行為金融學入門 井澤教授Lecture

1、Axiomatic Systems of the Expected Utility TheoryThe Expected utility theory was proposed by Bernoulli (1738) ,Ramsey (1931) , von Neumann-Morgenstern (1947) and fully developed by Savage (1954). in which choice over subjectively uncertain prospects is characterized by expected utility risk preferenc

2、es and standard probabilistic beliefs. The difference of these earlier studies and Ramsey is the manner which represents uncertainty. In the earlier studies, the objective framework, uncertainty comes prepackaged in terms of numerical probabilities. On the other hand, Ramsey treats the choice of pro

3、bability distributions over outcomes or lotteriesA similar and relatively simple formulation, that provides insights into the most important assumptions of expected utility for statistical purposes, was proposed by Anscombe and Aumann (1963). The Expected Utility Maximization Theorem (Savage,1954)Su

4、re-Thing Principle is also called the strong independence assumption. It states that if outcome x and outcome x are indifferent in themselves, then for any outcome y, a probability mix of x and y must be indifferent to a probability mix of x and y.Denying the principle is a possible response to the

5、Allais paradox.Sure-Thing Principle w or LotteryLotterywzx or yTAILSHEADSPossible Decisions:Take w. (w)Refuse w, and take x if tails. (.5x+.5z)Refuse w, and take y if tails. (.5y+.5z)suppose an individual would prefer x over y, but he would also prefer .5y + .5z over .5x + .5z, in violation of subst

6、itution. Suppose that w is some other prize that he would consider better than .5x + .5z and worse than .5y + .5z. That is, x y but .5y + .5z w .5x + .5z.Therefore, the third strategy would be best. However, if she/he takes the third strategy, and the coin comes up Tails, then she/he would choose x!

7、 That is actually end up with the second strategy , which is worst.Roger B. Myerson, Game Theory: Analysis of Conflict,1997.Why we need Sure-Thing Principle? A Brief Proof of the TheoremSuppose special lotteries;a: always gives the best prize for every state.z: always gives the worst prize for every

8、 state.bt: gives a if state=t, and gives z otherwise.Define asx t a + (1 ) z Define asbt S a + (1 ) z We can show that is u(x|t), and is p(t|S), satisfying the axiom.NOTEt means “indifferent if a decision maker knew that state t occurs,”and S means “indifferent if a decision maker knew that true sta

9、te is in S.”x : a lottery that will gives the prize x with probability 1 . a + (1 ) z: a two stage lottery. An Heuristic ExampleExpected UtilitiesSubjective ProbabilityContingent (state-dependent) UtilityIn the same way, under the condition that B will occures surely, the utility level of the prize

10、of 5 is 1, and one of the prize of 0 is 0.; under the condition that Y will occure surely, the utility level of the prize 4 is 1, and one of the prize of 0 is 0. This is “contingent utility!”Under the condition that R will occurs surely, if you ask a expected utility maximizer what is R to satisfy,

11、VbR=RVa+(1R R)Vz, the answer should be R =1This means, under the condition that R will occurs surely, the utility level of the prize 10 is 1, and one of the prize 0 is 0.Therefore, all the utility levels of the lottery a are 1 for expected utility maximizers, and we get,Va=VaR+VaB+VaY=R1B1Y1And Vz=0

12、This assures that t can be interpreted as the subjective probability for each states.Questions:How can we know who is an expected utility maximizer?How is the behavior of non-expected utility maximizer different from one of the expected utility maximizer actually?V 1 = V 3 = V 4 V 2Expected value ma

13、ximizerV 2 V 1 = V 3 = V 4Maxmin Expected Utility(V 1 = V 4) (V 1 V 2V 1 V 3) and (V 3V 1V 2V 1)Recursive Nonexpected Utility(V 1 V 3 V 4V 2 V 4)(V 1 V 2V 1 V 3 V 4)(V 1 V 3 V 4V 1 V 2)Recursive Expected UtilityReservation price for urn i (i = 1, .4).Red (R)Black (B)Urn 155Urn 2UnknownUrn 3Determine

14、d by a Lottery 0, 1,2, ,1010 - RUrn 4Determined by a Lottery 0, 1010 - RHalevy, Econometrica 2007Because it is possible that the decision maker believes that urn 2 has more red or more black balls.Subjective expected utility (SEU；Savage,1954) V 1 = V 3 = V 4 V 2The (maximal) subjective expected util

15、ity of a bet on urn i (Li) is,Maxmin Expected utilityV 2 V 1 = V 3 = V 4Gilboa andSchmeidler (1989, J. Math. E.)a generalization of expected utilityV 1 = V 3 = V 4 the core contains the two “pessimistic”nonsymmetrical beliefs that all balls are red and that all balls are black.(within the AnscombeAu

16、mann (1963) framework）An Example of Evaluation of Urn 2 by Recursive Expected or Non-Expected Utility Maximizer 10 red ballswith probability a 0 red ballswith probability a 5 red ballswith probability (1-2a) Evaluation ofLottery($x;1, $0,0)Evaluation ofLottery($x;0, $x,1)Evaluation ofLottery($x;0.5,

17、 $x,0.5)Evaluation of Betting on RedIf bet on RedIf bet on RedIf bet on RedThe first stage lottery:Subjective probability model of the second stage lotteryApplied Quiggins (1982) Rank Dependent UtilityRecursive Nonexpected Utility (Segal ,1987, 1990) Let x1 x2 xn Segals approach(two-stage lottery):

18、the decision maker has a second-order subjective belief over the possible probability distributions over the states. Assume that the decision makers model of the ambiguous urn (L2) is, with probability , it contains 10 red balls; with probability , it contains 0 red balls; and with probability (12),

19、 it contains 5 red balls. That is, If the agent bets on red from urn 2, then she first evaluates the second-stage lotteries ($x,1; $0,0) ($x ,0; $0,1), and ($x,0.5; $0,0.5) . Then, she evaluates the ambiguous lottery by substituting the certainty equivalents as the prizes in RDU.where, f is convex.T

20、he decision maker, of the Ellsberg paradox , might fail to reduce compound lotteries into single lotteries.The time neutrality assumption in Segal (1990): the decision maker is indifferent between bets on urns 1 and 4.The decision maker may believe that there are more red or more black balls in the

21、second urn and prefer to bet on the (subjectively) more probable color. Hence, the decision maker will be indifferent between urns 1 and 4, and prefer them to a bet on urn 3. Indifference between the three objective urns results if f is the identity function (in which case RDU reduces to EU) and the

22、 reduction of compound lotteries holds. Hence, in terms of elicited valuations the theorys predictions are(V 1 = V 4), (V 1 V 2V 1 V 3), and (V 3V 1V 2V 1)That is, the recursive nonexpected utility model predicts a negative correlationbetween V 21 (= V 2V 1) and V 43 (= V 4V 3), because ambiguity av

23、ersion (V 1V 2) implies that (V 4V 3), and (V 4V 3) implies ambiguity seeking (V 1V 2).Recursive Nonexpected Utility (Segal ,1987, 1990) Recursive Expected UtilityKlibanoff, Marinacci, and Mukerji (2005, Econometrica)The decision maker has a subjective prior over the possible (derived from an underl

24、ying belief over the finer state space) and evaluates an act using subjective expected utility according to the utility index v with respect to substituting the certainty equivalents (calculated from a von NeumannMorgenstern utility index u) of the objective lotteries for every .An example: 1 = (1 0

25、)no blacks in urn 2, 2 = (0 1)no reds in urn 2. The subjective prior = (1 1/2 ;2 1/2 ) Then, the subjective probability that urn 2 has only red balls (1) is equal to the subjective probability that it has all blacks (2), which is equal to 1/2 (similar to the objective urn 4).The Choquet expected uti

26、lity model (Schmeidler(1989),Gilboa and Schmeidler (1989) evaluating lotteries (u) and second-order acts (v), which induce identical objective probability distributions over outcomes. In this case, v would be an affine transformation of u and ROCL would apply. Then,V 1 = V 3 = V 4v() would be more c

27、oncave than u() even when is objective. If this is the case, the decision maker will evaluate urns 1, 3, and4 in the manner(V 1 V 3 V 4V 2 V 4)(V 1 V 2V 1 V 3 V 4)(V 1 V 3 V 4V 1 V 2)That is, if the subjective prior belief over the composition of the second urn issymmetric around 0.5 and nondegenera

28、te, we would expect a positive correlationbetween V 43 (= V 4 V 3) and V 21 (= V 2 V 1), because then V 4 V 3if and only if V 1 V 2.Non Expected Utility Maximizers Behave as if Expected Utility Maximizers: An Experimental Test(with Hiroshi Kurata, and Makoto Okamura). Journal of Socio-Economics，38，2

29、009 (August),Our Findings“Under the conditions of the Becker- DeGroot-Marschak (BDM) method, Non expect utility maximizers (Non EU maximizers) behave as if they were expected utility maximizers (EU maximizers).21Background:Expected Utility Theory (EUT) and the BDM methodThe validity of the BDM depen

30、ds on EUT. EUTs validity? 1st ExperimentAny discrepancies between EU and non EU maximizers? 2nd Experiment22Outline of Our Experiments25 Subjects (male: 7; female: 19) Staff at Ritsumeikan Univ. with similar income levels.Two Experiments: Experiment 1: Experiment for the validity of Redution Axiom E

31、xperiment 2: the BDM experiment 23Mixture Independence Axiom (MIA)MIA: Comparison between one-stage lotteries Axiom of the expected utility theory24Compound Independence Axiom (CIA)CIA: Comparison between two-stage Lotteries25Reduction Axiom (RA)RA: Comparison between one-stage and two-stage lotteri

32、esRA indifference between one-stage and two-stage lotteries with the same return.26Theorem 1 (Segal, 1990, Econometrica)MIA, CIA and RA are pairwise independent; no one implies another.RA + CIA MIA (EUT)Our focus is on the validity of RA.Why we focus on RA:Necessary ConditionIndifferent Relation RA

33、has an experimental advantage.27Experiment #160 balls in a Bingo-game machine.Prizes of lotteries: 1,000 yen Lottery 1 (One-stage Lottery)win: 46-60 (15 balls); prize: win onceLottery 2 (Two-stage Lottery)win: 31-60 (30 balls); prize: win twice 28Experiment #1(Continued)Choose one of the following. (1) Lottery 1 is