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Introduction to Decision Making Theory


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Lecture slides on Decision Theory. The contents in large part come from the following excellent textbook.

Rubinstein, A. (2012). Lecture notes in microeconomic theory: the
economic agent, 2nd.

Published in: Economy & Finance
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Introduction to Decision Making Theory

  1. 1. Introduction to Decision Making Theory YASUDA, Yosuke Osaka University, Department of Economics September, 2015 Last updated: September 22 1 / 31
  2. 2. Readings Main Textbook Rubinstein, A. (2012). Lecture notes in microeconomic theory: the economic agent, 2nd. Lectures 1-3 and 7 are closely related. ← Can be downloaded FOR FREE from the author’s website: Other Related Books Binmore, K. (2008). Rational decisions. Chapters 1 and 3 are related. Gilboa, I. (2009). Theory of decision under uncertainty. Advanced Kreps, D. (1988). Notes on the Theory of Choice. Classic and popular Mas-Colell, A., Whinston, M. D., and Green, J. R. (1995). Microeconomic theory. Standard introduction for economics students My Lecture Website at Osaka U. For extensive references 2 / 31
  3. 3. Lecture Outline 1st. Decision Making over Certain Outcomes (Consequences) Preference, Choice, and Utility What Is Rationality? When Does Agent Look “As If” Rational? 2nd. Decision Making over Uncertain Outcomes Expected Value and Expected Utility (EU) Axiomatic Approach to EU Theory When Does EU Look Unrealistic? 3 / 31
  4. 4. Preferences The notion of preferences plays a central role in economic theory, which specifies the form of consistency or inconsistency in the person’s choices. is the mental attitude of an individual toward alternatives independent of any actual choice. Preferences require only that the individual make binary comparisons. Individual only examines two choice alternatives in the choice set X at a time and make a decision regarding those two. The description of preferences should provide an answer to the question of how the agent compares the two alternatives. Considering questionnaires P and R, we formulate the consistency requirements necessary to make the responses preferences. 4 / 31
  5. 5. Questionnaire P § ¦ ¤ ¥ P(x, y) For all distinct x and y in the set X. How do you compare x and y? Tick one and only one of the following three options. . 1 I prefer x to y, or x is strictly preferred to y: x y . 2 I prefer y to x, or y is strictly preferred to x: y x . 3 I am indifferent, or x is indifferent to y: x ∼ y A legal answer to the questionnaire P can be formulated as a function f which assigns to any pair (x, y) of distinct elements in X exactly one of the three values: x y, y x or x ∼ y. f(x, y) = 8 < : x y y x x ∼ y . 5 / 31
  6. 6. Preference P (1) Preferences are characterized by axioms that are intended to give formal mathematical expression to fundamental aspects of choice behavior and attitudes toward the objects of choice. The following basic axioms are (almost) always imposed. Definition 1 Preference P on a set X is a function f so that for any three different elements x, y and z in X, the following two properties hold: Axiom 1 — No order effect: f(x, y) = f(y, x). Axiom 2 — Transitivity: 1 if f(x, y) = x y and f(y, z) = y z, then f(x, z) = x z, and 2 if f(x, y) = x ∼ y and f(y, z) = y ∼ z, then f(x, z) = x ∼ z. 6 / 31
  7. 7. Preference P (2) The first property requires the answer to P(x, y) being identical to the answer to P(y, x), and the second requires that the answer to P(x, y) and P(y, z) are consistent with the answer to P(x, z) in a particular way. £ ¢   ¡Ex Violation of Transitivity For any x, y ∈ R, f(x, y) = x y if x − y ≥ 1 and f(x, y) = x ∼ y if |x − y| < 1. This is not a preference relation since transitivity is violated. For instance, suppose x = 10, y = 10.6, z = 11.2. Then, f(x, y) = x ∼ y and f(y, z) = y ∼ z, but f(x, z) = z x, which violates transitivity (2-2). £ ¢   ¡Rm What happens if transitivity (2-1) fails to be satisfied? 7 / 31
  8. 8. Questionnaire R § ¦ ¤ ¥ R(x, y) for all x, y ∈ X, not necessarily distinct. £ ¢   ¡Q Is x at least as preferred as y? Tick one and only one of the followings. . 1 Yes (or, x is at least as good as y): x y . 2 No (or, x is strictly worse than y): x y Definition 2 Preference R on a set X is a binary relation on X satisfying Axioms 1 & 2. Axiom 1’ — Completeness: ← Individual can make comparison. For any x, y ∈ X, x y or y x. Axiom 2’ — Transitivity: ← Individual choices are consistent. For any x, y, z ∈ X, if x y and y z, then x z. 8 / 31
  9. 9. Equivalence of the Two Preferences The following mapping (bijection) translates one formulation of preferences to another. Note that completeness guarantees “x y and y x” never happen. f(x, y) = x y ⇔ x y and y x. f(x, y) = y x ⇔ y x and x y. f(x, y) = x ∼ y ⇔ x y and y x. § ¦ ¤ ¥ Fg Table 1.1 in Rubinstein (pp.7) Most economics textbooks take the second definition, i.e., preference R, and denote x y when both x y and y x, and x ∼ y, when x y and y x. Definition 3 Preference R, a binary relation that satisfies Axioms 1’ and 2’, is called a rational preference or preference relation. 9 / 31
  10. 10. *Preferences on Sets (Menu) Problem 3 in Rubinstein (pp.10) based on Kannai and Peleg (1984). Let Z be a finite set and let X be the set of all nonempty subsets of Z. We consider a preference relation on X, interpreted as a menu (not Z). . 1 If A B and C is a set disjoint to both A and B, then A ∪ C B ∪ C, and if A B and C is a set disjoint to A and B, then A ∪ C B ∪ C. . 2 If x ∈ Z and {x} {y} for all y ∈ A, then A ∪ {x} A, and if x ∈ Z and {y} {x} for all y ∈ A, then A A ∪ {x}. § ¦ ¤ ¥Q1 Provide an example of a preference relation that satisfies one of the above two properties, but does not satisfy the other. § ¦ ¤ ¥Q2 Show that if there are x, y, and z ∈ Z such that {x} {y} {z}, then there is no preference relation satisfying both properties. 10 / 31
  11. 11. Utility Representation Definition 4 Function U : X → R represents the preference if for all x and y ∈ X, x y if and only if U(x) ≥ U(y). If the function U represents the preference relation , we refer to it as a utility function and we say that has a utility representation. £ ¢   ¡Q Under what conditions do utility representations exist? Theorem 1 If is a preference relation on a finite set X, then has a utility representation with values being natural numbers. Proof. There is a minimal (resp. maximal) element (an element a ∈ X is minimal (resp. maximal) if a x (resp. a x) for any x ∈ X) in any finite set A ⊂ X. We can construct a sequence of sets from the minimal to the maximal and can assign natural numbers according to their ordering. 11 / 31
  12. 12. *Continuous Preferences To guarantee the existence of a utility representation over consumption set, i.e., an infinite subset of Rn , we need some additional axiom. Definition 5 A preference relation on X is continuous (Axiom 3) if {xn } (a sequence of alternatives) with limit x satisfies the following two conditions for all y ∈ X. . 1 if x y, then for all n sufficiently large, xn y, and . 2 if y x, then for all n sufficiently large, y xn . The equivalent definition of continuity is that the “at least as good as” and “no better than” sets for each point x ∈ X are closed. Axiom 3 rules out certain discontinuous behavior and guarantees that sudden preference reversals do not occur: if y is preferred to z and x is a consumption bundle close enough to y, then x must be preferred to z. 12 / 31
  13. 13. *Continuous Utility Axioms 1-3 guarantee the existence of a (continuous) utility function. Theorem 2 Assume that X is a convex subset of Rn . If is a continuous preference relation on X, then is represented by a continuous utility function. Here are two remarks on continuity. . 1 If on X is represented by a continuous function U, then must be continuous (converse is not true: continuous preferences can be represented by a discontinuous function). 2 The lexicographic preferences are not continuous. Theorem 3 The lexicographic preference relation L on [0, 1] × [0, 1], (a1, a2) L (b1, b2) if a1 > b1 or both a1 = b1 and a2 ≥ b2, does not have a utility representation. 13 / 31
  14. 14. Choice Function We consider an agent’s behavior as a hypothetical response to the following questionnaire, one for each A ⊆ X: § ¦ ¤ ¥ Q(A) Assume you must choose from a set of alternatives A. Which alternative do you choose? A choice function C assigns to each set A ⊆ X a unique element of A. → C(A) is the chosen element from the set A. Here are a couple of remarks on choice functions. 1 We assume that the agent selects a unique element in A for every question Q(A). cf. choice corresponding 2 The choice function C need not to be observable. 3 The agent behaving in accordance with C will choose C(A) if she has to make a choice from a set A. 14 / 31
  15. 15. Rational Choice When the agent has in mind a preference relation on X, given any choice problem Q(A) for A ⊆ X, she chooses an element in A which is “ optimal”. Definition 6 An induced choice function C is the function that assigns every nonempty set A ⊆ X the -best element of A. A choice function C can be rationalized if there is a preference relation on X so that C = C . £ ¢   ¡Q Under what conditions any choice functions can be presented “as if” derived from some preference relation? Definition 7 Choice function C satisfies (Sen’s) condition α if for any A ⊂ B, C(B) ∈ A implies C(A) = C(B). § ¦ ¤ ¥ Fg Figure 3.1 in Rubinstein (pp.25) 15 / 31
  16. 16. Choice ⇐⇒ Preference ( ⇐⇒ Utility) Theorem 4 Assume C is a choice function with a domain containing at least all subsets of X of size 2 or 3. If C satisfies condition α, then there is a preference relation on X so that C = C . Proof. Define by x y if x = C({x, y}). Let us first show that satisfies completeness and transitivity. Completeness: Follows from that C = ({x, y}) is well-defined. Transitivity: If x y and y z, then by definition of we have C({x, y}) = x and C({y, z}) = y. If C({x, z}) = z, then, by condition α, C({x, y, z}) = x. Similarly, by C({x, y}) = x and condition α, C({x, y, z}) = y, and by C({y, z}) = y and condition α, C({x, y, z}) = z. A contradiction to C({x, y, z}) ∈ {x, y, z}. Next we show that C(A) = C (A) for all A ⊆ X. Suppose on contrary C(A) = C (A). That is, C(A) = x and C (A) = y(= x). By y x, this means C({x, y}) = y, contradicting condition α. 16 / 31
  17. 17. As If Rational Preferences? £ ¢   ¡Rm Any induced choice function satisfies condition α, and Theorem 4 establishes the converse. Condition α ⇐⇒ C = C Can each of the following procedures be rationalized? . 1 Choose the worst procedure . 2 Second-best procedure . 3 Satisficing procedure (by Herbert Simon) 4 Satisficing using two orderings The satisfying procedure seems unrelated to the maximization of a preference relation or utility function. Nevertheless, it can be rationalized, i.e., described as if the decision maker (DM) maximizes a preference relation. 17 / 31
  18. 18. Decision under Uncertainty We have so far not distinguished between individual’s actions and consequences, but many choices made by agents take place under conditions of uncertainty. We introduce an environment in which the correspondence between actions and consequences is not deterministic but stochastic. The domain of choice functions should be extended. The choice of an action is viewed as choosing a lottery where the prizes are the consequences. The DM is assumed not to care about the nature of the random factors but only about the distribution of consequences. 18 / 31
  19. 19. Lotteries We consider preferences and choices over the set of lotteries. Let S be a set of consequences or prizes. We assume that S is a finite set and the number of its elements (= |S|) is S. A lottery p is a function that assigns a nonnegative number to each prize s, where P s∈S p(s) = 1. ← p(s) is the probability of obtaining the prize s given the lottery p. Let α ◦ x ⊕ (1 − α) ◦ y denote the lottery in which the prize x is realized with probability α and the prize y with 1 − α. Let L(S) be the (infinite) space containing all lotteries with prizes in S. → {x ∈ RS +| P xs = 1}. We will discuss preferences over L(S). 19 / 31
  20. 20. St Petersburg Paradox (1) The most primitive way to evaluate a lottery is to calculate its mathematical expectation, i.e., E[p] = P s∈S p(s)s. Daniel Bernoulli first doubts this approach in the 18th century when he examined the St Petersburg paradox. £ ¢   ¡Ex St Petersburg Paradox A fair coin is tossed until it shows heads for the first time. If the first head appears on the k-th trial, a player wins $2k . £ ¢   ¡Q How much are you willing to pay to participate in this gamble? £ ¢   ¡Rm The expected value of the lottery is infinite: 2 2 + 22 22 + 23 23 + · · · = 1 + 1 + 1 + · · · = ∞. 20 / 31
  21. 21. St Petersburg Paradox (2) The St Petersburg paradox shows that maximizing your dollar expectation may not always be a good idea; an agent in risky situation might want to maximize the expectation of some utility function with decreasing marginal utility: E[u(x)] = u(2) 1 2 + u(4) 1 4 + u(8) 1 8 + · · ·, which can be a finite number. £ ¢   ¡Q Under what kinds of conditions can a DM be described as if she maximizes the expectation of some “utility function”? £ ¢   ¡Rm We know that for any preference relation defined on the space of lotteries that satisfies continuity, there is a utility representation U: L(S) → R, continuous in the probabilities, such that p q if and only if U(p) ≥ U(q). 21 / 31
  22. 22. Properties of Lotteries We impose the following three assumptions on the lotteries. . 1 1 ◦ x ⊕ (1 − 1) ◦ y ∼ x → Getting a prize with prob. 1 is the same as getting the prize for certain. . 2 α ◦ x ⊕ (1 − α) ◦ y ∼ (1 − α) ◦ y ⊕ α ◦ x → DM does not care about the order in which the lottery is described. . 3 β ◦ (α ◦ x ⊕ (1 − α) ◦ y) ⊕ (1 − β) ◦ y ∼ (βα) ◦ x ⊕ (1 − βα) ◦ y → A DM’s perception of a lottery depends only on the net probabilities of receiving the various prizes. The first two assumptions appear to be innocuous. The third assumption sometimes called “reduction of compound lotteries” is somewhat suspect. There is some evidence to suggest that DM treats compound lotteries different than one-shot lotteries. 22 / 31
  23. 23. Expected Utility Theory (1) We will use the following two axioms to isolate a family of preference relations which have a representation by a more structured utility function. Independence Axiom (I): For any p, q, r ∈ L(S) and any α ∈ (0, 1), p q ⇔ α ◦ p ⊕ (1 − α) ◦ r α ◦ q ⊕ (1 − α) ◦ r. Continuity Axiom (C): If p q r, then there exists α ∈ (0, 1) such that q ∼ [α ◦ p ⊕ (1 − α) ◦ r]. Theorem 5 Let be a preference relation over L(S) satisfying the I and C. There are numbers {v(s)}s∈S such that p q ⇐⇒ U(p) = X s∈S p(s)v(s) ≥ U(q) = X s∈S q(s)v(s). 23 / 31
  24. 24. Expected Utility Theory (2) Sketch of the proof. Let M and m be a best and a worst certain lotteries in L(S). When M ∼ m, choosing v(s) = 0 for all s we have P s∈S p(s)v(s) = 0 for all p ∈ L(S). Consider the case that M m. By I and C, there must be a single number v(s) ∈ [0, 1] such that v(s) ◦ M ⊕ (1 − v(s)) ◦ m ∼ [s] where [s] is a certain lottery with prize s, i.e., [s] = 1 ◦ s. In particular, v(M) = 1 and v(m) = 0. I implies that p ∼ X s∈S p(s)v(s) ! ◦ M ⊕ 1 − X s∈S p(s)v(s) ! ◦ m. Since M m, we can show that p q ⇐⇒ X s∈S p(s)v(s) ≥ X s∈S q(s)v(s). 24 / 31
  25. 25. vNM Utility Function (1) Note the function U is a utility function representing the preferences on L(S) while v is a utility function defined over S. → Building block for U(p) v is called a vNM (Von Neumann-Morgenstern) utility function. £ ¢   ¡Q How can we construct the vNM utility function? Let si(∈ S), i = 1, ..., K be a set of consequences and s1, sK be the best and the worst consequences. That is, for any i, [s1] [si] [sK ]. Then, construct a function v : S → [0, 1] in the following way: v(s1) = 1 and v(sK ) = 0, and [sj] ∼ v(sj) ◦ [s1] ⊕ (1 − v(sj) ◦ [sK ] for all j. By continuity axiom, we can find a unique value of v(sj) ∈ [0, 1]. 25 / 31
  26. 26. vNM Utility Function (2) £ ¢   ¡Q To what extent, vNM utility function is unique? The vNM utilities are unique up to positive affine transformation, i.e., multiplication by a positive number and adding any scalar. → Not invariant to arbitrary monotonic transformation Theorem 6 Let be a preference relation defined over L(S), v(s) be the vNM utilities representing the preference relation, and w(s) = αv(s) + β for all s (where α > 0). Then, the utility function W(p) = P s∈S p(s)w(s) also represents . Note that vNM utility functions do NOT (directly) attach numerical numbers to lotteries. v(s) is NOT a cardinal utility function, but a numerical function which is (intermediately) used to construct a utility representation U over L(S). 26 / 31
  27. 27. vNM Utility Function (3) Proof. For any lotteries p, q ∈ L(S), p q if and only if X s∈S p(s)v(s) ≥ X s∈S q(s)v(s). Now, the followings hold. X s∈S p(s)w(s) = X s∈S p(s)(αv(s) + β) = α X s∈S p(s)v(s) + β. X s∈S q(s)w(s) = X s∈S q(s)(αv(s) + β) = α X s∈S q(s)v(s) + β. Thus, X s∈S p(s)v(s) ≥ X s∈S q(s)v(s) holds if and only if X s∈S p(s)w(s) ≥ X s∈S q(s)w(s) (for α > 0). 27 / 31
  28. 28. Allais Paradox (1) Many experiments reveal systematic deviations from vNM assumptions. The most famous one is the Allais paradox. £ ¢   ¡Ex Allais paradox Choose first the between L1 = [3000] and L2 = 0.8 ◦ [4000] ⊕ 0.2 ◦ [0] and then choose between L3 = 0.5 ◦ [3000] ⊕ 0.5 ◦ [0] and L4 = 0.4 ◦ [4000] ⊕ 0.6 ◦ [0]. Note that L3 = 0.5 ◦ L1 ⊕ 0.5 ◦ [0] and L4 = 0.5 ◦ L2 ⊕ 0.5 ◦ [0]. Axiom I requires that the preference between L1 and L2 be the same as that between L3 and L4. However, a majority of people express the preferences L1 L2 and L3 ≺ L4, violating the axiom. 28 / 31
  29. 29. Allais paradox (2) Assume L1 L2 but α ◦ L ⊕ (1 − α) ◦ L1 ≺ α ◦ L ⊕ (1 − α) ◦ L2. (In our example of Allais paradox, α = 0.5 and L = [0].) Then, we can perform the following trick on the DM: . 1 Take α ◦ L ⊕ (1 − α) ◦ L1. . 2 Take instead α ◦ L ⊕ (1 − α) ◦ L2, which you prefer (and you pay me something...). . 3 Let us agree to replace L2 with L1 in case L2 realizes (and you pay me something now...). 4 Note that you hold α ◦ L ⊕ (1 − α) ◦ L1. 5 Let us start from the beginning... This argument may make the independence axiom looks somewhat reasonable (and Allais paradox unreasonable). 29 / 31
  30. 30. Zeckhouser’s Paradox (1) The following paradox also shows that many people do not necessarily follow the expected utility maximization behavior. £ ¢   ¡Ex Zeckhauser’s paradox Some bullets are loaded into a revolver with six chambers. The cylinder is then spun and the gun pointed at your head. Would you be prepared to pay more to get one bullet removed when only one bullet was loaded, or when four bullets were loaded? £ ¢   ¡Q People usually say they would pay more in the first case, because they would then be buying their lives for certain. Is this decision reasonable? £ ¢   ¡Rm Note that you cannot use your money once you die... 30 / 31
  31. 31. Zeckhouser’s Paradox (2) Suppose $X (resp. $Y ) is the most that you are willing to pay to get one bullet removed from a gun containing one (resp. four) bullet. Let L mean death, and W mean being alive after paying nothing. Let C mean being alive after paying $X, and D alive after paying $Y . Note that L is the worst and W is the best consequences, and u(C) > u(D) ⇐⇒ C D ⇐⇒ X < Y . Let u(L) = 0 and u(W) = 1. Then, u(C) and u(D) can be calculated by u(C) = 1 6 u(L) + 5 6 u(W) = 5 6 , and 1 2 u(L) + 1 2 u(D) = 2 3 u(L) + 1 3 u(W) ⇒ u(D) = 2 3 . Since u(C) > u(D), you must be ready to pay less to get one bullet removed when only one bullet was loaded than when four bullets were loaded. 31 / 31