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## Decision Theory

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**Decision Theory**Lecture 8**Reduction of compound lotteries**1 A 1/2 A 1/3 1/4 A 1/2 1/2 1/3 3/8 B B 3/8 1/3 C 1/2 1/2 A 1/4 A 3/8 1/2 B 3/8 C C 1/2 A 1/4 B 1/4 C**Fallacy (2)**• Fire insurance: • Fire: paythepremium, house rebuilt (C) • No Fire: paythepremium, house untouched (B) • No insurance: • Fire: house burnt, no compansation (D) • No Fire: house the same (A) • A≻B≻C≻D • Suppose the probability of fireis 0.5 • And an individualisindifferentbetweenbuying and not buyingthe insurance • AlthoughFire insurance hassmallervariance and ½u(A) + ½u(D) = ½u(B) + ½u(C), itdoes not meanthatFire insurance should be chosenover No insurance Loss = $70K Loss = $60K Loss = $K Loss = $0 Riskaverse, and EL(insurance) = $65K EL(no insurance) = $50K**Fallacy (3)**• Fire insurance: • Fire: paythepremium, house rebuilt (C) • No Fire: paythepremium, house untouched (B) • No insurance: • Fire: house burnt, no compansation (D) • No Fire: house the same (A) • A≻B≻C≻D • Supposethattheprobability of fireis ½ and an individualprefers not buyingfire insurance and hence ½u(A) + ½u(D) > ½u(B) + ½u(C) ⇒ u(A) - u(B) > u(C) - u(D) • Howeveritdoes not meanthatthechangefrom B to A ismorepreferredthanthechangefrom D to C. • Preferencesaredefinedoverpairs of alternatives not pairs of pairs of alternatives**Crucial axiom - independence**• Our version • The general version • Why the general version implies our version?**Independence – examples**• If I prefer to go to the moviesthan to go for a swim, I mustprefer: • to toss a coin and: • heads: go to the movies • tails: vacuumclean • than to toss a coin and: • heads: go for a swim • tails: vacuumclean • If I prefer to bet on red than on even in roulette, then I mustprefer: • to toss a coin and • heads: bet on 18 • tails: bet on red • than to toss a coin and: • heads: bet on 18 • tails: bet on even**Machina triangle**p2 x2 1 1-a P a aP+(1-a)R x1 R x3 1 p1**Independenceassumption in the Machina triangle**Suppose that A1 is better than A2 is better than A3 p2 1 αP+(1-α)R P αQ+(1-α)R R Q 1 p1**17.1 and 17.2**17.1) Choose one lottery: P=(1 mln, 1) Q=(5 mln, 0.1; 1 mln, 0.89; 0 mln, 0.01) 17.2) Choose one lottery: P’=(1 mln, 0.11; 0 mln, 0.89) Q’=(5 mln, 0.1; 0 mln, 0.9) Kahneman, Tversky (1979) [commonconsequenceeffectviolation of independence] Many peoplechoose P over Q and Q’ over P’**Common consequence graphically**P = (1 mln, 1) P’= (1 mln, 0.11; 0, 0.89) Q = (5 mln, 0.1; 1 mln, 0.89; 0, 0.01) Q’= (5 mln, 0.1; 0, 0.9) • If we plug c = 1mln, we get P and Q respectively • If we plug c = 0, we get P’ and Q’ respectively**18.1 i18.2**18.1) Choose one lottery: P=(3000 PLN, 1) Q=(4000 PLN, 0.8; 0 PLN, 0.2) 18.2) Choose one lottery: P’=(3000 PLN, 0.25; 0 PLN, 0.75) Q’=(4000 PLN, 0.2; 0 PLN, 0.8) Kahneman, Tversky (1979) [common ratio effect, violation of independence] Many peoplechoose P over Q and Q’ over P’**Common ratio graphically**P=(3000 PLN, 1) P’=(3000 PLN, 0.25; 0 PLN, 0.75) Q=(4000 PLN, 0.8; 0 PLN, 0.2) Q’=(4000 PLN, 0.2; 0 PLN, 0.8)**Monotonicity of utilityfunction**• prefersmore to less • currentwealth x • probability p of bankruptcy (u(0)=0) • how much to pay to avoidit?**Monotonicity of utilityfunction**F(x) 1 x**First Order StochasticDominance (FOSD)**cdf cdf 1 1 1 1 t t**FOSD**• Assume X and Y aretwodifferentlotteries (FX(.), FY(.) are not the same) • LotteryX FOSD Y if: For all a, hence: Thosewhoprefermore to less willneverchooselotterythatisdominated in the abovesense. • Theorem: X FOSD Y if and onlyif Eu(X) ≥ Eu(Y), for allinreasing u**Marginalutility**x+d Todaychancenodessplit 50:50 x-d x**Marginalutility**utility payoff**Certaintyequivalent and riskpremium**4,5 1 2 0,5 0,5 utility 10 6 1 **Marginalutility**• x – initialwealth (number) • l – lottery with zero exp. value (randomvariable) • k –multiplier (we tak k close to zero) • d – riskpremium (number) • x-d – certaintyequivalent for x+l**Second Order StochasticDominance**cdf 1 t 1 1 1 t**SOSD**• Assume X and Y aretwodifferentlotteries (FX(.), FY(.) are not the same) • LotteryX SOSD Y if: For all a hence: • Thosewhoareriskaversewillneverchoose a lotterythatisdominated in the abovesense. • Theorem: X SOSD Y if and onlyif Eu(X) ≥ Eu(Y), for allinreasing and concave u**Mean-variancecriterium**• Riskaversiondoesn’tmeanthatalways: A betterthan B, ifonly E(A)=E(B) and Var(A)<Var(B) • somelotteries do not result from another with meanpreservingspread • istruewhenVar(A)=0 • Mean-variancecriteriumworkse.g. for normallydistributedrandomvariables (lotteries)**Measures of riskaversion**• Riskpremiummeasuresriskaversion with respect to a givenlottery • As a function of payoffvaluesriskaversionismeasured by Arrow, Prattmeasures of (local) riskaversion**Exercise 1**• From now on let’sassume X is a set of monetarypay-offs (decisionmakerprefersmoremoneythan less) • Decisionmaker with vNMutilityfunctionpreferslottery (100, ¼; 1000, ¾) to (500, ½; 1000, ½) • Whatis a realtionbetween(100, ½; 500, ¼; 1000, ¼) and (100, ¼; 500, ¾)? • Suggestion – we canarbitrarily set utilityfunction for twooutcomes**Exercise 2**• Decisionmakerisindifferentbetweenpairs of lotteries:(500, 1) and (0, 0,4; 1000, 0,6)and (300, 1) and (0, ½; 500, ½) • Can we guess the preferencerelationbetween(0, 0,2; 300, 0,3; 1000, ½) and (500, 1)?**Exercise 3**• Decisionmaker with vNMutilityisriskaverse and indifferentbetween the followingpairs(400, 1) and (0, 0,3; 1000, 0,7)and(0, ½; 200, ½) and (0, 5/7; 400, 2/7) • Can we guess the preferencerelationbetween(200, ½; 600, ½) and (0, 4/9; 100, 5/9)? • (suggestion – remamberthanvNMutilityisconcave)