)(0.3)
(2/3)(0.4)
(1/3X0.7) + (2/3X0.6)
= 19/30.
reveals a simple and generally
reliable indicator
The above example
of
individuals
all
the epistemic
When
significance
independence:
in order to assign probabilities,
make use of independence
when
i.e.,
of
certain probabilities
other prob
by multiplying
they actually calculate
a
to
be
have
theoretical
commitment
assumed
abilities,
prior
they may
to incorporate
to such independence.
Failure
such instances
of in
a
in
of
consensual
their
individual
dis
dependence
aggregation
of
tributions would thus ignore an epistemically
feature
their
significant
that one
We have seen in the above example,
however,
cases
of epistemically
of in
take account
significant
our
consensus.
to
within
the
framework
of
The
dependence
approach
can
a
our
also
shows
how
method
accommodate
example
easily
consensus
that some random variable is binomially
distributed,
contrary
to the claim of Laddaga
and Loewer.
of independence
is in
One
further remark about the preservation
of
and Loewer make
order. Laddaga
the strong claim that a method
must
(even in
independence
probabilities
always preserve
aggregating
cases where
to
in
it is of no epistemic
avoid
order
significance)
to
But
the
of
which
sequence
they employ
inconsistency.
propositions
are bi-level
not
which
do
admit
argue for this assertion
propositions
distributions.
can easily
this fact by
however
assignments,
they may seek to disguise
probability
I
label
H
&
K
&
after
their choice of notation. What
is,
all, the
they
=
assertion H81K&
Since the first two prop
P(H & K)
P(H)P(K).
ositions are amenable only to first order probability
and the
assignments
a
to
second
order
third is amenable
probability
assignment
only
in higher order probabilities),
the entire com
(assuming one believes
can
no
be
assigned
meaningful
probability.
pound proposition
it could easily be
if Laddaga
and Loewer's
Indeed,
gambit worked,
to show that aggregation
methods must preserve
modified
every feature
no matter how esoteric. Thus,
common
to all individual distributions,
for example,
nonindependence
and nonequality
of probabilities
assig
WEIGHTED
AVERAGING
103
to be preserved
in consensual
dis
have
ned to two events would
in every individual distribution.
if these properties
obtained
tributions
the
that Lehrer
To
take an even wilder
suppose
assigns
example,
and
and
I
them
A
B
the
and
1/5
4/5
assign
propositions
probabilities
that the prob
the probabilities
2/5 and 3/5. Let F be the assertion
in lowest terms, have a
abilities of A and B are rational numbers which,
to
to
and Loewer's
denominator
5. According
Laddaga
equal
our
set
of
matrix
for
the
argument,
expanded
propositions
probability
A &F,
and B& F would be
A&F,
B&F,
A&F
1/5
(L)
(W) 2/5
A8iF
0
0
B&F
4/5
3/5
B&F
0
0
assuming that any reasonable method would assign the second and
a consensual
of zero, consensual
fourth propositions
prob
probability
?
abilities for the above four events must take the form p, 0, 1 p, and 0.
of F equals 1 and any p not of the form
Thus the consensual
probability
for
be
with
would
inconsistent
this fact. But
this
l<fc<5,
fc/5,
and I based our initial
conclusion
is absurd. If for example, Lehrer
on separate
of disjoint
examinations
samples of 5, he
probabilities
case
A
1
I
in
in
and
2
of
the
cases, etc., it would be
presence
detecting
to
A
B
and
the
consensual
reasonable
assign
probabilities
eminently
Now
and 7/10.
conclude
and Loewer
?111 of their paper with an example
Laddaga
two
of a decision
probabilities,
problem where,
prior to aggregating
on
course
of
and
afterwards
individuals
the
best
action,
agree
they
reply that that prior agreement
disagree. To our (correctly) anticipated
is shown to be ill founded by the consensual
aggregated
probabilities,
based on weighted
averaging
probabilities
they reply that consensual
3/10
are not rationally compelling.
Not ever? Suppose A has seen Si happen
6 times each following
dx and d2 and s2 4 times each. B, on the other
hand, has seen (in a disjoint sample) Si happen 4 times each following dx
their probabilities
and d2 and s2 6 times each. They aggregate
using the
on
vector
fact
that
based
the
consensual
(|,^)
they have
weight
in
of the
This
results
their
each
size
observed
samples.
assigning
equal
outcomes
Such
consensual
Si and s2 equal consensual
probability.
new
basis for calculating
strike me as an impeccable
probabilities
a
on
If
them
the
utilities.
changes
prior agreement
expected
employing
104
best
CARL
course
agreement
of action, so be
at any price.
3.
WAGNER
it.We
ISAAC
are after
informed
agreement,
not
LEVI
Levi endorses, with some qualifications,
cism of our method's
failure to preserve
consensus
as the convex
representing
and Loewer's
criti
asserts
that
He
independence.
hull of a set of competing
Laddaga
vectors
for example,
avoids
this problem.
that
Suppose,
I assign the respective
2/9) and
(4/9, 2/9,1/9,
probabilities
a
to
of
exhaustive
(1/9, 2/9, 4/9, 2/9)
sequence
pairwise contradictory,
the propositions
s1? s2, s3, s4. On each of our assignments
propositions
u = 5i v s2 and t= s2 v s3 turn out to be independent.
If we come to
on
our
t
true
that
condition
is
and
this
evidence,
agree
respective
will
conditional
be
and
(0,2/3,1/3,0)
probability
assignments
and
Levi
would
that
both
after
before
(0,1/3,2/3,0).
emphasize
to u by either of
conditionalization
the maximum
probability
assigned
probability
Levi and
us is 2/3 and the minimum
of
is 1/3. This is not preservation
probability
sense
in
the
demanded
and
Loewer,
but,
independence
by Laddaga
of interval
estimates
under
conditionalization
rather, preservation
relative to an independent
item of evidence.
(i.e., irrelevant)
What Laddaga
and Loewer
demand
is that any consensual
assign
to
ment
of unconditional
the propositions
Su s2, s3y s4
probabilities
u and t turn out to be independent.
should be such that the propositions
To the extent that Levi regards each weighted
arithmetic mean of our
a
as
of
resolution
the conflict" between
"potential
original assignments
our original assignments,
of in
he is potentially
violating
preservation
sense
in
of
and
the
Loewer.
the
Indeed,
dependence
Laddaga
only
actual resolution
of our conflict which preserves
in this
independence
or for him to
sense would be for me to adopt Levi's original distribution
no actual
To
and
mine
Lehrer
be
sure,
1983).
(see
adopt
Wagner,
of independence
of preservation
will occur under Levi's
if our disagreement
is not resolved,
scheme
but if this is virtue,
it is
virtue by default.
In any case, as I have argued in my reply to Laddaga
and Loewer,
as
our model,
of
in
is
independence
rarely
epistemic
significance where,
are directed at the probabilities
of a set of
the initial acts of assessment
violation
exhaustive
the
neither
Thus,
propositions.
contradictory,
violation
of
of
Levi's
model,
independence
by
preservation
potential
nor its actual violation
is cause for concern.
by our model
pairwise
WEIGHTED
105
AVERAGING
of consensus
is its
very positive
aspect of Levi's
conception
area
to
in
of
restrain
the
premature
Particularly
tendency
compromise.
to
of
utilities
relative
all
group decisionmaking,
expected
calculating
in the convex hull of a set of competing
the probability
distributions
seems to me to be a salutary exercise. Of course,
distributions
these
One
two courses of action will rarely unanimously
rival ways of evaluating
establish
the superiority of one action over another.
beat another in this sense
Indeed, one course of action will decisively
just when the smallest expected
utility attributed by anyone
(using his
to
initial subjective
the
former
the
exceeds
expec
largest
probabilities)
ted utility attributed by anyone to the latter. In some cases requiring a
to adopt one course of action over another to be supported
decision
in
on
this way may be excessively
But
issues
like
nuclear
demanding.
(1980) has persuasively
safety, as Levi
argued, an extreme modesty
regarding
our probability
estimates
4.
HANNU
is entirely
appropriate.
NURMI
at alleged
of Nurmi's
is directed
of the
deficiencies
critique
as a social choice mechanism.
of weighted
method
He
averaging
observes
first that our method
does not always pick the Condorcet
or exclude
winner
the Condorcet
loser. Condorcet
criteria may be
reasonable
restrictions on social welfare functions
(methods of produc
from a profile of individual orderings)
since
ing a consensual
ordering
the inputs contain no information
about depth of preference.
They are
on social welfare
not reasonable
restrictions
(methods of
functional
a
from a profile of individual utilities).
consensual
ordering
producing
to restrict our advocacy
We
have been very careful
of weighted
cases
to
arithmetic
where
differences
in utilities are inter
averaging
Most
formal
(Consensus,
p. 118). In such cases,
personally
comparable
results
Theorem
show
that
arithmetic
(Consensus,
6.9)
weighted
is the precisely
of aggregation.
In
method
averaging
appropriate
are
it
is
if
that
differences
inter
clear, however,
formally
already
to take them into
then it is not unreasonable
personally
comparable,
account. The aforementioned
theorem strengthens
this to the assertion
that they must be taken into account. Thus, A's much
greater pref
erence for x over y can justifiably defeat the more modest
preferences
of B and C for y over x, even when A, B, and C all receive equal
weight.
106
CARL
WAGNER
next that our method
observes
violates
the weak axiom of
as well as path-independence.
The examples
used
preference
this involve
of von Neumann-Morgenstern
the aggregation
indicates. But differences
in such utilities
utilities, as Nurmi forthrightly
are not interpersonally
and we specifically
forbid the
comparable,
Nurmi
revealed
to show
of such utilities by weighted
arithmetic
aggregation
except
averaging,
case where
one person
in the dictatorial
receives
all the weight
(Consensus,
pp. 119-120).
Nurmi observes
further that, under our method,
it is possible
that x is
for a given assignment
ranked highest
of utilities, but loses first place
for another assignment
of utilities which differs from the first only in
that one individual assigns x an even higher utility than he did before.
are used in the two
This is only possible, of course,
if different weights
cases. I see nothing remarkable or disturbing
about this. Weights
make
a difference.
to do.
That's what they're designed
In addition, Nurmi remarks that, under our method,
two groups may
separately rank x highest, but jcmay lose first place when the groups are
even though no one changes his
The
amalgamated,
utility assignments.
for this lies in a change in the weights. Again, weights make
explanation
a difference.
In Nurmi's
individual
1, who prefers
jci, reduces
example
his own weight
to individual 4, who prefers
and assigns the difference
individual 3, who prefers xu reduces his own weight
x2. Similarly,
and
to individual 2, who prefers x2. Individuals
2 and
assigns the difference
4 do not, however,
is it surprising or "inconsistent,"
reciprocate. Why
as Nurmi would have it, that x2 might now be ranked first?
to the foregoing
In addition
as a social
of our method
criticisms
choice mechanism,
Nurmi
identifies what he takes to be a defect in the
of arriving at consensual weights
general method
by iterated weighted
of weight
matrices.
His
i.e., by repeated multiplication
averaging,
observation
is that a modest
shift in the weights
assigned by a single
can produce
a substantial
individual
as
shift in consensual
weights,
illustrated by the matrices
"
"0.50
0.05
0.05
0.00
0.05
0.00
0.45
0.05
0.00
0.00
0.00
0.85
0.50
0.05
0.05
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.50
0.05
0.10
0.00
0.40
0.80
0.40
0.00
0.10
0.10
0.50
0.45
107
AVERAGING
WEIGHTED
that its fifth row is
is identical with Wx except
and W2, which
consensual
0.05. The corresponding
0.05
0.80
0.00
0.00
are
vectors
weight
0.10
d=
[0.078
0.371
0.041
0.054
0.371
0.085]
[0.108
0.514
0.057
0.032
0.257
0.032].
and
C2
=
an
and Nurmi
is right in demanding
example
interesting
to
is
two-fold.
the
The
First,
proper way
explanation
explanation.
terms
in
of
than
is
rather
differences
ratios,
(see
compare weights
of how apparently
for examples
substantial
Consensus,
pp. 131-132,
no change
in the consensual weight
shifts in a weight matrix produce
vector because
ratios of off-diagonal
elements are unchanged;
also the
to
at
of
it
is
the
p. 134). Second,
top
important
keep in mind
example
of our elementary
the applicability
that Nurmi
is assuming
model,
This
is an
at different
levels are precluded.
where different weight matrices
Thus
in the manner
he is not simply shifting a first order weight matrix
as well. (See also in this
indicated above, but all higher order matrices
connection my reply to Baird.) Combining
the above remarks, I would
as follows:
describe
Nurmi's
"Individual
doubles
the
#5
example
to
which
he
and
halves
the
the
which
#1
he
assigns
weight
weight
an
to
in
of
of
each
infinite
matrices.
#6
assigns
hierarchy
weight
Individual #1, on the other hand, assigns nearly half of his available
at all to #5. Conversely,
to #2
at all levels, and no weight
weight
to #5 at all
individual #6 assigns nearly half of his available weight
levels, and no weight at all to #2. The result is that individual #5, who
a consensual
now
received
equal to that of #2,
weight
previously
receives only half of the consensual weight
received by #2." This does
at all levels, the
not strike me as being counter-intuitive.
If I double,
no
to
someone
I
for
which
who
has
me, and halve
respect
give
weight
the weight which I give to someone with substantial respect for me, is it
that I find my position undermined?
any wonder
5.
FREDERICK
SCHMITT
Schmitt objects to the fact that we do not allow an individual to receive
on different
different weights
in probability
competing
propositions
an
assessment
From
the
observation
that
individual
may be
problems.
108
CARL
WAGNER
or
than about another, he
less informed about one proposition
are
allowed.
that
"variant"
should
be
But probabilities
infers
weights
a
to a set of mutually
not assigned
in
exhaustive
exclusive,
propositions
a
one.
acts.
The
is
of
isolated
unified
In
to
order
sequence
enterprise
assess the probability
that Native Dancer will win the race, we need to
formal
will be. A
result
know who
his competitors
(Consensus,
this
of
intuition
the possibility
Theorem
6.4) supports
by granting
to
different
different
assigned
probabilities
by
propositions
aggregating
that this apparent flexibility
is
functions
(Axiom IA) but then showing
a
set
of
because
consensual
(of
probabilities
largely
precluded,
must sum to one. Schmitt
is exer
exhaustive
exclusive,
propositions)
more
cised to the point of exclamation
("But the rejection of this assumption
is of a piece with the rejection of invariance! Consensual
probabilities
not sum to one ...") by the fact that this basic property
must obviously
of probabilities
is used in the proof. The reasons for his agitation escape
me.
NOTE
1
at the Center
a participant
for Advanced
our work
by providing
I was
at the
Lehrer
supported
time
on Freedom
and Causality
which
Sciences,
Study in the Behavioral
me with a Fellowship
1978-79.
during
in the Institute
directed
by
subsequently
REFERENCES
and Wagner,
Dordrecht-Boston.
Keith
Lehrer,
Reidel,
Keith
Lehrer,
dependence
Isaac:
Levi,
1981,
Rational
Consensus
in Science
Carl:
and Wagner,
1983,
'Probability
Amalgamation
to Laddaga',
Issue: A Reply
55, 339-346.
Synthese
MIT
Press, Cambridge,
1980, The Enterprise
of Knowledge,
Raiffa,
Howard:
Wagner,
Notre
Dame
Carl:
Department
Carl:
1968,
Decision
1984,
'Aggregating
Journal
of Formal
of Mathematics
University
of Tennessee
Knoxville,
U.S.A.
TN
37996-1300
Analysis,
Addison-Wesley,
Probabilities:
Subjective
Logic
25,
233-240.
and
Society,
and
the
Mass.
Mass.
Reading,
Some Limitative
Theorems',
In