Expressive Rationality

Transcription

1 Expressive Rationality Larry Blume Cornell University & The Santa Fe Institute & IHS, Vienna

2 What Caused the Financial Crisis? Blume Expressive Rationality 1

3 What Caused the Financial Crisis? MATH! Blume Expressive Rationality 1

4 What Caused the Financial Crisis? When it comes to the all-too-human problem of recessions and depressions, economists need to abandon the neat but wrong solution of assuming that everyone is rational and markets work perfectly. Paul Krugman, NYT Blume Expressive Rationality 1

5 What Caused the Financial Crisis? When it comes to the all-too-human problem of recessions and depressions, economists need to abandon the neat but wrong solution of assuming that everyone is rational and markets work perfectly. Paul Krugman, NYT Copula image source: quant.

6 What is Probability? Coin Flip Blume Expressive Rationality 2

7 What is Probability? Horse Race Blume Expressive Rationality 2

8 What is Probability? Horse Race This is the Objective Subjective Distinction

9 What is Probability? Electron Cloud Blume Expressive Rationality 3

10 What is Probability? Galton Board Blume Expressive Rationality 3

11 What is Probability? f (x) = { µx if 0 x 1/2, µ(1 x) if 1/2 < x 1. Tent Map Blume Expressive Rationality 3

12 What is Probability? Financial Market Blume Expressive Rationality 3

13 What is Probability? Financial Market The ontic/aleatory epistemic distinction. Quantum mechanics gives a probalistic description of the location of an electron around the nucleus of an atom. It is inherently probalistic. The Dalton board would be deterministic if we knew the initial conditions, but we don t. Same with Coin flips. For 2 µ 2 the Tent map has a unique invariant measure ν µ which is absolutely continuous with respect to Lebesgue measure, and that measure is ergodic. If you do not know the initial condition, and have any prior belief p 0 on x 0, then your prior belief about x 1000 should be p 1000 ν µ. So even though beliefs about one run of the tent map should be thought of as epistemic, the system forces your beliefs about x There are physical systems with this property the location of a gas molecule, solutions to the n-body problem, etc. Financial Markets?

14 Classification of Probability Types Ontic Epistemic Objective Electron Cloud Coin Flip Subjective Tent Map Horse Race Blume Expressive Rationality 4

15 Measurement and Meaning Definition: A measurement structure M =< S, R 1,..., R n > is a set of objects S together with n m i -ary relations R i on S. Definition: A real-valued representation of M is a set Σ of real numbers and n m i -ary relations R i on Σ, together with a function φ : S Σ such that (s 1,..., s mi ) R i iff ( φ(s 1 ),..., φ(s mi ) ) R i. The problem of measurement is to find a representation. The problem of meaning is to determine which properties of < Σ, R 1,..., R n > have meaning for M. Blume Expressive Rationality 5

16 Measurement and Meaning Definition: A measurement structure M =< S, R1,..., Rn > is a set of objects S together with n mi-ary relations Ri on S. Definition: A real-valued representation of M is a set Σ of real numbers and n mi-ary relations Ri on Σ, together with a function φ : S Σ such that (s1,..., smi ) Ri iff ( φ(s1),..., φ(smi )) Ri. The problem of measurement is to find a representation. The problem of meaning is to determine which properties of < Σ, R1,..., Rn > have meaning for M. What does it mean to measure something? What are the experiments. The relation being measured: warmer than, better than, more likely than; together with operations on objects, such as concatenation, piling up on one side of a balance beam,... These relations collectively define experiments. Frequentism 1. Probability is assigned only to collectives. These are repetitive events or mass phenomena. 2. Collectives are modeled as infinite sequences. 3. Relative frequencies converge not just for the infinite sequence, but for any infinite subsequence. So what is the experiment that we can perform on, say, financial markets?

17 Probability as a Theory of Measurement An algebra A of sets of elements of a ground set S (of states). A complete relation B on A: A B means A is at least as likely as B. S A for all A A, and S. A representation of is a function p : A [0, 1] such that A B iff p(a) p(b). Blume Expressive Rationality 6

18 Probability as a Theory of Measurement: Finite States When is a representation a probability? A C = p(a C) = p(a) + p(c). Disjoint Union Property: Suppose C is disjoint from A, B. A B iff A C B C. Blume Expressive Rationality 7

19 Probability as a Theory of Measurement: Finite States When is a representation a probability? A C = p(a C) = p(a) + p(c). Cancellation: If A 1,..., A N and B 1,..., B N are sets such that for all s, #{n : s A n } = #{n : s B n } and for all n N 1, A n B n, then B N A N. Blume Expressive Rationality 7

20 Probability as a Theory of Measurement: Many States When is a representation a probability? 1. is complete and transitive on the power set S of S. 2. For A, B disjoint from C, A B iff A C B C. 3. If A B, there is a finite partition {C 1,..., C M } of S such that for all m, A B C. Theorem (Savage): If so there is a unique probability measure p on S such that A B iff p(a) p(b). Furthermore, for all A and 0 ρ 1 there is a B A such that p(b) = ρp(a). Blume Expressive Rationality 8

21 Probability as a Theory of Measurement: Many States When is a representation a probability? 1. is complete and transitive on the power set S of S. 2. For A, B disjoint from C, A B iff A C B C. 3. If A B, there is a finite partition {C1,..., CM} of S such that for all m, A B C. Theorem (Savage): If so there is a unique probability measure p on S such that A B iff p(a) p(b). Furthermore, for all A and 0 ρ 1 there is a B A such that p(b) = ρp(a). If you accept the continuum hypothesis, then p must be only finitely additive. S can be a σ-algebra, but not just an algebra. Savage s axioms does not imply that the state space be uncountable.

22 Alternative Measures of Probability Sets of Probabilities Non-additive Probabilities Belief Functions Inner and Outer Measure Lexicographic Probabilities Possibility Measures Plausibility Measures Ranking Functions Blume Expressive Rationality 9

23 Why Axiomatic Decision Theory? Are representations compelling? Axioms characterize preferences in terms of choice behavior. Important Models SEU LSEU Probabilistic Sophistication Wald citerion Minimax regret CEU MMEU Prospect Theory Hurwicz α rule Blume Expressive Rationality 10

24 Why Axiomatic Decision Theory? Are representations compelling? Axioms characterize preferences in terms of choice behavior. Important Models SEU LSEU Probabilistic Sophistication Wald citerion Minimax regret CEU MMEU Prospect Theory Hurwicz α rule Representations per se are not compelling. Characterize representations in terms of choice behavior. A decision model is normatively appropriate iff its characterizing axioms have normative appeal A decision model is descriptively appropriate iff its characterizing axioms have descriptive appeal Axioms give us a handle on verifying or falsifying, justifying or criticizing given models. vn-m - preferences over probability distributions SEU

25 A Common Framework States: A finite state space S. Outcomes: A finite set O, with best and worst outcomes x and x. Roulette Wheels: The set R of probability distributions on O. Horse Lotteries: The set H of functions h : S O. Preferences: A preference relation on H. Anscombe and Aumann (1963). Blume Expressive Rationality 11

26 Subjective Expected Utility SEU 1. is complete and transitive. SEU 2. Independence: If h k then for all g and 0 α 1, αg + (1 α)h αg + (1 α)k. SEU 3. Archimedean axiom. SEU 4. State independence. There is a payoff function u : O R and a unique probability distribution p on S which define a functional V on H, V (h) = s p(s) o u(o)h(s)(o) such that h k iff V (h) V (k). Blume Expressive Rationality 12

27 Probabilistic Sophistication Savage framework: S, O, acts f : s o. P 0 set of finite-support probabilities on O Definition: An individuali s said to be probabilistically sophisticated if there exists a probability measure p on S and a preference functional V (x 1, p 1,..., x m, p m ) on P 0 satisfying mixture continuity and monotonicity with respect to stochastic dominance, such that preferences on acts are represented by the functional f V ( x 1, p(f 1 (x 1 )),..., x n, p(f 1 (x n ) ) where {x 1,..., x n } is the range of f. Blume Expressive Rationality 13

28 Probabilistic Sophistication Savage framework: S, O, acts f : s o. P0 set of finite-support probabilities on O Definition: An individuali s said to be probabilistically sophisticated if there exists a probability measure p on S and a preference functional V (x1, p1,..., xm, pm) on P0 satisfying mixture continuity and monotonicity with respect to stochastic dominance, such that preferences on acts are represented by the functional f V ( x1, p(f 1 (x1)),..., xn, p(f 1 (xn) ) where {x1,..., xn} is the range of f. The separation of tastes and beliefs. Mixture-closed, Stochastic dominance: p stochastically dominates q iff for every increasing function f, R f dp R f dq.

29 Why Non-Additive Probabilities? Ellsberg s Urns Schmeidler s Coins Blume Expressive Rationality 14

30 Why Non-Additive Probabilities? Ellsberg s Urns Schmeidler s Coins Image:

31 Choquet Expected Utility Definition: Acts f and g are comonotonic if f (s) f (t) implies g(s) g(t) for all s, t S. CEU 1. is complete and transitive. CEU 2. Comonotonic independence. CEU 3. Archimedean axiom. CEU 4. State independence. There is a payoff function u : O R and a unique capacity φ on S which define a functional V on H, V (h) = u(o)h(s)(o)dφ(s) such that h k iff V (h) V (k). S o Blume Expressive Rationality 15

32 MMEU MMEU 1. is complete and transitive. MMEU 2. Certainty independence. MMEU 3. Archimedean axiom. MMEU 4. State independence. MMEU 5. f g implies that for all 0 < α < 1, αf + (1 α)g f. There is a payoff function u : O R and a set P of probability distributions on S which define a functional V on H, V (h) = inf p(s) u(o)h(s)(o) p P o such that h k iff V (h) V (k). s Blume Expressive Rationality 16

33 MMEU MMEU 1. is complete and transitive. MMEU 2. Certainty independence. MMEU 3. Archimedean axiom. MMEU 4. State independence. MMEU 5. f g implies that for all 0 < α < 1, αf + (1 α)g f. There is a payoff function u : O R and a set P of probability distributions on S which define a functional V on H, V (h) = inf u(o)h(s)(o) p P s such that h k iff V (h) V (k). p(s) o Connection between MMEU, CEU. The core of a capacity is the set of all probability measures that dominate it. If φ is convex, φ(a B) + φ(a B) φ(a) + φ(b), then the Choquet integral of any real-valued function f with respect to φ is the minimum of the integrals with respect to probability distributions in the core. Notice that with equality, this is inclusion/exclusion, and is always satisfied by any probability measure.

34 The Problem with Conditioning Savage Conditioning: f A g iff for some h, f A h g A h. In SEU, any h gives the same answer, and A is represented by u and p( A). Blume Expressive Rationality 17

35 The Problem with Conditioning Savage Conditioning: f A g iff for some h, f A h g A h. S = {x, y, z}. φ(x) = 1/4, φ(y) = φ(z) = 1/8, φ(x, z) = 1/2, φ(y, z) = 3/4, φ(x, y) = 1/2. Let A = {x, y} f (x) = 1 on x, else 0, g(x) = 1 on y, else 0. Blume Expressive Rationality 17

36 The Problem with Conditioning Savage Conditioning: f A g iff for some h, f A h g A h. S = {x, y, z}. φ(x) = 1/4, φ(y) = φ(z) = 1/8, φ(x, z) = 1/2, φ(y, z) = 3/4, φ(x, y) = 1/2. Let A = {x, y} f (x) = 1 on x, else 0, g(x) = 1 on y, else 0. f dφ = 0 + 1φ(x) = 1/4 g dφ = 0 + 1φ(y) = 1/8 Blume Expressive Rationality 17

37 The Problem with Conditioning Savage Conditioning: f A g iff for some h, f A h g A h. S = {x, y, z}. φ(x) = 1/4, φ(y) = φ(z) = 1/8, φ(x, z) = 1/2, φ(y, z) = 3/4, φ(x, y) = 1/2. Let A = {x, y} f (x) = 1 on x, else 0, g(x) = 1 on y, else 0. f dφ = 0 + 1φ(x) = 1/4 g dφ = 0 + 1φ(y) = 1/8 h = 0 f A h dφ = 0 + 1φ(x) = 1/4 g A h dφ = 0 + 1φ(y) = 1/8 Blume Expressive Rationality 17

38 The Problem with Conditioning Savage Conditioning: f A g iff for some h, f A h g A h. S = {x, y, z}. φ(x) = 1/4, φ(y) = φ(z) = 1/8, φ(x, z) = 1/2, φ(y, z) = 3/4, φ(x, y) = 1/2. Let A = {x, y} f (x) = 1 on x, else 0, g(x) = 1 on y, else 0. f dφ = 0 + 1φ(x) = 1/4 g dφ = 0 + 1φ(y) = 1/8 h = 0 f A h dφ = 0 + 1φ(x) = 1/4 h = 1 f A h dφ = 0+1φ(x, z) = 1/2 g A h dφ = 0 + 1φ(y) = 1/8 g A h dφ = 0+1φ(y, z) = 3/4 Blume Expressive Rationality 17

39 The Problem with Conditioning Savage Conditioning: f A g iff for some h, f A h g A h. MMEU version µ 1 = (3/8, 1/8, 1/2) µ 2 = (1/4, 1/2, 1/4) µ 3 = (1/4, 5/8, 1/8) f A 0 1/4 g A 0 1/8 f A 1 3/8 g A 1 5/8 Blume Expressive Rationality 17

40 The Problem with Conditioning Savage Conditioning: f A g iff for some h, f Ah g Ah. MMEU version µ1 = (3/8, 1/8, 1/2) µ2 = (1/4, 1/2, 1/4) µ3 = (1/4, 5/8, 1/8) f A0 1/4 g A0 1/8 f A1 3/8 g A1 5/8 Updating is the problem of defining conditional preference. Define f A h. Same issue arises with MMEU. This is what I will talk about most.

41 The Conditioning Tradeoff Dynamic Consistency: If f A g in all ex post situations A, then f g. Consequentialism: Conditional preferences given A A only depend on what happens in A. Fact of Life: If the CEU updating rule satisfies dynamic consistency and conequentialism, then the capacity is a probability (and updating is Bayes). Blume Expressive Rationality 18

42 f -Bayesian Updating Definition: An updating rule is a map (, A) A that assigns to an unconditional preference relation and an event A the preference relation given A. Definition: For act f, the f -Bayesian updating rule is g f A h iff g A f h A f. Blume Expressive Rationality 19

43 f -Bayesian Updating Definition: An updating rule is a map (, A) A that assigns to an unconditional preference relation and an event A the preference relation given A. Definition: For act f, the f -Bayesian updating rule is g f A h iff g A f h A f. The optimistic rule: f o. A is good news. The Bayesian rule. φ A (B) = φ(b A)/φ(A). Select and update all priors assigning maximal probability to A. Blume Expressive Rationality 19

44 f -Bayesian Updating Definition: An updating rule is a map (, A) A that assigns to an unconditional preference relation and an event A the preference relation given A. Definition: For act f, the f -Bayesian updating rule is g f A h iff g A f h A f. The pessimistic rule: f o. A is bad news. φ A (B) = [ φ ( (B A) A c) φ(a c ) ] / ( 1 φ(a c ) ). The Dempster-Shafer rule for belief functions. Blume Expressive Rationality 19

45 f -Bayesian Updating Definition: An updating rule is a map (, A) A that assigns to an unconditional preference relation and an event A the preference relation given A. Definition: For act f, the f -Bayesian updating rule is g f A h iff g Af haf. The pessimistic rule: f o. A is bad news. φa(b) = [ φ ( (B A) A c) φ(a c ) ] / ( 1 φ(a c ) ). The Dempster-Shafer rule for belief functions. Affine u. S 4. Fagin-Halpern is full updating.

46 MMEU: Full Bayesian Updating F1. The updating rules (, A) A takes MMEU preferences ino MMEU preferences. F2. For all non-null events A and outcomes o, if f o then f A o o. Theorem: If u and P are an MMEU representation for and p(a) > 0 for all p P, then u and {q = p( A), p P} give an MMEU representation for A. Fagin and Halpern (1989), Pires (2002). Blume Expressive Rationality 20

47 MMEU: Full Bayesian Updating F1. The updating rules (, A) A takes MMEU preferences ino MMEU preferences. F2. For all non-null events A and outcomes o, if f o then f Ao o. Theorem: If u and P are an MMEU representation for and p(a) > 0 for all p P, then u and {q = p( A), p P} give an MMEU representation for A. Fagin and Halpern (1989), Pires (2002). MMEU is inherently pessimist, selecting priors which put the most weight on the worst outcomes. F2 guarantees that no matter the weights on E vs. E c, the relative weights on states within E have to cohere with the unconditional weights.

48 Easy Implications: Portfolios 0 net positions in portfolios over a range of prices. An asset pays off 1 in state H and 3 instate T. Trader beliefs are φ(h) = 0.3 and φ(t ) = 0.4. The expected payoff of a unit long position at price p is v b = (1 p) + 0.4(2) = 1.8 p. The value of a unit short position is v s = p (2) = p 2.4. For 1.8 < p < 2.4, the 0 position is better than both. Home Bias Paradox Equity Premium Puzzle Blume Expressive Rationality 21

49 Easy Implications: Portfolios 0 net positions in portfolios over a range of prices. An asset pays off 1 in state H and 3 instate T. Trader beliefs are φ(h) = 0.3 and φ(t ) = 0.4. The expected payoff of a unit long position at price p is vb = (1 p) + 0.4(2) = 1.8 p. The value of a unit short position is vs = p (2) = p 2.4. For 1.8 < p < 2.4, the 0 position is better than both. Home Bias Paradox Equity Premium Puzzle WIth SEU preferences, the 0 zone is a point, and there is indifference between buying and selling.

50 FINIS! Blume Expressive Rationality 22

51 Definition: Non-Additive Probability Definition: A non-additive probability ν on S is a function mapping subsets of S to [0, 1] such that N.1. ν( ) = 0, N.2. ν(s) = 1, N.3. If A B, then ν(a) ν(b). For example, S = {s 1, s 2 }. ν α ( ) = 0 ν α (s 1 ) = ν α (s 2 ) = α ν α (S) = 1 Blume Expressive Rationality 23

52 Definition: Non-Additive Probability Definition: A non-additive probability ν on S is a function mapping subsets of S to [0, 1] such that N.1. ν( ) = 0, N.2. ν(s) = 1, N.3. If A B, then ν(a) ν(b). Integration Suppose the values of f are x 1 < < x n. Then E ν f = x 1 + (x 2 x 1 )ν(f > x 1 ) + (x n x n 1 )ν(f > x n 1 ). Blume Expressive Rationality 23

53 Dempster-Shafer Belief Functions Definition: A belief function β on S is a function mapping subsets of S to [0, 1] such that B.1. β( ) = 0, B.2. β(s) = 1, B.3. β( n i=1 A i) n i=1 I {1,...,n}: I =i ( 1)i+1 β( j I A j ). Blume Expressive Rationality 24

54 Dempster-Shafer Belief Functions Definition: A belief function β on S is a function mapping subsets of S to [0, 1] such that B.1. β( ) = 0, B.2. β(s) = 1, B.3. β( n i=1 A i) n i=1 I {1,...,n}: I =i ( 1)i+1 β( j I A j ). B.3 is like inclusion-exclusion: Pr(A B) = Pr(A) + Pr(B) Pr(A B), Pr(A B C) = Pr(A) + Pr(B) + Pr(C) Pr(A B) Pr(A C) Pr(B C) + Pr(A B C). Blume Expressive Rationality 24

55 Dempster-Shafer Belief Functions Definition: A belief function β on S is a function mapping subsets of S to [0, 1] such that B.1. β( ) = 0, B.2. β(s) = 1, B.3. β( n i=1 A i) n i=1 I {1,...,n}: I =i ( 1)i+1 β( j I A j ). β(a) = inf{p(a) : p β} Blume Expressive Rationality 24

56 Dempster-Shafer Belief Functions Definition: A belief function β on S is a function mapping subsets of S to [0, 1] such that B.1. β( ) = 0, B.2. β(s) = 1, B.3. β( n i=1 Ai) n i=1 β(a) = inf{p(a) : p β} I {1,...,n}: I =i ( 1)i+1 β( j I Aj). Belief functions are tight capacities. m(a) is the weight of evidence for A not assigned to any of its subsets. Theorem due to Shafer.

57 Mass Functions & Belief Functions Definition: A mass function m on S is a function mapping subsets of S to [0, 1] such that M.1. m( ) = 0, M.2. A S (A) = 1. Theorem: If S is finite and S = 2 S, then beta is a belief function if and only if there is a (unique) mass function m such that for all A, β(a) = B A m(b). Blume Expressive Rationality 25

58 Mass Functions & Belief Functions Definition: A mass function m on S is a function mapping subsets of S to [0, 1] such that M.1. m( ) = 0, M.2. A S (A) = 1. Theorem: If S is finite and S = 2 S, then beta is a belief function if and only if there is a (unique) mass function m such that for all A, β(a) = B A m(b). Belief functions are tight capacities. m(a) is the weight of evidence for A not assigned to any of its subsets. Theorem due to Shafer.

59 Definition: Lexicographic Probability Definition: A lexicographic probability on S is a vector of probabilities µ = (µ 1,..., µ n ) on S such that A B iff the vector µ(a) lexicographically dominates µ(b). LSEU ( { } ) n U(f ) = E µi u(o)f (s)(o). i=1 f g if and only if U(f ) U(g). o Blume Brandenburger and Dekel (1993a,b) Blume Expressive Rationality 26

60 References Blume, L., A. Brandenburger and E. Dekel (1993). Lexicographic probabilities and choice under uncertainty. Econometrica. Cerreia-Vioglio,S., et. al., (2008). Uncertainty averse preferences. Unpublished. Fagin, R. and Halpern, J. (1991). A new approach to updating belief, in Uncertainty in Artificial Intelligence 6. Gelman, A. (2006). The boxer, the wrestler, and the coin flip: A paradox of robust Bayesian inference and belief functions. The American Statistician. Gilboa, I. and D. Schmeidler (1989). Maxmin expected utility with a non-unique prior. Journal of Mathematical Economics. Gilboa, I. and D. Schmeidler (1993). Updating ambiguous beliefs. Journal of Economic Theory 59. Kast, R., A. Lapied and P. Toquebeuf (2008). Updating Choquet integrals, consequentialism and dynamic consistency. Unpublished. Machina, M. and D. Schmeidler (1992). A more robust definition of subjective probability Econometrica. Marinacci, M. and L. Montrucchio (2004). Introduction to the mathematics of ambiguity. In Uncertainty in Economic Theory: A collection of essays in honor of David Schmeidlers 65th birthday. Pires, C (2002). A rule for updating ambiguous beliefs. Theory and Decision Schmeidler, D. (1989). Subjective probability and expected utility without additivity, Econometrica. Blume Expressive Rationality 27