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Decision Theory

Decision Theory

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Decision Theory

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  1. DecisionTheory Lecture 2

  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 • Revealedpreference and utilitytheory Choice Choicefunction Utility U(TheTruth) > U(TheMatrix)

  4. Individualdecisionmaking • under Certainty • Choicefunctions Weakaxiomof revealedpreference (WARP) NOT ALLOWED

  5. ??? You go to a restaurantinwhileyouare on vacationinTuscany and youaregiventhefollowing menu: • bistecca • pollo The cookanouncesthat he canalsoserve • trippaalla fiorentina

  6. Individualdecisionmaking • under Certainty • Reveleadpreference and ordinalutilitytheory Choice Preferencerelation Utilityfunction ≻ U(TheTruth) > U(TheMatrix) If u() is a utilityfunction, thenanystrictlyincreasingtransformationg∘u() is a utilityfunctionrepresentingthe same preferences

  7. The doctrine of utilitarianism saw the maximization of utility as a moral criterion for the organization of society. According to utilitarians, such as Jeremy Bentham (1748–1832) and John Stuart Mill (1806–1873), society should aim to maximize the total utility of individuals, aiming for "the greatest happiness for the greatest number of people". Another theory forwarded by John Rawls (1921–2002) would have society maximize the utility of those with the lowest utility, raising them up to create a more equitable distribution across society

  8. Choicefunction Preferencerelation ≻

  9. Individualdecisionmaking • under Certainty • Operations Research, Management Science

  10. 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

  11. Individualdecisionmaking • under risk • Objectiveprobabilities (ExpectedUtility) • Subjectiveprobabilities (SubjectiveExpectedUtility)

  12. ExpectedutilityTheory • Cardinalutilityfunction • Thisisthefoundation of gametheory – mixedstrategies • Thisisthefoundation of decisiontheory under risk – enablesmodelingriskattitudes If u(.) is a utilityfunction, thenanyaffinetransformation(au(.)+b, where a>0) isalso a utilityfunctionrepresentingthe same preferences

  13. Normativevspositivedecisiontheory • Behavioral (positive) economics • Experiments • Psychology • Empiricalresults • Behavioraltheories • Traditional (normative) economics • Mathematics • TraditionalMacro and Micro

  14. 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

  15. Individualdecisionmaking • under uncertainty • DecisionRules(in a while) • Uncertainty/ambiguityaversionmodels, e.g. Multipleprior/maximin model of Gilboa, Schmeidler Subjectiveprobabilitymay not exist

  16. 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

  17. Individualdecisiontheoryvsgametheory

  18. Zero-sum games • In zero-sum games, payoffsineachcell sum up to zero • Movement diagram

  19. Zero-sum games • Minimax = maximin = value of thegame • Thegamemayhavemultiplesaddlepoints

  20. Zero-sum games • Or itmayhave no saddlepoints • To findthevalue of suchgame, considermixedstrategies

  21. Zero-sum games • Ifthereismorestrategies, youdon’tknowwhich one will be part of optimalmixedstrategy. • LetColumnmixedstrategy be (x,1-x) • Then Raw will try to maximize

  22. Zero-sum games • Column will try to choose x to minimizetheupperenvelope

  23. Zero-sum games • TranformintoLinearProgramming

  24. Fishing on Jamaica • In the fifties, Davenport studied a village of 200 people on thesouthshore of Jamaica, whoseinhabitantsmadetheirliving by fishing.

  25. Twenty-six fishing crews in sailing, dugout canoes fish this area [fishing grounds extend outward from shore about 22 miles] by setting fish pots, which are drawn and reset, weather and sea permitting, on three regular fishing days each week … The fishing grounds are divided into inside and outside banks. The inside banks lie from 5-15 miles offshore, while the outside banks all lie beyond … Because of special underwater contours and the location of one prominent headland, very strong currents set across the outside banks at frequent intervals … These currents are not related in any apparent way to weather and sea conditions of the local region. The inside banks are almost fully protected from the currents. [Davenport 1960]

  26. Jamaica on a map

  27. Strategies • Therewere 26 woodencanoes. Thecaptains of thecanoesmightadopt 3 fishingstrategies: • IN – putallpots on theinside banks • OUT – putallpots on theoutside banks • IN-OUT) – putsomepots on theinside banks, somepots on theoutside

  28. Advantages and disadvantages of fishingintheopensea Disadvantages • Ittakesmore time to reach, so fewerspotscan be set • Whenthecurrentisrunning, itisharmful to outsidepots • marksaredraggedaway • potsmay be smashedwhilemoving • changesintemeperaturemaykillfishinsidethepots Advanatages • Theoutside banks producehigherqualityfishbothinvariaties and insize. • If many outsidefishareavailable, theymaydrivetheinsidefishoffthe market. • The OUT and IN-OUT strategiesrequirebettercanoes. • Theircaptainsdominatethe sport of canoe racing, whichisprestigious and offerslargerewards.

  29. Collecting data • Davenport collected the data concerning the fishermenaveragemonthly profit depending on the fishingstrategiestheyused to adopt.

  30. OUT Strategy

  31. Zero-sum game? Thecurrent’s problem • Thereis no saddle point • Mixedstrategy: • Assumethatthecurrentisvicious and playsstrategy FLOW withprobability p, and NO FLOW withprobability 1-p • Fishermen’sstrategy: IN with prob. q1, OUT with prob. q2, IN-OUT with prob. q3 • For every p, fishermenchoose q1,q2 and q3 thatmaximizes: • And theviciouscurrentchooses p, so thatthefishermenget min

  32. Graphicalsolution of thecurrent’s problem Solution: p=0.31 Mixedstrategy of thecurrent

  33. Thefishermen’s problem • Similarly: • For everyfishermen’sstrategy q1,q2 and q3, theviciouscurrentchooses p so thatthefishermenearntheleast: • Thefishermen will try to choose q1,q2 and q3 to maximizetheirpayoff:

  34. Maximin andminimax Optimalstrategy for thefishermen Value of thegame Optimalstrategy for thecurrent

  35. Forecast and observation Gametheorypredicts Observationshows No fishermenrisksfishingoutside Strategy69% IN, 31% IN-OUT [Payoff: 13.38] Current’s „strategy”: 25% FLOW, 75% NO FLOW • No fishermenrisksfishingoutside • Strategy67% IN, 33% IN-OUT [Payoff: 13.31] • Optimalcurrent’sstrategy31% FLOW, 69% NO FLOW The similarity is striking Davenport’s finding went unchallenged for several years Until …

  36. Currentis not vicious • Kozelka 1969 and Read, Read 1970 pointed out a seriousflaw: • The currentis not a reasoningentityand cannotadjust to fishermenchangingtheirstrategies. • HencefishermenshoulduseExpected Value principle: • Expectedpayoff of the fishermen: • IN: 0.25 x 17.3 + 0.75 x 11.5 = 12.95 • OUT: 0.25 x (-4.4) + 0.75 x 20.6 = 14.35 • IN-OUT: 0.25 x 5.2 + 0.75 x 17.0 = 14.05 • Hence, all of the fishermenshouldfishOUTside. • Maybe, theyare not welladaptedafterall

  37. Currentmay be viciousafterall • The currentdoes not reason, but itisveryrisky to fishoutside. • Evenif the currentruns 25% of the timeON AVERAGE, itmight run considerablymoreor less in the short run of a year. • Suppose one yearit ran 35% of the time. Expectedpayoffs: • IN: 0.35 x 17.3 + 0.65 x 11.5 = 13.53 • OUT: 0.35 x (-4.4) + 0.65 x 11.5 = 11.85 • IN-OUT: 0.35 x 5.2 + 0.65 x 17.0 = 12.87. • By treating the current as theiropponent, fishermenGUARANTEEthemselvespayoff of atleast13.31. • Fishermenpay 1.05 pounds as insurancepremium

  38. Decisionmaking under uncertainty

  39. Decisionmaking under uncertainty

  40. Decisionmaking under uncertainty Regretmatrix

  41. Decisionmaking under uncertainty Regretmatrix

  42. Decisionmaking under uncertainty

  43. Father: “I want you to marry a girl of my choice” Son: “I will choose my own bride!” Father: “But the girl is Bill Gates’ daughter.” Son: “Well, in that case…ok” Next, the father approaches Bill Gates. Father: “I have a husband for your daughter.” Bill Gates: “But my daughter is too young to marry!” Father: “But this young man is a vice‐president of the World Bank.” Bill Gates: “Ah, in that case…ok” Finally the father goes to see the president of the World Bank. Father: “I have a young man to be recommended as a vicepresident.” President: “But I already have more vice‐ presidents than I need!” Father: “But this young man is Bill Gates’s son‐in‐law.” President: “Ah, in that case…”