Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Slide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Slide 58Slide 59Slide 60Slide 61Slide 62Slide 63Slide 64Slide 65Slide 66Slide 67Slide 68Slide 69Slide 70Slide 71Slide 72Slide 73Slide 74Slide 75Slide 76Slide 77Slide 78Slide 79Slide 80Slide 81Slide 82Group-Clicker Question (1 pt): What is the Expected Value of the lottery (0.25, 0.25, 0.50) over the monetary prizes($4, $16, $50)?0%0%ECON 410 Preferences Over Risk“If life is free of failures you are not taking enough risks.”–Jackson Brown, Jr.1.302.452HW3 Due Thursday, Feb. 14http://www.unceminor.org/ Applications due 3/8 Class 16 - Uncertainty Fundamentals3Class 16 - Uncertainty FundamentalsExisting TheoryLottery DefinitionExpected ValuePreference Over LotteriesIndependence AxiomExpected Utility TheoremCardinality4Class 16 - Uncertainty Fundamentals�5Class 16 - Uncertainty FundamentalsComplete•A decision can be madeComplete•A decision can be madeSymmetric•An individual is indifferent to all optionsSymmetric•An individual is indifferent to all optionsTransitive•Preference rankings are “consistent”Transitive•Preference rankings are “consistent”Local nonsatiation•No bliss pointsLocal nonsatiation•No bliss pointsMonotonicity•More is betterMonotonicity•More is betterStrict Convexity•Preference for diversityStrict Convexity•Preference for diversityContinuity•No unexpected shifts in preferencesContinuity•No unexpected shifts in preferences6Class 16 - Uncertainty FundamentalsComplete:A preference relation over lotteries is complete if, for each lottery x and y, either x y or y x or both, where x and y are lotteries.��7Class 16 - Uncertainty FundamentalsTransitive:A preference relation over lotteries is transitive if, for every x, y, and z, if x y and y z then x z where x, y, and z are lotteries.���8Class 16 - Uncertainty FundamentalsContinuityLoosely, preferences over lotteries are continuous if, whenever (p1, p2,…pn) (q1, q2,…qn) and (p1, p2,…pn) (t1, t2,…tn) then (t1, t2,…tn) (q1, q2,…qn) ���9Class 16 - Uncertainty FundamentalsExisting TheoryLottery DefinitionExpected ValuePreference Over LotteriesIndependence AxiomExpected Utility TheoremCardinality10Class 16 - Uncertainty FundamentalsMixingGiven two lotteries, (p1, p2,…pn) and (t1, t2,…tn), a mixture of these two lotteries is the new lottery(ap1+(1-a)t1, ap2+(1-a)t2,…, apn+(1-a)tn)11Class 16 - Uncertainty Fundamentals$0 $0(0.8, 0.2) (0.25, 0.75)12Class 16 - Uncertainty FundamentalsIndependencePreferences over lotteries display independence if, whenever(p1, p2,…pn) (q1, q2,…qn) then (p1, p2,…pn) mixed with (t1, t2,…tn)(q1, q2,…qn) mixed with (t1, t2,…tn)when the mixing is done in the same manner.��13Class 16 - Uncertainty Fundamentals�14Class 16 - Uncertainty Fundamentals15Class 16 - Uncertainty Fundamentals16Class 16 - Uncertainty Fundamentals��Group-Clicker Question (P): You are offered a lottery over ($5 million, $1 million, $0). Which lottery would you select?1. (0, 1, 0)2. (0.10, .89, .01)Group-Clicker Question (P): You are offered a lottery over ($5 million, $1 million, $0). Which lottery would you select?1. (0, .11, .89)2. (0.10, 0, .90)19Class 16 - Uncertainty FundamentalsOption 1Lottery 1 Lottery 2Prize Probability Prize Probability$1 89% $1 89%$1 11% $5 10%$0 1%($5 million, $1 million, $0)Lottery 1: (0, 1, 0)vsLottery 2: (0.10, .89, .01)20Class 16 - Uncertainty FundamentalsOption 2Lottery 1 Lottery 2Prize Probability Prize Probability$0 89% $0 89%$1 11% $5 10%$0 1%($5 million, $1 million, $0)Lottery 1: (0, .11, .89)vsLottery 2: (0.10, 0, .90)21Class 16 - Uncertainty FundamentalsOption 2Lottery 1 Lottery 2Prize Probability Prize Probability$0 89% $0 89%$1 11% $5 10%$0 1%Option 1Lottery 1 Lottery 2Prize Probability Prize Probability$1 89% $1 89%$1 11% $5 10%$0 1%Group-Clicker Question (P): You are offered a lottery over(a trip to Cancun with your friends, staying home and watching a movie of your friends having a great time in Cancun, staying home and watching no movie). Which lottery would you select?1. (.99, 0.01, 0)2. (0.99, 0, .01)23Class 16 - Uncertainty FundamentalsExisting TheoryLottery DefinitionExpected ValuePreference Over LotteriesIndependence AxiomExpected Utility TheoremCardinality24Class 16 - Uncertainty FundamentalsWhy Economists Assume IndependenceWhy Economists Assume IndependenceNormativeIndependence is how people should behave. “The night of the fight, you might feel a slight sting. That’s pride messing with you. Forget pride. Pride only hurts. It never helps. You fight through that nonsense.”-Marcellus Wallace, Pulp Fiction25Class 16 - Uncertainty FundamentalsWhy Economists Assume IndependenceWhy Economists Assume IndependenceLarge Organizations Should Be “Rational”When looking at “large” questions, assuming independence is very reasonable.26Class 16 - Uncertainty FundamentalsWhy Economists Assume IndependenceWhy Economists Assume IndependenceMoney PumpIf an individual’s preferences over lotteries do not display Independence, there is an opportunity for a money pump.27Class 16 - Uncertainty FundamentalsWhy Economists Assume IndependenceWhy Economists Assume IndependenceTractabilityIf we assume Independence, we can do amazing things.28Class 16 - Uncertainty FundamentalsExpected Utility TheoremIf an individual’s preferences over lotteries are complete, transitive, continuous, and satisfy the independence axiom, then(p1, p2,…pn) (q1, q2,…qn) if and only ifp1u1+p2u2+…+pnun >= q1u1+q2u2+…+qnun where the ui’s are known as von Neumann-Morgenstern (vNM) utilities and can be interpreted as the cardinal utility received from good i.�29Class 16 - Uncertainty FundamentalsThings:Lottery: 0.25 0.35 0.40vNM Utility:10 2 5Expected Utility of the Lottery =30Class 16 - Uncertainty FundamentalsThings:Lottery: 0.25 0.35 0.40vNM Utility:10 2 5Expected Utility of the Lottery = (10)(.25) +31Class 16 - Uncertainty FundamentalsThings:Lottery: 0.25 0.35 0.40vNM Utility:10 2 5Expected Utility of the Lottery