Social Choice Inspired Ordinal Measurement Alexis Tsoukis LAMSADE - - PowerPoint PPT Presentation

What is the problem? Prototypes or borders? Borda and Condorcet An example Social Choice Inspired Ordinal Measurement Alexis Tsoukiàs LAMSADE - CNRS, Université Paris-Dauphine DIMACS, 19/09/2013 Alexis Tsoukiàs Social Choice Inspired Ordinal Measurement

What is the problem? Prototypes or borders? Borda and Condorcet An example Outline Alexis Tsoukiàs Social Choice Inspired Ordinal Measurement

What is the problem? Prototypes or borders? Borda and Condorcet An example What is the problem? Suppose a device can bee in 4 states: BUS: Business as usual BAW: Be Aware CTC: Call The Cavalry RHA: Rush Away monitoring 100 sensors providing binary information Alexis Tsoukiàs Social Choice Inspired Ordinal Measurement

What is the problem? Prototypes or borders? Borda and Condorcet An example Can we go the hard way? There are 2 100 possible combinations There is no way we can produce an exhaustive association of each combination to each state. What if we had 4 sensors providing an analogical signal? Computationally the problem remains very hard, the combinations becoming infinite. Alexis Tsoukiàs Social Choice Inspired Ordinal Measurement

What is the problem? Prototypes or borders? Borda and Condorcet An example Can we go the hard way? There are 2 100 possible combinations There is no way we can produce an exhaustive association of each combination to each state. What if we had 4 sensors providing an analogical signal? Computationally the problem remains very hard, the combinations becoming infinite. Alexis Tsoukiàs Social Choice Inspired Ordinal Measurement

What is the problem? Prototypes or borders? Borda and Condorcet An example Notation A = { a 1 , · · · a j , · · · a n } objects; C = { c 1 , · · · a i , · · · a m } classes; c i ⊲ c i + 1 ; X = X 1 × X 2 × · · · X n an attribute space; a j = � a 1 j · · · a n ¯ j � ∈ X c i = � c 1 i · · · c n ¯ i � inX Alexis Tsoukiàs Social Choice Inspired Ordinal Measurement

What is the problem? Prototypes or borders? Borda and Condorcet An example Option 1 a i ∼ ¯ ¯ a j ∈ c i ⇔ c i ∼ being a symmetric and reflexive binary relation (with an indifference or similarity meaning). In this case ¯ c i is a “prototype” Alexis Tsoukiàs Social Choice Inspired Ordinal Measurement

What is the problem? Prototypes or borders? Borda and Condorcet An example Option 2 a i ≻ ¯ ¯ a j ∈ c i ⇔ c i ≻ being an asymmetric and irreflexive binary relation (with a strict indifference or dissimilarity meaning). In this case ¯ c i is the “the minimum frontier” separating c i c m = � inf ( X 1 ) , · · · inf ( X n ) � from c i + 1 and ¯ Alexis Tsoukiàs Social Choice Inspired Ordinal Measurement

What is the problem? Prototypes or borders? Borda and Condorcet An example The Borda path: counting values � � x � y ⇔ r j ( x ) ≥ r j ( y ) j j What do we need to know? Alexis Tsoukiàs Social Choice Inspired Ordinal Measurement

What is the problem? Prototypes or borders? Borda and Condorcet An example The Borda path: counting values � � x � y ⇔ r j ( x ) ≥ r j ( y ) j j What do we need to know? the primitives: � j ⊆ A × A Differences of preferences: - ( xy ) 1 � ( zw ) 1 - ( xy ) 1 � ( zw ) 2 Alexis Tsoukiàs Social Choice Inspired Ordinal Measurement

What is the problem? Prototypes or borders? Borda and Condorcet An example The Condorcet path: counting preferences x � y ⇔ H xy ≥ H yx What do we need to know? Alexis Tsoukiàs Social Choice Inspired Ordinal Measurement

What is the problem? Prototypes or borders? Borda and Condorcet An example The Condorcet path: counting preferences x � y ⇔ H xy ≥ H yx What do we need to know? the primitives: � j ⊆ A × A An ordering relation on 2 � j Alexis Tsoukiàs Social Choice Inspired Ordinal Measurement

What is the problem? Prototypes or borders? Borda and Condorcet An example Positive and Negative reasons � w j ± � w j a j � c i ⇔ ≥ γ ∧ ¬ v ( c i , a j ) w j relative importance of each attribute (“weighted majority”); { X j : a j j � c j J ± = i } ; γ a threshold; ∃ X j : c j i ≫≫ a j v ( c i , a j ) ⇔ j ; Alexis Tsoukiàs Social Choice Inspired Ordinal Measurement

What is the problem? Prototypes or borders? Borda and Condorcet An example Remarks From a computational point of vue this is much easier: For 1 each a j we need at most m comparisons ( m being the number of categories) which implies at most n × m comparisons in order to classify a whole set of n objects. It is much more complicated to learn the various 2 parameters such as w j , γ , ¯ c i etc... Alexis Tsoukiàs Social Choice Inspired Ordinal Measurement

What is the problem? Prototypes or borders? Borda and Condorcet An example Remarks From a computational point of vue this is much easier: For 1 each a j we need at most m comparisons ( m being the number of categories) which implies at most n × m comparisons in order to classify a whole set of n objects. It is much more complicated to learn the various 2 parameters such as w j , γ , ¯ c i etc... Alexis Tsoukiàs Social Choice Inspired Ordinal Measurement

What is the problem? Prototypes or borders? Borda and Condorcet An example Conclusions A reasonable way to perform ordinal measurement. 1 Nice axiomatisations. 2 Open preference learning problems 3 Alexis Tsoukiàs Social Choice Inspired Ordinal Measurement