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CHAPTER 3

CHAPTER 3. Certainty Equivalents from Utility Theory. 3.1. Certainty Equivalents. Suppose you had 2 choices: a 1 . Flip a fair coin Heads: you get $500, Tails: you get nothing a 2 . $200 for certain: price for selling a 1  p(  ) a 1 a 2 Heads 0.5 500 200 Tails 0.5 0 200

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CHAPTER 3

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  1. CHAPTER 3 Certainty Equivalents from Utility Theory

  2. 3.1. Certainty Equivalents • Suppose you had 2 choices: • a1. Flip a fair coin Heads: you get $500, Tails: you get nothing • a2. $200 for certain: price for selling a1 •  p() a1 a2 Heads0.5 500 200 Tails 0.5 0 200 • Your sale price of a1 is its certainty equivalence

  3. Certainty Equivalents Definition • A certain payoff which has the same value to the decision maker as an uncertain payoff • a1. Uncertain payoff: Heads: +$500, Tails: 0 • a2. Certain payoff: $X • If X = 0, you will prefer a1 • If X = 500, you will prefer a2 • Certainty Equivalent: Value of X (0  X  500) that makes a1 = a2

  4. Determining & Using CE In general, given 2 choices: • a1. $X with probability p, or $Y with prob. (1 – p) • a2. $Z for certain (X  Z  Y) • The value of Z that makes the 2 options equal to decision maker is the: Certainty equivalent of a1 : CE(a1) • Stochastic problems can be transformed to deterministic equivalent • Criterion: select option aj with maximum CE(aj)

  5. Making Decisions based on CE • Given 4 choices depending on flipping a fair coin •  p() a1 a2 a3 a4 Heads0.5 550 700 400 300 Tails 0.5 0 –100 100 150 CE 200 150 230 220 • a1 is chosen under minimax regret • a2 is chosen under max EV (but highest risk) • a3 is chosen under max certainty equivalent (combines expected value with risk) • a4 is chosen under maximin

  6. Certainty Equivalents & Coherence • The concept of CE provides a coherent approach for evaluating (ranking) decisions • A valid criterion must recommend ranking consistent with the CE options • A coherent criterion must provide the same score for an uncertain option and its certainty equivalent

  7. CE & Coherence Counter-Example • Given: • Option aj has CE(aj) = y*, but • Under criterion C, score C(aj)  C(y*) • We can find y’ such that: • C(aj) = C(y’), y’ > y* • Decision maker (DM) will pay to replace y* by y’ • Next, since C(aj) = C(y’), DM will not mind switching from y’ to aj. • Next, since y* = CE(aj), DM will not mind switching from aj to y*

  8. CE & Coherence Counter-Example • Example shows that an incoherent criterion makes DM a perpetual money-making machine • For coherence: y’ = y* • Any evaluation criterion must be subjected to this coherence test • Can we use only CE criterion for all decision problems? No, only for simple 2-outcome problems

  9. CE for complex problems • Given:  p() a1 Excellent0.1 10,000 Good 0.3 5,000 Average 0.3 1,000 Poor 0.2 – 400 Terrible 0.1 – 3,000 • Evaluating CE(a1) is extremely difficult • Utility theory is used for complex problems

  10. 3.2. Utility Functions • Utility: Relative value (worth) of each payoff to the decision maker • Utility Theory: Transform payoffs into utility scale (0  1) • Utility & Coherence: Expected utility criterion EU(aj) ranking of options is consistent with DM certainty equivalents EU(aj)

  11. Evaluating utility functions • Given:  p() a1 a2 Good 0.3 $1000 $800 Average 0.4 $500 $600 Poor 0.3 $300 $400 • Min payoff = $300, Max payoff = $1000 • Range of payoffs (300  1000) U(300) = 0 U(1000) = 1

  12. Evaluating utility functions • What is CE for: (p = 0.5 of $300, and p = 0.5 of $1000)? • Assume CE = $500 U(500) = 0.5*U(300) + 0.5*U(1000) = 0.5 • For (p = 0.5 of $300, and p = 0.5 of $500) Assume CE = $375 U(375) = 0.5*U(300) + 0.5*U(500) = 0.25 • If equal prob of 500 & 1000 has CE = 700, we get: U(700) = 0.75

  13. Evaluating utility functions • y 300 375 500 700 1000 • U(y) 0 0.25 0.5 0.75 1.0 1 300 375 500 700 1000 y

  14. Converting payoffs to utilities • Utility matrix, using interpolation:  p() a1 a2 Good 0.3 1 0.85 Average 0.4 0.5 0.65 Poor 0.3 0 0.33 EU 0.5 0.61 • Since U(375) = 0.25 & U(500) = 0.5 U(400) = 0.25 + [(400-375)/(500-375)]*(0.5-0.25) = 0.3 • Based on EU, choose a2

  15. Steps in using utility functions • Derive the utility function using simple CE questions • Transform payoffs into utilities • Choose decision with max expected utility

  16. Utility Ex 1: Oil exploration • Decisions: Alternative investment strategies in oil exploration • To evaluate utility, 2 options: a1. Invest $X to explore for oil prob p: you get $Y, prob (1 – p): you get 0 a2. Do not invest • What probability p would make you indifferent?

  17. Utility Ex 2: Education planning • Decisions: Alternative reading improvement programs • Payoff: Average reading performance • Utility function changes slope around national average (50%) Risk = doing worse than national average Shape of utility function indicates risk attitude

  18. 3.3. Risk Attitudes • Given 2 choices: •  p() a1 a2 Heads0.5 500 200 Tails 0.5 0 200 • If 2 options are equivalent to you, i.e., CE(a1) = 200, then CE(a1) = 200 < EV(a1) = 250 • You considered are risk averse (avoider)

  19. Risk Premium • Risk Premium Money DM is willing to pay to avoid uncertainty (risk) RP(y) = EV(y) – CE(y) = 250 – 200 = 50 • 3 risk attitudes: • Risk-Averse: RP(y) > 0 • Risk-Neutral: RP(y) = 0 • Risk-Seeking: RP(y) < 0

  20. Risk-Neutral Utility Function Straight line: EV(y) = CE(y) RP(y) = constant U’(y) = 1, U’’(y) = 0 U(y) 1 0 y ymin ymax

  21. Risk-Averse Utility Function Concave line: EV(y) > CE(y) RP(y) > 0 U’(y) > 0, U’’(y)  0 U(y) 1 y ymin ymax

  22. Risk-Seeking Utility Function Convex line: EV(y) < CE(y) RP(y) < 0 U’(y) > 0, U’’(y)  0 U(y) 1 y ymin ymax

  23. Risk Attitude Example • Given 2 options: • a1. Uncertain payoff: Heads: +$500, Tails: 0 • a2. Certain payoff: $X What value of X would make 2 options equivalent? • Risk averse: X = 200 RP = 50 • Risk neutral: X = 250 RP = 0 • Risk seeking: X = 300 RP = – 50

  24. Applications of Risk Attitude • Risk Aversion Most common approach in significant decisions • Risk neutrality Corresponds to expected value criterion. Should be used in routine, non-significant decisions • Risk attitude may: - change over time - increase with increasing capital

  25. Risk Attitude vs. payoff range y • A payoff consists of both: • Certain amount y • Uncertain amount   << y, mean = 0, variance = 2, • RP( + y) = EV( + y) – CE( + y) = y – CE( + y) • Risk attitude is: • Decreasing if RP(+y) decreases as y increases • constant if RP(+y) is constant as y increases • Increasing if RP(+y) increases as y increases

  26. Risk Attitude vs. payoff range y • Constant risk attitude (premium) Constantly risk-averse U(y) = a – be– ry, r > 0, a & b constants Constantly risk-neutral U(y) = a + by, a & b constants Constantly risk-averse U(y) = a + be– ry, r < 0, a & b constants

  27. Risk Attitude vs. payoff range y • Decreasing risk attitude Risk aversion (premium) decreases with increasing capital U(y) = – e– ay – be– cy, a > 0, bc > 0 • Decreasing risk attitude Risk aversion (premium) proportional to y RP( + y) = a + by

  28. Risk Aversion Function • r(y) = – U’’(y)/U’(y) • RP( + y)  0.5 2 r(y) • Example Given: U(y) = a + by – cy2, b, c > 0, 0 < y < b/2c U’(y) = b – 2cy U’’(y) = – 2c r(y) = 2c/(b – 2cy) RP( + y) = c2/(b – 2cy) > 0 (increasing risk attitude)

  29. 3.4. Theoretical Assumptions of Utility • Preceding sections: • How utility works • This section: • Why utility works • Theoretical basis • Basic assumptions

  30. Notation • Prospect Aj n payoffs, Yi, each with probability pji, i = 1…n payoff Y1 Y2 … Yn probability pj1 pj2 … pjn Aj = (pj1, Y1; pj2, Y2; … ; pjn, Yn)

  31. Notation • Compound Prospect Ck m prospects, Aj, each with probability qkj, j = 1…m prospect A1 A2 … Am probability qk1 qk2 … qkm Ck = (qk1, A1; qk2, A2; … ; qkm, Am)

  32. Notation example • A1: fair coin Heads (p11 = 0.5)  Y1 = 20 Tails: (p12 = 0.5)  Y2 = – 10 • A2: bent coin Heads (p21 = 0.3)  Y1 = 20 Tails: (p22 = 0.7)  Y2 = – 10 • C1: fair die even: 2, 4, 6 (q11 = 0.5)  A1 Odd: 1, 3, 5 (q12 = 0.5)  A2

  33. Assumption 1 (Structure) • It is sufficient to describe the choices open to the decision maker in terms of payoff values and their associated probabilities • Reducing the problem to prospects and compound prospects captures all that is essential to the decision maker • Temporal resolution of uncertainty: The decision maker may choose between 2 alternatives with exactly the same payoffs and probabilities based on different payoff times

  34. Assumption 2 (Ordering) • The decision maker may express preference or indifference between any pair of payoffs • Notation Y1> Y2 Y1 is preferred to Y2 Y1 Y2 Y1 is preferred to or same as Y2 Y2 is not preferred to Y1 Y* = best payoff, Y* = worst payoff • Transitivity: If A1 A2 and A2 A3 then A1 A3

  35. Assumption 3 (Reduction of Compound Prospects) • Any compound prospect should be indifferent to its equivalent simple prospect Ck (qk1, A1; qk2, A2; … ; qkm, Am)  [qk1(p11, Y1; p12, Y2; … ; p1n, Yn); qk2(p21, Y1; p22, Y2; … ; p2n, Yn); . . . qkm(pm1, Y1; pm2, Y2; … ; pmn, Yn)]  (p'k1, Y1; p'k2, Y2; … ; p'km, Ym) Where p'kj = qk1p1j + qk2p2j + . . . + qkmpmj

  36. Assumption 3 example • C1: fair die q1j Aj p1j Y1 p2j Y2 q11 = 0.5  A1: fair coin0.5 20 0.5 –10 q12 = 0.5  A2: bent coin0.3 20 0.7 –10 • C1  (0.5, A1; 0.5, A2)  [0.5(0.5, 20; 0.5, –10); 0.5(0.3, 20; 0.7, –10)]  [(0.25 + 0.15), 20; (0.25 + 0.35), –10]  [0.4, 20; 0.6, –10]

  37. Assumption 3 & Coherence • Assumption 3 indicates ideal level of coherence • No preference for single or multiple steps • Assumption 3 does not apply if • Preference for multiple steps, game atmosphere • Special type of risk in a particular business

  38. Assumption 4 (Continuity) • Every payoff Yi can be considered a certainty equivalent for a prospect: [ui, Y*; (1 – ui), Y*], 0  ui 1 Y* = best payoff, Y* = worst payoff • Since each uncertain prospect has an equivalent certain payoff (CE), • then each certain payoff has an equivalent uncertain prospect

  39. Assumption 4 (Continuity) • Since Yi = CE of: Ai [ui, Y*; (1 – ui), Y*], 0  ui 1 Y* = best payoff, Y* = worst payoff • ui(Yi) = probability of Y* that makes Ai Yi • ui(Y*) = 1 for max payoff • ui(Y*) = 0 for min payoff • ui(Yi) = utility of payoffYi

  40. Assumption 5 (Substitutability) • In any prospect, Yi can be substituted by its a uncertain equivalent: [ui, Y*; (1 – ui), Y*] • Yi and [ui, Y*; (1 – ui), Y*] are indifferent, not only when considered alone, but also when considered part of a more complicated prospect • Similar to coherence related to minimax regret: ranking of alternatives should not change if other alternatives are added

  41. Assumption 6 (Transitivity of Prospects) • The decision maker can express preference or indifference between all pairs of prospects. • Extension of Assumption 2 (payoff preference) • Any prospect can be expressed in terms of Y* & Y* A1 (p11, Y1; p12, Y2; … ; p1n, Yn)  (p11, [u1, Y*; (1 – u1), Y*]; . . . )  (p1, Y*; p2, Y*) Where p1 = p11u1 + p12u2 + . . . + p1nun

  42. Assumption 7 ( Monotonicity) • A prospect Ar [pr, Y*; (1 – pr), Y*] is preferred or indifferent to (  ) prospect As [ps, Y*; (1 – ps), Y*] iff: pr ps • Given 2 options with the same 2 alternative payoffs, we prefer the option with higher probability of the better payoff • For options with several payoffs: Ar Asiff: pr1 u1 + pr2 u2 + . . . + pr1 un ps1 u1 + ps2 u2 + . . . + ps1 un EU(Ar)  EU(As)

  43. 3.5. Some Caveats in Interpreting Utility • Utility theory is normative: • It suggests what people should do to be coherent • Does not describe what they actually do • In practice, people violate expected utility criterion depending on circumstances

  44. Utilities do not add up • Expected utility of a sum of payoff is not equal to sum of expected utilities U(A + B)  U(A) + U(B) • Unless the decision maker is risk-neutral

  45. Utility differences do not express strength of preferences • Given: Y1 > Y2 > Y3 > Y4 , and U(Y1 – Y2 ) > U(Y3 – Y4) This does not imply moving from Y2 to Y1 is preferable to moving from Y4 to Y3. • Utility provides an “ordinal” scale, not an “interval” scale • Ordinal: teacher evaluation, (7 – 6)  (9 – 8) • Interval: weight in kilograms, (60 – 50) = (80 – 70)

  46. Utilities are not comparable from person to person • If 2 people assign the same utility to a prospect, we cannot say it has the same worth to each • Utility values are completely subjective • Utilities of different people cannot be added to determine group preferences

  47. 3.6. Issues in the assessment of risk • Utility assessment is not a natural activity for DM • Unnatural setup may results in wrong utility values, and wrong decisions • Method of assessment must be as close as possible to real problem

  48. Basic utility assessment process • Given 2 options: • X  certain payoff • Y  probability p of payoff G (gain) probability (1 – p) of payoff L (loss) • Four variables • X, Y, G, L • Fix any 3 variables, ask DM to supply the 4th

  49. 4 Response modes • Certainty equivalence: DM gives X • Probability equivalence: DM gives p • Gain equivalence: DM gives G • Loss equivalence: DM gives L • First 2 methods most common

  50. Level of probabilty • 4 variables: X, p, G, L • Except in probability equivalence methods, p is given • Small probabilities get distorted • p = 1 – p = 0.5 seems to be least biased

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