Preferences Outline 1 Informal introduction to fair division Resource allocation problems: six examples Resource allocation and fair division: taxonomy 2 Preferences Preference structures A brief incursion into multi-attribute utility theory Combinatorial spaces and compact representation 3 Languages for compact preference representation 4 Ordinal preference representation Ranking single objects Conditional importance networks Prioritized goals 5 Cardinal preference representation k -additive utilities Generalized Additive Independence Weighted Goals Preferences for Fair Division 19 / 92 �
Preferences – Preference structures Admissible bundles From now on we focus on indivisible goods. O = { o 1 ,..., o m } indivisible objects 2 O set of all bundles of objects X ⊆ 2 O set of admissible bundles that an agent may receive Examples of admissible bundles: cardinality constraint: each agent receives exactly k objects: X = { S ⊆ O , | S | ≤ k } categorized items (Mackin and Xia, 15): objects are clustered in categories and each agent receives exactly one item from each category: X = D 1 × ... × D p where D i is the set of all objects of category i . Example: one first dish + one main dish + one drink per agent Preferences for Fair Division 20 / 92 �
Preferences – Preference structures Preferences over bundles N sets of agents O = { o 1 ,..., o m } indivisible objects Notation: [ o 1 o 2 | o 3 | o 4 o 5 ] is the allocation where that agent 1 receives { o 1 o 2 } , 2 receives { o 3 } , 3 receives { o 4 , o 5 } . “No externality” assumption: an agent’s preferences bear only on the bundle she receives 1 is indifferent between [ o 1 o 2 | o 3 | o 4 o 5 ] and [ o 1 o 2 | o 3 o 5 | o 4 ] 2 is indifferent between [ o 1 o 2 | o 3 | o 4 o 5 ] and [ ∅ | o 3 | o 1 o 2 o 4 o 5 ] etc. Therefore: it is sufficient to know each agent’s preferences over bundles (as opposed to her preferences over all allocations). Preferences for Fair Division 21 / 92 �
Preferences – Preference structures Preference structures Specifying preferences on X : comparing, ranking, evaluating bundles. Preferences for Fair Division 22 / 92 �
Preferences – Preference structures Preference structures Ordinal preferences Preference relation on X : reflexive and transitive relation � x � y x is at least as good as y x ≻ y ⇔ x � y and not y � x x is preferred to y ( strict preference ) x ∼ y ⇔ x � y and y � x x and y are equally preferred ( indifference ) ⇔ neither x � y nor y � x x Q y x and y are ( incomparable ) � is often assumed to be complete (no incomparabilities) More sophisticated models: interval orders, semi-orders etc. Preferences for Fair Division 23 / 92 �
Preferences – Preference structures Preference structures Cardinal preferences Utility function u : X → R More generally u : X → V ordered scale; example: V = { unacceptable , bad , medium , good , excellent } From cardinal preferences to ordinal preferences: x � u y ⇔ u ( x ) ≥ u ( y ) Preferences for Fair Division 24 / 92 �
Preferences – Preference structures Preference structures Dichotomous preferences A ⊆ X set of acceptable bundles dichotomous preferences are cardinal preferences: V = { 0 , 1 } ; u ( S ) = 1 ⇔ S ∈ A . dichotomous preferences are ordinal preferences: S � S ′ ⇔ ( S ∈ A ) or ( S ′ / ∈ A ) . Preferences for Fair Division 25 / 92 �
Preferences – Preference structures Preference structures Fuzzy preferences µ R : X 2 → [ 0 , 1 ] µ R ( x , y ) degree to which x is preferred to y . more general than both cardinal and ordinal preferences Preferences for Fair Division 26 / 92 �
Preferences – Preference structures Preference structures fuzzy preferences cardinal preferences ordinal preferences dichotomous preferences Preferences for Fair Division 27 / 92 �
Preferences – Preference structures Monotonicity O = { o 1 ,..., o m } indivisible objects 2 O set of all bundles of objects X ⊆ 2 O set of admissible bundles that an agent may receive Typically, preferences over bundles are monotonic : receiving one more good never makes an agent less happy. ordinal preferences: if S ⊇ S ′ then S � S ′ cardinal preferences: if S ⊇ S ′ then u ( S ) ≥ u ( S ′ ) Strict monotonicity: ordinal preferences: if S ⊃ S ′ then S ≻ S ′ cardinal preferences: if S ⊃ S ′ then u ( S ) > u ( S ′ ) Preferences for Fair Division 28 / 92 �
Preferences – Preference structures Preferential dependencies Existence of preferential dependencies between variables: I’d like to have two consecutive time slots for my lectures (but not three) if I don’t get the shared custody of the children then at least I’d like to keep the cat I want Ann or Charles or Daphne in my team, each of whom would be an excellent goal keeper if I receive the left shoe then I’m ready to pay more for the right shoe Preferences for Fair Division 29 / 92 �
Preferences – A brief incursion into multi-attribute utility theory An incursion into multi-attribute utility theory N = { 1 , 2 ,..., n } set of attributes D i : set of values for the i th attribute X = D 1 × ... × D n set of all conceivable alternatives. Here: in general, X = 2 O : attribute X i is object o i , binary domains { in , out } (in categorized domains) attributes are categories. J ⊆ N subset of attributes D J = Π j ∈ J D j , D − J = Π j �∈ J D j , ( x J , y − J ) ∈ X : contains x j for each i ∈ J and y i for each i / ∈ J ( x i , y − i ) ∈ X : identical to y except for the value of attribute i . Example: X = 2 { o 1 , o 2 , o 3 , o 4 , o 5 } x = ( in , out , out , in , in ) = { o 1 , o 4 , o 5 } ; y = ( out , in , in , in , out ) = { o 2 , o 3 , o 4 } ; ( x 1 , y − 1 ) = ( in , in , in , in , out ) = { o 1 , o 2 , o 3 , o 4 } Preferences for Fair Division 30 / 92 �
Preferences – A brief incursion into multi-attribute utility theory An incursion into multi-attribute utility theory The simplest model: representing preferences via additively decomposable utilities (a) for all x , y ∈ X , x � y ⇔ u ( x ) ≥ u ( y ) (b) for all x = ( x 1 ,..., x n ) ∈ X , u ( x ) = � n i = 1 u i ( x i ) ≥ � n i = 1 u i ( y i ) x = ( x 1 ,..., x n ) , y = ( y 1 ,..., y n ) : alternatives x i value of x on attribute i u i ( x i ) marginal utility value of x on attribute i When does an agent have an additively decomposable utility function? Preferences for Fair Division 31 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Additive decompositions Start with two attributes: X = D 1 × D 2 An agent’s preference relation on X is representable by an additively decomposable utility function iff for all x , y ∈ X , x � y ⇔ u 1 ( x 1 )+ u 2 ( x 2 ) ≥ u 1 ( y 1 )+ u 2 ( y 2 ) where u 1 : D 1 → R ; u 2 : D 2 → R A first necessary condition (Debreu, 1954): � must be a weak order, i.e. , a relation satisfying completeness: for all x , y ∈ X , either x � y or y � x . transitivity: for all x , y ∈ X , x � y and y � z implies x � z . From now on we assume that � is a weak order. Preferences for Fair Division 32 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Additive decompositions: 2-dimensional spaces Assume there exists u representing � . Then for every x 1 , y 1 ∈ D 1 and x 2 , y 2 ∈ D 2 , ( x 1 , x 2 ) � ( y 1 , x 2 ) ⇔ u 1 ( x 1 )+ u 2 ( x 2 ) ≥ u 1 ( y 1 )+ u 2 ( x 2 ) ⇔ u 1 ( x 1 ) ≥ u 1 ( y 1 ) ⇔ u 1 ( x 1 )+ u 2 ( y 2 ) ≥ u 1 ( y 1 )+ u 2 ( y 2 ) ⇔ ( x 1 , y 2 ) � ( y 1 , y 2 ) This property expresses some independence between the attributes: the decision maker takes into account the attributes separately. Preferences for Fair Division 33 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Preferential independence for two attributes Preferential independence (Keeney & Raiffa, 76): Attribute 1 is preferentially independent from attribute 2 (w.r.t. � ) if for all x 1 , y 1 ∈ D 1 and x 2 , y 2 ∈ D 2 , ( x 1 , x 2 ) � ( y 1 , x 2 ) ⇔ ( x 1 , y 2 ) � ( y 1 , y 2 ) The preferences over the possible values of D 1 are independent from the value of D 2 Example a } , { b , ¯ Two binary attributes A , B with domains { a , ¯ b } Preference relation: ab ≻ a ¯ a ¯ b ≻ ¯ b ≻ ¯ ab Preferences for Fair Division 34 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Preferential independence for two attributes Preferential independence (Keeney & Raiffa, 76): Attribute 1 is preferentially independent from attribute 2 (w.r.t. � ) if for all x 1 , y 1 ∈ D 1 and x 2 , y 2 ∈ D 2 , ( x 1 , x 2 ) � ( y 1 , x 2 ) ⇔ ( x 1 , y 2 ) � ( y 1 , y 2 ) The preferences over the possible values of D 1 are independent from the value of D 2 Example a } , { b , ¯ Two binary attributes A , B with domains { a , ¯ b } Preference relation: ab ≻ a ¯ a ¯ b ≻ ¯ b ≻ ¯ ab A preferentially independent from B Preferences for Fair Division 34 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Preferential independence for two attributes Preferential independence (Keeney & Raiffa, 76): Attribute 1 is preferentially independent from attribute 2 (w.r.t. � ) if for all x 1 , y 1 ∈ D 1 and x 2 , y 2 ∈ D 2 , ( x 1 , x 2 ) � ( y 1 , x 2 ) ⇔ ( x 1 , y 2 ) � ( y 1 , y 2 ) The preferences over the possible values of D 1 are independent from the value of D 2 Example a } , { b , ¯ Two binary attributes A , B with domains { a , ¯ b } Preference relation: ab ≻ a ¯ a ¯ b ≻ ¯ b ≻ ¯ ab A preferentially independent from B B preferentially dependent on A Preferences for Fair Division 34 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Separability for two attributes Separability A preference relation � on X = D 1 × D 2 is separable if 1 is independent from 2 and 2 is independent from 1 w.r.t. � . ab ≻ a ¯ a ¯ b ≻ ¯ b ≻ ¯ ab not separable ab ≻ a ¯ a ¯ b ≻ ¯ ab ≻ ¯ b separable Preferences for Fair Division 35 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Preferential independence for n attributes N set of attributes; { U , V , W } partition of N . D U = × i ∈ U D i etc. Conditional preferential independence (Keeney & Raiffa, 76) U is preferentially independent from V (given W ) iff for all u , u ′ ∈ D U , v , v ′ ∈ D V , w , w ′ ∈ D W , ( u , v , w ) � ( u ′ , v , w ) iff ( u , v ′ , w ) � ( u ′ , v ′ , w ) given any fixed value w of W , the preferences over the possible values of U are independent from the value of V c ≻ a ¯ c ≻ a ¯ a ¯ a ¯ abc ≻ ab ¯ b ¯ bc ≻ ¯ b ¯ c ≻ ¯ bc ≻ ¯ abc ≻ ¯ ab ¯ c Preferences for Fair Division 36 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Preferential independence for n attributes N set of attributes; { U , V , W } partition of N . D U = × i ∈ U D i etc. Conditional preferential independence (Keeney & Raiffa, 76) U is preferentially independent from V (given W ) iff for all u , u ′ ∈ D U , v , v ′ ∈ D V , w , w ′ ∈ D W , ( u , v , w ) � ( u ′ , v , w ) iff ( u , v ′ , w ) � ( u ′ , v ′ , w ) given any fixed value w of W , the preferences over the possible values of U are independent from the value of V c ≻ a ¯ c ≻ a ¯ a ¯ a ¯ abc ≻ ab ¯ b ¯ bc ≻ ¯ b ¯ c ≻ ¯ bc ≻ ¯ abc ≻ ¯ ab ¯ c a independent from { b , c } ? Preferences for Fair Division 36 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Preferential independence for n attributes N set of attributes; { U , V , W } partition of N . D U = × i ∈ U D i etc. Conditional preferential independence (Keeney & Raiffa, 76) U is preferentially independent from V (given W ) iff for all u , u ′ ∈ D U , v , v ′ ∈ D V , w , w ′ ∈ D W , ( u , v , w ) � ( u ′ , v , w ) iff ( u , v ′ , w ) � ( u ′ , v ′ , w ) given any fixed value w of W , the preferences over the possible values of U are independent from the value of V c ≻ a ¯ c ≻ a ¯ a ¯ a ¯ abc ≻ ab ¯ b ¯ bc ≻ ¯ b ¯ c ≻ ¯ bc ≻ ¯ abc ≻ ¯ ab ¯ c a independent from { b , c } ? yes { b , c } independent from a ? Preferences for Fair Division 36 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Preferential independence for n attributes N set of attributes; { U , V , W } partition of N . D U = × i ∈ U D i etc. Conditional preferential independence (Keeney & Raiffa, 76) U is preferentially independent from V (given W ) iff for all u , u ′ ∈ D U , v , v ′ ∈ D V , w , w ′ ∈ D W , ( u , v , w ) � ( u ′ , v , w ) iff ( u , v ′ , w ) � ( u ′ , v ′ , w ) given any fixed value w of W , the preferences over the possible values of U are independent from the value of V c ≻ a ¯ c ≻ a ¯ a ¯ a ¯ abc ≻ ab ¯ b ¯ bc ≻ ¯ b ¯ c ≻ ¯ bc ≻ ¯ abc ≻ ¯ ab ¯ c a independent from { b , c } ? yes { b , c } independent from a ? no b independent from { a , c } ? Preferences for Fair Division 36 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Preferential independence for n attributes N set of attributes; { U , V , W } partition of N . D U = × i ∈ U D i etc. Conditional preferential independence (Keeney & Raiffa, 76) U is preferentially independent from V (given W ) iff for all u , u ′ ∈ D U , v , v ′ ∈ D V , w , w ′ ∈ D W , ( u , v , w ) � ( u ′ , v , w ) iff ( u , v ′ , w ) � ( u ′ , v ′ , w ) given any fixed value w of W , the preferences over the possible values of U are independent from the value of V c ≻ a ¯ c ≻ a ¯ a ¯ a ¯ abc ≻ ab ¯ b ¯ bc ≻ ¯ b ¯ c ≻ ¯ bc ≻ ¯ abc ≻ ¯ ab ¯ c a independent from { b , c } ? yes { b , c } independent from a ? no b independent from { a , c } ? no b independent from a given c ? Preferences for Fair Division 36 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Preferential independence for n attributes N set of attributes; { U , V , W } partition of N . D U = × i ∈ U D i etc. Conditional preferential independence (Keeney & Raiffa, 76) U is preferentially independent from V (given W ) iff for all u , u ′ ∈ D U , v , v ′ ∈ D V , w , w ′ ∈ D W , ( u , v , w ) � ( u ′ , v , w ) iff ( u , v ′ , w ) � ( u ′ , v ′ , w ) given any fixed value w of W , the preferences over the possible values of U are independent from the value of V c ≻ a ¯ c ≻ a ¯ a ¯ a ¯ abc ≻ ab ¯ b ¯ bc ≻ ¯ b ¯ c ≻ ¯ bc ≻ ¯ abc ≻ ¯ ab ¯ c a independent from { b , c } ? yes { b , c } independent from a ? no b independent from { a , c } ? no b independent from a given c ? no b independent from c given a ? Preferences for Fair Division 36 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Preferential independence for n attributes N set of attributes; { U , V , W } partition of N . D U = × i ∈ U D i etc. Conditional preferential independence (Keeney & Raiffa, 76) U is preferentially independent from V (given W ) iff for all u , u ′ ∈ D U , v , v ′ ∈ D V , w , w ′ ∈ D W , ( u , v , w ) � ( u ′ , v , w ) iff ( u , v ′ , w ) � ( u ′ , v ′ , w ) given any fixed value w of W , the preferences over the possible values of U are independent from the value of V c ≻ a ¯ c ≻ a ¯ a ¯ a ¯ abc ≻ ab ¯ b ¯ bc ≻ ¯ b ¯ c ≻ ¯ bc ≻ ¯ abc ≻ ¯ ab ¯ c a independent from { b , c } ? yes { b , c } independent from a ? no b independent from { a , c } ? no b independent from a given c ? no b independent from c given a ? yes Preferences for Fair Division 36 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Separability and weak separability U ⊆ N is independent for � if U is preferentially independent from N \ U � is separable if for every U ⊆ N , U is independent for � � is weakly separable if for every i ∈ N , { i } is independent for � (Remark: both notions coincide for n = 2) c ≻ a ¯ a ¯ bc ≻ a ¯ a ¯ abc ≻ ab ¯ bc ≻ ¯ abc ≻ ¯ b ¯ c ≻ ¯ b ¯ c ≻ ¯ ab ¯ c Preferences for Fair Division 37 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Separability and weak separability U ⊆ N is independent for � if U is preferentially independent from N \ U � is separable if for every U ⊆ N , U is independent for � � is weakly separable if for every i ∈ N , { i } is independent for � (Remark: both notions coincide for n = 2) c ≻ a ¯ a ¯ bc ≻ a ¯ a ¯ abc ≻ ab ¯ bc ≻ ¯ abc ≻ ¯ b ¯ c ≻ ¯ b ¯ c ≻ ¯ ab ¯ c � is not weakly separable ( b not independent from c given a ) c ≻ a ¯ a ¯ bc ≻ a ¯ a ¯ abc ≻ ab ¯ bc ≻ ¯ abc ≻ ¯ c ≻ ¯ c ≻ ¯ b ¯ ab ¯ b ¯ c Preferences for Fair Division 37 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Separability and weak separability U ⊆ N is independent for � if U is preferentially independent from N \ U � is separable if for every U ⊆ N , U is independent for � � is weakly separable if for every i ∈ N , { i } is independent for � (Remark: both notions coincide for n = 2) c ≻ a ¯ a ¯ bc ≻ a ¯ a ¯ abc ≻ ab ¯ bc ≻ ¯ abc ≻ ¯ b ¯ c ≻ ¯ b ¯ c ≻ ¯ ab ¯ c � is not weakly separable ( b not independent from c given a ) c ≻ a ¯ a ¯ bc ≻ a ¯ a ¯ abc ≻ ab ¯ bc ≻ ¯ abc ≻ ¯ c ≻ ¯ c ≻ ¯ b ¯ ab ¯ b ¯ c � is weakly separable Preferences for Fair Division 37 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Separability and weak separability U ⊆ N is independent for � if U is preferentially independent from N \ U � is separable if for every U ⊆ N , U is independent for � � is weakly separable if for every i ∈ N , { i } is independent for � (Remark: both notions coincide for n = 2) c ≻ a ¯ a ¯ bc ≻ a ¯ a ¯ abc ≻ ab ¯ bc ≻ ¯ abc ≻ ¯ b ¯ c ≻ ¯ b ¯ c ≻ ¯ ab ¯ c � is not weakly separable ( b not independent from c given a ) c ≻ a ¯ a ¯ bc ≻ a ¯ a ¯ abc ≻ ab ¯ bc ≻ ¯ abc ≻ ¯ c ≻ ¯ c ≻ ¯ b ¯ ab ¯ b ¯ c � is weakly separable � is not strongly separable Preferences for Fair Division 37 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Additive decompositions Question : is a strongly separable weak order � always representable by an additively decomposable utility function? X = D 1 × D 2 with D 1 = { a , b , c } and D 2 = { d , e , f } ad ≻ bd ≻ ae ≻ af ≻ be ≻ cd ≻ ce ≻ bf ≻ cf � separable however � cannot be represented bu u = u 1 + u 2 ( 1 ) af ≻ be ⇒ u 1 ( a )+ u 2 ( f ) > u 1 ( b )+ u 2 ( e ) ( 2 ) be ≻ cd ⇒ u 1 ( b )+ u 2 ( e ) > u 1 ( c )+ u 2 ( d ) ( 3 ) ce ≻ bf ⇒ u 1 ( c )+ u 2 ( e ) > u 1 ( b )+ u 2 ( f ) ( 4 ) bd ≻ ae ⇒ u 1 ( b )+ u 2 ( d ) > u 1 ( a )+ u 2 ( e ) ( 1 )+( 2 ) u 1 ( a )+ u 2 ( f ) > u 1 ( c )+ u 2 ( d ) ( 3 )+( 4 ) u 1 ( c )+ u 2 ( d ) > u 1 ( a )+ u 2 ( f ) Preferences for Fair Division 38 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Additive independence We need a stronger notion of independence. N = { 1 ,..., n } attributes Von Neumann - Morgenstern lottery over X : [( p , x );( 1 − p , x ′ )] where x , x ′ ∈ X Additive independence ≻ satisfies additive independence if for every pair of lotteries L , L ′ over X such that for every attribute i , L and L ′ have the same marginal probabilities over D i , we have L ∼ L ′ . Preferences for Fair Division 39 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Additive independence n = 2; X = D A × D B . Example : let � on the set of lotteries over X defined by L � L ′ if u ( L ′ ) where u defined as follows: u ( L ) ≥ ¯ ¯ u ( a 0 , b 0 ) = 10 u ( a 0 , b 1 ) = 7 u ( a 0 , b 2 ) = 5 u ( a 1 , b 0 ) = 9 u ( a 1 , b 1 ) = 6 u ( a 1 , b 2 ) = 4 u ( a 2 , b 0 ) = 5 u ( a 2 , b 1 ) = 2 u ( a 2 , b 2 ) = 0 [ 0 . 5 , ( a 1 , b 1 ); 0 . 5 , ( a 0 , b 0 )] ∼ [ 0 . 5 , ( a 1 , b 0 ); 0 . 5 , ( a 0 , b 1 )] [ 0 . 5 , ( a 2 , b 1 ); 0 . 5 , ( a 0 , b 0 )] ∼ [ 0 . 5 , ( a 2 , b 0 ); 0 . 5 , ( a 0 , b 1 )] etc. � satisfies additive independence. Preferences for Fair Division 40 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Additive independence n = 2; X = D A × D B . Let � satisfying additive independence. For any a , a ′ ∈ D A , b , b ′ ∈ D B we have 0 . 5 u ( a , b )+ 0 . 5 u ( a ′ , b ′ ) = 0 . 5 u ( a , b ′ )+ 0 . 5 u ( a ′ , b ) therefore fix a 0 ∈ D A , b 0 , b 1 ∈ D B ; u ( a , b 0 ) − u ( a , b 1 ) = u ( a 0 , b 0 ) − u ( a 0 , b 1 ) = C u ( a , b 0 ) = u ( a , b 1 )+( u ( a 0 , b 0 ) − u ( a 0 , b 1 )) = u ( a , b 1 )+ C All marginal utility functions u A ( ., b ) : D A → R are the same up to a translation. fix u ( a 0 , b 0 ) = 0. u ( a , b ) = u ( a , b 0 )+ u ( a 0 , b ) = u A ( a )+ u B ( b ) u is additively decomposable! Preferences for Fair Division 41 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Additive independence Characterization of additively decomposable utilities (Fishburn): A weak order � satisfies additive independence if and only if there exists an additively decomposable utility function u such that for all lotteries L , L ′ over X, we have L � L ′ if and only if ¯ u ( L ′ ) u ( L ) ≥ ¯ ¯ u ( L ) expected utility of L Remark : this is a characterization theorem for preference relations over lotteries . Can we find a characterization theorem for preferences over alternatives? Preferences for Fair Division 42 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Additive independence A characterization when X is finite . X = D 1 × ... × D n where each D i is a finite set. Let m be an integer ≥ 2 and let x 1 ,..., x m , y 1 ,..., y m ∈ X . We say that ( x 1 ,..., x m ) E m ( y 1 ,..., y m ) if for all attributes i ∈ N , ( x 1 i ,..., x m i ) is a permutation of ( y 1 i ,..., y m i ) . Suppose that ( x 1 ,..., x m ) E m ( y 1 ,..., y m ) ; u is additively decomposable then m n m n � � u i ( x j � � u i ( y j i ) = i ) j = 1 i = 1 j = 1 i = 1 Therefore, if x j � y j for all j = 1 ,..., m − 1 then x m � y m . Condition C m Let m ≥ 2. C m holds if for all x 1 ,..., x m , y 1 ,..., y m ∈ X such that ( x 1 ,..., x m ) E m ( y 1 ,..., y m ) , we have x j � y j for all j = 1 ,..., m − 1 implies x m � y m Preferences for Fair Division 43 / 92 �
Preferences – A brief incursion into multi-attribute utility theory Additive independence Theorem (Fishburn) Let � be a weak order on a finite set X = D 1 × ... D n . There are real-valued functions u i on D i such that u ( x ) = � n i = 1 u i ( x i ) for all x ∈ X if and only if � satisfies C m for all m . Remark : for a set X of given cardinality, only a finite number of values of m have to be checked. Preferences for Fair Division 44 / 92 �
Preferences – Combinatorial spaces and compact representation Combinatorial spaces. . . O = { o 1 ,..., o m } indivisible objects 2 O set of all bundles of objects X ⊆ 2 O set of admissible bundles that an agent may receive Each agent has to express her preferences over X : Sometimes, this is not a problem (for instance: one-to-one allocation) However, generally X has a heavy combinatorial structure Preferences for Fair Division 45 / 92 �
Preferences – Combinatorial spaces and compact representation Combinatorial spaces. . . The combinatorial trap. . . Two objects. . . o 1 o 2 ≻ o 2 ≻ o 1 ≻ ∅ → 4 subsets to compare Preferences for Fair Division 46 / 92 �
Preferences – Combinatorial spaces and compact representation Combinatorial spaces. . . The combinatorial trap. . . Four objects. . . o 1 o 2 o 3 o 4 ≻ o 1 o 2 o 4 ≻ o 1 o 3 o 4 ≻ o 2 o 3 o 4 ≻ o 1 o 2 o 3 ≻ o 1 o 3 ≻ o 2 o 4 ≻ o 3 o 4 ≻ o 1 o 4 ≻ o 1 ≻ o 2 ≻ o 4 ≻ o 3 ≻ ∅ → 16 subsets Preferences for Fair Division 46 / 92 �
Preferences – Combinatorial spaces and compact representation Combinatorial spaces. . . The combinatorial trap. . . Twenty binary variables. . . o 8 o 5 ≻ o 5 o 3 o 9 ≻ o 8 ≻ ∅ ≻ o 5 ≻ o 8 o 5 o 3 o 9 ≻ o 8 o 3 ≻ o 5 o 9 ≻ o 3 o 9 ≻ o 8 o 9 ≻ o 8 o 3 o 9 ≻ o 5 o 3 ≻ o 9 ≻ o 3 ≻ o 8 o 5 o 9 ≻ o 8 o 5 o 3 o 1 o 2 o 5 o 8 o 9 ≻ o 1 o 5 o 6 ≻ o 7 ≻ o 2 o 3 o 4 o 5 o 6 o 7 o 8 ≻ o 1 o 2 o 3 o 4 o 5 ≻ o 1 o 3 ≻ o 2 ≻ o 1 o 3 o 7 o 9 ≻ o 1 o 5 ≻ o 1 o 7 o 8 o 9 ≻ o 2 ≻ o 4 ≻ o 6 ≻ o 1 o 7 ≻ o 1 o 2 o 3 ≻ o 1 o 2 ≻ o 2 o 5 o 4 ≻ o 1 ≻ o 2 ≻ o 1 o 2 o 5 o 4 ≻ o 1 o 5 ≻ o 2 o 4 ≻ o 5 o 4 ≻ o 1 o 4 ≻ o 1 o 5 o 4 ≻ o 2 o 5 ≻ o 4 ≻ o 5 ≻ o 1 o 2 o 4 ≻ o 1 o 2 o 5 ≻ o 1 o 5 ≻ o 5 o 3 o 9 ≻ o 1 ≻ ∅ ≻ o 5 ≻ o 1 o 5 o 3 o 9 ≻ o 1 o 3 ≻ o 5 o 9 ≻ o 3 o 9 ≻ o 1 o 9 ≻ o 1 o 3 o 9 ≻ o 5 o 3 ≻ o 9 ≻ o 3 ≻ o 1 o 5 o 9 ≻ o 1 o 5 o 3 o 9 o 6 o 5 o 1 o 9 ≻ o 9 o 5 o 6 ≻ o 7 ≻ o 6 o 3 o 4 o 5 o 6 o 7 o 1 ≻ o 9 o 6 o 3 o 4 o 5 ≻ o 9 o 3 ≻ o 6 ≻ o 9 o 3 o 7 o 9 ≻ o 9 o 5 ≻ o 9 o 7 o 1 o 9 ≻ o 6 ≻ o 4 ≻ o 6 ≻ o 9 o 7 ≻ o 9 o 6 o 3 ≻ o 9 o 6 ≻ o 6 o 5 o 4 ≻ o 9 ≻ o 6 ≻ o 9 o 6 o 5 o 4 ≻ o 9 o 5 ≻ o 6 o 4 ≻ o 5 o 4 ≻ o 9 o 4 ≻ → 1048575 subsets → the expression takes more than 12 days. Preferences for Fair Division 46 / 92 �
Preferences – Combinatorial spaces and compact representation The dilemma The expression of preferential dependencies is often necessary. but . . . Representing and eliciting � or u in extenso is unfeasible in practice. Preferences for Fair Division 47 / 92 �
Languages for compact preference representation Outline 1 Informal introduction to fair division Resource allocation problems: six examples Resource allocation and fair division: taxonomy 2 Preferences Preference structures A brief incursion into multi-attribute utility theory Combinatorial spaces and compact representation 3 Languages for compact preference representation 4 Ordinal preference representation Ranking single objects Conditional importance networks Prioritized goals 5 Cardinal preference representation k -additive utilities Generalized Additive Independence Weighted Goals Preferences for Fair Division 48 / 92 �
Languages for compact preference representation Combinatorial spaces: the dilemma n attributes, each with d possible values ⇒ d n alternatives [In fair division: alternatives are bundles of objects] Way 1 Assume preferential independence elicitation and optimization are made easier (e.g. using decomposable utilities) but weak expressivity (impossibility to express preferential dependencies). Way 2 Allow the user to express any possible preference over the alternatives full expressivity but representing and eliciting � or u in extenso is unfeasible in practice. Preferences for Fair Division 49 / 92 �
Languages for compact preference representation Combinatorial spaces: the dilemma n attributes, each with d possible values ⇒ d n alternatives [In fair division: alternatives are bundles of objects] Way 1 Assume preferential independence elicitation and optimization are made easier (e.g. using decomposable utilities) but weak expressivity (impossibility to express preferential dependencies). Way 2 Allow the user to express any possible preference over the alternatives full expressivity but representing and eliciting � or u in extenso is unfeasible in practice. ⇓ Half-way: languages for compact preference representation Preferences for Fair Division 49 / 92 �
Languages for compact preference representation Representation languages for fair division O = { o 1 ,..., o m } set of objects X = 2 O Representation language : � L , I L � , where L language I L : Φ ∈ L �→ preference relation � Φ or utility function u Φ induced by Φ Preferences for Fair Division 50 / 92 �
Languages for compact preference representation Representation languages for fair division Example 1: a language for dichotomous preferences : L prop : set of all propositional formulas built from the propositional symbols { o 1 ,..., o n } ϕ ∈ L �→ u Φ defined by u ( S ) = 1 if S � ϕ , = 0 otherwise. Preferences for Fair Division 51 / 92 �
Languages for compact preference representation Representation languages for fair division Example 1: a language for dichotomous preferences : L prop : set of all propositional formulas built from the propositional symbols { o 1 ,..., o n } ϕ ∈ L �→ u Φ defined by u ( S ) = 1 if S � ϕ , = 0 otherwise. Example O = { , , , , , , } . � � ∧ ∧ ) ∨ Goal: ( Preferences for Fair Division 51 / 92 �
Languages for compact preference representation Representation languages for fair division O = { o 1 ,..., o m } set of objects X = 2 O Representation language : � L , I L � , where L language I L : Φ ∈ L �→ preference relation � Φ or utility function u Φ induced by Φ Example 2: (obvious) language for additive utility functions : L add : set of all collections of real numbers W = { u i , 1 ≤ i ≤ m } for all S ⊆ O , u W ( S ) = � i , o i ∈ S u i Preferences for Fair Division 52 / 92 �
Languages for compact preference representation Representation languages for fair division O = { o 1 ,..., o m } set of objects X = 2 O Representation language : � L , I L � , where L language I L : Φ ∈ L �→ preference relation � Φ or utility function u Φ induced by Φ Example 3: “explicit” representations for utility functions: L exp = set of all collections of pairs {� S , u ( S ) �| S ∈ X} for preference relations: L ′ exp = list S 1 ≻ S 2 ≻ S 3 ≻ ... representing a ranking over X . Preferences for Fair Division 53 / 92 �
Languages for compact preference representation Representation languages On which criteria can we evaluate the different languages? Expressive power : what is the set of all preference structures expressible in the language? Preferences for Fair Division 54 / 92 �
Languages for compact preference representation Representation languages On which criteria can we evaluate the different languages? Expressive power : what is the set of all preference structures expressible in the language? Succinctness : (informally) � L 1 , I L 1 � is at least as succinct as language � L 2 , I L 2 � is any preference structure expressible in � L 2 , I L 2 � can be expressed in � L 1 , I L 1 � without any exponential growth of size . Preferences for Fair Division 54 / 92 �
Languages for compact preference representation Representation languages On which criteria can we evaluate the different languages? Expressive power : what is the set of all preference structures expressible in the language? Succinctness : (informally) � L 1 , I L 1 � is at least as succinct as language � L 2 , I L 2 � is any preference structure expressible in � L 2 , I L 2 � can be expressed in � L 1 , I L 1 � without any exponential growth of size . Computational complexity : how hard is it to compare two alternatives or to find an optimal alternative when the preferences are represented in � L , I L � ? Preferences for Fair Division 54 / 92 �
Languages for compact preference representation Representation languages On which criteria can we evaluate the different languages? Expressive power : what is the set of all preference structures expressible in the language? Succinctness : (informally) � L 1 , I L 1 � is at least as succinct as language � L 2 , I L 2 � is any preference structure expressible in � L 2 , I L 2 � can be expressed in � L 1 , I L 1 � without any exponential growth of size . Computational complexity : how hard is it to compare two alternatives or to find an optimal alternative when the preferences are represented in � L , I L � ? Easiness of elicitation Preference elicitation = interaction with a user, so as to acquire her preferences, encoded in a language � L , I L � . Is it easy to construct protocols for eliciting the agent’s preferences in � L , I L � ? Preferences for Fair Division 54 / 92 �
Languages for compact preference representation Expressive power Representation language: � L , I L � Expressive power of a language = set of all preference structures that can be expressed in the language = I L ( L ) . � L , I L � at least as expressive as � L ′ , I L ′ � iff I L ( L ) ⊇ I L ′ ( L ′ ) . Examples : expressive power of L add : all additive utility functions over X ; expressive power of L exp : all utility functions over X . � L exp , I L exp � is more expressive than � L add , I L add � . Preferences for Fair Division 55 / 92 �
Languages for compact preference representation Succinctness Relative notion: � L 1 , I L 1 � is at least as succinct as � L 2 , I L 2 � if there exists F : L 2 → L 1 and a polynomial function p such that for all Φ ∈ L 2 : I L 2 (Φ) = I L 1 ( F (Φ)) : Φ and F (Φ) induce the same preferences | F (Φ) | ≤ p ( | Φ | ) : the translation is succinct Example: � L exp , add , I exp , add � = explicit representation restricted to additive utility functions = set of all collections of pairs U = {� x , u ( x ) �| x ∈ X} such that u is additively decomposable � L add , I L add � is strictly more succinct than L exp , add ; but � L exp , I L exp � and � L add , I L add � are incomparable because � L exp , I L exp � is more expressive than � L add , I L add � . Preferences for Fair Division 56 / 92 �
Languages for compact preference representation Computational complexity What is the computational complexity of the following problems when the preferences on X are represented in the language � L , I L � : Given an input Φ in the language � L , I L � , ... dominance : and x , y ∈ X , do we have x � Φ y ? optimisation : find the preferred alternative (or one of the preferred alternatives) (trivial for monotonic preferences) constrained optimisation : and a subset C , possibly defined succinctly, find the preferred option (or one of the preferred options) x ∈ C . Measuring hardness uses computational complexity notions. Preferences for Fair Division 57 / 92 �
Languages for compact preference representation Elicitation Preference elicitation = interaction with a user, so as to acquire her preferences, encoded in a language L (or more generally, so as to acquire enough information about her preferences) Construction of elicitation protocols for some families of languages: exploiting preferential independencies so as to reduce the amount of information to elicit and the cognitive effort spent in communication; trade-off expressivity vs. elicitation complexity. Preferences for Fair Division 58 / 92 �
Ordinal preference representation Outline 1 Informal introduction to fair division Resource allocation problems: six examples Resource allocation and fair division: taxonomy 2 Preferences Preference structures A brief incursion into multi-attribute utility theory Combinatorial spaces and compact representation 3 Languages for compact preference representation 4 Ordinal preference representation Ranking single objects Conditional importance networks Prioritized goals 5 Cardinal preference representation k -additive utilities Generalized Additive Independence Weighted Goals Preferences for Fair Division 59 / 92 �
Ordinal preference representation – Ranking single objects Ranking single objects O = { o 1 ,..., o m } set of objects X = 2 O L sing : set of all rankings over O for each ranking ⊲ over O , I ( ⊲ ) = ≻ is the monotonic and separable extension of ⊲ to 2 O , that is, the smallest preference relation ≻ over 2 O such that ≻ extends ⊲ : for all o i , o j ∈ O , o i ⊲ o j implies { o i } ≻ ′ { o j } ≻ is separable ≻ is monotonic ≻ sometimes called the Bossong-Schweigert extension, or the responsive extension of ⊲ . Preferences for Fair Division 60 / 92 �
Ordinal preference representation – Ranking single objects Ranking single objects m = 2, o 1 ⊲ o 2 o 1 o 2 o 1 o 2 ∅ Preferences for Fair Division 61 / 92 �
Ordinal preference representation – Ranking single objects Ranking single objects m = 3, o 1 ⊲ o 2 ⊲ o 3 o 1 o 2 o 3 o 1 o 2 o 1 o 3 o 1 o 2 o 3 o 2 o 3 ∅ Preferences for Fair Division 62 / 92 �
Ordinal preference representation – Ranking single objects Ranking single objects m = 3, o 1 ⊲ o 2 ⊲ o 3 ⊲ o 4 o 1 o 2 o 3 o 4 o 1 o 2 o 3 o 1 o 2 o 4 o 1 o 2 o 1 o 3 o 4 o 1 o 3 o 2 o 3 o 4 o 1 o 4 o 2 o 3 o 1 o 2 o 4 o 2 o 3 o 4 o 3 o 4 ∅ Preferences for Fair Division 63 / 92 �
Ordinal preference representation – Ranking single objects Ranking single objects Pros: communication complexity: O ( m . log m ) . Cons: assumes separability: what will an agent report if she prefers o 2 over o 3 when she has o 1 and o 3 over o 2 if not? o 1 o 2 o 3 ≻ o 1 o 2 ≻ o 2 o 3 ≻ o 1 ≻ o 3 ≻ o 2 ≻ ∅ o 1 ⊲ o 3 ⊲ o 2 or o 1 ⊲ o 2 ⊲ o 3 ? produces a (very) partial order Preferences for Fair Division 64 / 92 �
Ordinal preference representation – Conditional importance networks Conditional importance networks (Bouveret, Endriss, Lang, 09) allow to express conditional importance statements such as ab : cde ⊲ fg if I have a and I do not have b then I prefer to have { c , d , e } rather than { f , g } all other things being equal Preferences for Fair Division 65 / 92 �
Ordinal preference representation – Conditional importance networks Conditional importance networks Conditional importance statement S + , S − : S 1 ⊲ S 2 (with S + , S − , S 1 and S 2 pairwise-disjoint). ≻ is compatible with S + , S − : S 1 ⊲ S 2 if for every A , B ⊆ O such that A ⊇ S + and B ⊇ S + A ∩S − = ∅ and B ∩S − = ∅ A ⊇ S 1 and B �⊇ S 1 B ⊇ S 2 and A �⊇ S 2 for each o ∈ O \ ( S + ∪S − ∪S 1 ∪S 2 ) , we have o ∈ A iff o ∈ B then A ≻ B Example: ad : b ⊲ ce implies for example ab ≻ ace , abfg ≻ acefg , . . . CI-net A CI-net is a set N of conditional importance statements. Preferences for Fair Division 66 / 92 �
Ordinal preference representation – Conditional importance networks Conditional importance networks Conditional importance statement S + , S − : S 1 ⊲ S 2 (with S + , S − , S 1 and S 2 pairwise-disjoint). CI-net A CI-net is a set N of conditional importance statements on V . Preference relation induced from a CI-net ≻ N is the smallest preference relation over 2 O such that ≻ N is compatible with every conditional importance statement in N ≻ N is monotonic Preferences for Fair Division 67 / 92 �
Ordinal preference representation – Conditional importance networks Conditional importance networks A CI-net of 4 objects { a , b , c , d } : { a : d ⊲ bc , ad : b ⊲ c , d : c ⊲ b } abcd abc abd acd bcd ac ab ad bc bd cd a c b d ∅ Preferences for Fair Division 68 / 92 �
Ordinal preference representation – Conditional importance networks Conditional importance networks A CI-net of 4 objects { a , b , c , d } : { a : d ⊲ bc , ad : b ⊲ c , d : c ⊲ b } abcd abc abd acd bcd ac ab ad bc bd cd a c b d ∅ Induced preference relation ≻ N : the smallest preference monotonic relation compatible with all CI-statements . Preferences for Fair Division 68 / 92 �
Ordinal preference representation – Conditional importance networks Conditional importance networks we recover the singleton ranking form when the CI-net is of the form ∅ , ∅ : o 1 ⊲ o 2 ∅ , ∅ : o 2 ⊲ o 3 ; ... ∅ , ∅ : o m − 1 ⊲ o m CI-nets can express all strict monotonic preference relations on 2 O . dominance and satisfiability: PSPACE-complete (existence of exponentially long irreducible dominance sequences) in P for precondition-free, singleton-comparing CI-statements (such as { a ⊲ c , b ⊲ c , e ⊲ d } ). Preferences for Fair Division 69 / 92 �
Ordinal preference representation – Prioritized goals Prioritized goals Φ = { ϕ 1 ,...,ϕ q } + a weak order � on { ϕ 1 ,...,ϕ q } equivalently, Φ = � Φ 1 ,..., Φ q � where Φ 1 is the set of highest priority formulas, etc. leximin semantics A ≻ B if there is a k ≤ q such that |{ ϕ ∈ Φ i , A � Φ k }| = |{ ϕ ∈ Φ i , B � Φ k }| ; for each i < k : |{ ϕ ∈ Φ i , A � Φ i }| = |{ ϕ ∈ Φ i , B � Φ i }| . discriimin semantics A ≻ B if there is a k ≤ q such that { ϕ ∈ Φ i , A � Φ k } ⊃ { ϕ ∈ Φ i , B � Φ k } ; for each i < k : { ϕ ∈ Φ i , A � Φ i } = { ϕ ∈ Φ i , B � Φ i } . Particular case: conditionally lexicographic preferences Preferences for Fair Division 70 / 92 �
Cardinal preference representation Outline 1 Informal introduction to fair division Resource allocation problems: six examples Resource allocation and fair division: taxonomy 2 Preferences Preference structures A brief incursion into multi-attribute utility theory Combinatorial spaces and compact representation 3 Languages for compact preference representation 4 Ordinal preference representation Ranking single objects Conditional importance networks Prioritized goals 5 Cardinal preference representation k -additive utilities Generalized Additive Independence Weighted Goals Preferences for Fair Division 71 / 92 �
Cardinal preference representation – k -additive utilities k -additive utilities A utility function over X = 2 O is k -additive if it can be expressed as the sum of sub-utilities over subsets of objects of cardinality ≤ k . Φ : u : { S ⊆ O , | S | ≤ k } → R � u ( x ) = u ( S ) S ⊆ O , | S |≤ k Example: O = { a , b , c , d } , k = 2 u ( a , b , d ) = u ( ab )+ u ( ad )+ u ( bd )+ u ( a )+ u ( b )+ u ( d ) Preferences for Fair Division 72 / 92 �
Cardinal preference representation – k -additive utilities k -additive utilities u is 1-additive ⇔ u is additive every utility function is m -additive ( m = | O | ) a k -additive function can be also expressed as the sum of sub-utilities over subsets of attributes of cardinality exactly k . � u ( x ) = v ( S ) S ⊆ O , | S | = k � � m can be specified by values v ( S ) for all | S | = k : values k polynomially large if k is a constant, otherwise exponentially large Preferences for Fair Division 73 / 92 �
Cardinal preference representation – k -additive utilities k -additive utilities An example O consists of 10 pairs of shoes u ( S ) = 10 p + s if S contains a total of p matching pairs and in addition s single shoes u is 2-additive: u ( { left i } ) = u ( { right i } ) = 1 for all i u ( { left i , right i } ) = 8 for all i Exercise: express u as the sum of local values of sets of exactly two shoes. Preferences for Fair Division 74 / 92 �
Cardinal preference representation – k -additive utilities k -additive utilities Another example Categorized domain: three attributes N = { main , dessert , wine } , and X = { Meat , Fish , Veggie }×{ Apple , Cake }×{ Red , White } u main u dessert u wine u main , wine u main , dessert u dessert , wine r w a c a c 8 5 − 1 2 0 m m m 1 1 0 0 a r r 10 − 1 5 0 0 f f f 5 0 0 0 c w w 12 0 0 0 3 v v v u ( vrc ) = u M ( v )+ u D ( c )+ u W ( r )+ u MW ( vr )+ u MD ( vc )+ u WD ( rc ) = 12 + 5 + 0 + 0 + 3 + 0 = 18 Exercise: find the optimal alternative Preferences for Fair Division 75 / 92 �
Cardinal preference representation – k -additive utilities Incursion into computational complexity Two key notions from computational complexity theory: a problem is in the class P if it can be solved by an algorithm running in an amount of time bounded by a polynomial function of the size of the input data. a decision problem (= checking that a property holds) is in NP ( nondeterministic polynomial time ) if given a solution of the problem, this solution can be verified in polynomial time a problem is NP-hard if it is “at least as difficult” as all problems in NP a decision problem is NP-complete if (a) it is in NP and (b) it is NP-hard is is strongly believed that P is strictly contained in NP (therefore: for solving an NP-complete problem, so far we only have exponential-time algorithms). Preferences for Fair Division 76 / 92 �
Cardinal preference representation – k -additive utilities k -additive form: complexity For any k ≥ 2: given a k -additive representation... and an alternative x , computing u ( x ) is in P and a number α , checking that there exists an alternative x such that u ( x ) ≥ α is NP-complete finding x with u ( x ) maximal is NP-hard (except of course if we know beforehand that preference are monotonic...) Preferences for Fair Division 77 / 92 �
Cardinal preference representation – Generalized Additive Independence Generalized Additive Independence GAI-decomposability Let X 1 ,..., X k be a family of subsets of N such that � i X i = N . u is GAI-decomposable with respect to X 1 ,..., X k if there exist k subu- tility functions u i : X i → R such that k � u ( x ) = u i ( x X i ) i = 1 k -additivity = GAI-decomposability, with |X i | ≤ k for all i . Preferences for Fair Division 78 / 92 �
Cardinal preference representation – Generalized Additive Independence Generalized Additive Independence N = { first , main , dessert , wine } X = { Soup , Pasta }×{ Meat , Fish , Veggie }×{ Apple , Cake }×{ Red , White } X 1 ,..., X k = {{ first } , { main , wine } , { main , dessert }} u first u main , wine u main , dessert r w a c m 13 7 m 2 0 s 3 f 9 15 f 0 0 p 1 v 12 12 v 0 3 Dominance is in P Optimisation is NP-hard in the general case Preferences for Fair Division 79 / 92 �
Cardinal preference representation – Weighted Goals Background on propositional logic Let ATM be a set of propositional symbols . The propositional language generated from PS is the set of formulas L PS defined as follows: every propositional symbol is a formula; ⊤ and ⊥ are formulas; if ϕ is a formula then ¬ ϕ is a formula; if ϕ and ψ are formulas then ϕ ∧ ψ , ϕ ∨ ψ , ϕ → ψ , and ϕ ↔ ψ are formula; ⊤ ( true ) and ⊥ ( false ): logical constants ¬ (not): unary connective ∧ (and), ∨ (or), → (implies), ↔ (equivalent) : binary connectives. Preferences for Fair Division 80 / 92 �
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