Quantum Decision Theory

Transcription

1 Quantum Decision Theory V.I. Yukalov and D. Sornette Department of Management, Technology and Economics\ ETH Zürich

2 Plan 1. Classical Decision Theory 1.1. Notations and definitions 1.2. Typical paradoxes 1.3. Absence of solution 2. Quantum Decision Theory 2.1. Definitions and axioms 2.2. General properties 2.3. Solution of paradoxes

3 Classical Decision Theory Set of outcomes, set of payoffs, consumer set, field of events X = {x n : n = 1, 2,... N }, Utility function elementary utility function, satisfaction function (i) nondecreasing (ii) concave u x : X R u x 1 u x 2 x 1 x 2 x n R u x 1 1 x 2 u x 1 1 u x 2 0 1, ü x 0

4 Risk aversion Coefficient of absolute risk aversion Pratt (1964) r x Coefficient of relative risk aversion Pratt (1964), Arrow (1965) Exponential utility function Risk aversion ü x u x R x x r x = x ü x u x Portfolio problem u x = 1 e kx r x = k, R x = kx

5 Lottery Expected Utility Bernoulli (1738) Von Neumann Morgenstern (1944) Probability measure over X Linear combination {p x n : n = 1,2,..., N }, N n=1 p x n = 1 L = {x n, p x n : n = 1,2,..., N } L 1 = {x n, p 1 x n }, L 2 = {x n, p 2 x n } L 1 1 L 2 = {x n, p 1 X n 1 p 2 x n } 0 1

6 Lottery mean N x = 1 N n=1 p x n x n Lottery dispersion N 2 L = 1 N n=1 p x n x n 2 x 2 L dispersion, measure of uncertainty Expected utility of lottery N U L = n=1 Comparison of lotteries p x n u x n Indifference: L 1 = L 2 U L 1 = U L 2 Preference: L 1 L 2 U L 1 U L 2 L 1 L 2 U L 1 U L 2

7 Properties of expected utility (1) completeness for L 1 and L 2, one of relations L 1 = L 2, L 1 L 2, L 1 L 2, L 1 L 2, L 1 L 2 (2) transitivity (3) continuity if L 1 L 2 and L 2 L 3, then L 1 L 3 for L 1 L 2 L 3, there exist [0,1] L 1 1 L 3 = L 2 (4) independence for L 1 L 2 and any L 3, 0 1, L 1 1 L 3 L 2 1 L 3

8 Classical decision-making scheme Set of lotteries {L j : j = 1, 2,...,} Expected utility U L j L j = {x n, p j x n } compare U L j Optimal lottery L U L sup j U L j

9 Allais Paradox Allais (1953) Compatibility violation: Several choices are not compatible with utility theory Payoff set X = {x 1, x 2, x 3 } units of x n millions of dollars $10 6 x 1 = 0, x 2 = 1, x 3 = 5 Set of 4 lotteries {L j : j = 1, 2,3, 4} Probability measures {p j x n } Balance conditions p 1 x n p 3 x n = p 2 x n p 4 x n for all n = 1,2, 3.

10 {p 1 x n } = {0,1,0}, {p 2 x n } = {0.01,0.89, 0.10} {p 3 x n } = {0.9,0,0.1}, {p 4 x n }= {0.89,0.11,0} p 1 x 1 p 3 x 1 = p 2 x 1 p 4 x 1 = 0.9 p 1 x 2 p 3 x 2 = p 2 x 2 p 4 x 2 = 1 p 1 x 3 p 3 x 3 = p 2 x 3 p 4 x 3 = 0.1

11 Lotteries L 1 = { $0, 0 $1, 1 $5, 0 } 2 L 1 = 0 { $0, 0.01 } L 2 = $1, L 2 = $5, 0.10 L 1 L 2 L 2 is more uncertain

12 { $0, 0.9 } L 3 = $1, 0 2 L 3 = $5, 0.1 L 4 = { $ 0, 0.89 $1, 0.11 $5, 0 } 2 L 4 = 0 L 3 L 4 L 3 is more uncertain, but stake is larger U L 1 = u 1 U L 2 = 0.01u u u 5 U L 3 = 0.9 u u 5 U L 4 = 0.89u u 1

13 L 1 L 2 U L 1 U L u u 0 0.1u 5 L 3 L 4 U L 3 U L u u 0 0.1u 5 Contradiction! For any definition of u(x)!

14 Independence Paradox Allais (1953) Independence axiom: if L 1 L 2 and L 3 L 4, then for any [0,1] L 1 1 L 3 L 2 1 L 4

15 Lotteries as in the Allais paradox {L j : j = 1, 2,3, 4} take = 1 2 { $0, 0.45 } 1 2 L 1 L 3 = $1, 0.50 $5, 0.05 { $0, 0.45 } 1 2 L 2 L 4 = $1, 0.50 $5, 0.05

16 U L 1 L 3 2 U L 2 L 4 2 but by the independence axiom it should be U L 1 L 3 2 U L 2 L 4 2 Contradiction! For any definition of u(x)!

17 Ellsberg Paradox Ellsberg (1961) 1 urn: 50 red balls + 50 black balls 2 urn: 100 balls, red or black in an unknown proportion x 1 x 2 50 red + 50 black Payoffs x 1 prize for getting a red ball x 2 prize for getting a black ball 100 Units of x n, say, $1000

18 Get a red ball from 1-st urn {$0, L 1 = $1, 1 } Get a red ball from 2-nd urn { L 2 = $0, } 0 1 $1, 1 preference: L 1 L 2

19 Get a black ball from 1-st urn {$0, L 3 = $1, 1 } Get a black ball from 2-nd urn { L 4 = $0, 1 } 0 1 $1, Preference: L 3 L 4 Indifference: L 2 = L 4

20 L 1 L u U L 1 U L 2 u 1 u 0 1 u 1 L 3 L u No such [0,1] Contradiction! U L 3 U L 4 u 1 1 u 0 u 1

21 Also: L 2 = L 4 U L 2 = U L 4 hence = 1 2 p j x n = 1 2 = const then L 1 = L 2 = L 3 = L 4 U L j = const for any definition of U L

22 Kahneman-Tversky Paradox Kahneman-Tversky (1979) Invariance violation: Preference instead of indifference Set of payoffs {x n }= {1,1.5, 2} Units of x n, thousands of dollars $1000 u 1.5 = 1 2 [ u 1 u 2 ]

23 After winning, one gets { $1, 0.5 L = 1 $1.5, 0 $ 2, 0.5 }, L 2 = { $1, 0 } $1.5, 1 $ 2, 0 2 L 1 = L 2 = 0 L 2 is more certain L 2 > L 1

24 After loosing, one gets { $1, 0.5 L = 3 $1.5, 0 $ 2, 0.5 }, L 4 = { $1, 0 } $1.5, 1 $ 2, 0 2 L 3 = L 4 = 0 L 4 is more certain, but L 3 > L 4

25 L 2 L 1 L 3 L 4 U L 2 U L 1 U L 3 U L 4 However U L j = 1 2 u u 2 = u 1.5 for all j = 1,2,3,4 U L j = const Contradiction! For any definition of u(x)!

26 Rabin Paradox Rabin (2000) payoffs: X 1 = {x l 1, x, x g 1 } l 1 loss, g 1 gain x l 1, l 1 0, g 1 0 Small difference between gain and loss g 1 l 1

27 { { L 1 = $ x l 1, 0.5 } $ x, 0, $ x g 1, 0.5 L 2 = $ x l 1, 0 } $ x, 1 $ x g 1, 0 2 L L 2 = 0 L 1 is more uncertain L 2 > L 1

28 { { Payoffs: X 2 = {x l 2, x, x g 2 } x l 2, l 2 0, g 2 0 Large difference between gain and loss: g 2 l 2 L 3 = $ x l 2, 0.5 } $ x, 0, $ x g 2, 0.5 L 4 = $ x l 2, 0 } $ x, 1 $ x g 2, 0 2 L L 4 = 0 L 3 L 4

29 Although L 3 is more uncertain But the stake is much larger L 2 L 1 U L 2 U L 1 u x 1 2 u x l u x g 1. L 3 L 4 U L 3 U L 4 u x 1 2 u x l u x g 2

30 Rabin theorem (2000) If for some l > 0, g > 0 u x 1 2 u x l 1 2 u x g, then it is so for all l, g, because of the concavity of u(x). Contradiction with above! For any concave u(x)!

31 Tversky-Shafir (1992) Two-step gambles Disjunction Effect 1-st step: - st gamble won (B 1 ) -st gamble lost (B 2 ) 2-nd step: -nd gamble accepted (A 1 ) -nd gamble refused (A 2 ) People accept the 2 -nd gamble independently whether they won the first, p A 1 B 1 p A 2 B 1, or they lost the first gamble, p A 1 B 2 p A 2 B 2.

32 But, when the results of the 1-st gamble are not known, B = B 1 B 2 B 1 B 2 = 0, people restrain from the 2-nd gamble, By probability theory, p A 2 B p A 1 B. p A 1 B = p A 1 B 1 p A 1 B 2, p A 2 B = p A 2 B 1 p A 2 B 2. If p A 1 B j p A 2 B j for j = 1,2, then p A 1 B p A 2 B. Contradiction!

33 Savage (1954) Sure-thing principle Humans respect probability theory: p A 1 B j p A 2 B j p A 1 B p A 2 B However, disjunction effect: Humans do not abide to probability theory!

34 Another example of Disjunction Effect 1-st step: 2-nd step: Students go to vacation in any case of known results: p A 1 B 1 p A 2 B 1, p A 1 B 2 p A 2 B 2. When results are not known, students forgo vacations: p A 1 B p A 2 B Contradiction with sure-thing principle! B = B 1 B 2

35 Conjunction Fallacy Tversky-Kahneman (1983) One event (A). Another event (B = B 1 + B 2 ), which may happen (B 1 ), or does not happen (B 2 ).

36 People often judge: p A B p A B 1. But, by probability theory, hence, conjunction rule: p A B = p A B 1 p A B 2, p A B p A B j j = 1, 2. Contradiction! Examples: description of a person, of a subject, of an event,... Decide on the existence of one feature (A). Decide on the existence of another feature (B 1 ) or absence of it (B 2 ). p A B p A B 1 B = B 1 B 2.

37 Save utility theory? Non-expected utility functionals. For a lottery L = {x n, p x n } Instead of expected utility U(L), utility functionals F L = F [ x n, p x n, u x n ] Minimal requirements: Risk aversion Between two lotteries L 1 and L 2, with the same mean x L 1 = x L 2 the lottery L 1 is preferred to L 2 ( L 1 > L 2 ) if 2 L 1 2 L 2. Then F(L 1 ) > F(L 2 ). Safra and Segal Safra and Segal (2008): Non-expected utility functionals do not remove paradoxes!

38 What to do? 1. Realistic problems are complicated, consisting of many parts. 2. Different parts of a problem interact and interfere with each other. 3. Several thoughts of mind can be intricately interconnected (entangled). Life is complex!

39 Quantum Decision Theory 1. Action ring Main definitions = {A n : n=1, 2,..., N } Intended actions A n addition A m A n associative: A 1 A 2 A 3 = A 1 A 2 A 3 reversible: A 1 A 2 = A 3 A 1 = A 3 A 2

40 multiplication: A m A n distributive: A 1 A 2 A 3 = A 1 A 2 A 1 A 3 idempotent: A n A n A n 2 = A n noncommutative: A m A n A n A m (generally) empty action: A n 0 = 0 A n = 0 disjoint actions: A m A n = A n A m = 0

41 2. Action Modes Composite actions action modes, representations A n A n = A n 3. Action prospects conjunction, composite or simple, composite and simple prospects

42 4. Elementary prospects binary multi-index = {i n, n : n=1,2,..., N } number of α, cardinality conjunction of modes e e = e

43 5. Prospect lattice L = { j : j=1,2,..., N L } ordering: i j or i j 6. Mode states A n complex function scalar product A n A n =

44 7. Mode space closed linear envelope Hilbert space 8. Basic states elementary prospect e

45 9. Mind space 10. Prospect states 11. Strategic states reference states

46 12. Mind strategy Person character, basic beliefs and principles 13. Prospect operators Involutive bijective algebra 14. Operator averages

47 15. Prospect probability 16. Prospect ordering π 1 indifferent to π 2 : π 1 preferred to π 2 : Decisions are probabilistic

48 17. Partial probabilities π j e α conjunction prospects 18. Attraction factor Quantifies the attractiveness of the project with respect to risk, uncertainty, biases. Caused by action interference.

49 19. Attraction ordering π 1 is more attractive than π 2 : (less risky, less uncertain) π 1 and π 2 are equally attractive: (equally risky, equally uncertain) 20. Attraction conditions π 1 is more attractive than π 2 if it is connected with: (a) more certain gain, (b) less certain loss, (c) higher activity under certainty, (d) lower activity under uncertainty. Aversion to risk, uncertainty, and loss.

50 General properties Proposition 1. Proposition 2. Attraction alternation

51 Proposition 3. π 1 preferred to π 2 if and only if Return to classical decision theory:

52 Binary mind Two actions Two mode spaces Mind space

53 Elementary prospects e j = A j B Basic states Action prospects: Prospect probabilities: Conditional probability

54 Correspondence lottery payoffs normalized measure of probability of the payoffs in the lottery normalized utility of q A j B? No equivalent

55 Allais paradox Balance condition for all is more attractive:

56 more attractive: Balance condition in classical utility theory contradiction In QDT no contradiction!

57 Tversky Shafir (1992) A 1 : second gamble accepted A 2 : second gamble refused B 1 : first gamble won B 2 : first gamble lost Experiment Disjunction Effect 1-st gamble won + 2-nd accepted: 1-st gamble won + 2-nd refused: p A 1 B 1 = p A 2 B 1 = p A 1 B 1 = = p A 2 B 1

58 1-st gamble lost + 2-nd accepted: 1-st gamble lost + 2-nd refused: p A 1 B 2 = p A 2 B 2 = p A 1 B 2 = = p A 2 B 2 B = B 1 B 2 1-st gamble not known + 2-nd accepted: 1-st gamble not known + 2-nd refused: p A 1 B = 0.36 p A 2 B = 0.64 p A 1 B = = p A 2 B

59 Theory Active under uncertainty: A 1 B attraction factor q(a 1 B) Passive under uncertainty: A 2 B attraction factor q(a 2 B) Alternation theorem q A 2 B q A 1 B q A 2 B = q A 1 B 0 q A 2 B 0.25, q A 1 B 0.25

60 Prediction p A 1 B = p A 1 B 1 p A 1 B 2 q A 1 B p A 2 B = p A 2 B 1 p A 2 B 2 q A 2 B p A 1 B = 0.39, p A 2 B = 0.61 Agreement with experiment! Theory: P A 1 B = = p A 2 B Experiment P A 1 B = = p A 2 B

61 Conclusions Novel approach to decision making is developed based on a complex Hilbert space over a lattice of composite prospects. Risk and uncertainty are taken into account. Paradoxes of classical decision theory are explained. Good quantitative agreement with empirical data. Conjunction fallacy is a sufficient condition for disjunction effect. References: V.I. Yukalov and D. Sornette, Quantum Decision Theory, arxiv.org (2008); Mathematical Basis of Quantum Decision Theory, ssrn.com/abstract= (2008).