Week of May 5, lecture 1: Expected utility theory

Transcription

1 Microeconomics 3 Andreas Ortmann, Ph.D. Summer 2003 (420 2) andreas.ortmann@cerge-ei.cz Week of May 5, lecture 1: Expected utility theory Key readings: MWG 6.A., 6.B. Supplementary sources: Binmore (1992), Kreps (1990) Assignments: [for May 13] MWG 6.C., 6.D., 6.E., 6.F. [for May 13] Problem set # 1 ~ downloadable Wednesday noon from home.cergeei.cz/babicky/micro3. There you will also find around the same time the lecture notes for the week. [for May 15] Cox and Grether (1996), The preference reversal phenomenon: Response mode, markets and incentives. Economic Theory 7, [for May 20] Holt and Laury (2002), Risk Aversion and Incentive Effects. American Economic Review 92, [for May 20] Harrison, Johnson, McInnes, and Rutstroem (2003), Risk Aversion and Incentive Effects: Comment. Manuscript. [for May 20] Brown Kruse and Thompson (2001), A comparison of salient rewards in experiments: money and class points. Economics Letters 74, [for May 20] Berg, Dickhaut, and Rietz (1999), On the performance of the Lottery Procedure for Controlling Risk Preferences. Manuscript. [leisure reading] Hertwig and Ortmann (2001), Experimental practices in economics: A methodological challenge for psychologists? Behavioral and Brain Sciences 24, [Includes commentaries of 34 economists and psychologists and our response to them.] [You can pick up copies of the articles from the boxes to the right of my office door.]

2 2 A little bit of history never hurt anyone: Decision theory, game theory, experimental economics, and design economics 1944 (1947) Von Neumann & Morgenstern (Games and Economic Behavior) - cooperative game theory -> assumes communication & cooperation - non-cooperative game theory -> models interactive decision problems (This strategic interaction may reflect conflict or cooperation ) -> assumes rationality (self-interest?) as primitive ( Nash program ) - both require choice under risk and uncertainty -> risk and uncertainty are sometimes used synonymously. -> Risk is assumed to be quantifiable while uncertainty is not. - decision theory is special case of game theory, with nature as other player 1950's 's Savage, Edwards, Nash, Selten, Harsanyi, and Aumann (N,S, also EE)] For a readable account of the JDM/experimental part of this research, see Goldstein and Hogarth (1997), Judgment and decision research: Some historical context. In: Goldstein and Hogarth (1997), Research on judgment and decision making. Currents, connections, and controversies. Cambridge: CUP, For a readable account of the game theoretic/experimental part of this research see Roth (1993) On the Early History of Experimental Economics, Journal of the History of Economic Thought 15, Also available at Maynard Smith (Evolution and the Theory of Games) Tirole (The Theory of Industrial Organization) Kreps (Microeconomic Theory), Barro (Macroeconomic Policy) Nash, Selten, Harsanyi -> Nobel prize for their work in game theory Mas-Colell, Whinston, Green (Microeconomic Theory) Rabin -> John Bates Clark medal for his attempt to ground economic theory (models) on insights from psychology Kahneman, Smith -> Nobel prize for their experimental work 200? - Alvin E. Roth et al. for their work on design economics

3 3 Definition 6.B.1. (simple lottery): A simple lottery L is a list L = (p 1,, p N ) with p n $ 0 for all n and 3 n p n = 1 where p n is interpreted as the probability of outcome n occuring. Note 0: Lotteries are sometimes (e.g., Starmer 2000) called prospects = a list of consequences with associated probabilities Note 1: A lottery is a formal device to represent risky alternatives. Mathematically speaking, a lottery is a random variable whose possible realizations are defined by the (for now, finite) set of possible outcomes C (which can take various forms including monetary payoffs, then often called prizes.) If the possible outcomes have numerical values (as in the case of prizes), then the expectation of L can be computed. [Example] Note 2: The set of all possible outcomes is assumed to be known, probabilities likewise are assumed to be (objectively) known. Note 3: Simple lotteries can be represented geometrically in the N-1 dimensional simplex ) = {p 0 œ N +: p p N = 1}. Each vertex of a simplex stands for a degenerate lottery. Note 4: When n = 3, then the simplex can be represented in two-dimensional space. Specifically, it is represented by an equilateral triangle with altitude 1. (Why? See footnote 2 in MWG p. 169) When n = 2, then the simplex is obviously a line. [Discussion Figure 6.B.1] Note 5: For another representation see Starmer (2000), Developments in Non-Expected Utility Theory: The Hunt for a Descriptive Theory of Choice under Risk. Journal of Economic Literature 38,

4 Definition 6.B.2 (compound lottery): Given K simple lotteries L K = (p k 1,..., p k N); k = 1,..., K, and probabilities " k $ 0 with 3 k = 1, the compound lottery (L 1,..., L K ; " 1,..., " K ) is the risky alternative that yields the simple lottery L k with probability " k for k = 1,..., K. Note 1: A compound lottery is a lottery in which the prizes are themselves lotteries. What one wins in effect is the chance of winning in a second lottery. Note 2: All lotteries involved are assumed to be independent. Note 3: Any compound lottery (L 1,..., L K ; " 1,..., " K ) can be reduced to the simple lottery L = (p 1,, p N ), the so-called reduced lottery: compute the total probability of each final outcome by multiplying the probability that each lottery L k arises, " k, by the probability p k n that outcome n arises in lottery L k, and then add over k. [Example] Note 4: Any reduced lottery L of any compound lottery (L 1,..., L K ; " 1,..., " K ) can by obtained by vector addition: L = " 1 L " K L K ; 0 ). The probability of outcome n in the reduced lottery is p n = " 1 p 1 n " K p K n. The linear structure of the space of lotteries is central to the theory choice under uncertainty. [Example] [Discussion Figures 6.B.2 and 6.B.3 ] Note 5: We assume that for any risky alternative, only the reduced lottery over all final outcomes is of relevance to the decision maker ( consequentialist premise ). 4

5 5 Definition 6.B.3 (Continuity of preference relation) The preference relation on the space of simple lotteries is continuous if for any L, L, L 0, the sets and {" 0 [0,1]: " L + (1-" )L L } d [0,1] {" 0 [0,1]: L " L + (1-" )L } d [0,1] are closed. Note: The continuity axiom is necessary to guarantee that L L if and only if U(L) U(L ), i.e., that a utility function can be constructed. It is equivalent to the continuity requirement for the theory of consumer demand. See definition 3.C.1. Definition 6.B.4. (Independence axiom) The preference relation on the space of simple lotteries satisfies the independence axiom if for all L, L, L 0 and " 0 (0,1) we have L L if and only if " L + (1-") L " L + (1-") L. Note: There is no equivalent requirement in the theory of consumer demand. [Discussion Figure 6.B.4]

6 Definition 6.B.5. (VNM expected utility function) The utility function U: -> œ has an expected utility form if there is an assignment of numbers (u 1,..., u N ) to the N outcomes such that for every simple lottery L = (p 1,..., p N ) 0 we have U(L) = u 1 p u N p N. A utility function U: -> œ with the expected utility form is called a Von Neumann- Morgenstern (VNM) expected utility function. Note 1: Let L n denote the lottery that yields outcome n with probability 1, then U(L n ) = u n. Hence expected utility: the utility of a lottery is the expected value of the utilities u n of the N outcomes. Note 2: U(L) is a linear function in the probabilities or linear in probabilities. This means it satisfies property (6.B.1) U(3 k " k L k ) = 3 k " k U(L k ) for any K lotteries L k 0, k = 1,..., K, and probabilities (" 1,..., " K ) $ 0, 3 k " k = 1. 6

7 7 Proposition 6.B.1 (linearity of utility functions) A utility function U: -> œ has an expected utility form if and only if it is linear. Proof: See MWG p Proposition 6.B.2 (utility scales) Suppose that U: -> œ is a VNM expected utility function for the preference relation on. Then U*: -> œ is another vnm expected utility function for if and only if there are scalars $ > 0 and ( such that U*(L) = $U(L) + ( for every L 0, i.e., if U*(L) is an affine transformation of U(L). Proof: See MWG pp Note 1: One consequence is that differences of utilities have meaning; they are quasicardinal but not cardinal as MWG claim). Specifically, the origin and utility scale of a VNM utility scale can be chosen in an arbitrary fashion, but that s it. Think of the measurement of temperatures where Fahrenheit is an affine (linear) transformation of Celsius (e.g., f = (9/5)c + 32.) Note 2: Note that VNM expected utility functions are not ordinal (as those utility functions over commodity bundles that you met earlier and for which we only required that they were invariant under strictly increasing transformations.) Note 3: The fact that VNM utility scales are quasi-cardinal does not necessarily mean that we can compare utility across individuals. (Make sure to understand why that is and come up with an example of a utility specification that would allow us to compare utilities across people!)

8 Proposition 6.B.3 (Expected Utility Theorem) Suppose that the rational preference relation on the space of lotteries satisfies the continuity and independence axioms. Then admits a utility representation of the expected utility form. That is, we can assign a number of u n to each outcome n = 1,..., N in such as manner that for any two lotteries L = (p 1,..., p N ) and L = (p 1,..., p N ), we have L L if and only if 3 n u n p n $ 3 n u n p n. Proof: MWG pp Note 1: The expected utility theorem summarizes under what conditions (namely, continuity and independence axiom) a decision maker s preferences can be represented by a utility function. This is similar to proposition 3.C.1 in MWG but obviously not the same. Note 2: Recall that the relevant state space for the case of 3 outcomes can be represented by a simplex in two-dimensional space. Hence the map of indifference curves (contour lines), or indifference map can be represented on this space too. Note 3: One key difference is the shape that indifference curves (contour lines) typically take. Specifically, they are straight lines. That shouldn t come as a surprise since the indifference curves are characterized by the marginal rate of substitution (MRS) of one good (one lottery) for another good (another lottery). By proposition 6.B.1 and property 6.B.1 a utility function that represents preferences on lotteries is linear. Hence the MRS is linear. (Have you seen a situation like that before in the theory of consumer demand? You should have. How, for example, would an indifference map for dimes and nickels look like?) [Discussion of Figure 6.B.5 a] Note 4: Figures 6.B.5 b,c illustrate that the linearity property of the indifference map for lotteries results from the independence axiom. The argument: whenever a decision maker is indifferent between two lotteries (whenever two lotteries lie on the same indifference curve), then the decision maker is also indifferent between any linear combination of these two, with the constant coefficients in front of the lotteries being from the open unit interval and adding up to 1. [Discussion of Figures 6.B.5 b,c] The problem with Figure 6.B.5b: L - L but "L + (1 - ")L L = "L + (1 - ")L. By the independence axiom we need to have "L + (1 - ")L - "L + (1 - ")L. Contradiction. The problem with Figure 6.B.5c: L - L but "L + (1 -")L "L + (1 -")L. By the independence axiom we need "L + (1 -")L -"L + (1 -")L. Contradiction. 8

9 9 [Decision questionnaire 1] Decision questionnaire 1 Your Name: Note: A couple of the decisions below will be played out, i.e. you can earn (or lose) real money. Decision situation 1: You are being offered the following six pairs of gambles. Please determine in each instance which you prefer! [Circle your choice!] Gamble 1 Gamble 2 A 10% chance of winning CK 100; and CK 10 90% chance of winning CK 0 B 10% chance of winning CK 1,000; and CK % chance of winning CK 0 C 20% chance of winning CK 1,000; and CK % chance of winning CK D 10% chance of winning CK 2,000; and CK % chance of winning CK 0 E 1% chance of winning CK 10,000; and CK % chance of winning CK 0 F 20% chance of winning CK 2,000; and CK % chance of winning CK - 250

10 10 Decision situation 2 (Allais paradox): Choose between two gambles. The first gives you a.33 chance of CK 1,375, a.66 chance of CK 1,200, and a.01 chance of nothing. The second gives you CK 1,200 for sure. You take the... first... second [Mark your choice!] Choose between two gambles. The first gives you a.33 chance of CK 1,375 and a.67 chance of nothing. The second gives you a.34 chance of winning CK1,200 and.66 chance of nothing. You take the... first... second [Mark your choice!] Decision situation 3 (Ellsberg paradox): An urn contains 300 colored marbles; 100 of the marbles are red, and 200 are some mixture of blue and green. We will reach into this urn and select a marble at random: You receive KC 1,000 if the selected marble is of a specified color. Would you rather that color be red or blue? You receive KC 1,000 if the marble selected is not of a specified color. Would you rather that color be blue or red?