Intrinsic Uncertainty in Financial Economics
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MICHAEL A. S. GUTH, Ph.D., J.D. |
Intrinsic Uncertainty in Financial Economics
`I saw you take his kiss!' "Tis true.'
`Oh, modesty!' "Twas strictly kept:
He thought me asleep; at least, I knew
He thought I thought he thought I slept.'
¾ Coventry Patmore
4.1
Introduction[1]
Individuals in financial markets typically do
not know the beliefs, preferences, income constraints, and other exogenous
parameters of the rest of the market.
This form of uncertainty is intrinsic to the market participants and
arises in the absence of common knowledge.
`Intrinsic uncertainty' and higher-order beliefs ¾ i.e., beliefs about other people's beliefs
¾ are important in financial economics, game
theory, and any field where the beliefs and actions of others determine payoffs.
Indeed, intrinsic uncertainty may be the single most important determinant of
stock market volatility.
A theorist can impose common
knowledge in a model to eliminate this intrinsic uncertainty. When the value of some parameter
is common
knowledge among agents, it means that each agent knows the value of
, knows everyone knows the value, knows everyone knows
everyone knows the value, ad infinitum.
The finance literature often
assumes individuals have `homogeneous beliefs' to circumvent the aggregation
problems associated with heterogeneous probability beliefs. Yet a homogeneous beliefs assumption can
still permit disagreement and intrinsic uncertainty over state-contingent
variables. Some models may actually
require a stronger `agreed common knowledge beliefs'
assumption.
This article proves that if
two people have the same priors, receive private signals, and have posteriors
for an event that are common knowledge, then ¾ contrary to the widely held view
¾ these posteriors may be unequal. The two people can agree to
disagree. We have obtained the
opposite conclusion of the common knowledge literature,[2]
even under that literature's rigid assumptions. Furthermore, this state of disagreement
arises even under the homogeneous beliefs assumption prevalent throughout
finance theory.
In financial markets,
investors' collective behavior determines equilibrium prices. Investors concerned with capital gains
must therefore consider the beliefs and trading positions of other market
participants. They need to know if
others intend to buy more or sell their holdings. Consequently, assumptions about investor
beliefs will determine the existence and nature of the market equilibrium. By showing the possibility of multiple
equilibria under a homogeneous beliefs regime, we clarify the fundamental
assumptions that underpin the existence and uniqueness of an equilibrium in
financial economics theory.
Financial agents can also
have differing priors that are common knowledge: they can disagree and know they
disagree (on the correct prior distribution). Section 4.5 covers heterogeneous common
knowledge beliefs.
4.2
Intuition Distinguishing Common and Common Knowledge
Priors
People with common priors[3]
agree on the likelihood of some outcome by assigning it the same probability
a priori. However, they can
agree without knowing they agree.
With agreed common knowledge priors, individuals not only have the same
priors but know they have the same priors.
Suppose an individual has homogeneous beliefs, receives a private signal,
and hears someone call out a (posterior) probability for a subset of states of
the world that differs from his own posterior for that subset. He can attribute the difference between
the posteriors to either different information contained in the private signals,
different priors, or a combination of both. In short, market traders could not
distinguish between (1) someone who has the same prior beliefs but different
information and (2) someone with different prior beliefs.
If prior beliefs for all
market traders were the same and were common knowledge, then, in theory, market
traders could attribute any difference in posterior beliefs to informational
asymmetries. This deduction would
enable market traders to achieve a pooling equilibrium. Thus, in contingent claims analysis,
market participants who have the same and common knowledge prior beliefs cannot
`agree to disagree' over the probability that individual states will be
revealed.[4]
An event is common knowledge
between traders i and j when both traders know that event has
occurred, i knows that j knows the event has occurred, j
knows that i knows that j knows the event has occurred, ad
infinitum. Beliefs are common
knowledge among market traders when everyone knows everyone knows ....
everyone's beliefs; however, assuming beliefs are common knowledge need not
imply they also agree.
Consider an economy operating
under uncertainty with n people.
Define the state of the world for each individual i on the
probability space
. The subjective
probability distributions
over
represent the
prior beliefs of the n individuals as to the likelihood of each
possible state. We refer to the
distribution
as the
first-order beliefs of i, and hence, write the probability
distribution with `1' as a superscript.
Let
denote an event
or subset of possible states of the world.
Then
denotes the
(prior) probability i assigns to the outcome that the true state
.
Definition 1:
The individuals have common (same) first-order beliefs when
, for all individuals i, j, over events
.
The `homogeneous beliefs'
assumption of finance theory refers to Definition 1. Let
denote the
information partition of individual i that separates the elements of
into disjoint
subsets and completely spans the space.
In game theoretic terms, the information partitions correspond to
information sets comprising indistinguishable nodes for a given
player.
When the actual state of
nature is
, individual i receives a private signal that the true
state belongs to a particular class of his partition,
where
. For example,
suppose
{
},
{
},
{
},
, and
.[5] Let both i and j
have the same prior probability
distribution over the four
events
in
:
= 0.4,
= 0.1,
= 0.2,
= 0.3. These priors may be based on a history
of ten trials, where the state
was revealed
four times,
was revealed
once, and so on. Then
= .4/.5 = 4/5,
and
=
4/6.
At this point some theorists
have erroneously inferred that by hearing trader i call out the posterior
4/5 for the event E, trader j can somehow tell that trader
i received the signal
. Yet, unless
trader j knows something about trader i's prior beliefs, he has no
way of knowing whether the posterior 4/5 applies to i's conditional event
or
.
Individual i knows the
event E occurs with probability 1 only if
Depending
on his prior probability distribution and how he updates his prior beliefs,
i can potentially assign any probability in the interval [0,1] to
E when ![]()
The first-order beliefs of
the other n-1 traders are typically unknown to a trader, and standard
Bayesian reasoning encourages the treatment of any unknown by assigning a
subjective probability distribution to it.
Therefore, in addition to his own distribution
for
, each trader i must also have a subjective
probability distribution for the unknown first-order beliefs (
) of the other n-1 market participants. Let
denote this
subjective probability distribution over the first-order beliefs of the other
n-1 traders.[6]
Definition 2:
The second-order beliefs of individual i,
, are a probability distribution for individual i over
the possible values of (
).
Returning to the example, if
trader j assigns positive probability (in his second-order belief) to
trader i employing any of the following priors: (.2, .3, .1, .4); (.1, .4, .1, .4); (.4,
.1, .1, .4); (.35, .6, .01, .04), then the posterior
= 4/5 could mean
i received the signal
=
, rather than
=
.
The same intrinsic
uncertainty holds for trader i.
Upon hearing the posterior
= 4/6, trader
i might conclude that trader j received the signal
. But since he
does not know how trader i assigns prior probability to the elements of
, trader j could assign some positive probability in his second-order belief
distribution
to the possible
priors (.3, .2, .1, .4); (.2, .2, .2, .4).
For both of these possible priors, trader j's posterior for
E of 4/6 would indicate he received the signal
![]()
Finally, the traders could
draw the correct inference about the signals from erroneous assumptions about
priors. For example, if trader
j assigns positive probability to trader i using the prior (.1,
.4, .3, .2) instead of trader i's actual prior (.4, .1, .2, .3), trader
j might still correctly infer that i's posterior for E of
4/5 means trader i received the signal ![]()
In principle, it is possible
to define third-order beliefs, which would be a probability distribution
over second-order beliefs, and so on for an infinite regression. Third- and higher-order beliefs in
finance theory date back to Keynes' classic beauty contest analogy to stock
market investment:
[P]rofessional investment may be likened to those newspaper competitions
in which the competitors have to pick out the six prettiest faces from a hundred
photographs, the prize being awarded to the competitor who choice most nearly
corresponds to the average preferences of the competitors as a whole; so that
each competitor has to pick, not those faces which he himself finds prettiest,
but those which he thinks likeliest to catch the fancy of other competitors, all
of whom are looking at the problem from the same point of view. It is not a case of choosing those
which, to the best of one's judgment, are really the prettiest, nor even those
which average opinion genuinely thinks the prettiest. We have reached the third degree where
we devote our intelligences to anticipating what average opinion expects the
average opinion to be. Keynes
(1936), p.156.
Chapter 6 develops a Cartesian product space
framework to represent higher-order beliefs more
rigorously.
In the two-person example,
with traders i and j, the second-order beliefs for trader i
have been reduced from a probability distribution over the (n-1)-tuple
(![]()
), to simply a probability distribution over
. Similarly,
is now a
probability distribution over
. By induction
the third-order beliefs of trader i,
, distribute probability over the possible values of
.
Common knowledge is the
opposite extreme of complete intrinsic uncertainty. For the mathematics of higher-order
beliefs, common knowledge first-order beliefs imply mass points on the
higher-order beliefs: all
probability will be assigned to the known value of the first-order
belief.
If trader i assigns
probability to more than one probability distribution in his second-order belief
for trader j, then he must not know the first-order probability
distribution employed by trader j.
Hence, traders i and j cannot `know' they agree if their
respective second-order beliefs for each other assign positive probability to
more than one first-order belief. The following definitions apply to a
two-person model with traders i and j, and
.
Definition 3: j
is said to know
if and only if
![]()
Definition 4:
i is said to know j knows
if and only if
![]()
Definition 5:
j is said to know i knows j knows
if and only if
![]()
Definition 6:
The first-order posterior
is common
knowledge between i and j at
if and only if
for any even integer K, and any odd integer L,
![]()
=
c.
Note that in Definition 6 the
conditioning event
is actually
known only to trader i. The
posterior for trader j conditioned on trader i's signal
corresponds to the posterior j would assign to the event given what
j believes i has been informed.
4.3
Illustration of Intrinsic Uncertainty with Homogeneous
Beliefs
This section presents some additional numerical
examples that point out the distinction between homogeneous beliefs, agreed
common knowledge beliefs, and common knowledge (posterior) beliefs that are not
equal.
Example 1:
Homogeneous Beliefs, Homogeneous Posteriors
Suppose the numerical values of the common
(homogeneous) prior for i and j is set equal among the four
possible states of the world. Take
{
},
= {
},
{
},
and
But now
let both i and j assign equal prior probability of 1/4 to all four
states. Then
and ![]()
The traders would have equal posteriors without
any further revising, unlike the numerical example of Section 4.2 in which the
posteriors remained unequal. Yet,
even for this case, the traders cannot know whether their equal posteriors stem
from having the same priors or having different priors and different
information. The traders still face
intrinsic uncertainty and agree upon the posterior probability by
chance.
Example 2:
Homogeneous Beliefs, Different Posteriors
Suppose i and j have the same
prior belief for the state of the world in a model with only two possible states
S1 and S2; the prior probability for
S1 is 1/2 and the prior for S2 is 1/2. Let E represent the event that
S1 will appear in the next trial. Suppose that each person can observe one
previous outcome, and that these trials came up S1 for trader
i and S2 for trader j. If each trader's information consists
solely of his endowed prior distribution and the outcome of his one observation,
then the posteriors for E will be 2/3 for trader i and 1/3 for
trader j.
If each one then informs the
other of his posterior, what will they conclude about the previous
outcomes? The common knowledge
literature suggests that trader j would can immediately infer that
i's observation came up S1 and that i would
infer j's trial came up S2, so that both people would
revise their posteriors to 1/2.
But, in the face of intrinsic
uncertainty, how could trader i infer anything from trader j's
posterior without knowing j's prior as well? Neither trader would know
how many trials the other observed, let alone the outcome of these trials. Therefore, the 2/3 and 1/3 posteriors
may remain divergent.[7]
Example 3:
Same Prior, Different Posterior, Further
Refinement
When the agents' posteriors are announced and
common knowledge, then each agent has three pieces of information. He knows his own prior, his own private
signal (in the form of a class of his partition), and he knows both his and the
other trader's posterior is common knowledge. For many problems this amount of
information may suffice to lead to posterior revision and equality, even when
the priors are only the same but not common
knowledge.
Consider
= {1,2,3},
{4,5}, {6,7,8,9};
= {8,1,4},
{2,5,7,9}, {3,6};
= 6;
= {6,7,8,9};
= {3,6};
E = (3,7); and both i and j assign equal (1/9) probability
to the nine events in
.
Both trader i's and
trader j's posterior for E,
and