4-1Interval Utility Scales • 4-2Monetary Values vs. Utilities • 4-3Von Neumann-Morgenstern Utility Theory • 4-4Criticisms of Utility Theory • 4-5The Predictor Paradox • 4-6Causal Decision Theory
4-1 Interval Utility Scales • Why we need interval utility scales to make decisions under risk? • What is interval utility scales？ • Prove interval utility scales suffice for decisions under risk.
Why Interval Utility Scales(1/3) • Ordinal utility scales do not suffice for making decisions under risk. • A1=9/4 A2=11/4
Why Interval Utility Scales(2/3) • Transform table as below • A1=23/4 A2=11/4
Why Interval Utility Scales(3/3) • Ordinal scales only tell us what is ranked first and second, above and below. • It not enough to know that you prefer one outcome to another. • Interval scales are all we require for decisions under risk.
What is interval utility scales • In addition to recording an agent’s ranking of outcomes, we need measure only the relative lengths of the “preference intervals” between them.
What is interval utility scales • Sample: • Cola (C) • Ice cream (I) • Apple (A) • Popcorn (P) least preferred most preferred Ｐ A C I
Conditions of Interval Utility • If I use an interval scale, I must be sure that the relative lengths of the intervals on the line are reflected as will. • Utility numbers on an interval scales must satisfy the following conditions. xPy if and only if u(x) > u(y). xIy if and only if u(x) = u(y). The preference interval between x and y is greater than or equal to that between z and w if and only if | u(x) – u(y) | >= | u(z) - u(w) |
Positive Linear Transformation Positive linear transformation 10u(x) + 5 least preferred least preferred most preferred most preferred 1 15 2 25 75 7 105 10 Ｐ Ｐ A A C C I I
Ratio Scales • Sample • Lengths(yards , feet, meters) • Weights(pounds, grams, ounces ) • Two important features： • Natural zero point.(no length , no weight) • In converting frome one ratio scale to another we multiply by a positive constant. ( ex. 1公尺 = 100公分) • Ratio scales are a tighter kind of interval scale.
Interval Scales vs. Ratio Scales • One way to appreciate the difference between ration and interval scales is to think of changing scales in terms of changing the labels on our measuring instruments. • Ratio scales have the same zero point. • Ex. Yard & Feet (1 yard = 3 feet) • Interval scales have no fixed zero point. • Ex. Fahrenheit & Celsius (F=C*9/5+32)
Interval Scales vs. Ratio Scales • Ex1. • Suppose that at noon yesterday , the outdoor temperature was at the freezing point of water and that today at noon it measured 64degrees on the Fahrenheit scale. • Was it twice as hot today at noon as it was yesterday?
Interval Scales vs. Ratio Scales • Ex2. • Suppose I am driving at 60 miles per hour and you are driving at 30. Now convert out speeds to kilometers per hour. • I would still be driving twice as fast as you.
Prove Interval Utility Scales Suffice for Decisions Under Risk • u’s and v’s are utility numbers • p’s and q’s are probabilities. • The probabilities across for each row sum to 1.
Prove Interval Utility Scales Suffice for Decisions Under Risk • The expected utilities for Ai and A j : • EU(Ai)=u1p1+ u2p2 +…+ unpn • EU(Ai)=v1q1+ v2q2 +…+ vnqn • Positive linear transformation by au+b and av+b (with a>0). • EUnew(Ai)=(au1+b) p1+(au2+b) p2+…+(aun+b) pn • EUnew(Aj)=(av1+b) q1+(av2+b) q2+…+(avn+b) qn
Prove Interval Utility Scales Suffice for Decisions Under Risk • 展開算式 • EUnew(Ai)=[au1p1+au2p2+…+aunpn]+ [bp1+bp2+…+bpn] • EUnew(Aj)=[av1q1+av2q2+…+avnqn]+ [bq1+bq2+…+bqn] • 將a、b提出 • EUnew(Ai)=a[u1p1+u2p2+…+unpn]+ b[p1+p2+…+pn] • EUnew(Aj)=a[v1q1+v2q2+…+vnqn]+ b[q1+q2+…+qn]
Prove Interval Utility Scales Suffice for Decisions Under Risk • 結果 • EUnew(Ai)=aEU(Ai) + b • EUnew(Aj)=aEU(Aj) + b • EUnew(Ai) and EUnew(Aj) will stand in the same order as EU(Ai) AND EU(Aj) • a > 0 • EU(Ai) AND EU(Aj) both add b • The same conclusion holds for all expected utility rankings of any acts using these two scales.
Monetary Scale • A popular and often convenient method for determining how strongly a person prefers something is to find out how much money he or she will pay for it. • Be expected to be at least an ordinal scale. • It often works as an interval scale too – at least over a limited range of items.
Monetary Scale • Ex. • If I sense no difference between $5 increases (e.g., from $100 to $105) for prices between $100 and $200. • It can adequately function as an interval scale for my preferences for items in the range of $100 ~ $200. • With this range it would make sense for me to make decisions under risk on the basis of expected monetary vales (EMVs).
Business Decisions Base on EMVs • We are used to valuing things in terms of money – we even price intangibles. • Our own labor and time • Beautiful sunset • EMVs are often used as a basis for decisions under risk.
Nonbusiness Decisions Base on EMVs • Ex. • The federal government’s decision in 1976 to vaccinate the population against an expected epidemic of swine flu. • Suppose a flu epidemic will remove 5 million people from the work force for a period of five days. • Further, suppose the average worker contributes $200 per five days of work to the GNP. • Finally, suppose the GNP cost of the vaccine program is $40 million, that without it there is a 90% chance of an epidemic and with it only a 10% chance.
Nonbusiness Decisions Base on EMVs • The EMV of having the program • -$1040 * 0.1 + -40 * 0.9 = -140 million • The EMV of not having the program • -$1000 * 0.9 + 0 =- $900 million
The Contravention of EMV • Few people would find EMVs a satisfactory basis for every decision. • Sometimes making greater profits is not worth the effort. • Some apparently rational businesspeople gladly sacrifice profits for special considerations. • On a personal level the “true value” of an alternative is often above or below its EMV.
True Value Below EMV • After ten years of work you have saved $15,000 as a down payment on your dream house. • If you can invest your $15,000 for one month, he can assure you an 80% chance at a $50,000 return . Unfortunately, if the investment fails, you lose everything.
True Value Below EMV • The EMV of this investment • 35000 * 0.8 – 15000 * 0.2 =$ 25,000 • You choice to buy your dream house , because you feel you cannot afford to risk the $15,000.
True Value Above EMV • Suppose you have been trying to purchase a ticket for a championship basketball game. Tickets are available at $20 but you have only $10 on you. • A fellow comes along and offers to match your $10 on a single roll of the dice. If you roll snake eyes, you get the total pot; otherwise he takes it.
True Value Above EMV • The EMV of rolling is -$7.5 • Your take the bet, since having the $20 is worth the risk to you.
Practical Problems with EMVS • You alone have been given a ticket for the one and only lottery your state will have . The ticket gives you one chance in a million of winning $1,000,000. • The EMV of the ticket is $1 ? • After the lottery is drawn you win nothing or $1,000,000 – but never $1.
Practical Problems with EMVS • Assuming money is all that counts • Would it be rational for you to sell your ticket for $2? • With a one-shot decision there is nothing to average. • We have still failed to connect EMVs with cash values.
Von Neumann-Morgenstern Utility Theory • John Von Neumann, a mathematician AND Oskar Morgenstern, an economist, developed an approach to utility that avoids the objections we raised to EMVs. • It more entrenched among social scientists. • It separates utility from probability, whereas Ramsey’s approach generates utility and subjective probability functions simultaneously.
Von Neumann-Morgenstern Utility Theory • This theory bases on the idea of measuring the strength of a person’s preference for a thing by the risks he or she is willing to take to receive it.
Von Neumann-Morgenstern Utility Theory • Sample: • Suppose that we know that you prefer a trip to Washington to one to New York to one to Los Angeles. We still do not know how much more you prefer going to Washington to going to New York. • We don’t know the strength of a person’s preference.
Measuring the Strength of a Person’s Preference • We can measure that by asking you the following question: • Suppose you were offered a choice between a trip to New York and a lottery that will give you a trip to Washington if you “win” and one to Los Angeles if you “lose”. How great a chance of winning would you need to have in order to be indifferent between these two choices ?
Measuring the Strength of a Person’s Preference • Presumably, • If you prefer New York quite a bit more than Los Angeles. • You need high chance at Washington. • If your only slightly prefer New York to Los Angeles. • A small chance will suffice.
Measuring the Strength of a Person’s Preference • Suppose that you reply that you would need a 75% chance at Washington – no more and no less. • We should conclude that the New York trip occurs ¾ of the way between Washington and Los Angeles on your scale.
Expression Lottery • L(a, x, y) • This stand for the lottery that gives you a chance equal to a at the prize x and a chance equal to 1 – a at the prize y. • Your ranking can be represented as • Washington • New York, L(3/4, Washington, Los Angeles) • Los Angeles
Advantage of Von Neumann-Morgenstern Utility Theory • Forge a link between utilities and expected utilities. • This permits us to avoid our previous problems with monetary values and EMVs. • We can sensibly set a price for any kind of item. • It yields an agent’s personal utilities rather than monetary values generated by the marketplace.
Demands of Von Neumann-Morgenstern Utility Theory • Agents’ abilities to fix their preferences is much stronger demand than do our previous conditions of rationality. • The ordering condition: • Reduction-of-compound-lotteries condition • Continuity condition • Better-prizes condition • Better-chances condition
Reduction-of-Compound-Lotteries Condition • Agents must evaluate compound lotteries in agreement with the probability calculus. • Ex. • A • B, L(3/4, A , D) • C, L(2/5, B , D),L(2/5,L(3/4,A,D),D),L(6/20,A,D)2/5*u(L(3/4,A,D))=2/5(3/4+1/4*0)=2/5*3/4 • D
Continuity Condition • Given three alternatives A, B, C with B ranked between A and C, agents must be indifferent between B and some lottery yielding A and C as prize. • L(a, A ,C)
Better-Prizes condition • Given two other lotteries agents will prefer the one giving the better “first” prize – if everything else is equal. • Ex. • A > B > C >D • L(3/4, A , D) • L(3/4, B , D)
Better-Chances Condition • Given two otherwise identical lotteries, agents will prefer the one that gives them the best chance at the “first” prize • Ex. • A > B > C >D • L(3/4, A , D) • L(1/4, A , D)
Expected Utility Theorem • u(x) > u(y) if and only if xPy • u(x) = u(y) if and only if xIy • u[L(a, x, y)] =au(x) + (1-a)u(y) • Any u’ also satisfying (1) ~ (3) is a positive linear transformation of u. • (3) states that the utility of a lottery is equal to its expected utility. • (1)~(4) is known as the expected utility theorem.
Specify Lotteries More Precisely • Definition: • The agent is concerned with determining the utilities for some set of outcomes, alternatives , or prizes. Let us call these basic prizes. • Assume: • The number of basic prizes is a finite number greater than 1 and that the agent is not indifferent between all of them.
Specify Lotteries More Precisely • Select a top-ranked prize and label it “B” • Select a bottom-ranked prize and label it “W” • Rule for constructing Lotteries: • Every basic prize is a lottery. • If L1 and L2 are lotteries, so is L(a, L1, L2),where 0 ≤ a ≤ 1 • Something is a lottery if and only if it can be constructed according to conditions 1 and 2
Proof Continuity Condition • For any lotteries x, y, and z,if xPy and yPz, then there is some real number a such that 0 ≤ a ≤ 1 andyIL(a, x ,z) • One of the its consequences is that the ordering of the lotteries is continuous.
Proof the Better-Prizes Condition • For any lotteries x, y, and z and any number a(0 ≤ a ≤ 1),xPy if and only if L(a, z, x) P L(a, z, y) and L(a, x ,z) P L(a, y, z)