Decisions with multiple attributes

Transcription

1 Decisions with multiple attributes A brief introduction Denis Bouyssou CNRS LAMSADE Paris, France JFRO December 2006

2 Introduction Aims mainly pedagogical present elements of the classical theory position some extensions wrt this classical theory

3 Introduction Comparing holiday packages cost # of days travel time category of hotel distance to beach Wifi cultural interest A 200 e h *** 45 km Y ++ B 425 e h **** 0 km N C 150 e 4 7 h ** 250 km N + D 300 e 5 10 h *** 5 km Y Central problems helping a DM choose between these packages helping a DM structure his preferences

4 Introduction Comparing holiday packages cost # of days travel time category of hotel distance to beach Wifi cultural interest A 200 e h *** 45 km Y ++ B 425 e h **** 0 km N C 150 e 4 7 h ** 250 km N + D 300 e 5 10 h *** 5 km Y Central problems helping a DM choose between these packages helping a DM structure his preferences

5 Introduction Two different contexts 1 decision aiding careful analysis of objectives careful analysis of attributes careful selection of alternatives availability of the DM 2 recommendation systems no analysis of objectives attributes as available alternatives as available limited access to the user

6 Introduction Two different contexts 1 decision aiding careful analysis of objectives careful analysis of attributes careful selection of alternatives availability of the DM 2 recommendation systems no analysis of objectives attributes as available alternatives as available limited access to the user

7 Introduction Two different contexts 1 decision aiding careful analysis of objectives careful analysis of attributes careful selection of alternatives availability of the DM 2 recommendation systems no analysis of objectives attributes as available alternatives as available limited access to the user

8 Introduction Basic model additive value function model n n x y v i (x i ) v i (y i ) i=1 i=1 x, y : alternatives x i : evaluation of alternative x on attribute i v i (x i ) : number underlies most existing MCDM techniques Underlying theory: conjoint measurement Economics (Debreu, 1960) Psychology (Luce & Tukey, 1964) tools to help structure preferences

9 Introduction Basic model additive value function model n n x y v i (x i ) v i (y i ) i=1 i=1 x, y : alternatives x i : evaluation of alternative x on attribute i v i (x i ) : number underlies most existing MCDM techniques Underlying theory: conjoint measurement Economics (Debreu, 1960) Psychology (Luce & Tukey, 1964) tools to help structure preferences

10 Introduction Basic model additive value function model n n x y v i (x i ) v i (y i ) i=1 i=1 x, y : alternatives x i : evaluation of alternative x on attribute i v i (x i ) : number underlies most existing MCDM techniques Underlying theory: conjoint measurement Economics (Debreu, 1960) Psychology (Luce & Tukey, 1964) tools to help structure preferences

11 Outline: Classical theory 1 An aside: measurement in Physics 2 An example: even swaps 3 Notation 4 Additive value functions: outline of theory

12 Outline: Classical theory 1 An aside: measurement in Physics 2 An example: even swaps 3 Notation 4 Additive value functions: outline of theory

13 Outline: Classical theory 1 An aside: measurement in Physics 2 An example: even swaps 3 Notation 4 Additive value functions: outline of theory

14 Outline: Classical theory 1 An aside: measurement in Physics 2 An example: even swaps 3 Notation 4 Additive value functions: outline of theory

15 Outline: Extensions 5 Models with interactions 6 Ordinal models

16 Outline: Extensions 5 Models with interactions 6 Ordinal models

17 Part I Classical theory: conjoint measurement

18 Outline 1 An aside: measurement in Physics 2 An example: even swaps 3 Notation 4 Additive value functions: outline of theory

19 Aside: measurement of physical quantities Lonely individual on a desert island no tools, no books, no knowledge of Physics wants to rebuild a system of physical measures A collection a rigid straight rods problem: measuring the length of these rods pre-theoretical intuition length softness, beauty 3 main steps comparing objects creating and comparing new objects creating standard sequences

20 Aside: measurement of physical quantities Lonely individual on a desert island no tools, no books, no knowledge of Physics wants to rebuild a system of physical measures A collection a rigid straight rods problem: measuring the length of these rods pre-theoretical intuition length softness, beauty 3 main steps comparing objects creating and comparing new objects creating standard sequences

21 Aside: measurement of physical quantities Lonely individual on a desert island no tools, no books, no knowledge of Physics wants to rebuild a system of physical measures A collection a rigid straight rods problem: measuring the length of these rods pre-theoretical intuition length softness, beauty 3 main steps comparing objects creating and comparing new objects creating standard sequences

22 Step 1: comparing objects experimental to conclude which rod has more length rods side by side on the same horizontal plane a a b b a a b b

23 Step 1: comparing objects experimental to conclude which rod has more length rods side by side on the same horizontal plane a a b b a a b b

24 Step 1: comparing objects experimental to conclude which rod has more length rods side by side on the same horizontal plane a a b b a a b b

25 Comparing objects Results a b: extremity of rod a is higher than extremity of rod b a b: extremity of rod a is as high as extremity of rod b Expected properties a b, a b or b a is asymmetric is symmetric is transitive is transitive and combine nicely a b and b c a c a b and b c a c

26 Comparing objects Results a b: extremity of rod a is higher than extremity of rod b a b: extremity of rod a is as high as extremity of rod b Expected properties a b, a b or b a is asymmetric is symmetric is transitive is transitive and combine nicely a b and b c a c a b and b c a c

27 Comparing objects Results a b: extremity of rod a is higher than extremity of rod b a b: extremity of rod a is as high as extremity of rod b Expected properties a b, a b or b a is asymmetric is symmetric is transitive is transitive and combine nicely a b and b c a c a b and b c a c

28 Comparing objects Results a b: extremity of rod a is higher than extremity of rod b a b: extremity of rod a is as high as extremity of rod b Expected properties a b, a b or b a is asymmetric is symmetric is transitive is transitive and combine nicely a b and b c a c a b and b c a c

29 Comparing objects Results a b: extremity of rod a is higher than extremity of rod b a b: extremity of rod a is as high as extremity of rod b Expected properties a b, a b or b a is asymmetric is symmetric is transitive is transitive and combine nicely a b and b c a c a b and b c a c

30 Comparing objects Results a b: extremity of rod a is higher than extremity of rod b a b: extremity of rod a is as high as extremity of rod b Expected properties a b, a b or b a is asymmetric is symmetric is transitive is transitive and combine nicely a b and b c a c a b and b c a c

31 Comparing objects Summary of experiments binary relation = that is a weak order complete (a b or b a) transitive (a b and b c a c) Consequences associate a real number Φ(a) to each object a the comparison of numbers faithfully reflects the results of experiments a b Φ(a) > Φ(b) a b Φ(a) = Φ(b) the function Φ defines an ordinal scale applying an increasing transformation to Φ leads to a scale that has the same properties any two scales having the same properties are related by an increasing transformation

32 Comparing objects Summary of experiments binary relation = that is a weak order complete (a b or b a) transitive (a b and b c a c) Consequences associate a real number Φ(a) to each object a the comparison of numbers faithfully reflects the results of experiments a b Φ(a) > Φ(b) a b Φ(a) = Φ(b) the function Φ defines an ordinal scale applying an increasing transformation to Φ leads to a scale that has the same properties any two scales having the same properties are related by an increasing transformation go faster

33 Comments Nature of the scale Φ is quite far from a full-blown measure of length... useful though since it allows the experiments to be done only once Hypotheses are stringent highly precise comparisons several practical problems any two objects can be compared connections between experiments comparisons may vary in time idealization of the measurement process

34 Comments Nature of the scale Φ is quite far from a full-blown measure of length... useful though since it allows the experiments to be done only once Hypotheses are stringent highly precise comparisons several practical problems any two objects can be compared connections between experiments comparisons may vary in time idealization of the measurement process

35 Step 2: creating and comparing new objects use the available objects to create new ones concatenate objects by placing two or more rods in a row b d a c a b c d a b c d

36 Step 2: creating and comparing new objects use the available objects to create new ones concatenate objects by placing two or more rods in a row b d a c a b c d a b c d

37 Concatenation we want to be able to deduce Φ(a b) from Φ(a) and Φ(b) simplest requirement Φ(a b) = Φ(a) + Φ(b) monotonicity constraints a b and c d a c b d

38 Concatenation we want to be able to deduce Φ(a b) from Φ(a) and Φ(b) simplest requirement Φ(a b) = Φ(a) + Φ(b) monotonicity constraints a b and c d a c b d go faster

39 Example five rods: r 1, r 2,..., r 5 we may only concatenate two rods (space reasons) we may only experiment with different rods data: r 1 r 5 r 3 r 4 r 1 r 2 r 5 r 4 r 3 r 2 r 1 all constraints are satisfied: weak ordering and monotonicity

40 Example r 1 r 5 r 3 r 4 r 1 r 2 r 5 r 4 r 3 r 2 r 1 Φ Φ Φ r r r r r Φ, Φ and Φ are equally good to compare simple rods only Φ and Φ capture the comparison of concatenated rods going from Φ to Φ does not involve a change of units it is tempting to use Φ or Φ to infer comparisons that have not been performed... disappointing Φ : r 2 r 3 r 1 r 4 Φ : r 2 r 3 r 1 r 4

41 Example r 1 r 5 r 3 r 4 r 1 r 2 r 5 r 4 r 3 r 2 r 1 Φ Φ Φ r r r r r Φ, Φ and Φ are equally good to compare simple rods only Φ and Φ capture the comparison of concatenated rods going from Φ to Φ does not involve a change of units it is tempting to use Φ or Φ to infer comparisons that have not been performed... disappointing Φ : r 2 r 3 r 1 r 4 Φ : r 2 r 3 r 1 r 4

42 Example r 1 r 5 r 3 r 4 r 1 r 2 r 5 r 4 r 3 r 2 r 1 Φ Φ Φ r r r r r Φ, Φ and Φ are equally good to compare simple rods only Φ and Φ capture the comparison of concatenated rods going from Φ to Φ does not involve a change of units it is tempting to use Φ or Φ to infer comparisons that have not been performed... disappointing Φ : r 2 r 3 r 1 r 4 Φ : r 2 r 3 r 1 r 4

43 Example r 1 r 5 r 3 r 4 r 1 r 2 r 5 r 4 r 3 r 2 r 1 Φ Φ Φ r r r r r Φ, Φ and Φ are equally good to compare simple rods only Φ and Φ capture the comparison of concatenated rods going from Φ to Φ does not involve a change of units it is tempting to use Φ or Φ to infer comparisons that have not been performed... disappointing Φ : r 2 r 3 r 1 r 4 Φ : r 2 r 3 r 1 r 4

44 Step 3: creating and using standard sequences choose a standard rod be able to build perfect copies of the standard concatenate the standard rod with its perfects copies a S(k) s 8 s 7 s 6 s 5 s 4 s 3 s 2 s 1 S(8) a S(7) Φ(s) = 1 7 < Φ(a) < 8

45 Step 3: creating and using standard sequences choose a standard rod be able to build perfect copies of the standard concatenate the standard rod with its perfects copies a S(k) s 8 s 7 s 6 s 5 s 4 s 3 s 2 s 1 S(8) a S(7) Φ(s) = 1 7 < Φ(a) < 8

46 Step 3: creating and using standard sequences choose a standard rod be able to build perfect copies of the standard concatenate the standard rod with its perfects copies a S(k) s 8 s 7 s 6 s 5 s 4 s 3 s 2 s 1 S(8) a S(7) Φ(s) = 1 7 < Φ(a) < 8

47 Convergence First method choose a smaller standard rod repeat the process Second method prepare a perfect copy of the object concatenate the object with its perfect copy compare the doubled object to the original standard sequence repeat the process

48 Convergence First method choose a smaller standard rod repeat the process Second method prepare a perfect copy of the object concatenate the object with its perfect copy compare the doubled object to the original standard sequence repeat the process

49 Summary Extensive measurement Krantz, Luce, Suppes & Tversky (1971, chap. 3) 4 Ingredients 1 well-behaved relations and 2 concatenation operation 3 consistency requirements linking, and 4 ability to prepare perfect copies of some objects in order to build standard sequences Neglected problems many!

50 Summary Extensive measurement Krantz, Luce, Suppes & Tversky (1971, chap. 3) 4 Ingredients 1 well-behaved relations and 2 concatenation operation 3 consistency requirements linking, and 4 ability to prepare perfect copies of some objects in order to build standard sequences Neglected problems many!

51 Summary Extensive measurement Krantz, Luce, Suppes & Tversky (1971, chap. 3) 4 Ingredients 1 well-behaved relations and 2 concatenation operation 3 consistency requirements linking, and 4 ability to prepare perfect copies of some objects in order to build standard sequences Neglected problems many!

52 Question Can this be applied outside Physics? no concatenation operation (intelligence!)

53 What is conjoint measurement? Conjoint measurement mimicking the operations of extensive measurement when there are no concatenation operation readily available when several dimensions are involved Seems overly ambitious let us start with a simple example

54 What is conjoint measurement? Conjoint measurement mimicking the operations of extensive measurement when there are no concatenation operation readily available when several dimensions are involved Seems overly ambitious let us start with a simple example

55 Outline 1 An aside: measurement in Physics 2 An example: even swaps 3 Notation 4 Additive value functions: outline of theory

56 Example: Hammond, Keeney & Raiffa Choice of an office to rent five locations have been identified five attributes are being considered Commute time (minutes) Clients: percentage of clients living close to the office Services: ad hoc scale A (all facilities), B (telephone and fax), C (no facility) Size: square feet ( 0.1 m 2 ) Cost: $ per month Attributes Commute, Size and Cost are natural attributes Clients is a proxy attribute Services is a constructed attribute

57 Example: Hammond, Keeney & Raiffa Choice of an office to rent five locations have been identified five attributes are being considered Commute time (minutes) Clients: percentage of clients living close to the office Services: ad hoc scale A (all facilities), B (telephone and fax), C (no facility) Size: square feet ( 0.1 m 2 ) Cost: $ per month Attributes Commute, Size and Cost are natural attributes Clients is a proxy attribute Services is a constructed attribute

58 Data a b c d e Commute Clients Services A B C A C Size Cost Hypotheses and context a single cooperative DM choice of a single office ceteris paribus reasoning seems possible Commute: decreasing Clients: increasing Services: increasing Size: increasing Cost: decreasing dominance has meaning

59 Data a b c d e Commute Clients Services A B C A C Size Cost Hypotheses and context a single cooperative DM choice of a single office ceteris paribus reasoning seems possible Commute: decreasing Clients: increasing Services: increasing Size: increasing Cost: decreasing dominance has meaning

60 Data a b c d e Commute Clients Services A B C A C Size Cost Hypotheses and context a single cooperative DM choice of a single office ceteris paribus reasoning seems possible Commute: decreasing Clients: increasing Services: increasing Size: increasing Cost: decreasing dominance has meaning

61 Data a b c d e Commute Clients Services A B C A C Size Cost Hypotheses and context a single cooperative DM choice of a single office ceteris paribus reasoning seems possible Commute: decreasing Clients: increasing Services: increasing Size: increasing Cost: decreasing dominance has meaning

62 a b c d e Commute Clients Services A B C A C Size Cost b dominates alternative e d is close to dominating a divide and conquer: dropping alternatives drop a and e

63 a b c d e Commute Clients Services A B C A C Size Cost b dominates alternative e d is close to dominating a divide and conquer: dropping alternatives drop a and e

64 a b c d e Commute Clients Services A B C A C Size Cost b dominates alternative e d is close to dominating a divide and conquer: dropping alternatives drop a and e

65 b c d Commute Clients Services B C A Size Cost no more dominance assessing tradeoffs all alternatives except c have a common evaluation on Commute modify c in order to bring it to this level starting with c, what is the gain on Clients that would exactly compensate a loss of 5 min on Commute? difficult but central question

66 b c d Commute Clients Services B C A Size Cost no more dominance assessing tradeoffs all alternatives except c have a common evaluation on Commute modify c in order to bring it to this level starting with c, what is the gain on Clients that would exactly compensate a loss of 5 min on Commute? difficult but central question

67 b c d Commute Clients Services B C A Size Cost no more dominance assessing tradeoffs all alternatives except c have a common evaluation on Commute modify c in order to bring it to this level starting with c, what is the gain on Clients that would exactly compensate a loss of 5 min on Commute? difficult but central question

68 b c d Commute Clients Services B C A Size Cost no more dominance assessing tradeoffs all alternatives except c have a common evaluation on Commute modify c in order to bring it to this level starting with c, what is the gain on Clients that would exactly compensate a loss of 5 min on Commute? difficult but central question

69 c c Commute Clients δ Services C C Size Cost find δ such that c c Answer for δ = 8, I am indifferent between c and c replace c with c

70 c c Commute Clients δ Services C C Size Cost find δ such that c c Answer for δ = 8, I am indifferent between c and c replace c with c

71 b c d Commute Clients Services B C A Size Cost all alternatives have a common evaluation on Commute divide and conquer: dropping attributes drop attribute Commute b c d Clients Services B C A Size Cost

72 b c d Commute Clients Services B C A Size Cost all alternatives have a common evaluation on Commute divide and conquer: dropping attributes drop attribute Commute b c d Clients Services B C A Size Cost

73 b c d Commute Clients Services B C A Size Cost all alternatives have a common evaluation on Commute divide and conquer: dropping attributes drop attribute Commute b c d Clients Services B C A Size Cost

74 b c d Clients Services B C A Size Cost check again for dominance unfruitful assess new tradeoffs neutralize Service using Cost as reference

75 b c d Clients Services B C A Size Cost check again for dominance unfruitful assess new tradeoffs neutralize Service using Cost as reference

76 b c d Clients Services B C A Size Cost check again for dominance unfruitful assess new tradeoffs neutralize Service using Cost as reference

77 b c d Clients Services B C A Size Cost Questions what maximal increase in monthly cost would you be prepared to pay to go from C to B on service for c? answer: 250 $ what minimal decrease in monthly cost would you ask if we go from A to B on service for d? answer: 100 $ b c c d d Clients Services B C B A B Size Cost

78 b c d Clients Services B C A Size Cost Questions what maximal increase in monthly cost would you be prepared to pay to go from C to B on service for c? answer: 250 $ what minimal decrease in monthly cost would you ask if we go from A to B on service for d? answer: 100 $ b c c d d Clients Services B C B A B Size Cost

79 b c d Clients Services B C A Size Cost Questions what maximal increase in monthly cost would you be prepared to pay to go from C to B on service for c? answer: 250 $ what minimal decrease in monthly cost would you ask if we go from A to B on service for d? answer: 100 $ b c c d d Clients Services B C B A B Size Cost

80 b c d Clients Services B C A Size Cost Questions what maximal increase in monthly cost would you be prepared to pay to go from C to B on service for c? answer: 250 $ what minimal decrease in monthly cost would you ask if we go from A to B on service for d? answer: 100 $ b c c d d Clients Services B C B A B Size Cost

81 replacing c with c replacing d with d dropping Service b c d Clients Size Cost checking for dominance: c is dominated by b c can be dropped

82 replacing c with c replacing d with d dropping Service b c d Clients Size Cost checking for dominance: c is dominated by b c can be dropped

83 dropping c b d Clients Size Cost no dominance question: starting with b what is the additional cost that you would be prepared to pay to increase size by 250? answer: 250 $ b b d Clients Size Cost

84 dropping c b d Clients Size Cost no dominance question: starting with b what is the additional cost that you would be prepared to pay to increase size by 250? answer: 250 $ b b d Clients Size Cost

85 dropping c b d Clients Size Cost no dominance question: starting with b what is the additional cost that you would be prepared to pay to increase size by 250? answer: 250 $ b b d Clients Size Cost

86 dropping c b d Clients Size Cost no dominance question: starting with b what is the additional cost that you would be prepared to pay to increase size by 250? answer: 250 $ b b d Clients Size Cost

87 replace b with b drop Size b d Clients Size Cost b d Clients Cost check for dominance d dominates b Conclusion Recommend d as the final choice

88 replace b with b drop Size b d Clients Size Cost b d Clients Cost check for dominance d dominates b Conclusion Recommend d as the final choice

89 replace b with b drop Size b d Clients Size Cost b d Clients Cost check for dominance d dominates b Conclusion Recommend d as the final choice

90 Summary Remarks very simple process process entirely governed by and no question on intensity of preference notice that importance plays absolutely no rôle why be interested in something more complex? Problems set of alternative is small many questions otherwise output is not a preference model if new alternatives appear, the process should be restarted what are the underlying hypotheses?

91 Summary Remarks very simple process process entirely governed by and no question on intensity of preference notice that importance plays absolutely no rôle why be interested in something more complex? Problems set of alternative is small many questions otherwise output is not a preference model if new alternatives appear, the process should be restarted what are the underlying hypotheses?

92 Summary Remarks very simple process process entirely governed by and no question on intensity of preference notice that importance plays absolutely no rôle why be interested in something more complex? Problems set of alternative is small many questions otherwise output is not a preference model if new alternatives appear, the process should be restarted what are the underlying hypotheses?

93 Summary Remarks very simple process process entirely governed by and no question on intensity of preference notice that importance plays absolutely no rôle why be interested in something more complex? Problems set of alternative is small many questions otherwise output is not a preference model if new alternatives appear, the process should be restarted what are the underlying hypotheses?

94 Summary Remarks very simple process process entirely governed by and no question on intensity of preference notice that importance plays absolutely no rôle why be interested in something more complex? Problems set of alternative is small many questions otherwise output is not a preference model if new alternatives appear, the process should be restarted what are the underlying hypotheses?

95 Summary Remarks very simple process process entirely governed by and no question on intensity of preference notice that importance plays absolutely no rôle why be interested in something more complex? Problems set of alternative is small many questions otherwise output is not a preference model if new alternatives appear, the process should be restarted what are the underlying hypotheses?

96 Summary Remarks very simple process process entirely governed by and no question on intensity of preference notice that importance plays absolutely no rôle why be interested in something more complex? Problems set of alternative is small many questions otherwise output is not a preference model if new alternatives appear, the process should be restarted what are the underlying hypotheses?

97 Monsieur Jourdain doing conjoint measurement Similarity with extensive measurement : preference, : indifference we have implicitly supposed that they combine nicely Recommendation: d we should be able to prove that d a, d b, d c and d e dominance: b e and d a tradeoffs + dominance: b c, c c, c c, d d, b b, d b d a, b e c c, c c, b c b c d d, b b, d b d b

98 Monsieur Jourdain doing conjoint measurement Similarity with extensive measurement : preference, : indifference we have implicitly supposed that they combine nicely Recommendation: d we should be able to prove that d a, d b, d c and d e dominance: b e and d a tradeoffs + dominance: b c, c c, c c, d d, b b, d b d a, b e c c, c c, b c b c d d, b b, d b d b

99 Monsieur Jourdain doing conjoint measurement Similarity with extensive measurement : preference, : indifference we have implicitly supposed that they combine nicely Recommendation: d we should be able to prove that d a, d b, d c and d e dominance: b e and d a tradeoffs + dominance: b c, c c, c c, d d, b b, d b d a, b e c c, c c, b c b c d d, b b, d b d b

100 Monsieur Jourdain doing conjoint measurement Similarity with extensive measurement : preference, : indifference we have implicitly supposed that they combine nicely Recommendation: d we should be able to prove that d a, d b, d c and d e dominance: b e and d a tradeoffs + dominance: b c, c c, c c, d d, b b, d b d a, b e c c, c c, b c b c d d, b b, d b d b

101 Monsieur Jourdain doing conjoint measurement Similarity with extensive measurement : preference, : indifference we have implicitly supposed that they combine nicely Recommendation: d we should be able to prove that d a, d b, d c and d e dominance: b e and d a tradeoffs + dominance: b c, c c, c c, d d, b b, d b d a, b e c c, c c, b c b c d d, b b, d b d b

102 Monsieur Jourdain doing conjoint measurement Similarity with extensive measurement : preference, : indifference we have implicitly supposed that they combine nicely Recommendation: d we should be able to prove that d a, d b, d c and d e dominance: b e and d a tradeoffs + dominance: b c, c c, c c, d d, b b, d b d a, b e c c, c c, b c b c d d, b b, d b d b

103 Monsieur Jourdain doing conjoint measurement Similarity with extensive measurement : preference, : indifference we have implicitly supposed that they combine nicely Recommendation: d we should be able to prove that d a, d b, d c and d e dominance: b e and d a tradeoffs + dominance: b c, c c, c c, d d, b b, d b d a, b e c c, c c, b c b c d d, b b, d b d b

104 Monsieur Jourdain doing conjoint measurement Similarity with extensive measurement : preference, : indifference we have implicitly supposed that they combine nicely Recommendation: d we should be able to prove that d a, d b, d c and d e dominance: b e and d a tradeoffs + dominance: b c, c c, c c, d d, b b, d b d a, b e c c, c c, b c b c d d, b b, d b d b

105 Monsieur Jourdain doing conjoint measurement OK... but where are the standard sequences? hidden... but really there! standard sequence for length: objects that have exactly the same length tradeoffs: preference intervals on distinct attributes that have the same length c c [25, 20] on Commute has the same length as [70, 78] on Client c c f f Commute Clients Services C C C C Size Cost [70, 78] has the same length [78, 82] on Client

106 Monsieur Jourdain doing conjoint measurement OK... but where are the standard sequences? hidden... but really there! standard sequence for length: objects that have exactly the same length tradeoffs: preference intervals on distinct attributes that have the same length c c [25, 20] on Commute has the same length as [70, 78] on Client c c f f Commute Clients Services C C C C Size Cost [70, 78] has the same length [78, 82] on Client

107 Monsieur Jourdain doing conjoint measurement OK... but where are the standard sequences? hidden... but really there! standard sequence for length: objects that have exactly the same length tradeoffs: preference intervals on distinct attributes that have the same length c c [25, 20] on Commute has the same length as [70, 78] on Client c c f f Commute Clients Services C C C C Size Cost [70, 78] has the same length [78, 82] on Client

108 Monsieur Jourdain doing conjoint measurement OK... but where are the standard sequences? hidden... but really there! standard sequence for length: objects that have exactly the same length tradeoffs: preference intervals on distinct attributes that have the same length c c [25, 20] on Commute has the same length as [70, 78] on Client c c f f Commute Clients Services C C C C Size Cost [70, 78] has the same length [78, 82] on Client

109 Monsieur Jourdain doing conjoint measurement OK... but where are the standard sequences? hidden... but really there! standard sequence for length: objects that have exactly the same length tradeoffs: preference intervals on distinct attributes that have the same length c c [25, 20] on Commute has the same length as [70, 78] on Client c c f f Commute Clients Services C C C C Size Cost [70, 78] has the same length [78, 82] on Client

110 Outline 1 An aside: measurement in Physics 2 An example: even swaps 3 Notation 4 Additive value functions: outline of theory

111 Setting N = {1, 2,..., n} set of attributes X i : set of possible levels on the ith attribute X = n i=1 X i: set of all conceivable alternatives X include the alternatives under study... and many others J N: subset of attributes X J = j J X j, X J = j / J X j (x J, y J ) X (x i, y i ) X : binary relation on X: at least as good as x y x y and Not[y x] x y x y and y x

112 Setting N = {1, 2,..., n} set of attributes X i : set of possible levels on the ith attribute X = n i=1 X i: set of all conceivable alternatives X include the alternatives under study... and many others J N: subset of attributes X J = j J X j, X J = j / J X j (x J, y J ) X (x i, y i ) X : binary relation on X: at least as good as x y x y and Not[y x] x y x y and y x

113 Setting N = {1, 2,..., n} set of attributes X i : set of possible levels on the ith attribute X = n i=1 X i: set of all conceivable alternatives X include the alternatives under study... and many others J N: subset of attributes X J = j J X j, X J = j / J X j (x J, y J ) X (x i, y i ) X : binary relation on X: at least as good as x y x y and Not[y x] x y x y and y x

114 Preference relations on Cartesian products Applications Economics: consumers comparing bundles of goods Decision under uncertainty: consequences in several states Inter-temporal decision making: consequences at several moments in time Inequality measurement: distribution of wealth across individuals Decision making with multiple attributes in all other cases, the Cartesian product is homogeneous

115 Preference relations on Cartesian products Applications Economics: consumers comparing bundles of goods Decision under uncertainty: consequences in several states Inter-temporal decision making: consequences at several moments in time Inequality measurement: distribution of wealth across individuals Decision making with multiple attributes in all other cases, the Cartesian product is homogeneous

116 Preference relations on Cartesian products Applications Economics: consumers comparing bundles of goods Decision under uncertainty: consequences in several states Inter-temporal decision making: consequences at several moments in time Inequality measurement: distribution of wealth across individuals Decision making with multiple attributes in all other cases, the Cartesian product is homogeneous

117 What will be ignored today Ignored structuring of objectives from objectives to attributes adequate family of attributes risk, uncertainty, imprecision Keeney s view fundamental objectives: why? means objectives: how?

118 What will be ignored today Ignored structuring of objectives from objectives to attributes adequate family of attributes risk, uncertainty, imprecision skip examples Keeney s view fundamental objectives: why? means objectives: how?

119 What will be ignored today Ignored structuring of objectives from objectives to attributes adequate family of attributes risk, uncertainty, imprecision skip examples Keeney s view fundamental objectives: why? means objectives: how?

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126 Marginal preference and independence Marginal preferences J N: subset of attributes J marginal preference relation induced by on X J x J J y J (x J, z J ) (y J, z J ), for all z J X J Independence J is independent for if [(x J, z J ) (y J, z J ), for some z J X J ] x J J y J common levels on attributes other than J do not affect preference Separability J is separable for if [(x J, z J ) (y J, z J ), for some z J X J ] x J J y J varying common levels on attributes other than J do reverse strict preference

127 Marginal preference and independence Marginal preferences J N: subset of attributes J marginal preference relation induced by on X J x J J y J (x J, z J ) (y J, z J ), for all z J X J Independence J is independent for if [(x J, z J ) (y J, z J ), for some z J X J ] x J J y J common levels on attributes other than J do not affect preference Separability J is separable for if [(x J, z J ) (y J, z J ), for some z J X J ] x J J y J varying common levels on attributes other than J do reverse strict preference

128 Marginal preference and independence Marginal preferences J N: subset of attributes J marginal preference relation induced by on X J x J J y J (x J, z J ) (y J, z J ), for all z J X J Independence J is independent for if [(x J, z J ) (y J, z J ), for some z J X J ] x J J y J common levels on attributes other than J do not affect preference Separability J is separable for if [(x J, z J ) (y J, z J ), for some z J X J ] x J J y J varying common levels on attributes other than J do reverse strict preference

129 Independence Definition for all i N, {i} is independent, is weakly independent for all J N, J is independent, is independent Proposition Let be a weakly independent weak order on X = n i=1 X i. Then: i is a weak order on X i [x i i y i, for all i N] x y [x i i y i, for all i N and x j j y j for some j N] x y for all x, y X Dominance as soon as I have a weakly independent weak order dominance arguments apply

130 Independence Definition for all i N, {i} is independent, is weakly independent for all J N, J is independent, is independent Proposition Let be a weakly independent weak order on X = n i=1 X i. Then: i is a weak order on X i [x i i y i, for all i N] x y [x i i y i, for all i N and x j j y j for some j N] x y for all x, y X Dominance as soon as I have a weakly independent weak order dominance arguments apply

131 Independence Definition for all i N, {i} is independent, is weakly independent for all J N, J is independent, is independent Proposition Let be a weakly independent weak order on X = n i=1 X i. Then: i is a weak order on X i [x i i y i, for all i N] x y [x i i y i, for all i N and x j j y j for some j N] x y for all x, y X Dominance as soon as I have a weakly independent weak order dominance arguments apply

132 Independence in practice Independence it is easy to imagine examples in which independence is violated Main course and Wine example it is nearly hopeless to try to work if weak independence (at least weak separability) is not satisfied some (e.g., R. L. Keeney) think that the same is true for independence in all cases if independence is violated, things get complicated decision aiding vs AI May be excessive much more on independence this afternoon

133 Independence in practice Independence it is easy to imagine examples in which independence is violated Main course and Wine example it is nearly hopeless to try to work if weak independence (at least weak separability) is not satisfied some (e.g., R. L. Keeney) think that the same is true for independence in all cases if independence is violated, things get complicated decision aiding vs AI May be excessive much more on independence this afternoon

134 Independence in practice Independence it is easy to imagine examples in which independence is violated Main course and Wine example it is nearly hopeless to try to work if weak independence (at least weak separability) is not satisfied some (e.g., R. L. Keeney) think that the same is true for independence in all cases if independence is violated, things get complicated decision aiding vs AI May be excessive much more on independence this afternoon

135 Independence in practice Independence it is easy to imagine examples in which independence is violated Main course and Wine example it is nearly hopeless to try to work if weak independence (at least weak separability) is not satisfied some (e.g., R. L. Keeney) think that the same is true for independence in all cases if independence is violated, things get complicated decision aiding vs AI May be excessive much more on independence this afternoon

136 Independence in practice Independence it is easy to imagine examples in which independence is violated Main course and Wine example it is nearly hopeless to try to work if weak independence (at least weak separability) is not satisfied some (e.g., R. L. Keeney) think that the same is true for independence in all cases if independence is violated, things get complicated decision aiding vs AI May be excessive much more on independence this afternoon

137 Outline 1 An aside: measurement in Physics 2 An example: even swaps 3 Notation 4 Additive value functions: outline of theory The case of 2 attributes More than 2 attributes

138 Outline of theory: 2 attributes Question suppose I can observe on X = X 1 X 2 what must be supposed to guarantee that I can represent in the additive value function model v 1 : X 1 R v 2 : X 2 R (x 1, x 2 ) (y 1, y 2 ) v 1 (x 1 ) + v 2 (x 2 ) v 1 (y 1 ) + v 2 (y 2 ) must be an independent weak order Method try building standard sequences and see if it works!

139 Outline of theory: 2 attributes Question suppose I can observe on X = X 1 X 2 what must be supposed to guarantee that I can represent in the additive value function model v 1 : X 1 R v 2 : X 2 R (x 1, x 2 ) (y 1, y 2 ) v 1 (x 1 ) + v 2 (x 2 ) v 1 (y 1 ) + v 2 (y 2 ) must be an independent weak order Method try building standard sequences and see if it works!

140 Outline of theory: 2 attributes Question suppose I can observe on X = X 1 X 2 what must be supposed to guarantee that I can represent in the additive value function model v 1 : X 1 R v 2 : X 2 R (x 1, x 2 ) (y 1, y 2 ) v 1 (x 1 ) + v 2 (x 2 ) v 1 (y 1 ) + v 2 (y 2 ) must be an independent weak order Method try building standard sequences and see if it works!

141 Why an additive model? Answer v 1 and v 2 will be built so that additivity holds equivalent multiplicative model (x 1, x 2 ) (y 1, y 2 ) w 1 (x 1 )w 2 (x 2 ) w 1 (y 1 )w 2 (y 2 ) w 1 = exp(v 1 ) w 2 = exp(v 2 )

142 Why an additive model? Answer v 1 and v 2 will be built so that additivity holds equivalent multiplicative model (x 1, x 2 ) (y 1, y 2 ) w 1 (x 1 )w 2 (x 2 ) w 1 (y 1 )w 2 (y 2 ) w 1 = exp(v 1 ) w 2 = exp(v 2 )

143 Why an additive model? Answer v 1 and v 2 will be built so that additivity holds equivalent multiplicative model (x 1, x 2 ) (y 1, y 2 ) w 1 (x 1 )w 2 (x 2 ) w 1 (y 1 )w 2 (y 2 ) w 1 = exp(v 1 ) w 2 = exp(v 2 )

144 Uniqueness Important observation Suppose that there are v 1 and v 2 such that (x 1, x 2 ) (y 1, y 2 ) v 1 (x 1 ) + v 2 (x 2 ) v 1 (y 1 ) + v 2 (y 2 ) If α > 0 w 1 = αv 1 + β 1 w 2 = αv 2 + β 2 is also a valid representation Consequences fixing v 1 (x 1 ) = v 2 (x 2 ) = 0 is harmless fixing v 1 (y 1 ) = 1 is harmless if y 1 1 x 1

145 Uniqueness Important observation Suppose that there are v 1 and v 2 such that (x 1, x 2 ) (y 1, y 2 ) v 1 (x 1 ) + v 2 (x 2 ) v 1 (y 1 ) + v 2 (y 2 ) If α > 0 w 1 = αv 1 + β 1 w 2 = αv 2 + β 2 is also a valid representation Consequences fixing v 1 (x 1 ) = v 2 (x 2 ) = 0 is harmless fixing v 1 (y 1 ) = 1 is harmless if y 1 1 x 1

146 Standard sequences Preliminaries choose arbitrarily two levels x 0 1, x 1 1 X 1 make sure that x x 0 1 choose arbitrarily one level x 0 2 X 2 (x 0 1, x 0 2) X is the reference point (origin) the preference interval [x 0 1, x 1 1] is the unit

147 Building a standard sequence on X 2 find a preference interval on X 2 that has the same length as the reference interval [x 0 1, x 1 1] find x 1 2 such that (x 0 1, x 1 2) (x 1 1, x 0 2) v 1 (x 0 1) + v 2 (x 1 2) = v 1 (x 1 1) + v 2 (x 0 2) so that v 2 (x 1 2) v 2 (x 0 2) = v 1 (x 1 1) v 1 (x 0 1) the structure of X 2 has to be rich enough

148 Building a standard sequence on X 2 find a preference interval on X 2 that has the same length as the reference interval [x 0 1, x 1 1] find x 1 2 such that (x 0 1, x 1 2) (x 1 1, x 0 2) v 1 (x 0 1) + v 2 (x 1 2) = v 1 (x 1 1) + v 2 (x 0 2) so that v 2 (x 1 2) v 2 (x 0 2) = v 1 (x 1 1) v 1 (x 0 1) the structure of X 2 has to be rich enough

149 Building a standard sequence on X 2 find a preference interval on X 2 that has the same length as the reference interval [x 0 1, x 1 1] find x 1 2 such that (x 0 1, x 1 2) (x 1 1, x 0 2) v 1 (x 0 1) + v 2 (x 1 2) = v 1 (x 1 1) + v 2 (x 0 2) so that v 2 (x 1 2) v 2 (x 0 2) = v 1 (x 1 1) v 1 (x 0 1) the structure of X 2 has to be rich enough

150 Building a standard sequence on X 2 find a preference interval on X 2 that has the same length as the reference interval [x 0 1, x 1 1] find x 1 2 such that (x 0 1, x 1 2) (x 1 1, x 0 2) v 1 (x 0 1) + v 2 (x 1 2) = v 1 (x 1 1) + v 2 (x 0 2) so that v 2 (x 1 2) v 2 (x 0 2) = v 1 (x 1 1) v 1 (x 0 1) the structure of X 2 has to be rich enough

151 Standard sequences Consequences it can be supposed that (x 0 1, x 1 2) (x 1 1, x 0 2) v 2 (x 1 2) v 2 (x 0 2) = v 1 (x 1 1) v 1 (x 0 1) v 1 (x 0 1) = v 2 (x 0 2) = 0 v 1 (x 1 1) = 1 v 2 (x 1 2) = 1

152 Standard sequences Consequences it can be supposed that (x 0 1, x 1 2) (x 1 1, x 0 2) v 2 (x 1 2) v 2 (x 0 2) = v 1 (x 1 1) v 1 (x 0 1) v 1 (x 0 1) = v 2 (x 0 2) = 0 v 1 (x 1 1) = 1 v 2 (x 1 2) = 1

153 Going on (x 0 1, x 1 2) (x 1 1, x 0 2) (x 0 1, x 2 2) (x 1 1, x 1 2) (x 0 1, x 3 2) (x 1 1, x 2 2)... (x 0 1, x k 2) (x 1 1, x k 1 2 ) v 2 (x 1 2) v 2 (x 0 2) = v 1 (x 1 1) v 1 (x 0 1) = 1 v 2 (x 2 2) v 2 (x 1 2) = v 1 (x 1 1) v 1 (x 0 1) = 1 v 2 (x 3 2) v 2 (x 2 2) = v 1 (x 1 1) v 1 (x 0 1) = 1... v 2 (x k 2) v 2 (x k 1 2 ) = v 1 (x 1 1) v 1 (x 0 1) = 1 v 2 (x 2 2) = 2, v 2 (x 3 2) = 3,..., v 2 (x k 2) = k

154 Going on (x 0 1, x 1 2) (x 1 1, x 0 2) (x 0 1, x 2 2) (x 1 1, x 1 2) (x 0 1, x 3 2) (x 1 1, x 2 2)... (x 0 1, x k 2) (x 1 1, x k 1 2 ) v 2 (x 1 2) v 2 (x 0 2) = v 1 (x 1 1) v 1 (x 0 1) = 1 v 2 (x 2 2) v 2 (x 1 2) = v 1 (x 1 1) v 1 (x 0 1) = 1 v 2 (x 3 2) v 2 (x 2 2) = v 1 (x 1 1) v 1 (x 0 1) = 1... v 2 (x k 2) v 2 (x k 1 2 ) = v 1 (x 1 1) v 1 (x 0 1) = 1 v 2 (x 2 2) = 2, v 2 (x 3 2) = 3,..., v 2 (x k 2) = k

155 Going on (x 0 1, x 1 2) (x 1 1, x 0 2) (x 0 1, x 2 2) (x 1 1, x 1 2) (x 0 1, x 3 2) (x 1 1, x 2 2)... (x 0 1, x k 2) (x 1 1, x k 1 2 ) v 2 (x 1 2) v 2 (x 0 2) = v 1 (x 1 1) v 1 (x 0 1) = 1 v 2 (x 2 2) v 2 (x 1 2) = v 1 (x 1 1) v 1 (x 0 1) = 1 v 2 (x 3 2) v 2 (x 2 2) = v 1 (x 1 1) v 1 (x 0 1) = 1... v 2 (x k 2) v 2 (x k 1 2 ) = v 1 (x 1 1) v 1 (x 0 1) = 1 v 2 (x 2 2) = 2, v 2 (x 3 2) = 3,..., v 2 (x k 2) = k

156 X 2 x 4 2 x 3 2 x 2 2 x 1 2 x 0 2 x 0 1 x 1 1 X 1

157 X 2 x 4 2 x 3 2 x 2 2 x 1 2 x 0 2 x 0 1 x 1 1 X 1

158 X 2 x 4 2 x 3 2 x 2 2 x 1 2 x 0 2 x 0 1 x 1 1 X 1

159 X 2 x 4 2 x 3 2 x 2 2 x 1 2 x 0 2 x 0 1 x 1 1 X 1

160 X 2 x 4 2 x 3 2 x 2 2 x 1 2 x 0 2 x 0 1 x 1 1 X 1

161 Standard sequence Archimedean implicit hypothesis for length the standard sequence can reach any the length of any object x, y R, n N : x > ny a similar hypothesis has to hold here rough interpretation there are not infinitely liked or disliked consequences

162 Building a standard sequence on X 1 (x 2 1, x 0 2) (x 1 1, x 1 2) (x 3 1, x 0 2) (x 2 1, x 1 2)... (x k 1, x 0 2) (x k 1 1, x 1 2) v 1 (x 2 1) v 1 (x 1 1) = v 2 (x 1 2) v 2 (x 0 2) = 1 v 1 (x 3 1) v 1 (x 2 1) = v 2 (x 1 2) v 2 (x 0 2) = 1... v 1 (x k 1) v 1 (x k 1 1 ) = v 2 (x 1 2) v 2 (x 0 2) = 1 v 1 (x 2 1) = 2, v 1 (x 3 1) = 3,..., v 1 (x k 1) = k

163 Building a standard sequence on X 1 (x 2 1, x 0 2) (x 1 1, x 1 2) (x 3 1, x 0 2) (x 2 1, x 1 2)... (x k 1, x 0 2) (x k 1 1, x 1 2) v 1 (x 2 1) v 1 (x 1 1) = v 2 (x 1 2) v 2 (x 0 2) = 1 v 1 (x 3 1) v 1 (x 2 1) = v 2 (x 1 2) v 2 (x 0 2) = 1... v 1 (x k 1) v 1 (x k 1 1 ) = v 2 (x 1 2) v 2 (x 0 2) = 1 v 1 (x 2 1) = 2, v 1 (x 3 1) = 3,..., v 1 (x k 1) = k

164 Building a standard sequence on X 1 (x 2 1, x 0 2) (x 1 1, x 1 2) (x 3 1, x 0 2) (x 2 1, x 1 2)... (x k 1, x 0 2) (x k 1 1, x 1 2) v 1 (x 2 1) v 1 (x 1 1) = v 2 (x 1 2) v 2 (x 0 2) = 1 v 1 (x 3 1) v 1 (x 2 1) = v 2 (x 1 2) v 2 (x 0 2) = 1... v 1 (x k 1) v 1 (x k 1 1 ) = v 2 (x 1 2) v 2 (x 0 2) = 1 v 1 (x 2 1) = 2, v 1 (x 3 1) = 3,..., v 1 (x k 1) = k

165 X 2 x 1 2 x 0 2 x 0 1 x 1 1 x 2 1 x 3 1 x 4 1 X 1

166 X 2 x 4 2 x 3 2 x 2 2 x 1 2 x 0 2 x 0 1 x 1 1 x 2 1 x 3 1 x 4 1 X 1

167 X 2 x 4 2 x 3 2 x 2 2? x 1 2 x 0 2 x 0 1 x 1 1 x 2 1 x 3 1 x 4 1 X 1

168 Thomsen condition (x 1, x 2 ) (y 1, y 2 ) and (y 1, z 2 ) (z 1, x 2 ) (x 1, z 2 ) (z 1, y 2 ) X 2 z 2 y 2 x 2 y 1 x 1 z 1 X 1 Consequence there is an additive value function on the grid

169 X 2 x 4 2 x 3 2 x 2 2 x 1 2 x 0 2 x 0 1 x 1 1 x 2 1 x 3 1 x 4 1 X 1

170 X 2 x 4 2 x 3 2 x 2 2 x 1 2 x 0 2 x 0 1 x 1 1 x 2 1 x 3 1 x 4 1 X 1

171 Thomsen condition (x 1, x 2 ) (y 1, y 2 ) and (y 1, z 2 ) (z 1, x 2 ) (x 1, z 2 ) (z 1, y 2 ) X 2 z 2 y 2 x 2 y 1 x 1 z 1 X 1 Consequence there is an additive value function on the grid

172 X 2 x 4 2 x 3 2 x 2 2 x 1 2 x 0 2 x 0 1 x 1 1 x 2 1 x 3 1 x 4 1 x 5 1 X 1

173 Summary we have defined a grid there is an additive value function on the grid iterate the whole process with a denser grid

174 Summary we have defined a grid there is an additive value function on the grid iterate the whole process with a denser grid

175 Hypotheses Archimedean: every strictly bounded standard sequence is finite essentiality: both 1 and 2 are nontrivial restricted solvability

176 X 2 x 2 (x 1, x 2 ) (y 1, x 2 ) x 1 y 1 X 1

177 X 2 (z 1, z 2 ) x 2 (x 1, x 2 ) (y 1, x 2 ) x 1 y 1 } (y 1, x 2 ) (z 1, z 2 ) (z 1, z 2 ) (x 1, x 2 ) X 1

178 X 2 (z 1, z 2 ) x 2 (w 1, x 2 ) (x 1, x 2 ) (y 1, x 2 ) (y 1, x 2 ) (z 1, z 2 ) (z 1, z 2 ) (x 1, x 2 ) X 1 x 1 y 1 } w 1 such that (z 1, z 2 ) (w 1, x 2 )

179 Basic result Theorem (2 attributes) If restricted solvability holds each attribute is essential then the additive value function model holds if and only if is an independent weak order satisfying the Thomsen and the Archimedean conditions The representation is unique up to scale and location

180 General case Good news entirely similar... with a very nice surprise: Thomsen can be forgotten if n = 2, independence is identical with weak independence if n > 3, independence is much stronger than weak independence X 1 X 2 X 3 a b c d X 1 : % of nights at home X 2 : attractiveness of city X 3 : salary increase weak independence holds a b and d c is reasonable

181 General case Good news entirely similar... with a very nice surprise: Thomsen can be forgotten if n = 2, independence is identical with weak independence if n > 3, independence is much stronger than weak independence X 1 X 2 X 3 a b c d X 1 : % of nights at home X 2 : attractiveness of city X 3 : salary increase weak independence holds a b and d c is reasonable