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Decision Theory

Decision Theory. Lecture 4. Decision Theory – the foundation of modern economics. Individual decision making under Certainty C hoice functions Revelead preference and ordinal utility t heory Operations Research , Management Science under Risk

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Decision Theory

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  1. DecisionTheory Lecture4

  2. DecisionTheory – thefoundation of modern economics • Individualdecisionmaking • under Certainty • Choicefunctions • Reveleadpreference and ordinalutilitytheory • Operations Research, Management Science • under Risk • ExpectedUtilityTheory (objectiveprobabilities) • Bayesiandecisiontheory • ProspectTheory and otherbehavioraltheories • SubjectiveExpectedUtility (subjectiveprobabilities) • under Uncertainty • Decisionrules • Uncertaintyaversionmodels • Interactivedecisionmaking • Non-cooperativegametheory • Cooperativegametheory • Matching • Bargaining • Group decisionmaking (Socialchoicetheory) • Group decisions (Arrow, Maskin, etc.) • Votingtheory • Welfarefunctions

  3. Individualdecisionmaking • under Certainty • Choicefunctions Choice Choicefunction Weakaxiomof revealedpreference (WARP)

  4. Exemplary choice functions Pick the cheapest (e.g. public tenders) Pick the secondcheapest (wine for a party) Maximize the IRR (investment projects) Pickwhoevergetsmajority of votes (Talent shows on TV) …

  5. Choice functions – someintuition (1) good 2. Out of the gray set, A was chosen (a unique choice) Out of the blue set, B was chosen (a unique choice) B Do we findthesechoicesconfusing? (whenconsideredcollectively) A good 1.

  6. Choice functions – someintuition(2) good 2. Out of the gray set, A was chosen (a unique choice) Out of the blue set, B was chosen (a unique choice) Do we findthesechoicesconfusing? (whenconsideredcollectively) A B good 1.

  7. Choice functions – someintuition(3) good 2. Out of the gray set, A was chosen (a unique choice) Out of the blue set, B was chosen (a unique choice) Do we findthesechoicesconfusing? (whenconsideredcollectively) A B good 1.

  8. Choice functions – someintuition(4) Good 2. Out of the gray set, A was chosen (a unique choice) A Out of the blue set, B was chosen (a unique choice) Out of the golden set, C was chosen (a unique choice) B Do we findthesechoicesconfusing? (whenconsideredcollectively) C Good 1.

  9. Homework Can we, usingonlylinearbudgetconstraints, constructsuchanexample for twogoods, thatthereis a „consistency problem” whenconsideringmorethantwoalternatives, and no problem whenconsideringonlyeachtwoalternativesseparately? And whenconsideringthreegoods?

  10. Choice functions – a formaldefinition Notation: (Technical) properties: If C(B) contains a single element  thisis the choice Ifmoreelements  thesearepossiblechoices (not simultaneously, the decisionmakerpicks one in the waywhichis not describedhere)

  11. Anexercise • Let X={a,b,c}, B=2X • Write down the following choice functions: • C1: always a (ifpossible), if not – itdoesn’tmatter • C2: always the first one in the alphabetical order • C3: whatever but not the last one in the alphabetical order (unlessthereisjust one alternativeavailable) • C4: secondfirstalphabetically(unlessthereisjust one alternative) • C5: disregard c (iftechnicallyitispossible), and ifyou do disregard c, alsodisregard b (iftechnicallypossible)

  12. The solution

  13. The solution

  14. Desirableproperties • Sometimesaninternalconsistencyispostulated • Whyso? • positiveapproach – non-consistentwill go bankrupt • normative – in order not to go bankrupt • We’lldiscuss the following: • weakaxiom of revealedpreferences • a property • b property • g property

  15. WARP – weakaxiom of revealedpreferences Definition (WARP): A pair(B,C()) satisfies WARP, if the followingholds: if for some B from B, s.t. x,yB, we havexC(B), than for every B’ from B, s.t. x,yB’, ifyC(B’), thenxC(B’). Intuitively: ifx was shown to be atleast as willinglypicked as y (for a menu B), then for every menu B’containingx,y, ifyispicked, sodoesxhave to be.

  16. WARP – anintuition good 2. Out of the gray set, A was chosen (a unique choice) Out of the blue set, B was chosen (a unique choice) B Do we findthesechoicesconfusing? (whenconsideredcollectively) A good 1.

  17. WARP – anintuition good 2. Out of the gray set, A was chosen (a unique choice) Out of the blue set, B was chosen (a unique choice) Do we findthesechoicesconfusing? (whenconsideredcollectively) A B good 1.

  18. WARP – anintuition good 2. Out of the gray set, A was chosen (a unique choice) Out of the blue set, B was chosen (a unique choice) Do we findthesechoicesconfusing? (whenconsideredcollectively) A B good 1.

  19. Anexercise Checkwhichfunctions C1-C5 do not fulfill WARP, prove by givingexemplarymenus

  20. The solution C1 – fulfils C2 – fulfils C3 – doesn’t! b picked from {a,b,c} and not from {a,b} C4 – doesn’t! b picked from {a,b,c} and not from {b,c} C5 – doesn’t! b picked from {a,b} and not from {a,b,c}, while a picked

  21. aproperty (Chernoffproperty) Definition (aproperty): AssumeB=2X. C() meetsa, if the followingholds: if for some B out of B we havexC(B), then for every B’B, s.t. xB’, we havexC(B’). Intuitively: ifxpicked from menu B, thenshall be picked from eachsmaller menu B’ (ifpresent in it).

  22. apropertydifferently Ifsomething not picked from menu B’, shan’t be picked from a bigger one: If we add to B1somenewalternatives B2, then the choice will either not change, orsomething out of newalternativesshould be picked

  23. Homework Provethat the previousdefinitionsareequivalent

  24. Anexercise – check the aproperty for C1-C5

  25. bproperty Conclusion for the previousexercise – a and WARP differ (let’slook for otherproperties) Definition (bproperty): Take B=2X. C() meetsbproperty, if the followingholds: if form some B’ in B we havex,yC(B’), than for each B, B’B, we havexC(B) yC(B). Intuitively: ifx and yarepicked in a menu B’, thentheir status isequal in everygreater menu B.

  26. Anexercise – checkbproperty for C1-C5

  27. gproperty Definition (gproperty): AssumeB=2X. C() meetsg, if the followingholds: if for every menu Bi out of a family of menus we havexC(Bi), then for B=Bi we havexC(B). Intuitively: ifxispicked in every menu (in a family of menus), thanitisalsopicked in a joint menu

  28. Anexercise – checkgproperty for C1-C5

  29. The completesolution

  30. Properties and manipulation • Assume C1-C5 can be used in a public tender (a,b,cdenoteoffers) • Take C3({a,b})={a}, C3({b,c})={b}, C3({a,b,c})={a,b} • different choice for a complete problem (b may be selected), • differentwhenshort listing • … pairisecomparisonsalsochange the outcome – b „betterthan” c, a „betterthan” b, hence a • putting c on the tableimpacts the chocie (favours b – possiblealliance)

  31. Anexercise • Public tender • Alternatives – offersdescribed by: priceand time to deliver(qualityisconstant) • Rule #1: • minimize the expressiona  pricei + b  timei (for someweightsa>0,b>0determinedirrespectively of set of offers) • Rule #2: • calculated the minimalprice (MP) and minimaltime (MT) for alloffers (assume MP>0 and MT>0) • minimize the expressionpricei/MP + timei/MT • Whichrule do youlike?

  32. The solution Rule #1 – meets’emall: WARP, a, b, g(intuitively – the evaluationdoes not depend on the menu, will be formalizedlater)

  33. The solution • Rule #2 – doesn’tmeet a single one • Take B={x,y,z}, x=(4,4), y=(1,9), z=(16,1) • whatwill be selected? • Try to findsomemodifications in order to show howa, b, garebroken

  34. Summingup • Differentviews on decisionmaking • choice and choice functions • preferences • utilityfunction • We canjudge not onlyalternatives, but also choice rules • not meetingsomepropertiesyields a risk of beingmanipulated • differentproperties, not all of themequivalent

  35. Materials • Compulsory: • A. MasColell, M. Whinston, J. Green MicroeconomicTheory, Oxford University Press, 1995, rozdz. 1 • Supplementary: • A. Sen, ChoiceFunctions and RevealedPreference, TheReview of EconomicStudies, 1971, 38(3), s. 307-317

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