Ranking Specific Sets of Objects Jan Maly, Stefan Woltran PPI17 @ - PowerPoint PPT Presentation

Ranking Specific Sets of Objects Jan Maly, Stefan Woltran PPI17 @ BTW 2017, Stuttgart March 7, 2017

Lifting rankings from objects to sets Given A set S A linear order < on S A family X ⊆ P ( S ) \ {∅} of nonempty subsets of S The problem Is there a “good” ranking ≺ on X ? Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 1

An example - Basics S = { strawberry, vanilla, chocolate, lemon } Linear order: strawberry < chocolate < vanilla < lemon X = {{ strawberry } , { chocolate } , { strawberry, vanilla } , { strawberry, vanilla, lemon } , { strawberry, lemon }} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 2

The axiomatic approach What is a “good” ranking? The ranking should be based on the linear order < The ranking should be transitive, either reflexive or irreflexive, . . . “good” depends on the interpretation of X Possible interpretations Sets as final outcomes Opportunities Complete uncertainty etc. . . Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 3

Axioms for ranking sets under complete uncertainty Extension Rule For all x , y ∈ S if { x } , { y } ∈ X , then { x } ≺ { y } iff x < y Dominance For all A ∈ X and all x ∈ S if A ∪ { x } ∈ X , then y < x for all y ∈ A implies A ≺ A ∪ { x } x < y for all y ∈ A implies A ∪ { x } ≺ A If X = P ( S ) \ {∅} and ≺ is transitive, the extension rule is implied by dominance. Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 4

An example - extension rule and dominance S = { strawberry, vanilla, chocolate, lemon } Linear order: strawberry < chocolate < vanilla < lemon X = {{ strawberry } , { chocolate } , { strawberry, vanilla } , { strawberry, vanilla, lemon } , { strawberry, lemon }} Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 5

An example - extension rule and dominance S = { strawberry, vanilla, chocolate, lemon } Linear order: strawberry < chocolate < vanilla < lemon X = {{ strawberry } , { chocolate } , { strawberry, vanilla } , { strawberry, vanilla, lemon } , { strawberry, lemon }} Extension rule: { strawberry } ≺ { chocolate } Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 5

An example - extension rule and dominance S = { strawberry, vanilla, chocolate, lemon } Linear order: strawberry < chocolate < vanilla < lemon X = {{ strawberry } , { chocolate } , { strawberry, vanilla } , { strawberry, vanilla, lemon } , { strawberry, lemon }} Extension rule: { strawberry } ≺ { chocolate } Dominance: { strawberry } ≺ { strawberry, vanilla } ≺ { strawberry, vanilla, lemon } { strawberry } ≺ { strawberry, lemon } Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 5

Axioms for ranking sets under complete uncertainty Independence For all A , B ∈ X and for all x ∈ S \ ( A ∪ B ) if A ∪ { x } , B ∪ { x } ∈ X , then A ≺ B implies A ∪ { x } � B ∪ { x } Strict Independence For all A , B ∈ X and for all x ∈ S \ ( A ∪ B ) if A ∪ { x } , B ∪ { x } ∈ X , then A ≺ B implies A ∪ { x } ≺ B ∪ { x } Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 6

An example - all axioms S = { strawberry, vanilla, chocolate, lemon } Linear order: strawberry < chocolate < vanilla < lemon X = {{ strawberry } , { chocolate } , { strawberry, vanilla } , { strawberry, vanilla, lemon } , { strawberry, lemon }} Extension rule: { strawberry } ≺ { chocolate } Dominance: { strawberry } ≺ { strawberry, vanilla } ≺ { strawberry, vanilla, lemon } { strawberry } ≺ { strawberry, lemon } Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 7

An example - all axioms S = { strawberry, vanilla, chocolate, lemon } Linear order: strawberry < chocolate < vanilla < lemon X = {{ strawberry } , { chocolate } , { strawberry, vanilla } , { strawberry, vanilla, lemon } , { strawberry, lemon }} Extension rule: { strawberry } ≺ { chocolate } Dominance: { strawberry } ≺ { strawberry, vanilla } ≺ { strawberry, vanilla, lemon } { strawberry } ≺ { strawberry, lemon } Independence: { strawberry, lemon } � { strawberry, vanilla, lemon } Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 7

An example - all axioms S = { strawberry, vanilla, chocolate, lemon } Linear order: strawberry < chocolate < vanilla < lemon X = {{ strawberry } , { chocolate } , { strawberry, vanilla } , { strawberry, vanilla, lemon } , { strawberry, lemon }} Extension rule: { strawberry } ≺ { chocolate } Dominance: { strawberry } ≺ { strawberry, vanilla } ≺ { strawberry, vanilla, lemon } { strawberry } ≺ { strawberry, lemon } Independence: { strawberry, lemon } � { strawberry, vanilla, lemon } Strict independence: { strawberry, lemon } ≺ { strawberry, vanilla, lemon } Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 7

Classic impossibility results Kannai and Peleg (1984) Assume X = P ( S ) \ {∅} and | S | ≥ 6 , then there exists no order on X satisfying dominance and independence. Barberà and Pattanaik (1984) Assume X = P ( S ) \ {∅} and | S | ≥ 3 , then there exists no binary relation on X satisfying dominance and strict independence. Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 8

Proof of Barberà and Pattanaik Assume S = { 1 , 2 , 3 } (1) { 1 } ≺ { 1 , 2 } and (2) { 2 , 3 } ≺ { 3 } by dominance { 1 , 3 } ≺ { 1 , 2 , 3 } by (1) and strict independence { 1 , 2 , 3 } ≺ { 1 , 3 } by (2) and strict independence Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 9

Proof of Kannai and Peleg Observation: dominance and independence imply A ∼ { max ( A ) , min ( A ) } Assume S = { 1 , 2 , . . . , 6 } { 3 } ? { 2 , 5 } Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg Observation: dominance and independence imply A ∼ { max ( A ) , min ( A ) } Assume S = { 1 , 2 , . . . , 6 } { 3 } ≺ { 2 , 5 } Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg Observation: dominance and independence imply A ∼ { max ( A ) , min ( A ) } Assume S = { 1 , 2 , . . . , 6 } { 3 } ≺ { 2 , 5 } { 3 , 6 } � { 2 , 5 , 6 } by independence Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg Observation: dominance and independence imply A ∼ { max ( A ) , min ( A ) } Assume S = { 1 , 2 , . . . , 6 } { 3 } ≺ { 2 , 5 } { 3 , 6 } � { 2 , 5 , 6 } by independence { 3 , 4 , 5 , 6 } � { 2 , 3 , 4 , 5 , 6 } by observation Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg Observation: dominance and independence imply A ∼ { max ( A ) , min ( A ) } Assume S = { 1 , 2 , . . . , 6 } { 3 } ≺ { 2 , 5 } { 3 , 6 } � { 2 , 5 , 6 } by independence { 3 , 4 , 5 , 6 } � { 2 , 3 , 4 , 5 , 6 } by observation This contradicts dominance! Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg Observation: dominance and independence imply A ∼ { max ( A ) , min ( A ) } Assume S = { 1 , 2 , . . . , 6 } { 2 , 5 } � { 3 } Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg Observation: dominance and independence imply A ∼ { max ( A ) , min ( A ) } Assume S = { 1 , 2 , . . . , 6 } { 2 , 5 } � { 3 } ≺ { 4 } Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg Observation: dominance and independence imply A ∼ { max ( A ) , min ( A ) } Assume S = { 1 , 2 , . . . , 6 } { 2 , 5 } ≺ { 4 } Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg Observation: dominance and independence imply A ∼ { max ( A ) , min ( A ) } Assume S = { 1 , 2 , . . . , 6 } { 2 , 5 } ≺ { 4 } { 1 , 4 } � { 1 , 2 , 5 } by independence Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg Observation: dominance and independence imply A ∼ { max ( A ) , min ( A ) } Assume S = { 1 , 2 , . . . , 6 } { 2 , 5 } ≺ { 4 } { 1 , 4 } � { 1 , 2 , 5 } by independence { 1 , 2 , 3 , 4 } � { 1 , 2 , 3 , 4 , 5 } by observation Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg Observation: dominance and independence imply A ∼ { max ( A ) , min ( A ) } Assume S = { 1 , 2 , . . . , 6 } { 2 , 5 } ≺ { 4 } { 1 , 4 } � { 1 , 2 , 5 } by independence { 1 , 2 , 3 , 4 } � { 1 , 2 , 3 , 4 , 5 } by observation This contradicts dominance! Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Ditching the assumption X = P ( S ) \ {∅} In many applications X is subject to constraints. There are families X � = P ( S ) \ {∅} with | S | > 6 such that there is an order on X satisfying dominance and (strict) independence. It can be argued that dominance is too weak in the general case. Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 11

An example - all axioms S = { strawberry, vanilla, chocolate, lemon } Linear order: strawberry < chocolate < vanilla < lemon X = {{ strawberry } , { chocolate } , { strawberry, vanilla } , { strawberry, vanilla, lemon } , { strawberry, lemon }} Extension rule: { strawberry } ≺ { chocolate } Dominance: { strawberry } ≺ { strawberry, vanilla } ≺ { strawberry, vanilla, lemon } { strawberry } ≺ { strawberry, lemon } Independence: { strawberry, lemon } � { strawberry, vanilla, lemon } Strict independence: { strawberry, lemon } ≺ { strawberry, vanilla, lemon } Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 12