Decision Theory Preference relation and choice function

1. DecisionTheoryPreferencerelation and choicefunction

2. Agenda Binarny relations – properties Pre-orders and orders, relation of rationalpreferences Strictpreference and indifferencerelation

3. Today – anotherapproach to decisionmaking • Preferences: • capability of makingcomparisons • capability of deciding, which of twoalternativesisbetter/is not worse • Mathematically – binary relations in the set of decisionalternatives: • X – decisionalternatives • X2 – allpairs of decisionalternatives • RX2– binaryrelation in X, selectedsubset of orderedpairs of elements of X • ifxis in relationR with y, then we writexRyor(x,y)R

4. Examples of relations: • „Being a parent of” is a binary relation on a set of human beings • „Being a hat”is a binaryrelation on a set of objects • „x+y=z” is 3-ary relation on the set of numbers • „x is better than y more thanx’is better than y’ ” is a 4-ary relation on the set of alternatives.

5. Binary relations – example #1 • Example • X={1,2,3,4} • R – a relationdenoting „issmallerthan” • xRy – means „x issmallerthan y” • Thus: • (1,2)R; (1,3)R; (1,4)R; (2,3)R; (2,4)R; (3,4)R • 1R2, 1R3, 1R4, 2R3, 2R4, 3R4 • eg. (2,1) doesn’tbelong to R

6. Binary relations– example #2 • Example • X={1,2,3,4} • R – a relation with no (easy) interpretation • R={(1,2), (1,3), (2,3), (2,4), (3,2), (4,4)}

7. Binary relations – basicproperties • complete: xRyoryRx • reflexive: xRx (x) • antireflexive: not xRx (x) • transitive: ifxRy and yRz, thenxRz • symmetric: ifxRy, thenyRx • asymmetric: ifxRy, then not yRx • antisymmetric: ifxRy and yRx, thenx=y • negativelytransitive: if not xRyand not yRz, then notxRz • equivalent to: xRzimpliesxRyoryRz • acyclic: ifx1Rx2, x2Rx3, … , xn-1Rxnimplyx1≠xn

8. Exercise– check the properties of the following relations R1: (amongpeople), to have the same colour of the eyes R2: (amongpeople), to knoweachother R3: (in the family), to be anancestor of R4: (among real numbers), not to have the same value R5: (amongwords in English), to be a synonym R6: (amongcountries), to be atleast as good in a rank-table of summerolympics

9. Exercise– check the properties of the following relations R1: (amongpeople), to have the same colour of the eyes R2: (amongpeople), to knoweachother R3: (in the family), to be anancestor of R4: (among real numbers), not to have the same value R5: (amongwords in English), to be a synonym R6: (amongcountries), to be atleast as good in a rank-table of summerolympics

10. Preferencerelation • Preferences – capability of makingcomparisons, of selecting not worseanalternative out of a pair of alternatives • we’ll talk aboutselecting a strictlybetter(orjust as good) alternativelater on • Depending on itspreferenceswe’lluse one of the relations: • preorder • partial order • completepreorder (rationalpreferencerelation) • complete order (linear order)

11. Preorder • R isa preorder in X, ifitis: • reflexive • transitive • We do not want R to be: • Complete – we cannotcompareall the pairs of alternatives • Antisymmetric – ifxRy and yRX, then not necessarilyx=y

12. Preorder – anexample Michał isat a party and canpick from a buffetontohisplate: small tartares, cocktail tomatoes, sushi (maki), chunks of cheese A decisionalternativeis an orderdfour-tuple, denotingnumber of respectivepieces, therecan be at most 20 pcson the plate Michał preferesmorepcsthanfewer. At the same time, he prefersmoretartarethan less. Michał cannottell, if he wants to havemorepcsifitmean less tartare.

13. Preorder – anexample Michał isat a party and canpick from a buffetontohisplate: small tartares, cocktail tomatoes, sushi (maki), chunks of cheese A decisionalternativeis an orderdfour-tuple, denotingnumber of respectivepieces, therecan be at most 20 pcson the plate Michał preferesmorepcsthanfewer. At the same time, he prefersmoretartarethan less. Michał cannottell, if he wants to havemorepcsifitmean less tartare.

14. Partial order • R isa partial order in X, ifitis: • reflexive • transitive • antisymmetric(not needed in the preorder) • We do not want it to be: • Complete – we cannotcompareall the pairs of alternatives

15. Partial order – anexample Michał isat a party … Decisionalternativesareorderedpairs: # of pcs, # of tartares

16. Partial order – anexample Michał isat a party … Decisionalternativesareorderedpairs: # of pcs, # of tartares Conclusion – differentstructure (of the same problem), differentformalrepresentation

17. Completepreorder – rationalpreferencerelation • R is a completepreorderin X, ifitis: • transitive • complete • Completenessimpliesreflexivity • We do not want it to be: • antisymmetric – equallygoodalternativesareallowed to differ • In ourexample – if Michał didn’tvaluetartare (and justwanted to eat as much as possible)

18. Completepreorder – rationalpreferencerelation

19. Complete order (linear) • R is a complete order in X, ifitis: • transitive • complete • antisymmetric • In ourexample: • Michał wants to eat as much as possible • we representalternatives as # of pcs

20. Complete order (linear)

21. Preference relations

22. Preference and indifferencerelation • Let R be a completepreorder (transitive, complete) • xRymeans „xisatleast as good as y” • R generatesstrictpreferencerelation – P: • xPy, ifxRy and not yRx • xPymeans „xisbetterthany” • R generatesindifferencerelation – I: • xIy, ifxRy and yRx • xIymeans „xjust as good as y”

23. Anexercise X={a,b,c,d} R={(a,a), (a,b), (a,c), (a,d), (b,a), (b,b), (b,c), (b,d), (c,c), (c,d), (d,d)} Find P and I P={(a,c), (a,d), (b,c), (b,d), (c,d)} I={(a,a), (a,b), (b,a), (b,b), (c,c), (d,d)} R=PI

24. Properties of P and I (of previousslides) • Let P and I be generated by R – a completepreorder • P is: • asymmetric • negativelytransitive • antireflexive • acycylic • transitive • I isanequivalencerelation: • reflexive • transitive • symmetric

25. Proof of the properties of I (xIy xRy  yRx) • reflexive (xIx) • obvious – usingreflexivity of R we getxRx • transitive (xIy yIz  xIz) • predecessormeansthatxRy  yRx  yRz  zRy • usingtransitivity we getxRz  zRx, QED • symmetric(xIy yIx) • predecessormeansthatxRy  yRx, QED

26. Logicalpreliminary ⇔ ⇔ ⇔

27. P vs R (xPy xRy   yRx) R iscompleteiff P isasymmetric R istransitiveiff P isnegativelytransitive

28. Homework 1. ProvethatxRyyPzxPz 2. Show thatx,y: xPy  xIy  yPx

29. Anotherdefinition of rationalpreferences • Let’s start with relation P: • asymmetric • negativelytransitive • Then we saythat • xIy, ifxPyyPx • xRy, ifxPyxIy • Homework. Provethat with suchdefinitions: • I isanequivalencerelation • R is a completepreorder

30. Exercise X={a,b,c,d} P={(a,d), (c,d), (a,b), (c,b)} Find R and I I={(a,a), (a,c), (b,b), (b,d), (c,a), (c,c), (d,b), (d,d)} R=PI

31. Anotherdefinition of rationalpreferences • Canwe start with I? • reflexive • symmetric • transitive • No – we wouldn’t be able to order the abstractionclasses

32. Anotherdefinition of rationalpreferences • Isitenough to use P? • asymmetric • acyclic (not necessarilynegativelytransitive) • No – let’sseeanexample

33. P from the previousslide – anexample Mr X gotill and for years to comewillhave to takepillstwice a day in aninterval of exactly 12 hours. He canchoose the timehowever. All the decisionalternativesarerepresented by a circle with a circumference 12 (a clock). Let’sdenote the alternatives by the length of an arc from a given point (midnight/noon). Mr X hasverypeculiarpreferences – he prefersy to x, ify=x+p, otherwise he doesn’tcare ThusyPx, ifylies on the circlepunitsfarther (clockwise) thanx

34. Exercise • Whatpropertiesdoes P have? • asymmetry • negativetransitivity • transitivity • acyclicity • P generates „weird” preferences: • 1+2pbetterthan 1+p, 1+pbetterthan 1, 1+2pequallygood as 1 • 1 equallygood as 1+p/2, 1+p/2 equallygood as 1+p, 1 worsethan 1+p

35. Anotherdefinition of rationalpreferences • Whatif we take P? • asymmetric • transitive (not necessarilynegativelytransitive) • thusacyclic • First let’stry to findanexample • Then let’sthinkaboutsuchpreferences

36. Asymmetric, transitive, not negativelytransitiverelation – intuition

37. Asymmetric, transitive, not negativelytransitiverelation – example • X={R+}, xPy x>y+5 (I want more, but I aminsensitive to small changes) • Properties of P: • asymmetric – obviously • transitive – obviously • negativelytransitive? • 11 P 5, but • neither11 P 8, nor 8 P 5 • ThusIis not transitive: 11 I 8 and 8 I 5, but not 11 I 5 • Real example – non-inferioritytesting • H0: m1=m2 vs H1: m1≠m2 • H0: m1≤m2-dvs H1: m1>m2

38. Properties of preferences – a summary P („betterthan”) – asymmetric, negativelytransitive R („atleast as good as”) – transitive, complete colours, insensitiviness to small changes P („betterthan”) – asymmetric, transitive eg. Mr X P („betterthan”) – asymmetric, acyclic

39. Summary • Anotherway of talkingabout choice makingis to talk aboutbinary relations - preferences • Depending on the structure of a decision problem athand we canuse relations: preorder, totalpreorder, partial order, total order • the same problem can be sometimesdescribed in differentways • Relation of weakpreferencegeneratesstrictpreferencerelation and indifferencerelation. We canalso start from the strictpreference – asymmetric and negativelytransitive