Cyclical consistency and cyclical monotonicity Alexander Kolesnikov - PowerPoint PPT Presentation

Cyclical consistency and cyclical monotonicity Alexander Kolesnikov Higher School of Economics 2014 joint work with Olga Kudryavtseva, Tigran Nagapetyan

Rationalizability problem and revealed preferences P. Samuelson (1938), H.S. Houthakker (1955) We are given n goods and collection of 2 N vectors from R n + which are interpreted as Observations x 1 , · · · , x N Prices p 1 , · · · , p N Every observation i ) , x j x i = ( x 1 i , · · · , x n i ≥ 0 corresponds to a choice of goods made by customer

Rationalizability problem and revealed preferences P. Samuelson (1938), H.S. Houthakker (1955) We are given n goods and collection of 2 N vectors from R n + which are interpreted as Observations x 1 , · · · , x N Prices p 1 , · · · , p N Every observation i ) , x j x i = ( x 1 i , · · · , x n i ≥ 0 corresponds to a choice of goods made by customer

Rationalizability problem and revealed preferences P. Samuelson (1938), H.S. Houthakker (1955) We are given n goods and collection of 2 N vectors from R n + which are interpreted as Observations x 1 , · · · , x N Prices p 1 , · · · , p N Every observation i ) , x j x i = ( x 1 i , · · · , x n i ≥ 0 corresponds to a choice of goods made by customer

Rationalizability problem and revealed preferences P. Samuelson (1938), H.S. Houthakker (1955) We are given n goods and collection of 2 N vectors from R n + which are interpreted as Observations x 1 , · · · , x N Prices p 1 , · · · , p N Every observation i ) , x j x i = ( x 1 i , · · · , x n i ≥ 0 corresponds to a choice of goods made by customer

Rationalizability problem and revealed preferences P. Samuelson (1938), H.S. Houthakker (1955) We are given n goods and collection of 2 N vectors from R n + which are interpreted as Observations x 1 , · · · , x N Prices p 1 , · · · , p N Every observation i ) , x j x i = ( x 1 i , · · · , x n i ≥ 0 corresponds to a choice of goods made by customer

Rationalizability problem and revealed preferences P. Samuelson (1938), H.S. Houthakker (1955) We are given n goods and collection of 2 N vectors from R n + which are interpreted as Observations x 1 , · · · , x N Prices p 1 , · · · , p N Every observation i ) , x j x i = ( x 1 i , · · · , x n i ≥ 0 corresponds to a choice of goods made by customer

Rational choice The choice of goods ( x i , p i ) is rational if there exists utility function u satisfying u ( y ) < u ( x i ) for all i and every y ∈ R n + such that � y , p i � > � x i , p i � Observation: u must have convex superlevel sets { u > c } . Problem Find necessary and sufficient condition for rationalizability of { ( x i , p i ) } .

Rational choice The choice of goods ( x i , p i ) is rational if there exists utility function u satisfying u ( y ) < u ( x i ) for all i and every y ∈ R n + such that � y , p i � > � x i , p i � Observation: u must have convex superlevel sets { u > c } . Problem Find necessary and sufficient condition for rationalizability of { ( x i , p i ) } .

Rational choice The choice of goods ( x i , p i ) is rational if there exists utility function u satisfying u ( y ) < u ( x i ) for all i and every y ∈ R n + such that � y , p i � > � x i , p i � Observation: u must have convex superlevel sets { u > c } . Problem Find necessary and sufficient condition for rationalizability of { ( x i , p i ) } .

Cyclical consistency axiom Choose a subset of the data (denote again x 1 , x 2 , · · · ) x i is directly prefered to x j x i ≻ x j if � x j , p i � > � x i , p i � Equivalently a ij = � x j − x i , p i � > 0 . Cyclical consistency axiom The following cycle is not possible x 1 ≻ x 2 ≻ x 3 ≻ · · · ≻ x n ≻ x 1 .

Cyclical consistency axiom Choose a subset of the data (denote again x 1 , x 2 , · · · ) x i is directly prefered to x j x i ≻ x j if � x j , p i � > � x i , p i � Equivalently a ij = � x j − x i , p i � > 0 . Cyclical consistency axiom The following cycle is not possible x 1 ≻ x 2 ≻ x 3 ≻ · · · ≻ x n ≻ x 1 .

Cyclical consistency axiom Choose a subset of the data (denote again x 1 , x 2 , · · · ) x i is directly prefered to x j x i ≻ x j if � x j , p i � > � x i , p i � Equivalently a ij = � x j − x i , p i � > 0 . Cyclical consistency axiom The following cycle is not possible x 1 ≻ x 2 ≻ x 3 ≻ · · · ≻ x n ≻ x 1 .

In other words: assumption a 12 ≥ 0 , a 23 ≥ 0 , · · · , a k 1 ≥ 0 , implies a 12 = a 23 = · · · = a k 1 = 0 . This is the cyclical consistency axiom / strong axiom of revealed preference (SARP) Theorem (Houthakker) Cyclical consistency is equivalent to rationalizability.

In other words: assumption a 12 ≥ 0 , a 23 ≥ 0 , · · · , a k 1 ≥ 0 , implies a 12 = a 23 = · · · = a k 1 = 0 . This is the cyclical consistency axiom / strong axiom of revealed preference (SARP) Theorem (Houthakker) Cyclical consistency is equivalent to rationalizability.

In other words: assumption a 12 ≥ 0 , a 23 ≥ 0 , · · · , a k 1 ≥ 0 , implies a 12 = a 23 = · · · = a k 1 = 0 . This is the cyclical consistency axiom / strong axiom of revealed preference (SARP) Theorem (Houthakker) Cyclical consistency is equivalent to rationalizability.

Another assumption which implies cyclical consistency: there exists a positive function c on R + n satisfying c ( p 1 ) a 12 + c ( p 2 ) a 23 + · · · + c ( p k ) a k 1 ≤ 0 for every subset { x i , p i } of D . Rearranging the terms we get c ( p 1 ) � x 2 , p 1 � + c ( p 2 ) � x 3 , p 2 � + · · · + c ( p k ) � x 1 , p k � ≤ c ( p 1 ) � x 1 , p 1 � + c ( p 2 ) � x 2 , p 2 � + · · · + c ( p k ) � x k , p k � . This is exactly the cyclical monotonicity assumption for the cost function h ( x , y ) = − c ( y ) � x , y � .

Another assumption which implies cyclical consistency: there exists a positive function c on R + n satisfying c ( p 1 ) a 12 + c ( p 2 ) a 23 + · · · + c ( p k ) a k 1 ≤ 0 for every subset { x i , p i } of D . Rearranging the terms we get c ( p 1 ) � x 2 , p 1 � + c ( p 2 ) � x 3 , p 2 � + · · · + c ( p k ) � x 1 , p k � ≤ c ( p 1 ) � x 1 , p 1 � + c ( p 2 ) � x 2 , p 2 � + · · · + c ( p k ) � x k , p k � . This is exactly the cyclical monotonicity assumption for the cost function h ( x , y ) = − c ( y ) � x , y � .

Another assumption which implies cyclical consistency: there exists a positive function c on R + n satisfying c ( p 1 ) a 12 + c ( p 2 ) a 23 + · · · + c ( p k ) a k 1 ≤ 0 for every subset { x i , p i } of D . Rearranging the terms we get c ( p 1 ) � x 2 , p 1 � + c ( p 2 ) � x 3 , p 2 � + · · · + c ( p k ) � x 1 , p k � ≤ c ( p 1 ) � x 1 , p 1 � + c ( p 2 ) � x 2 , p 2 � + · · · + c ( p k ) � x k , p k � . This is exactly the cyclical monotonicity assumption for the cost function h ( x , y ) = − c ( y ) � x , y � .

Does cyclical consistency imply cyclical monotonicity for some function c ? Discrete case: yes Theorem (Afriat) Given a finite cyclically consistent vector field D = { x i , p i } , 1 ≤ i ≤ N there exist numbers c i such that { x i , c i · p i } is cyclically monotone h ( x , y ) = −� x , y � . By the Rockafellar theorem, there exists a concave utility function u such that u ( x j ) ≤ u ( x i ) + c i � x j − x i , p i � . Ekeland, Galichon (2012). Interpretation of the rationalizability problem as a dual to the housing problem of Shapley and Scarf.

Does cyclical consistency imply cyclical monotonicity for some function c ? Discrete case: yes Theorem (Afriat) Given a finite cyclically consistent vector field D = { x i , p i } , 1 ≤ i ≤ N there exist numbers c i such that { x i , c i · p i } is cyclically monotone h ( x , y ) = −� x , y � . By the Rockafellar theorem, there exists a concave utility function u such that u ( x j ) ≤ u ( x i ) + c i � x j − x i , p i � . Ekeland, Galichon (2012). Interpretation of the rationalizability problem as a dual to the housing problem of Shapley and Scarf.

Does cyclical consistency imply cyclical monotonicity for some function c ? Discrete case: yes Theorem (Afriat) Given a finite cyclically consistent vector field D = { x i , p i } , 1 ≤ i ≤ N there exist numbers c i such that { x i , c i · p i } is cyclically monotone h ( x , y ) = −� x , y � . By the Rockafellar theorem, there exists a concave utility function u such that u ( x j ) ≤ u ( x i ) + c i � x j − x i , p i � . Ekeland, Galichon (2012). Interpretation of the rationalizability problem as a dual to the housing problem of Shapley and Scarf.

Does cyclical consistency imply cyclical monotonicity for some function c ? Discrete case: yes Theorem (Afriat) Given a finite cyclically consistent vector field D = { x i , p i } , 1 ≤ i ≤ N there exist numbers c i such that { x i , c i · p i } is cyclically monotone h ( x , y ) = −� x , y � . By the Rockafellar theorem, there exists a concave utility function u such that u ( x j ) ≤ u ( x i ) + c i � x j − x i , p i � . Ekeland, Galichon (2012). Interpretation of the rationalizability problem as a dual to the housing problem of Shapley and Scarf.