4.1. Chapter 4. timing risk information utility

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2 4. Chapter 4 timing risk information utility

3 4.2. Money pumps Rationality in economics is identified with consistency. In particular, a preference relation must be total and transitive Transitivity implies that preferences don t go round in circles. If they did, you could pump money out of people: p

4 4.2. Money pumps Rationality in economics is identified with consistency. In particular, a preference relation must be total and transitive Transitivity implies that preferences don t go round in circles. If they did, you could pump money out of people: p lice s basket Bob s pocket

5 4.2. Money pumps Rationality in economics is identified with consistency. In particular, a preference relation must be total and transitive Transitivity implies that preferences don t go round in circles. If they did, you could pump money out of people: p Bob offers lice s basket Bob s pocket

6 4.2. Money pumps Rationality in economics is identified with consistency. In particular, a preference relation must be total and transitive Transitivity implies that preferences don t go round in circles. If they did, you could pump money out of people: p lice s preference p lice s basket Bob s pocket

7 4.2. Money pumps Rationality in economics is identified with consistency. In particular, a preference relation must be total and transitive Transitivity implies that preferences don t go round in circles. If they did, you could pump money out of people: p lice s basket Bob s pocket

8 4.2. Money pumps Rationality in economics is identified with consistency. In particular, a preference relation must be total and transitive Transitivity implies that preferences don t go round in circles. If they did, you could pump money out of people: p Bob offers lice s basket Bob s pocket

9 4.2. Money pumps Rationality in economics is identified with consistency. In particular, a preference relation must be total and transitive Transitivity implies that preferences don t go round in circles. If they did, you could pump money out of people: p lice s preference p lice s basket Bob s pocket

10 4.2. Money pumps Rationality in economics is identified with consistency. In particular, a preference relation must be total and transitive Transitivity implies that preferences don t go round in circles. If they did, you could pump money out of people: p lice s basket Bob s pocket

11 4.2. Money pumps Rationality in economics is identified with consistency. In particular, a preference relation must be total and transitive Transitivity implies that preferences don t go round in circles. If they did, you could pump money out of people: p Bob offers lice s basket Bob s pocket

12 4.2. Money pumps Rationality in economics is identified with consistency. In particular, a preference relation must be total and transitive Transitivity implies that preferences don t go round in circles. If they did, you could pump money out of people: p lice s preference p lice s basket Bob s pocket

13 4.2. Money pumps Rationality in economics is identified with consistency. In particular, a preference relation must be total and transitive Transitivity implies that preferences don t go round in circles. If they did, you could pump money out of people: lice is back where she started p Bob is 3 pennies better off

14 4.2. Money pumps lice loses out because her preferences aren t transitive p p p For transitive preferences a p b and b p c a p c

15 4.3 Utility functions If preferences are total and transitive, they can be described by a utility function: u(a) u(b) a p b ith a utility function, the problem of finding the feasible outcome x that Pandora likes best reduces to solving the maximization problem: u(x) = max u(s) s S x is the optimal outcome S is the feasible set

16 4.3.2 Utility functions Construct a utility function that describes the preference b p c d p a p e s b c d a e U(s)

17 4.3.2 Utility functions Construct a utility function that describes the preference b p c d p a p e s b c d a e U(s) 0

18 4.3.2 Utility functions Construct a utility function that describes the preference b p c d p a p e s b c d a e U(s) 0 /2 /2

19 4.3.2 Utility functions Construct a utility function that describes the preference b p c d p a p e s b c d a e U(s) 0 /2 /2 3/4

20 4.3.2 Utility functions Construct a utility function that describes the preference b p c d p a p e s b c d a e U(s) V(s) 0 /2 /2 3/ ,000

21 4.3.2 Utility functions Construct a utility function that describes the preference b p c d p a p e s b c d a e U(s) V(s) (s) 0 /2 /2 3/ ,

22 4.3.3 Utility functions Construct a utility function that describes the preference b p c d p a p e s b c d a e U(s) V(s) (s) 0 /2 /2 Causal 3/4 utility fallacy ,

23 4.4 Russian Roulette Olga Boris Vladimir

24 4.4 Russian Roulette single bullet six-shooter

25 4.4 Russian Roulette Boris s preferences: disgrace for Boris death for Boris Vladimir s preferences: L p p victory for Boris victory for Vladimir death for Vladimir L p 2 p 2 disgrace for Vladimir

26 root

27 root /6 /6 /6 / /6 /6

28 root /6 /6 /6 / /6 /6 I Boris is player I

29 root /6 fire 4.4. /6 chicken /6 fire /6 chicken /6 /6 I chicken fire chicken fire chicken fire chicken fire

30 root /6 L 4.4. /6 /6 /6 /6 /6 I

31 root /6 L 4.4. /6 /6 /6 /6 /6 I II Vladimir is player II

32 root /6 L 4.4. /6 /6 L /6 /6 /6 I II

33 root /6 L 4.4. /6 /6 L /6 /6 /6 I II I

34 root /6 L 4.4. /6 /6 L /6 /6 /6 I L II I

35 root /6 L 4.4. /6 /6 L /6 /6 /6 I L II II I

36 root /6 L 4.4. /6 /6 L /6 /6 /6 I II II L L I

37 root /6 L 4.4. /6 /6 L /6 /6 /6 I II II L L I I

38 root /6 L 4.4. /6 /6 L /6 /6 /6 I II I II L I L II L L

39 4.4.2 Russian Roulette: Version 2 I II I II I II L L L L L L

40 4.4.2 Russian Roulette: Version 2 I 5 II 4 I 3 II 2 I II L L L L L L

41 4.4.2 Russian Roulette: Version 2 I 5 II 4 I 3 II 2 I L L L L L II L Boris plays?

42 4.4.2 Russian Roulette: Version 2 I II 4 I 3 II 2 I L L L L L II L Vladimir plays?

43 4.4.2 Russian Roulette: Version 2 I 5 II 4 I 3 II 2 I L L L L L II L Boris plays? Vladimir plays?

44 4.4.2 Russian Roulette: Version 2 I 5 II 4 I 3 II 2 I II L L L L L L 0 /6 0 /6 0 /6 0 /6 0 /6 /6 Boris plays? Vladimir plays?

45 Russian Roulette: Version 2 I 5 II 4 I 3 II 2 I II L L L L L L 0 /6 0 /6 0 /6 0 /6 0 Boris plays? Boris gets /6 /6 Vladimir plays? /2 L /2

46 Russian Roulette: Version 2 I 5 II 4 I 3 II 2 I II L L L L L L Boris plays? Vladimir plays?

47 4.4.2 Russian Roulette: Version 2 I 5 II 4 I 3 II 2 I II L L L L L L Boris plays? Vladimir plays? Boris gets 0 0

48 4.4.2 Russian Roulette: Version 2 If Vladimir plays, should Boris prefer or?

49 4.4.2 Russian Roulette: Version 2 If Vladimir plays, should Boris prefer or? p L L p or /2 /2 /2 /2? e don t know, because we have only been told that L p p

50 4.5 Making risky choices hat preferences should rational players have between lotteries?

51 4.5. Making risky choices hat preferences should rational players have between lotteries? One possible answer is that rational players should work out how much money each prize is worth to them and then evaluate lotteries in terms of their expected dollar value.

52 4.5. Making risky choices St Petersburg paradox prize sequence probability $2 $4 $8 $6 $2 n H 2 TH TTH TTTH TTT TH 2 n

53 4.5. Making risky choices St Petersburg paradox prize sequence probability $2 $4 $8 $6 $2 n H 2 TH TTH TTTH TTT TH 2 n dollar expectation: 2prob(H) + 4 prob(th) + 8prob(TTH) +L = L =+++L

54 4.5. Making risky choices hat preferences should rational players have between lotteries? It is mistake to think that we can deduce players preferences over risky outcomes from their preferences over deterministic outcomes.

55 4.5. Making risky choices hat preferences should rational players have between lotteries? It is mistake to think that we can deduce players preferences over risky outcomes from their preferences over deterministic outcomes. s with preferences over different flavors of ice-cream, rational players may have different attitudes to risky prospects. ll we can sensibly ask is that their risk attitudes be consistent.

56 4.5.2 Von Neumann and Morgenstern utility The St Petersburg paradox shows that rational players won t necessarily always maximize the expected dollar value of a lottery, but VN&M showed that they will act as though maximizing the expected value of something. The something they act as though maximizing is called VN&M utility

57 4.5.2 Von Neumann and Morgenstern utility The St Petersburg paradox shows that rational players won t necessarily always maximize the expected dollar value of a lottery, but VN&M showed that they will act as though maximizing the expected value of something. The something they act as though maximizing is called VN&M utility To show this fact, VN&M needed some consistency postulates

58 4.5.2 Von Neumann and Morgenstern utility L = w w 2 w 3 w n pn p p 2 p 3

59 4.5.2 Von Neumann and Morgenstern utility L = w p p 2 p 3 w 2 w 3 w n pn Postulate 2: Each prize between the best prize and the worst prize L is equivalent to some lottery involving only and L.

60 4.5.2 Von Neumann and Morgenstern utility L = w p p 2 p 3 w 2 w 3 w n pn Postulate 2: Each prize between the best prize and the worst prize L is equivalent to some lottery involving only and L. So, for each prize w, there is a probability q with w L -q q

61 4.5.2 Von Neumann and Morgenstern utility L = w p p 2 p 3 w 2 w 3 w n pn q = qqq u(w) Postulate 2: Each prize between the best prize and the worst prize L is equivalent to some lottery involving only and L. So, for each prize w, there is a probability q with w L -q q

62 4.5.2 Von Neumann and Morgenstern utility L = w w 2 w 3 w n pn p p 2 p 3 w L -q q

63 4.5.2 Von Neumann and Morgenstern utility L = w p w 2 w 3 p 2 p 3 pn p 3 w n w L q -q Proposition 3: Rational players don t care if a prize in a lottery is replaced by another prize that they regard as equivalent. q

64 4.5.2 Von Neumann and Morgenstern utility L = w p w 2 p 2 w 3 p 3 w n pn p n L -q q L -q 2 q 2 L -q 3 q 3 L -q n q n p p 2 p 3 p n

65 4.5.2 Von Neumann and Morgenstern utility L = w p p 2 w 2 w 3 w n Postulate p 3 4: Rational pn p n players only care about the probability with which they get each prize. L L L -q q -q 2 q 2 -q 3 q 3 L -q n q n p p 2 p 3 p n

66 4.5.2 Von Neumann and Morgenstern utility L = w p w 2 p 2 w 3 p 3 w n pn p n L -q q L -q 2 q 2 L -q 3 q 3 L -q n q n p p 2 p 3 p n L -(p q +p 2 q 2 + +p n q n ) p q +p 2 q 2 + +p n q n

67 4.5.2 Von Neumann and Morgenstern utility L = w p w 2 w 3 p 2 w 3 p 3 w n pn L -(p q +p 2 q 2 + +p n q n ) p q +p 2 q 2 + +p n q n

68 4.5.2 Von Neumann and Morgenstern utility L = w p w 2 w 3 w n w 3 Postulate p 2 p 3 : rational pn player prefers whichever of two win-or-lose lotteries offers the larger probability of winning. L -(p q +p 2 q 2 + +p n q n ) p q +p 2 q 2 + +p n q n

69 4.5.2 Von Neumann and Morgenstern utility L = w p w 2 w 3 p 2 w 3 p 3 w n pn L -(p q +p 2 q 2 + +p n q n ) p q +p 2 q 2 + +p n q n rational player prefers whichever of two lotteries has the larger value of r = p q +p 2 q 2 + +p n q n

70 4.5.2 Von Neumann and Morgenstern utility L = w p w 2 w 3 p 2 w 3 p 3 w n pn L -(p q +p 2 q 2 + +p n q n ) p q +p 2 q 2 + +p n q n rational player therefore acts as though maximizing r = p q +p 2 q 2 + +p n q n = p u(w )+ p 2 u(w 2 )+ +p n u(w n ) = Eu(L)

71 4.5.2 Von Neumann and Morgenstern utility L = w p w 2 w 3 p 2 w 3 p 3 w n pn expected utility of the lottery L L -(p q +p 2 q 2 + +p n q n ) p q +p 2 q 2 + +p n q n rational player therefore acts as though maximizing r = p q +p 2 q 2 + +p n q n = p u(w )+ p 2 u(w 2 )+ +p n u(w n ) = Eu(L)

72 4.5.3 St Petersburg paradox again u(x) = 4 x Pandora s VN&M utility for money

73 4.5.3 St Petersburg paradox again Expected utility of the St Petersburg lottery u(x) = 4 Eu(L) = u(2) + ( ) 2 u(2 2 ) + ( ) 3 u(2 3 ) +L ) = x { ( ) ( ) L } 2 ) = ) = L

74 4.5.3 St Petersburg paradox again u(x) = 4 x Eu(L) = 2 u(2) + 2 { ( ) L } ) = ( 2 u(x) ) = Eu(L) 2 ) = X L X (2.42) 2 = 5.86 ) = Pandora will ( ) 2 just u(2 2 pay $X ) + ( ) 3 (no u(2 3 more) to participate in the St ) +L 2Petersburg lottery, if So Pandora will pay only $

75 4.5.3 Risk aversion u(x) = 4 x Pandora s VN&M utility for money

76 4.5.3 Risk aversion u(x) = 4 x M = $ $6 4/5 /5

77 4.5.3 Risk aversion u(x) = 4 x M = $ $6 4/5 /5 E(M) = = 4

78 4.5.3 Risk aversion u(x) = 4 x M = $ $6 4/5 /5 E(M) = = 4 dollar expectation

79 4.5.3 Risk aversion u(x) = 4 x M = $ $6 4/5 /5 E(M) = = 4 u(4) = 4 4 = 8 Eu(M) = 4 5 u() + 5 u(6) =

80 4.5.3 Risk aversion u(e M) > Eu(M) u(x) = 4 x M = $ $6 4/5 /5 E(M) = = 4 u(4) = 4 4 = 8 Eu(M) = 4 5 u() + 5 u(6) =

81 4.5.3 Risk aversion u(e M) > Eu(M) u(x) = 4 x M = $ $6 4/5 /5 Pandora is risk averse because she prefers getting the dollar expected value of a lottery for sure to the lottery itself.

82 4.5.3 Risk aversion u(x) = 4 x M = $ $6 4/5 /5

83 4.5.3 Risk aversion y u(x) = 4 x u(6) u(4) u() x

84 4.5.3 Risk aversion y u(x) = 4 x M = $ $6 4/5 /5 u(6) u(4) u() x

85 4.5.3 Risk aversion y u(x) = 4 x M = $ $6 4/5 /5 u(6) u(4) u() x

86 u() + u(6) 5 5 Risk aversion y u(x) = 4 x M = $ $6 4/5 /5 u(6) u(4) u() x

87 u() + u(6) 5 5 Risk aversion y u(x) = 4 x M = $ $6 4/5 /5 u(6) u(4) u() u(e M) > Eu(M) x

88 4.5.3 Risk aversion y Risk averse players have concave utility functions x

89 4.5.3 Risk aversion y Risk neutral players have affine utility functions x

90 4.5.3 Risk aversion y Risk loving players have convex utility functions x

91 4.6 Utility scales Celsius utility scale is like a temperature scale---you can choose the zero and the unit in any convenient way, but then you have no more room for maneuver. Fahrenheit

92 4.5.3 Utility scales f = 9 5 c + 32 utility scale is like a temperature scale---you can choose the zero and the unit in any convenient way, but then you have no more room for maneuver. affine transformation

93 4.7 Russian Roulette: Version 2 I II I II I II L L L L L L

94 4.7 Russian Roulette again e can now analyze Russian Roulette by assigning VN&M utilities to the outcomes u (L) = 0 u () = a u () = u 2 (L) = 0 u 2 () = b u 2 () = The smaller a or b, the readier the player will be to take a risk.

95 4.7 Russian Roulette again a = 0.25, b = 0.55 Boris is reckless and Vladimir mildly cautious I II I II I II

96 4.7 Russian Roulette again a = 0.25, b = 0.55 Boris is reckless and Vladimir mildly cautious.55 I II I II I II

97 4.7 Russian Roulette again a = 0.25, b = 0.55 Boris is reckless and Vladimir mildly cautious I II I II I 2 2 II

98 4.7 Russian Roulette again a = 0.25, b = 0.55 Boris is reckless and Vladimir mildly cautious I II I II I II

99 4.7 Russian Roulette again a = 0.25, b = 0.55 Boris is reckless and Vladimir mildly cautious I II I II 2 I 3 3 II

100 4.7 Russian Roulette again a = 0.25, b = 0.55 Boris is reckless and Vladimir mildly cautious I II I II I II

101 4.7 Russian Roulette again a = 0.25, b = 0.55 Boris is reckless and Vladimir mildly cautious Boris plays and Vladimir plays

102 4.7 Russian Roulette again a b Boris Vladimir Boris reckless Vladimir cautious both reckless both very cautious

103 llais paradox 4.8. J = $0m $m $5m 0 0 K= $0m $m $5m L = $0m $m $5m M = $0m $m $5m

104 llais paradox 4.8. J = $0m $m $5m 0 0 K= $0m $m $5m L = $0m $m $5m M = $0m $m $5m Is it consistent to say J f K and M f L?

105 u(0)=0 u()=x llais paradox u(5)= 4.8. J = $0m $m $5m 0 0 K= $0m $m $5m L = $0m $m $5m M = $0m $m $5m Is it consistent to say J f K and M f L?

106 u(0)=0 u()=x llais paradox u(5)= 4.8. J = $0m $m $5m 0 0 K= $0m $m $5m L = $0m $m $5m M = $0m $m $5m Is it consistent to say J f K and M f L? x >.89x >.x

107 u(0)=0 u()=x llais paradox u(5)= 4.8. J = $0m $m $5m 0 0 K= $0m $m $5m L = $0m $m $5m M = $0m $m $5m Is it consistent to say J f K and M f L? x > 0/ 0/ >x

108 u(0)=0 u()=x llais paradox u(5)= 4.8. J = $0m $m $5m 0 0 K= $0m $m $5m L = $0m $m $5m M = $0m $m $5m Is it consistent to say J f K and M f L? x > 0/ NO 0/ >x

109 Zeckhauser s paradox 4.8.2

110 4.8.2 Zeckhauser s paradox ould you pay more to get one bullet removed from a gun with one bullet than to get one bullet removed from a gun with four bullets?

111 4.8.2 Zeckhauser s paradox $X $Y

112 4.8.2 Zeckhauser s paradox X > Y? $X $Y

113 4.8.2 Zeckhauser s paradox alive after paying $X alive after paying nothing dead u(l) = 0 u(c) u() = $X

114 4.8.2 Zeckhauser s paradox u(l) = 0 u(c) u() = u(c) = 6 u(l) u( ) $X u(c) = 5 6

115 4.8.2 Zeckhauser s paradox alive after paying $Y alive after paying nothing dead u(l) = 0 u() u() = $Y

116 4.8.2 Zeckhauser s paradox alive after paying $Y alive after paying nothing dead u(l) = 0 u() u() = 2 u(l) + u( ) = u(l) + u() = u() 3 2 u() = 2 3 $Y

117 4.8.2 Zeckhauser s paradox u(c) = 5 6 u() = 2 3 C f

118 4.8.2 Zeckhauser s paradox u(c) = 5 6 u() = 2 3 C f f $X Y > X $Y