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 , are common knowledge.  The traders' partitions are also common knowledge.  Thus, when trader j calls out positive probability for the event (3,7) given his private signal, it becomes common knowledge that {8,1,4} since {8,1,4} = .  Similarly, it becomes common knowledge that  {4,5}.

    Define the set {} and analogously define the set  = {}.  It follows that {8,1,4}  and {4,5}  so these states would be removed from consideration when the posteriors are common knowledge.  From Definition 6, j knows i assigns zero probability to the set {4,5} implies   = 0, and the fact that i knows j knows i assigns zero posterior probability to {4,5} means  = 0, so that

where K is an even integer, and L is an odd integer.

    Previously, with only their private signals, the set of states that were common knowledge between i and j at  was given by {1,2,3,4,5,6,7,8,9}, which is the join of the classes of the two partitions   and  as defined in the Appendix.  Now that the posteriors for E are common knowledge as well, the set of possible states that are common knowledge has been refined to {2,3,6,7,9}.  Then i knows that the element `8' of his informed class {6,7,8,9} is not possible.  Trader i thus revises his posterior to

    At this stage, i's posterior remains at 1/3 conditioned on his original informed signal, the refinement of the state space, and a posterior for j of 1/2.  The two have moved closer to consensus but have not reached equality.  In some examples, the revision process may lead to consensus, but there is no guarantee.

    To illustrate how the traders reach agreement in the present example, suppose the traders not only have the same priors but that these priors are common knowledge as well:  the traders have agreed common knowledge prior beliefs.

    If trader i's prior was common knowledge, then by Definition 6

where K is any even integer and L is any odd integer, and condition (4.1) holds for all .  Similarly, if trader j's prior is common knowledge, then

The traders have `agreed' priors if their priors are equal:

again for all   Combining equations (4.1), (4.2), and (4.3) means that every first- and higher-order belief of trader i equals every first- and higher-order belief of trader j for a particular event under consideration.  The notation in the following definition applies to a two-person model.

Definition 7:  i and j have agreed common knowledge priors (ACKP) when for any pair of natural numbers, N and M, and

Alternatively, using set theory concepts, i and j have ACKP if condition (4.3) is met and both  and  are -measurable.

We will return to the -field measurability property in Chapter 6.

    With ACKP in the preceding example, hearing the posterior  immediately informs trader j that i received the informed signal  {6,7,8,9}.  Similarly, i knows j received the signal {3,6} when he hears the posterior  = 1/2.  Then each individual can determine that  = 6, in this particular example, by taking the intersection of {3,6} and {6,7,8,9}.  Thus, both traders i and j would then assign zero posterior probability to (3,7) conditioned on ACKP, their respective informed class of their partitions, common knowledge partitions, and the common knowledge posteriors.

4.4  The Consensus Propositions

Let J denote the join or `finest common coarsening' of the partitions   and let  be that element of the join containing .

Definition 8:  An event E is common knowledge at  if  [8]

Note that Definition 8 applies only to events, not to beliefs.

    The common knowledge literature's lattice partition framework requires additional constraints on the admissible set of elements for the join to develop the notion of common knowledge beliefs.  Define   {}, and let  be the element of the join of  and  that contains .  Geanakoplos and Polemarchakis (1982) have shown that a necessary condition for the posterior  to be common knowledge between i and j is that

    To see the severity of this restrictive condition in a contingent claims economy, suppose the model contains six states of the world numbered 1 through 6.  If  {1,2,3,4,5,6}, and {1,2}, {3}, {4,5,6}, then only a posterior such as  can be common knowledge before the agents start calling out their beliefs back and forth.  Surely contingent claims analysis requires a framework that can depict more than just the probability of the whole universe is 1.

    In Section 4.2, when the true state of the world is , trader i is given the private signal , and trader j is similarly informed the true state lies within the class .  It is then common knowledge between i and j that the true state lies within   When the agents receive further information about a posterior for some specific event E, they typically cannot refine the set  unless they know the prior upon which the posterior is based.  The following proposition states this fact more formally.

Proposition 1:  Let  and let  [0,1] and d  [0,1].  Define the set C = {and }, and similarly define the set D = {and }.  Let the posteriors  and  be common knowledge between i and j at .  Then , if

Proof:  See the Appendix.

    Proposition 1 shows that even when (two) traders have common priors and their posteriors for an event E are common knowledge, as when they both call out their posteriors,[9] then their posteriors may remain unequal.  Intrinsic uncertainty may lead to an equilibrium in which the traders disagree and have no basis for further iteration toward consensus.

    In contrast, the common knowledge literature has reached precisely the opposite conclusion.  The subtle distinction between `common priors' and common knowledge priors has somehow been largely overlooked in finance theory and, ironically, confused in the literature on common knowledge.  Aumann (1987, p.10) asserts that in all games of incomplete information, the priors are common knowledge.  `If it were not, then the description of  would be incomplete; we would be able to split  into several states, depending on the various possibilities for .'  Thus Aumann's treatment expands the state space to include a description of investor beliefs as well as a state of nature.  Aumann then seems to contradict himself on the same page when he asserts `neither a partition nor a prior is an event.'  Events are subsets of the state space.

    Brandenberger and Dekel (1993) found a more reasonable way of treating common knowledge priors as attributes of a particular type of person.  They suggest Aumann's expanded state space should be the product of the underlying states of nature times the possible type spaces of each individual.  `i is of a type that assigns probability 1 to j being of a type that assigns probability 1 to E, and so on.' (p.197)  Yet they go on to interpret common knowledge of the information structure ¾ including the prior and thus higher-order beliefs of other market participants ¾ as imposing no `loss of generality.'  I cannot agree.  Common knowledge prior beliefs is a highly restrictive assumption that was not intended and need not be imposed on many (game theoretic) models of differential information.  Chapter 6 proves that a large class of arbitrage pricing models with incomplete information only requires the weaker common priors or homogeneous beliefs assumption.  The long history of Arrow-Debreu contingent claims and securities models outlined in Chapter 2 contains common knowledge assumptions about future prices, but not necessarily about prior beliefs.

    Aumann argues that if game players know the structure of a game, then priors are automatically common knowledge.  As Proposition 1 shows, this argument is invalid.  We can have meaningful discussions of games of incomplete information in which the priors of players are not common knowledge.  The common knowledge literature's consensus results and no trading theorems depend critically and explicitly on a common knowledge priors assumption.  This assumption should be stated up front and not relegated to some murky discussion of `information structure,' on which reasonable people might disagree about what is and is not included.[10]

    A related article by Shin (1993) formalizes the notion that to claim one knows something implies that person can justify the claim by showing a proof.  Shin finds that a statement will be common knowledge at  only if there exists an event proving that statement in the (join) of the partitions.  It is difficult to conceive of such an event in the join of the partitions that could be used to prove a statement such as `individual i's prior is ....'  No such events are mentioned by Shin, although he does find some events related to common knowledge information partitions.

    By imposing ACKP as an assumption, we arrive at the following corollary to Proposition 1.

Proposition 2:  If i and j have ACKP, and their posteriors  and  are common knowledge be­tween i and j at , then .

Proof:  See the Appendix.

    Proposition 2 follows directly from Aumann (1976), but it settles a lingering question on the fundamental assumptions necessary to derive consensus.  Following Aumann's original proposition, a series of no-trading theorems were proposed by Milgrom and Stokey (1982), Tirole (1982), and Geanakoplos and Polemarchakis (1982).  These papers allege that with common priors, the mere willingness of others to trade should convince at least one trading partner that his own position must be disadvantageous.[11]  Hence all the required trading partners would allegedly never agree to the trade.  In general, the no trading results will only hold if the traders' priors are common knowledge.

4.5  Heterogeneous Common Knowledge Beliefs 

Section 4.4 emphasized agreed common knowledge priors.  Market participants could conceivably have unequal yet common knowledge priors.  It could be common knowledge between i and j that , for .  Differing common knowledge beliefs are a form of heterogeneous beliefs.  Market traders with heterogeneous beliefs that are not common knowledge still face intrinsic uncertainty over the unknown beliefs of others.

    It may be helpful to list possible individual belief assumptions in decreasing order of restrictiveness:

1. agreed common knowledge priors;

2. agreed priors  common priors  homogeneous priors

         homogeneous beliefs  equal priors;

3. heterogeneous common knowledge priors; and

4. heterogeneous priors.

    Does it make sense in financial markets to talk about beliefs that are common knowledge but heterogeneous, as in number 3 above?  In the real world, two equally intelligent and knowledgeable mutual fund managers may have different and common knowledge beliefs about the market's future performance.  Both mutual fund managers would seem to have access to similar financial reports, yet one may be bullish and one may be bearish on the market.  When these conflicting prognoses for the market are publicly announced, their heterogeneous beliefs become common knowledge.

    To maintain heterogeneous common knowledge beliefs in this pooling equilibrium, each mutual fund manager must consider himself either a better analyst or the recipient of better information than the party with whom he disagrees.  Each must believe that in the final analysis he will be proven correct.  This willingness to disagree openly also raises some interesting behavioral questions about aggressiveness and self-confidence (in one's own priors) that have not been addressed to date in the consensus literature.

4.6  Conclusions and Issues for Future Research

This chapter has focused on the standard `homogeneous beliefs' assumption that permeates finance theory.  Under this assumption, individuals typically do not know the beliefs of others in the market.  Several numerical examples were developed that employ homogeneous beliefs but focus on how intrinsic uncertainty over priors can still arise:  individuals can agree without knowing a priori they agree.

    Intrinsic uncertainty has nothing to do with `innate' differences in priors; it arises whether the priors are equal or unequal.  Intrinsic uncertainty depends on how much each individual knows about the rest of the market.

    This chapter has also presented two propositions.  The first states if (any) two people have common priors and their posteriors (after receiving private information) for an event are common knowledge, then these posteriors may be unequal.  The second proposition states that with agreed common knowledge priors (ACKP) and common knowledge posteriors, the posteriors must be equal.

    Future research should explore some of the interesting behavioral issues raised in the analysis.  First, we assumed that the individuals will honestly reveal their true posteriors.  In financial markets, traders have every incentive to conceal their true beliefs until after they have traded.  Furthermore, we assumed the individuals cooperate and actually want to reach consensus.  If one person adamantly refuses to update his beliefs, e.g., because he feels he has superior information or simply because he is stubborn, then another person listening to him repeatedly call out the same posterior may draw incorrect inferences.  Again, in financial markets, firms could obviously lose potential profits from sharing information with their competitors.

    Not all theories require highly restrictive market belief assumptions.  The common knowledge literature does for its key agreement results.  However, Chapter 6 shows that a large class of arbitrage pricing models actually need only the weaker assumption of homogeneous beliefs.  Therefore, the reader should exercise caution before replacing a homogeneous beliefs assumption with an ACKP assumption in his or her own models.

Appendix

This Appendix contains some illustrations of the lattice theory concepts used in the common knowledge literature, and proofs of Propositions 1 and 2.

Illustration of Meet and Join

The common knowledge literature uses relatively sophisticated notation and concepts from lattice theory.  This section attempts to explain some of these concepts.  Let J denote the join of the partitions , . . .  where the join of the partitions, as distinguished from the join of the elements of a set, is defined in lattice theory as the `finest common coarsening' of the partitions.  Let  be that element of the join containing .

    The basis for meets and joins of partitions in lattice theory becomes coarsenings and refinements of those partitions.  Donnellan (1968), pp.18-19, provides the following illustration.  If a set (abcd) is partitioned into  and  then the meet of  and , written , is (ab/c/d).  The meet of the partitions is the coarsest common refinement of the partitions.  The partition   contains two classes:   and (d). 

    If  and , then the join of  and , written  or , is .  The join of the partitions gives a coarsening of either partition.   is characterized as the most refined partition having the property that overlapping classes of   and  are wholly contained in just one of the classes:  the finest common coarsening.[12]

    Consider  = {1}, {2,3}, {4,5}, {6};  = {1,2}, {3,4,5}, {6};   = 3;  = {2,3}; and  = {3,4,5}.  The following set of statements applies to knowledge about what state of the world can occur:

    i knows {2,3};

    j knows {3,4,5};

    i knows j knows either {1,2} or {3,4,5}, but not both;

    j knows i knows either {2,3} or {4,5}, but not both;

    i knows j knows i knows {1} or {2,3} or {4,5}, but only one of them;      and

    j knows i knows j knows {1,2} or {3,4,5}, but not both.

By the third-order beliefs, we have reached a consensus over the range of possible states.  When the true state  = 3, each agent is informed the true state lies in that class of his partition that contains 3:  {2,3} for agent i and {3,4,5} for agent j.  Agent i does not know whether the true state is 2 or 3.  If the true state is 2, then i knows j would have been informed with the class {1,2}.  If the true state is 3, then i knows j would have been informed with the class {3,4,5}.  Therefore, as his second-order belief,  ({2,3}), agent i knows j was informed either {1,2} or {3,4,5}.  A similar logic applies to agent j's second-order belief.

    At the third-order, if j was in fact informed {1,2}, then i could reason that j would know i was informed either {1} if  = 1, and {2,3} if   = 2.  Remember, neither agent knows with certainty the exact value of .  If j was in fact informed {3,4,5}, then i could reason that j would know i was informed either {2,3} or {4,5}.  Thus agent i's third-order belief,  ({2,3}), comprises {1}, {2,3}, or {4,5}.

    In assessing the higher-order beliefs of each agent, we first look to each agent's informed signal.  The second-order beliefs depict those classes of the other agent's partition that share any element in common with the original agent's informed signal.  Once we have the second-order beliefs, we again iterate to find the range for the third-order beliefs.  This time we look for classes with any overlapping elements in the (expanded) second-order belief range.  Fourth-order beliefs would be defined over classes with overlapping elements of the third-order beliefs, and so on.

    At the third-order stage, we have defined a common range of {1,2,3,4,5} for both agents i and j.  It turns out that all higher-order beliefs will use this same range as well; this range must then be common knowledge by defintion.  This process of looking for classes with elements that overlap with some range of interest is merely the lattice theory concept of coarsening.  The finest common coarsening, or the join, of  and  is {1,2,3,4,5} in this example:  it is common knowledge between i and j that the true state of the world is in the set {1,2,3,4,5}.  Due to the way the partitions isolated the element {6}, the coarsening will not include it.  Thus it is also common knowledge between i and j that the true state will not be `6.'

Proof of Proposition 1

Since  is common knowledge between agents i and j at , then in general

where K is an even integer, and in particular

By definition,  and  and  and .  Let   Since the classes  are disjoint and  throughout C, then

                                                                             (A6)

    Similarly, it follows that   Therefore , if

and c = d if

Proof of Proposition 2

By statement in the proposition, the two traders i and j have ACKP and their posteriors  and  are common knowledge at  .  It follows that   and 

= c, and indeed  and  for any integer K.  Define the set C = {}.  By the agreed part of the ACKP assumption,  = c.  The partitions for i and j are common knowledge, so that j knows the elements of the set C.  Agent j knows what elements are contained in each of i's partitions; consequently, j knows which of i's partitions are contained in C since they must also satisfy the condition   Similarly, define the set D = {}, and both i and j will know all the elements of the set D (by the ACKP assumption).  The elements of C and D must be in , since the posteriors are common knowledge at  by assumption and it is common knowledge between i and j that the true event lies within .

    Write   Since the partitions  are disjoint and  throughout C, then   Similarly, through­out D, , so that    The intersection of C and D is nonempty, since it must include .  Since {}  and {}  then

and

The right-hand sides of (A7) and (A8) are equal from the `agreed' part of the ACKP assumption.  This result establishes Proposition 2.