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