1 CHAPTER 3 Certainty Equivalents from Utility Theory

Certainty Equivalents w Suppose you had 2 choices: w a 1. Flip a fair coin Heads: you get $500, Tails: you get nothing w a 2. $200 for certain: price for selling a 1 w  p(  )a 1 a 2 Heads Tails w Your sale price of a 1 is its certainty equivalence

3 Certainty Equivalents Definition w A certain payoff which has the same value to the decision maker as an uncertain payoff w a 1. Uncertain payoff: Heads: +$500, Tails: 0 w a 2. Certain payoff: $X w If X = 0, you will prefer a 1 w If X = 500, you will prefer a 2 w Certainty Equivalent: Value of X (0  X  500) that makes a 1 = a 2

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

5 Making Decisions based on CE w Given 4 choices depending on flipping a fair coin w  p(  )a 1 a 2 a 3 a 4 Heads Tails0.50 – CE w a 1 is chosen under minimax regret w a 2 is chosen under max EV (but highest risk) w a 3 is chosen under max certainty equivalent (combines expected value with risk) w a 4 is chosen under maximin

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

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

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

9 CE for complex problems w Given:  p(  )a 1 Excellent0.110,000 Good0.35,000 Average0.31,000 Poor0.2 – 400 Terrible0.1 – 3,000 w Evaluating CE(a 1 ) is extremely difficult w Utility theory is used for complex problems

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

11 Evaluating utility functions w Given:  p(  )a 1 a 2 Good0.3$1000$800 Average0.4$500$600 Poor0.3$300$400 w Min payoff = $300, Max payoff = $1000 w Range of payoffs (300  1000) U(300)= 0 U(1000)= 1

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

13 Evaluating utility functions w y w U(y) y

14 Converting payoffs to utilities w Utility matrix, using interpolation:  p(  )a 1 a 2 Good Average Poor EU w Since U(375) = 0.25 & U(500) = 0.5 U(400) = [( )/( )]*( ) = 0.3  w Based on EU, choose a 2

15 Steps in using utility functions 1. Derive the utility function using simple CE questions 2. Transform payoffs into utilities 3. Choose decision with max expected utility

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

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

Risk Attitudes w Given 2 choices: w  p(  )a 1 a 2 Heads Tails w If 2 options are equivalent to you, i.e., CE(a 1 ) = 200, then CE(a 1 ) = 200 < EV(a 1 ) = 250 w You considered are risk averse (avoider)

19 Risk Premium w Risk Premium Money DM is willing to pay to avoid uncertainty (risk) RP(y)= EV(y) – CE(y) = 250 – 200 = 50 w 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 y min y max

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

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

23 Risk Attitude Example w Given 2 options: w a 1. Uncertain payoff: Heads: +$500, Tails: 0 w a 2. Certain payoff: $X What value of X would make 2 options equivalent? w Risk averse:X = 200RP = 50 w Risk neutral:X = 250RP = 0 w Risk seeking:X = 300RP = – 50

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

25 Risk Attitude vs. payoff range y w A payoff consists of both: Certain amounty Uncertain amount   << y, mean = 0, variance =   2, w RP(  + y) = EV(  + y) – CE(  + y) = y – CE(  + y) w 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 w 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 w Decreasing risk attitude Risk aversion (premium) decreases with increasing capital U(y) = – e – ay – be – cy,a > 0,bc > 0 w Decreasing risk attitude Risk aversion (premium) proportional to y RP(  + y) = a + by

28 Risk Aversion Function w r(y) = – U’’(y)/U’(y) w RP(  + y)  0.5   2 r(y) w Example Given: U(y) = a + by – cy 2, 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)

Theoretical Assumptions of Utility w Preceding sections: How utility works w This section: Why utility works Theoretical basis Basic assumptions

30 Notation w Prospect A j n payoffs, Y i, each with probability p ji, i = 1…n payoffY 1 Y 2… Y n probabilityp j1 p j2… p jn A j = (p j1, Y 1 ; p j2, Y 2 ; … ; p jn, Y n )

31 Notation w Compound Prospect C k m prospects, A j, each with probability q kj, j = 1…m prospectA 1 A 2… A m probabilityq k1 q k2… q km C k = (q k1, A 1 ; q k2, A 2 ; … ; q km, A m )

32 Notation example w A 1 : fair coin Heads (p 11 = 0.5)  Y 1 = 20 Tails: (p 12 = 0.5)  Y 2 = – 10 w A 2 : bent coin Heads (p 21 = 0.3)  Y 1 = 20 Tails: (p 22 = 0.7)  Y 2 = – 10 w C 1 : fair die even: 2, 4, 6 (q 11 = 0.5)  A 1 Odd: 1, 3, 5 (q 12 = 0.5)  A 2

33 Assumption 1 (Structure) w It is sufficient to describe the choices open to the decision maker in terms of payoff values and their associated probabilities w Reducing the problem to prospects and compound prospects captures all that is essential to the decision maker w 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) w The decision maker may express preference or indifference between any pair of payoffs w Notation Y 1 > Y 2  Y 1 is preferred to Y 2 Y 1  Y 2  Y 1 is preferred to or same as Y 2 Y 2 is not preferred to Y 1 Y* = best payoff,Y * = worst payoff w Transitivity: If A 1  A 2 andA 2  A 3 thenA 1  A 3

35 Assumption 3 (Reduction of Compound Prospects) w Any compound prospect should be indifferent to its equivalent simple prospect C k  (q k1, A 1 ; q k2, A 2 ; … ; q km, A m )  [q k1 (p 11, Y 1 ; p 12, Y 2 ; … ; p 1n, Y n ); q k2 (p 21, Y 1 ; p 22, Y 2 ; … ; p 2n, Y n );... q km (p m1, Y 1 ; p m2, Y 2 ; … ; p mn, Y n )]  (p' k1, Y 1 ; p' k2, Y 2 ; … ; p' km, Y m ) Where p' kj = q k1 p 1j + q k2 p 2j q km p mj

36 Assumption 3 example w C 1 : fair die q 1j A j p 1j Y 1 p 2j Y 2 q 11 = 0.5  A 1 : fair coin –10 q 12 = 0.5  A 2 : bent coin –10 w C 1  (0.5, A 1 ; 0.5, A 2 )  [0.5(0.5, 20; 0.5, –10); 0.5(0.3, 20; 0.7, –10)]  [( ), 20; ( ), –10]  [0.4, 20; 0.6, –10]

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

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

39 Assumption 4 (Continuity) w Since Y i = CE of: A i  [u i, Y*; (1 – u i ), Y * ], 0  u i  1 Y* = best payoff, Y * = worst payoff w u i (Y i )= probability of Y* that makes A i  Y i w u i (Y*)= 1for max payoff w u i (Y * )= 0 for min payoff w u i (Y i )= utility of payoff Y i

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

41 Assumption 6 (Transitivity of Prospects) w The decision maker can express preference or indifference between all pairs of prospects. w Extension of Assumption 2 (payoff preference) w Any prospect can be expressed in terms of Y* & Y * A 1  (p 11, Y 1 ; p 12, Y 2 ; … ; p 1n, Y n )  (p 11, [u 1, Y*; (1 – u 1 ), Y * ];... )  (p 1, Y*; p 2, Y * ) Where p 1 = p 11 u 1 + p 12 u p 1n u n

42 Assumption 7 ( Monotonicity) w A prospect A r  [p r, Y*; (1 – p r ), Y * ] is preferred or indifferent to (  ) prospect A s  [p s, Y*; (1 – p s ), Y * ] iff: p r  p s w Given 2 options with the same 2 alternative payoffs, we prefer the option with higher probability of the better payoff w For options with several payoffs: A r  A s iff: p r1 u 1 + p r2 u p r1 u n  p s1 u 1 + p s2 u p s1 u n EU(A r )  EU(A s )

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

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

45 Utility differences do not express strength of preferences w Given: Y 1 > Y 2 > Y 3 > Y 4, and U(Y 1 – Y 2 ) > U(Y 3 – Y 4 ) This does not imply moving from Y 2 to Y 1 is preferable to moving from Y 4 to Y 3. w 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 w If 2 people assign the same utility to a prospect, we cannot say it has the same worth to each w Utility values are completely subjective w Utilities of different people cannot be added to determine group preferences

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

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

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

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

51 Levels of payoff w Initial G and L are Y max and Y min, but values inside interval are arbitrary w Moving from outside to inside creates bias to risk aversion w Adjustment bias A new assessment is made by adjusting the previous one, this adjustment is not enough w Solution: get assessments at different times

52 Assumptions or transfer of risk w If both G and L are losses, the question is: Facing either a loss of G (prob. p) or a loss of L (prob. 1 – p), how much would you pay to have it removed? Transfer uncertain (risky) loss to a certain loss w Inertia bias: Tendency to stay with current situation unless alternative is clearly better

53 Utility bias (Allais problem ) w Allais problem (Exercise 2.8) a 1 = $1M a 2 = (0.1, $5M; 0.89, $1M; 0.1, $0) a 3 = (0.1, $5M; 0.9, $0) a 4 = (0.11, $1M; 0.89, $0) w Compare a 1 to a 2, and a 3 to a 4,

54 Allais problem w Most people: a 1 > a 2, a 3 > a 4, w Incoherent based on expected utility U(0) = Y min = 0,U(5) = Y max = 1 EU(a 1 ) = U(1) EU(a 2 ) = U(1) w If a 1 > a 2, U(1) > U(1) 0.11U(1) > 0.1

55 Allais problem EU(a 3 ) = 0.1 EU(a 4 ) = 0.11U(1) w If a 3 > a 4, 0.1 > 0.11U(1) This contradicts the result for a 1 > a 2,

56 Certainty effect w Reason for inconsistency in Allais example w People overestimate a certain payoff compared to expected utility w Expected utility should not be taken for granted Consistency checks Sensitivity analysis Estimate utility to same no. of significant figures as payoff values Y