Modern Ways to Model Risk and Uncertainty Peter P. Topic: prospect theory (  classical expected utility) for modeling risk/uncertainty/ambiguity attitudes. Risk & uncertainty central in economics. Many economic models can be rewritten regarding risk modeling. Remove yellow comments before presentation. ALT-View-O Remove yellow comments before presentation. ALT-View-O

Outline: §1. Expected Utility and Risk Aversion; §2. The Valuable Intuition of Probabilistic Sensitivity (Deviating from EU); §3. The Old Theory on Probabilistic Sensitivity for Multiple Outcomes and why It Is Wrong; §4. Quiggin/Schmeidler Rank-Dependent Theory on Probabilistic Sensitivity for Multiple Outcomes and why It Is Natural; §5. Applications to Modeling Risk; §6. Making Ambiguity Operational Using Prospect Theory; §7. A Discussion of Framing and Losses. 2

Outline: §1. Expected Utility and Risk Aversion; §2. The Valuable Intuition of Probabilistic Sensitivity (Deviating from EU); §3. The Old Theory on Probabilistic Sensitivity for Multiple Outcomes and why It Is Wrong; §4. Quiggin/Schmeidler Rank-Dependent Theory on Probabilistic Sensitivity for Multiple Outcomes and why It Is Natural; §5. Applications to Modeling Risk; §6. Making Ambiguity Operational Using Prospect Theory; §7. A Discussion of Framing and Losses. 3

4 Expected value Simplest way to evaluate risky prospects: x1x1 xnxn p1p1 pnpn  p 1 x p n x n Violated by risk aversion: x1x1 xnxn p1p1 pnpn  p 1 x p n x n

5 Expected utility (EU) Bernoulli: x1x1 xnxn p1p1 pnpn  p 1 x p n x n Theorem. EU: Risk aversion  U concave U  U concave: Measure of risk aversion: –U´´/U´ (Pratt & Arrow). Other often-used index of risk aversion: –  U´´/U´. U( )

Intuitive problem: Risk aversion  U concave: travel back in time/memory when first heard. U reflects value of money; not risk !? U determined by specific nature of outcomes. Different for # hours of listening to music; # years to live; # liters of wine; … nonquantitative outcomes (health states) … 6 Saying that risk attitude should not depend on outcomes will not convince them.

Lopes (1987, Advances in Experimental  ): Risk attitude is more than the psychophysics of money. Empirical problems: Plentiful (Allais, Ellsberg) One more (Rabin 2000): For small amounts EU  EV. However, empirically not so! 7

Outline: §1. Expected Utility and Risk Aversion; §2. The Valuable Intuition of Probabilistic Sensitivity (Deviating from EU); §3. The Old Theory on Probabilistic Sensitivity for Multiple Outcomes and why It Is Wrong; §4. Quiggin/Schmeidler Rank-Dependent Theory on Probabilistic Sensitivity for Multiple Outcomes and why It Is Natural; §5. Applications to Modeling Risk; §6. Making Ambiguity Operational Using Prospect Theory; §7. A Discussion of Framing and Losses. 8

Rotate left and flipped horizontally: 1 = w(.10)100 U(1) = 0.10U(100) U(0) = (normalization) Psychology since 1950: Psychology: 9 = w(.30)100 Assume following data: 0 (e) (d) (c) p $0 $ $70 $30 $ (a) (b) 0.7 (c) $ p $0 $100 $70 $ (a) (b) (d) (e) $ ~ $1 0 (a) 0.50 $81 $ ~ $9 $ ~ $25 $ ~ $ ~ $ (b)(d)(c) (e) Psychology:  = w(p)100 Below is graph of w(p) (  100). EU: EU: U(9) = 0.30U(100) = EU: U(  ) = pU(100) = p. Below: is graph of U. next p.p. 11 (exercise) 9

Psychologists (Lopes etc.): What economists do with money, is better done with probabilities! Risk attitude has more to do with probs. 10 w increasing, w(0) = 0, w(1) = 1.  pU(  ) Economists At first, for simplicity, we consider linear U. Is OK for moderate amounts of money. p. 9 U/w graph p  1–p 0  w(p)  Psychologists p  1–p 0 Joint  0  w(p)U(  ) p 1–p

11 Assume data of p. 9. Exercise. Assume  = w(p)100; w(0) = 0; w(1) = 1. Question: What is w(0.10)? What is w(0.30)? What is w(0.50)? What is w(0.70)? What is w(0.90)? $ ~ $1 0 (a) 0.50 $81 $ ~ $9 $ ~ $25 $ ~ $ ~ $ (b)(d)(c) (e)

12 0 (e) (d) (c) p $0 $ $70 $30 $ (a) (b) 0.7 (c) $ p $0 $100 $70 $ (a) (b) (d) (e) $100  p  p Assume empirical relationship: p =  (  /100); so,  = 100p 2. Question 1: Assume EU ( U(  ) = pU(100)) + (1– p)U(0) ) with U(0) = 0 and U(100) = 1. What is formula of U(  ) in terms of  ? Question 2: Assume  = w(p)100 (and w(0) = 0; w(1) = 1). What is formula of w(p) in terms for p? Exercise.  ~ (p:100, 1–p:0) If blue  on abscissa comes say: For each  we can find the p s.t..if blue p on abscissa comes say reversed

Outline: §1. Expected Utility and Risk Aversion; §2. The Valuable Intuition of Probabilistic Sensitivity (Deviating from EU); §3. The Old Theory on Probabilistic Sensitivity for Multiple Outcomes and why It Is Wrong; §4. Quiggin/Schmeidler Rank-Dependent Theory on Probabilistic Sensitivity for Multiple Outcomes and why It Is Natural; §5. Applications to Modeling Risk; §6. Making Ambiguity Operational Using Prospect Theory; §7. A Discussion of Framing and Losses. 13

14 p1p1 x1x1 pnpn xnxn To explain what follows in a simple way, we assume U linear. (What follows holds in fact for general U.) Above formula is old (Edwards 1954). Revived by prospect theory (Kahneman & Tversky 1979). More general, more outcomes:  ?  w(p 1 )U(x 1 ) w(p n )U(x n )!?

However, problem: To explain, Say: w is not identity, so not w(p) = p for all p. Then w is nonlinear. Then, for some p 1, p 2 w(p 1 + p 2 )  w(p 1 ) + w(p 2 ) (take my word). 15

Case 1. w(p 1 + p 2 ) > w(p 1 ) + w(p 2 ). Something will go wrong (similar for Case 2 with <).  We will consider the theoretical value of a prospect;  Will change an outcome to see what happens;  An anomaly will result. First we repeat well-known things from expected value (EV) and expected utility (EU). 16

Figure * (for x 1 > … > x n  0; p 1 and p 2 as above). 17 EV, p 1 x p n x n is area. xnxn.. pnpn x1x1 x2x2 x3x3 p1p1 p2p2 p3p3 0. Next page input mode. Rotate figure left, then copy, then flip horizontally.

xnxn.. pnpn x1x1 x2x2 x3x3 p1p1 p2p2 p3p3 0. Rotating and flipping 18 EV:... During the lecture, first rotate left, then make copy, and then flip horizontally.

x2x2 xnxn pnpn p1p1 p2p2 p3p3 x3x3 x1x Flipping horizontally EV:.. x3x3 x1x1 x2x xnxn Rotating left pnpn p1p1 p2p2 p3p3  height: G(  ) = P(outcome >  ). called rank of , r(  ) 19 Thus, EV = ∫ G(  )d .

xnxn.. pnpn x1x1 x2x2 x3x3 p1p1 p2p2 p3p p 1 U(x 1 ) + p 2 U(x 2 ) p n U(x n ) EU is area : EU: we transform heights of columns (distances from x j "all the way down" to the x-axis). Figure ** U(x 2 ) U(x 3 ) U(x n ) U(x 1 ) (Preparatory illustration of EU)

x1x1 x2x2 xnxn pnpn p1p1 p2p2 p3p3 x3x w(p 3 )x 3 21 w(p n ) w(p 3 ) w(p 2 ) w(p 1 )x w(p n )x n Transforming probabi- lities of fixed outcomes (the old—wrong—way). Figure *** + w(p 2 )x 2 We have transformed height of each row/layer (distance from endpoint down to its lower neighbor). Value of prospect is Is EV (flipped & rotated) w(p 1 ) is area: If want to change some in the same figure appearing twice later, may have to start doing it here. And a change here implies corresponding change in later two figures. The first one following is to be used in input mode so that, obviously, animations don't matter. The second one after has this one grouped as starting point and then has animations added. Always say explicitly that the distance from a point to its lower neighbor is being transforme d.

22 Now we "play" with x 1 and see if the old evaluation behaves well. We reduce x 1. Next page input mode. Slowly move x1 leftwards to x2. When hit, go to next slide.

Transforming probabi- lities of fixed outcomes (the old—wrong—way). x2x2 xnxn pnpn p1p1 p2p2 p3p3 x3x w(p 3 )x 3 w(p n ) w(p 3 ) w(p 2 ) w(p 1 )x w(p n )x n Figure **** + w(p 2 )x 2 We have transformed height of each row/layer (distance from endpoint down to its lower neighbor). Value of prospect is Is EV (flipped & rotated) w(p 1 ) is area: 23 x1x1 Via escape go back to input mode. Then move x 1 leftward in 3 steps, until it hits x 2. Then move to next slide and go back to presentation mode.

Transforming probabi- lities of fixed outcomes (the old—wrong—way). x2x2 xnxn pnpn p1p1 p2p2 p3p3 x3x w(p 3 )x 3 w(p n ) w(p 3 ) w(p 2 ) w(p 1 )x w(p n )x n Figure **** + w(p 2 )x 2 We have transformed height of each row/layer (distance from endpoint down to its lower neighbor). Value of prospect is Is EV (flipped & rotated) w(p 1 ) is area: x1x1 w(p 1 + p 2 ) x1x1 x2x2 xnxn 0 x3x w(p 1 +p 2 )x 1 additional area!!! 24

25 Case 2. w(p 1 + p 2 ) < w(p 1 ) + w(p 2 ). Similar problems. Now move x 2 up towards x 1, until it hits x 1 : a sudden implosion of area, with - discontinuity; - increasing x 2 may decrease value. Conclusion. Transforming probabilities in old way is unsound.

Old way does not work. First discovered: Fishburn (1978). Also by: Kahneman & Tversky (1979). Taking stock of end 1970s: 1. Good psychological intuition that risk attitude  probabilistic sensitivity. But 2. No theory to do it. Then came the Quiggin & Schmeidler "rank-dependent" idea: 26

Outline: §1. Expected Utility and Risk Aversion; §2. The Valuable Intuition of Probabilistic Sensitivity (Deviating from EU); §3. The Old Theory on Probabilistic Sensitivity for Multiple Outcomes and why It Is Wrong; §4. Quiggin/Schmeidler Rank-Dependent Theory on Probabilistic Sensitivity for Multiple Outcomes and why It Is Natural; §5. Applications to Modeling Risk; §6. Making Ambiguity Operational Using Prospect Theory; §7. A Discussion of Framing and Losses. 27

xnxn.. pnpn x1x1 x2x2 x3x3 p1p1 p2p2 p3p3 0. U(x 1 ) U(x 2 ) U(x 3 ) U(x n ) pnpn p1p1 p2p2 p3p x2x2 xnxn pnpn p1p1 p2p2 p3p3 x3x3 x1x w(p n ) w(p 3 ) w(p 2 ) w(p 1 ) x2x2 xnxn 0 x3x x1x1 Economists' outcome sensitivity (EU)  28 Psychologists' probabilistic sensitivity ("OPT")  Quiggin (1982) & Schmeidler (1989): Why not do the same in the probability dimension as in the outcome dimension? OPT, in separable- outcomes form here, was not explained to them before. Explain here that EU transforms distances from points all the way down to the axes, but OPT transforms distances to neighbors below.

29 p1p1 x1x1 x2x2 xnxn x3x w(p n p 1 ) w(p n– p 1 ) w(p 1 ) w(p 2 +p 1 ) x2x2 xnxn 0 x3x3 x1x w(p 3 + p 2 + p 1 ) (1=)p n p 1... p n– p 1 p 3 + p 2 + p 1 p 2 +p 1 Psychologists' probabilistic sensitivity (PT)  (= 1!) At "=1" say that this is a crucial point, that weights add to 1.

x2x2 xnxn 0 x3x3 x1x w(p n p 1 ) w(p n– p 1 ) w(p 1 ) 1 = w(p 2 +p 1 ) = 30 w ( P( >  ) ) = ( w(p n p 1 )  w(p n  p 1 ) ) x n ( w(p 2 +p 1 )  w(p 1 ) ) x 2 w(p 1 )x Value is area : Using rank-notation simplifies. Then value is ( w(p n +r n )  w(r n ) ) x n ( w(p 2 +r 2 )  w(r 2 ) ) x 2 + ( w(p 1 +r 1 )  w(r 1 ) ) x 1 Figure ***** 

Preceding formula did with probabilities what EU did with outcomes. Best is to combine the two, with both w(p) and U(  ), resulting in (new) prospect theory: 31

Value is area : U(x 3 ) U(x n ) 0 U(x 1 )..... U(x 2 ) w(p n p 1 ) w(p n– p 1 ) w(p 1 ) w(p 2 +p 1 ) Figure ****** Value is ( w(p n +r n )  w(r n ) )U( x n ) ( w(p 2 +r 2 )  w(r 2 ) )U( x 2 ) + ( w(p 1 +r 1 )  w(r 1 ) )U( x 1 ) 32 This valuation method is called (cumulative) prospect theory (PT), or RDU.

U(x 3 ) U(x n ) 0 U(x 1 )..... U(x 2 )... w(p n p 1 ) w(p n–1 + … + p 1 ) w(p 1 ) w(p 2 +p 1 )... Flipped illustration of PT 33

Outline: §1. Expected Utility and Risk Aversion; §2. The Valuable Intuition of Probabilistic Sensitivity (Deviating from EU); §3. The Old Theory on Probabilistic Sensitivity for Multiple Outcomes and why It Is Wrong; §4. Quiggin/Schmeidler Rank-Dependent Theory on Probabilistic Sensitivity for Multiple Outcomes and why It Is Natural; §5. Applications to Modeling Risk; §6. Making Ambiguity Operational Using Prospect Theory; §7. A Discussion of Framing and Losses. 34

= w(¾)  w(½) = w(½)  w(¼) = 1  w(¾)  (60)  (40)  (20) w(p) =  p 1 ¾ 1 ½ ¼ 0 will give optimism = w(¼)  (80) w(p) = p 2 1 p  (80) = w(½)  w(¼) = 1  w(¾) ½ 35 Decision weights  (  ) of outcomes  ; consider (0.25:80, 0.25:60, 0.25:40, 0.25:20) = w(¼) = w(¾)  w(½) ¾ ¼ 0 1 will give pessimism  (60)  (40)  (20) Say that derivative of w says more.

36 inverse-S, (likelihood insensitivity) p w expected utility motivational cognitive pessimism extreme inverse-S ("fifty-fifty") prevailing finding pessimistic fifty-fifty Typical shapes of probability weighting Economists usually want pessimism for equilibria etc.

Outline: §1. Expected Utility and Risk Aversion; §2. The Valuable Intuition of Probabilistic Sensitivity (Deviating from EU); §3. The Old Theory on Probabilistic Sensitivity for Multiple Outcomes and why It Is Wrong; §4. Quiggin/Schmeidler Rank-Dependent Theory on Probabilistic Sensitivity for Multiple Outcomes and why It Is Natural; §5. Applications to Modeling Risk; §6. Making Ambiguity Operational Using Prospect Theory; §7. A Discussion of Framing and Losses. 37

We now turn to: Uncertainty/Ambiguity. Use prospect theory to make it operational. Can measure, predict, quantify completely. 38

Uncertainty: Outcomes contingent on uncertain events. Say a horse race takes place with n horses participating and exactly one winning. E i : horse i will win. (E 1 :x 1,…,E n :x n ): prospect yielding $x i if E i ; say x 1 > … > x n. If probabilities p i of E i are given, then PT =  1 U(x 1 ) + … +  n U(x n ) with  i = w(p i +r i ) – w(r i ). Write W(E) = w(P(E)). W is nonadditive. Then PT =  1 U(x 1 ) + … +  n U(x n ) with  i = W(E i  R i ) – W(R i ). This formula is applied in general, also if no probabilities p i of E i given. 39

U(x 3 ) U(x n ) 0 U(x 1 )..... U(x 2 )... W(E n ...  E 1 ) W(E n–1  …  E 1 ) W(E 1 ) W(E 2  E 1 )... Illustration of PT for uncertainty; prospect (E 1 :x 1,..., E n :x n ) 40

Now, for ambiguity, go to sources lecture (App. extra) starting at Ellsberg. 41

Outline: §1. Expected Utility and Risk Aversion; §2. The Valuable Intuition of Probabilistic Sensitivity (Deviating from EU); §3. The Old Theory on Probabilistic Sensitivity for Multiple Outcomes and why It Is Wrong; §4. Quiggin/Schmeidler Rank-Dependent Theory on Probabilistic Sensitivity for Multiple Outcomes and why It Is Natural; §5. Applications to Modeling Risk; §6. Making Ambiguity Operational Using Prospect Theory; §7. A Discussion of Framing and Losses. 42

Now for the other aspect of prospect theory, loss aversion. 43

46 How do people choose reference point, and frame gains and losses?...

47 Now medical example, App.4. Bring in nonEU twist at end.

xnxn.. pnpn x1x1 x2x2 x3x3 p1p1 p2p2 p3p3 0. Rotating and flipping 48 EV:... Reserve-copy if flipping etc. destroyed it. During the lecture, first rotate left, then make copy, and then flip horizontally.

Transforming probabi- lities of fixed outcomes (the old—wrong—way). x2x2 xnxn pnpn p1p1 p2p2 p3p3 x3x w(p 3 )x 3 w(p n ) w(p 3 ) w(p 2 ) w(p 1 )x w(p n )x n Figure **** + w(p 2 )x 2 We have transformed height of each row/layer (distance from endpoint down to its lower neighbor). Value of prospect is Is EV (flipped & rotated) w(p 1 ) is area: 49 x1x1 Via escape go back to input mode. Then move x 1 leftward in 3 steps, until it hits x 2. Then move to next slide and go back to presentation mode. Reserve-copy if flipping etc. destroyed it.