PREFERENCES AGGREGATION & DECISION THEORY A. ROLLAND, UNIVERSITE LUMIERE LYON II

Transcription

1 PREFERENCES AGGREGATION & DECISION THEORY A. ROLLAND, UNIVERSITE LUMIERE LYON II 2013

2 PLAN 1 Introduction & (Feb-05) 2 Preferences Aggregation : utility theory (Feb-15) 3 Preferences Aggregation : decision aiding theory (Feb-19) 4 Decision under uncertainty (Mar-12) 5 Tutorial (I & II) (Mar-19) 6 More about... and evaluation (Apr-02)

3 Introduction

4 DECISION MAKING Decision Making : the art of helping a decision maker to take a good decision Is deciding difficult?

5 DECIDING SHOULD BE DIFFICULT BECAUSE... there are too many possibilities to decide Examples which master should I choose? classical problems : Knapsack Problem (KP), Minimum Spanning Tree Problem, Traveller Salesman Problem (TSP)...

6 EXAMPLE : TSP How to visit 17 towns in Rhône-Alpes?

7 EXAMPLE : TSP How to visit 17 towns in Rhône-Alpes?

8 EXAMPLE : TSP How to visit 17 towns in Rhône-Alpes?

9 EXAMPLE : TSP How to visit 17 towns in Rhône-Alpes?

10 EXAMPLE : TSP How to visit 17 towns in Rhône-Alpes? (n 1)! possibilities!

11 EXAMPLE : TSP How to visit 17 towns in Rhône-Alpes? (n 1)! possibilities! possibilities in Rhône-Alpes

12 COMBINATORIAL OPTIMIZATION finding the best solution into a finite set of objects without any possibility to look at all of them!

13 DECIDING SHOULD BE DIFFICULT BECAUSE... there are too many possibilities to decide there are several decision makers to decide Examples where are we going to drink beer this evening? classical problems : voting theory

14 EXAMPLE : VOTING FOR SWEETS Three friends want to choose sweets together.

15 EXAMPLE : VOTING FOR SWEETS 1 2 3

16 EXAMPLE : VOTING FOR SWEETS 1 2 3

17 EXAMPLE : VOTING FOR SWEETS 1 2 3

18 EXAMPLE : VOTING FOR SWEETS 1 2 3

19 EXAMPLE : VOTING FOR SWEETS 1 2 3?

20 SOCIAL CHOICE finding the collective preferred solution knowing the preferences of every voter this solution sometimes should not exist!

21 DECIDING SHOULD BE DIFFICULT BECAUSE... there are too many possibilities to decide there are several decision makers to decide there are several criteria to be taken into consideration Examples Should I choose a bad movie with my favourite actor or a good movie without him? classical problems : multicriteria decision aiding

22 EXAMPLE : CHOOSING A CAMERA

23 EXAMPLE : CHOOSING A CAMERA

24 EXAMPLE : CHOOSING A CAMERA Mean Min Max σ Price Camera Camera

25 MULTICRITERIA finding the global preferred solution with possibly conflicting criteria

26 DECIDING SHOULD BE DIFFICULT BECAUSE... there are too many possibilities to decide there are several decision makers to decide there are several criteria to be taken into consideration consequences are uncertain Examples Should I take my umbrella? Expected Utility theory : basis of classical economic behaviour

27 EXAMPLE : UMBRELLA

28 EXAMPLE : UMBRELLA Score Probability

29 DO YOU WANT TO PLAY TOGETHER? GAME 1 You win 100 $ GAME 2 You win 0 $ with p=0.5 You win 250 $ with p=0.5 Which game do you choose?

30 DO YOU WANT TO PLAY TOGETHER? GAME 1 You win 10 $ GAME 2 You win 0 $ with p=0.5 You win 25 $ with p=0.5 Which game do you choose?

31 DO YOU WANT TO PLAY TOGETHER? GAME 1 You win $ GAME 2 You win 0 $ with p=0.5 You win $ with p=0.5 Which game do you choose?

32 DECISION UNDER UNCERTAINTY finding the global preferred solution without knowing the exact consequences

33 DECIDING SHOULD BE DIFFICULT BECAUSE... there are too many possibilities to decide Combinatorial optimization there are several decision makers to decide Social Choice Theory there are several criteria to be taken into consideration Multicriteria decision Making consequences are uncertain Decision under uncertainty

34 FORMAL FRAMEWORK Social Choice Multicriteria Uncertainty Candidates Alternatives Actions Voters Criteria States of the nature Ranks Values Consequences (Number) (Weight) (Probability) Social Choice : individual preferences global preferences Multicriteria : preferences on criteria preferences on the alternatives Uncertainty : preferences on the consequences preferences on the actions

35 Preference Modelling

36 BINARY RELATION BINARY RELATION : DEFINITION Let A a finite set of elements {a, b, c...}, where A = n. A binary relation R on A is a subset of the Cartesian product R A A, ie a set of ordered pairs (a; b) A A. If (a; b) R, noted arb, then element a is said to be in relation with element b for the relation R.

37 GRAPHICAL REPRESENTATION A C B D

38 MATRICIDAL REPRESENTATION One can also represent relation R by a squared binary matrix :

39 BINARY RELATION PROPERTIES If R is a binary relation on A A, then R is the complementary relation of R : arb bra I R is the symmetric part of R : ai R b arb and bra P R is the asymmetric part of R : ap R b arb and (bra) E R is an equivalence ( relation : ark brk ae R b kra krb k A

40 BINARY RELATION PROPERTIES R is said to be... reflexive if ara a A symmetric if arb bra asymmetric if arb bra anti-symmetric if arb bra b = a transitive if arb and brc arc semi-transitive if (arb brc) (ard drc) Ferrers if (arb crd) (ard crb) complete if arb bra, a b. Note that I R is symmetric and P R is asymmetric.

41 BINARY RELATION PROPERTIES R is said to be... reflexive if ara a A A C B D

42 BINARY RELATION PROPERTIES R is said to be... symmetric if arb bra A C B D

43 BINARY RELATION PROPERTIES R is said to be... asymmetric if arb bra A C B D

44 BINARY RELATION PROPERTIES R is said to be... anti-symmetric if arb bra b = a A C A B/C B D D

45 BINARY RELATION PROPERTIES R is said to be... transitive if arb and brc arc A C B D

46 BINARY RELATION PROPERTIES R is said to be... semi-transitive if (arb brc) (ard drc) A A B B D C D C A A B B D C D C

47 BINARY RELATION PROPERTIES R is said to be... Ferrers if (arb crd) (ard crb) A C A C B D B D

48 ORDERS DEFINITION A relation R is an order iff R is complete R is transitive R is antisymmetric Numerical representation : 7 6 A 5 B 4 C 3 D 2 E 1 F

49 PRE-ORDERS DEFINITION A relation R is an pre-order iff R is complete R is transitive Numerical representation : 7 6 A 5 B 4 3 C D 2 E 1 F

50 SEMI-ORDERS DEFINITION A relation R is a semi-order iff R is complete R is Ferrers R is semi-transitive Numerical representation :

51 INTERVAL ORDERS DEFINITION A relation R is an interval order iff R is complete R is Ferrers Numerical representation :

52 EXERCISE 1 Let B be the following binary relation : aba abb bba bbb bbc cba cbb cbc cbd cbe dba dbb dbc dbd dbe eba ebb ebc ebd ebe 1 propose a matricidal representation of B 2 propose a graphical representation of B 3 is B an order? a pre-order? a semi-order? an interval order?

53 EXERCISE 2 Let B be a binary relation on the set {a, b, c, d, e, f } defined by : aba abb abc abd abe abf bbb bbc bbd bbe bbf cbc cbd cbe cbf dbb dbe dbd dbe ebd ebe ebf fbe fbf 1 propose a matricidal representation of B 2 represent graphically the relation B 3 is the relation B reflexive? symmetric? asymmetric? transitive? semi-transitive? 4 how can you qualify the relation B?

54 EXERCISE 3 Word HandBall Championship Results of group A : All FRA BRA TUN ARG MON ALL FRA BRA TUN ARG 1 MON 1 how can you qualify this relation?

55 BIBLIOGRAPHY SOME USEFULL REFERENCES M. Pirlot and Ph. Vincke. Semi Orders. Kluwer Academic, Dordrecht, 1997 D. Bouyssou and Ph. Vincke, Binary Relations and Preference Modeling, in Concept and Methods for decision aiding, Ch. 2, 2009