Asset Pricing under Asym Information Epistomology Knowledge Partitions Knowledge Operator Group Knowledge Depth of Knowledge Public Events Asset Pricing under Asymmetric Information Knowledge No Trade Theorems Theorems Agree to Disagree No Trade Theorem I Efficiency No Trader Theorem II Markus K Brunnermeier Princeton University August 17 2007 Asset Pricing under Asym Information Overview Epistomology Knowledge Partitions Knowledge Operator Group Knowledge Depth of Knowledge Public Events Theorems Agree to Disagree No Trade Theorem I Efficiency No Trader Theorem II Modeling Information From Possibility Sets to Partitions Knowledge Operators Group Knowledge Mutual Common Knowledge No Trade Theorem Aumann s Agreeing to Disagree Geanakoplos generalization No Trade Theorems Net trades are observable Net trades are not observable Allocative Efficiency ex ante interim ex post Asset Pricing under Asym Information Epistomology From Possibility Sets to Partitions Knowledge Partitions Knowledge Operator Group Knowledge Depth of Knowledge Public Events Theorems Agree to Disagree No Trade Theorem I Efficiency No Trader Theorem II State Space Example 1 2 3 4 5 Five states 1 dhigh phigh 2 dhigh plow 3 dlow phigh 4 dlow plow and 5 d 0 p 0 event E set of states e g the dividend payment is high E 1 2 Illustration In 1 agent receives info that dividend is high agent can eliminate the states 3 4 and 5 In state 1 she thinks that only 1 and 2 are possible possibility sets Example possibility set P i00 1 1 2 if the true state is 1 and P i00 2 2 3 P i00 3 2 3 P i00 4 4 5 P i00 5 5 for the other states Individual i knows this information structure Asset Pricing under Asym Information Epistomology Knowledge Partitions Knowledge Operator Group Knowledge Depth of Knowledge Public Events Theorems Agree to Disagree No Trade Theorem I Efficiency No Trader Theorem II From Possibility Sets to Partitions Axiom of truth knowledge ensures that agent does not rule out the true state P i axiom of truth Positive introspection In 1 agent i thinks that 1 and 2 are both possible However by positive introspection she knows that in state 2 she would know that the true state of the world is either 2 or 3 Since 3 is not in her possibility set she can exclude 2 and hence she knows the true state in 1 Formally after positive introspection 0 P i P i 0 P i positive introspection P i0 1 1 P i0 2 2 3 P i0 3 2 3 P i0 4 4 5 P i0 5 5 Asset Pricing under Asym Information Epistomology Knowledge Partitions Knowledge Operator Group Knowledge Depth of Knowledge Public Events Theorems Agree to Disagree No Trade Theorem I Efficiency No Trader Theorem II From Possibility Sets to Partitions Negative introspection In state 4 i holds 4 and 5 as possible However in state 5 she knows that the true state of the world is not in 1 2 3 4 5 can infer that she must be in state 4 because she does not know whether the true state is in 5 or not Formally after negative introspection 0 P i P i 0 P i negative introspection After making use of positive and negative introspection individual i has the following information structure P i 1 1 P i 2 2 3 P i 3 2 3 P i 4 4 P i 5 5 partition In general Information structure becomes partition of a collection of subsets that are mutually disjoint and have a union Asset Pricing under Asym Information Epistomology Knowledge Operator Knowledge Partitions Knowledge Operator Group Knowledge Depth of Knowledge Public Events Theorems Agree to Disagree No Trade Theorem I Efficiency No Trader Theorem II Ki E P i E possibility set P i reports all states of the world individual i considers as possible given true state the knowledge operator reports all the states of the world i e an event in which agent i considers a certain event E possible That is it reports the set of all states in which agent i knows that the true state of the world is in the event E In our example individual i knows event E 0 dividend is high 1 2 only in state 1 i e Ki E 0 1 Asset Pricing under Asym Information 3 Properties of Knowledge Operator Epistomology Knowledge Partitions Knowledge Operator Group Knowledge Depth of Knowledge Public Events Theorems 1 Agent i always knows that one of the states is true Ki 2 Agree to Disagree No Trade Theorem I Efficiency No Trader Theorem II If i knows that the true state of the world is in event E1 then she also knows that the true state is in any E2 containing E1 i e Ki E1 Ki E2 for E1 E2 3 If i knows that the true state of the world is in event E1 and she knows that it is also in event E2 then she also knows that the true state is in event E1 E2 Ki E1 Ki E2 Ki E1 E2 Asset Pricing under Asym Information Restatement of Axiom Epistomology Knowledge Partitions Knowledge Operator Group Knowledge Depth of Knowledge Public Events Theorems Agree to Disagree No Trade Theorem I Efficiency No Trader Theorem II Axiom of Truth Ki E E That is if i knows E e g dividend is high then E is true i e the true state E Positive introspection knowing that you know KTYK axiom Ki E Ki Ki E KTYK This says that in all states in which individual i knows E she also knows that she knows E This refers to higher knowledge since it is a knowledge statement about her knowledge Asset Pricing under Asym Information Restatement of Axiom Epistomology Knowledge Partitions Knowledge Operator Group Knowledge Depth of Knowledge Public Events Theorems Agree to Disagree No Trade Theorem I Efficiency No Trader Theorem II Negative introspection knowing that you do not know KTYNK Ki E Ki Ki E KTYNK For any state in which individual i does not know whether the true state is in E or not she knows that she does not know whether the true state is in E or not requires a high degree of rationality It is the most demanding axiom of the three axioms Adding the last three axioms allows one to represent information in partitions Asset Pricing under Asym Information Epistomology Knowledge Partitions Knowledge Operator Group Knowledge Depth of Knowledge Public Events Theorems Agree to Disagree No Trade Theorem I Efficiency No Trader Theorem II Group Knowledge Common Knowledge Intersection of all events reported by the individual knowledge operators gives us the states in which all members of the group G know an event E KG E i G Ki E Mutual knowledge does not guarantee that all members of the group know that all the others know it too Knowledge about knowledge i e second order knowledge can easily be analyzed by applying the knowledge operator again e g Ki1 Ki2 E An event