Addressing Model Ambiguity in the Expected Utility Framework

Transcription

1 Addressing Model Ambiguity in the Expected Utility Framework Erick Delage Canada Research Chair in Decision Making under Uncertainty Associate Professor, Dept. of Decision Sciences, HEC Montréal Joint work with Yinyu Ye (Stanford) and Benjamin Armbruster (Northwestern) Monday, June 15th, E. Delage Model Ambiguity in the Expected Utility Framework

2 Introduction The Portfolio Selection Problem stock prices Boeing Motorola Dow Chemical Company Merck & Co., Inc year An individual meets with his financial advisor to tell him he wishes to invest in a given industrial sector, country, etc. Since uncertain factors affect performance, a good portfolio is one where the risks of losses are best justified by the potential gains How can one trade-off optimally the risks and the returns taking into account his own perception of what is a serious risk? 2 E. Delage Model Ambiguity in the Expected Utility Framework

3 Introduction Von Neumann-Morgenstern Expected Utility If the investor agrees with the following axioms: 1 Completeness : He can order any two gambles 2 Transitivity : H 1 H 2 H 3 H 1 H 3 3 Continuity : If H 1 H 2 H 3 then there is a p such that H 2 ph 1 + (1 p)h 3 4 Independence : If H 1 H 2 then ph 1 + (1 p)h 3 ph 2 + (1 p)h 3 for all p and H 3 then the preference he expresses between any two gambles must be representable by an expected utility measure: H 1 H 2 E[u(H 1 )] E[u(H 2 )] 3 E. Delage Model Ambiguity in the Expected Utility Framework

4 Introduction Expected Utility Framework When applying the expected utility framework to a decision problem: maximize E [u(h(x, ξ))], x X where x = decisions, ξ = uncertain parameters, h(x, ξ) = profit, 4 E. Delage Model Ambiguity in the Expected Utility Framework

5 Introduction Expected Utility Framework When applying the expected utility framework to a decision problem: maximize E [u(h(x, ξ))], x X where x = decisions, ξ = uncertain parameters, h(x, ξ) = profit, it is assumed that we know: 3 The distribution of the random vector ξ A utility function that matches investor s attitude to risk Retirement Fund Casino de Monte-Carlo 4 E. Delage Model Ambiguity in the Expected Utility Framework

6 Introduction Difficulties encountered in practice Difficulties of developing an accurate probabilistic model: Need to collect enough observations Need to consult with experts of the field of practice Need to make simplifying assumptions Unforeseen events (e.g., economic crisis) might occur 5 E. Delage Model Ambiguity in the Expected Utility Framework

7 Introduction Difficulties encountered in practice Difficulties of developing an accurate probabilistic model: Need to collect enough observations Need to consult with experts of the field of practice Need to make simplifying assumptions Unforeseen events (e.g., economic crisis) might occur Difficulties of developing an accurate utility function: Need to compare a large number of gambles Need to accept structural properties Perception might be biased 5 E. Delage Model Ambiguity in the Expected Utility Framework

8 Introduction Modern Robust Optimization Framework Generally attributed to Ben-Tal & Nemirovski (1998), this framework implements a worst-case approach to dealing with model ambiguity. max. {h(x, y)} y Y max. x X x X inf h(x, y) y Y 6 E. Delage Model Ambiguity in the Expected Utility Framework

9 Introduction Modern Robust Optimization Framework Generally attributed to Ben-Tal & Nemirovski (1998), this framework implements a worst-case approach to dealing with model ambiguity. max. {h(x, y)} y Y max. x X x X inf h(x, y) y Y Success of the robust optimization relies in part on applying duality to combine inner and outer problems max. x X min y T x max. y:ay b x X max. x X min y max λ 0 max λ 0 y T x + λ T (Ay b) min y y T x + λ T (Ay b) max. x X,λ 0 bt λ s.t. x + A T λ = 0 6 E. Delage Model Ambiguity in the Expected Utility Framework

10 Introduction Robust Expected Utility In this talk, we investigate how to address model ambiguity in the expected utility framework using robust optimization Distributionally robust optimization maximize x X inf E F [u(h(x, ξ))] F D Preference robust optimization maximize x X inf u U E F [u(h(x, ξ))] 7 E. Delage Model Ambiguity in the Expected Utility Framework

11 Outline Introduction 1 Introduction 2 Distributionally Robust Optimization 3 Preference Robust Optimization 4 Conclusion 8 E. Delage Model Ambiguity in the Expected Utility Framework

12 Outline Distributionally Robust Optimization 1 Introduction 2 Distributionally Robust Optimization 3 Preference Robust Optimization 4 Conclusion 9 E. Delage Model Ambiguity in the Expected Utility Framework

13 Distributionally Robust Optimization Dealing with model ambiguity: Ellsberg Paradox Consider an urn with 30 blue balls and 60 other balls that are either red or green (you don t know how many are red or green). 10 E. Delage Model Ambiguity in the Expected Utility Framework

14 Distributionally Robust Optimization Dealing with model ambiguity: Ellsberg Paradox Consider an urn with 30 blue balls and 60 other balls that are either red or green (you don t know how many are red or green). Experiment 1: Choose among the following two gambles Gamble A: If you draw a blue ball, then you win 100$ Gamble B: If you draw a red ball, then you win 100$ 10 E. Delage Model Ambiguity in the Expected Utility Framework

15 Distributionally Robust Optimization Dealing with model ambiguity: Ellsberg Paradox Consider an urn with 30 blue balls and 60 other balls that are either red or green (you don t know how many are red or green). Experiment 1: Choose among the following two gambles Gamble A: If you draw a blue ball, then you win 100$ Gamble B: If you draw a red ball, then you win 100$ Experiment 2: Choose among the following two gambles Gamble C: If you draw blue or green ball, then you win 100$ Gamble D: If you draw red or green ball, then you win 100$ 10 E. Delage Model Ambiguity in the Expected Utility Framework

16 Distributionally Robust Optimization Dealing with model ambiguity: Ellsberg Paradox Consider an urn with 30 blue balls and 60 other balls that are either red or green (you don t know how many are red or green). Experiment 1: Choose among the following two gambles Gamble A: If you draw a blue ball, then you win 100$ Gamble B: If you draw a red ball, then you win 100$ Experiment 2: Choose among the following two gambles Gamble C: If you draw blue or green ball, then you win 100$ Gamble D: If you draw red or green ball, then you win 100$ If you clearly prefer Gamble A & D, then you are averse to model ambiguity 10 E. Delage Model Ambiguity in the Expected Utility Framework

17 Distributionally Robust Optimization Distributionally Robust Optimization Let s consider that the choice of F is ambiguous Use available information to define D, such that F D 11 E. Delage Model Ambiguity in the Expected Utility Framework

18 Distributionally Robust Optimization Distributionally Robust Optimization Let s consider that the choice of F is ambiguous Use available information to define D, such that F D Distributionally Robust Optimization values the lowest performing possible stochastic model (DRO) maximize x X inf E F [u(h(x, ξ))] F D 11 E. Delage Model Ambiguity in the Expected Utility Framework

19 Distributionally Robust Optimization Distributionally Robust Optimization Let s consider that the choice of F is ambiguous Use available information to define D, such that F D Distributionally Robust Optimization values the lowest performing possible stochastic model (DRO) maximize x X inf E F [u(h(x, ξ))] F D Important milestones: 1958: H. Scarf introduces DRO 1989: I. Gilboa et al. introduces maxmin expected utility 2007: I. Popescu solves µ-σ portfolio prob. 2010: Bertsimas et al. solves linear µ-σ prob. 2010: Goh et al. develops library for Matlab 2010: Delage et al. solves concave-convexe S- µ- Σ max prob. 2014: Wiesemann et al. solves or approximates conic prob. 11 E. Delage Model Ambiguity in the Expected Utility Framework

20 Distributionally Robust Optimization A Classic Reduction I Let s make three assumptions about E [u(h(x, ξ))]. 1 The profit function is concave in x and convex in ξ In portfolio optimization, h(x, ξ) = ξ T x 2 The utility function is piecewise linear concave : u(y) = min 1 k K a k y + b k 3 The information about F is captured by D(γ) = F P(ξ S) = 1 E [ξ] ˆµ 2ˆΣ 1/2 γ 1 E [(ξ ˆµ)(ξ ˆµ) T ] (1 + γ 2 )ˆΣ 12 E. Delage Model Ambiguity in the Expected Utility Framework

21 Distributionally Robust Optimization A Classic Reduction II The DRO problem with D(γ) is equivalent to max. x X inf F E F [u(h(x, ξ))] s.t. P F (ξ S) = 1 E F [ξ] ˆµ 2ˆΣ 1/2 γ 1 E F [(ξ ˆµ)(ξ ˆµ) T ] (1 + γ 2 )ˆΣ For portfolio selection, if S = polyhedron or ellipsoid, then DRO equivalent to semi-definite program. E.g., when S = R m, constraint ( ) can be replaced by [ Q (q + a k C 2 y k )/2 (q + a k C 2 y k ) T /2 a k c T 1 x + b k r ] 0, k 13 E. Delage Model Ambiguity in the Expected Utility Framework

22 Distributionally Robust Optimization A Classic Reduction II The DRO problem with D(γ) is equivalent to max. x X max Q,q,r s.t. ( r γ 2 ˆΣ + ˆµˆµ T) Q ˆµ T q γ 1 ˆΣ 1/2 (q + 2Qˆµ) r u(h(x, ξ)) + ξ T q + ξ T Qξ ξ S Q 0 For portfolio selection, if S = polyhedron or ellipsoid, then DRO equivalent to semi-definite program. E.g., when S = R m, constraint ( ) can be replaced by [ Q (q + a k C 2 y k )/2 (q + a k C 2 y k ) T /2 a k c T 1 x + b k r ] 0, k 14 E. Delage Model Ambiguity in the Expected Utility Framework

23 Distributionally Robust Optimization A Classic Reduction II The DRO problem with D(γ) is equivalent to max. x, Q,q,r s.t. ( r γ 2 ˆΣ + ˆµˆµ T) Q ˆµ T q γ 1 ˆΣ 1/2 (q + 2Qˆµ) r a k (h(x, ξ)) + b k + ξ T q + ξ T Qξ ξ S, k ( ) Q 0 15 E. Delage Model Ambiguity in the Expected Utility Framework

24 Distributionally Robust Optimization A Classic Reduction II The DRO problem with D(γ) is equivalent to max. x, Q,q,r s.t. ( r γ 2 ˆΣ + ˆµˆµ T) Q ˆµ T q γ 1 ˆΣ 1/2 (q + 2Qˆµ) r a k (h(x, ξ)) + b k + ξ T q + ξ T Qξ ξ S, k ( ) Q 0 For portfolio selection, if S = polyhedron or ellipsoid, then DRO equivalent to semi-definite program. E.g., when S = R m, constraint ( ) can be replaced by [ ] Q (q + a k x)/2 (q + a k x) T 0, k /2 b k r 15 E. Delage Model Ambiguity in the Expected Utility Framework

25 Distributionally Robust Optimization Distributionally Robust Portfolio Optimization Let s consider the case of portfolio optimization: max. x X min E F [u(ξ T x)], F D where x i is how much is invested in stock i with future return ξ i. 16 E. Delage Model Ambiguity in the Expected Utility Framework

26 Distributionally Robust Optimization Distributionally Robust Portfolio Optimization Let s consider the case of portfolio optimization: max. x X min E F [u(ξ T x)], F D where x i is how much is invested in stock i with future return ξ i. Does the robust solution perform better than solution of expected utility problem with fixed ˆF? D = D(γ) vs. D = { ˆF } 16 E. Delage Model Ambiguity in the Expected Utility Framework

27 Distributionally Robust Optimization Experiments in Portfolio Optimization 30 stocks tracked over years using Yahoo! Finance stock price Boeing Motorola Dow Chemical Company Merck & Co., Inc year 17 E. Delage Model Ambiguity in the Expected Utility Framework

28 Distributionally Robust Optimization Wealth Evolution for 300 Experiments Distributionally robust approach Classical approach 1.5 Wealth Wealth Year Year 10% and 90% percentiles are indicated periodically. 18 E. Delage Model Ambiguity in the Expected Utility Framework

29 Distributionally Robust Optimization Wealth Evolution for 300 Experiments Distributionally robust approach Classical approach 1.5 Wealth Wealth Year Year 10% and 90% percentiles are indicated periodically. 79% of time, the DRO outperformed the classical approach 67% improvement on average using DRO with D(γ) 18 E. Delage Model Ambiguity in the Expected Utility Framework

30 Distributionally Robust Optimization Other applications Territory partitioning for multi-vehicle routing problem with J. G. Carlsson Details Fleet mix optimization problem with S. Arroyo and Y. Ye Details Quadratic knapsack problem with J. Cheng and A. Lisser Multi-item newsvendor problem with A. Ardestani-Jaafari Many others in scheduling, environmental policies, smart grid management, etc. 19 E. Delage Model Ambiguity in the Expected Utility Framework

31 Outline Preference Robust Optimization 1 Introduction 2 Distributionally Robust Optimization 3 Preference Robust Optimization 4 Conclusion 20 E. Delage Model Ambiguity in the Expected Utility Framework

32 Preference Robust Optimization How does One Assesses Risk Tolerance? Here are two questions from a survey proposed by [Grable & Lytton, Financial Services Review (1999)]. 21 E. Delage Model Ambiguity in the Expected Utility Framework

33 Preference Robust Optimization How does One Assesses Risk Tolerance? Here are two questions from a survey proposed by [Grable & Lytton, Financial Services Review (1999)]. You have just finished saving for a once-in-a-lifetime vacation. Three weeks before you plan to leave, you lose your job. You would: 1 Cancel the vacation 2 Take a much more modest vacation 3 Go as scheduled, reasoning that you need the time to prepare for a job search 4 Extend your vacation, because this might be your last chance to go first-class 21 E. Delage Model Ambiguity in the Expected Utility Framework

34 Preference Robust Optimization How does One Assesses Risk Tolerance? Here are two questions from a survey proposed by [Grable & Lytton, Financial Services Review (1999)]. You have just finished saving for a once-in-a-lifetime vacation. Three weeks before you plan to leave, you lose your job. You would: 1 Cancel the vacation 2 Take a much more modest vacation 3 Go as scheduled, reasoning that you need the time to prepare for a job search 4 Extend your vacation, because this might be your last chance to go first-class You are on a TV game show and can choose one of the following. Which would you take? 1 $1,000 in cash 2 A 50% chance at winning $ A 25% chance at winning $ 10,000 4 A 5% chance at winning $100, E. Delage Model Ambiguity in the Expected Utility Framework

35 Preference Robust Optimization Common Utility Function Estimation Techniques I Parametric approach: One assumes that the function has a specific parametric form Negative exponential utility (CARA) Power utility (CRRA) HARA (incr/decreasing absolute/relative risk aversion) u(y) = 1 η η ( ) η ay 1 η + b Ask enough questions to identify the parameters 22 E. Delage Model Ambiguity in the Expected Utility Framework

36 Preference Robust Optimization Common Utility Function Estimation Techniques I Parametric approach: One assumes that the function has a specific parametric form Negative exponential utility (CARA) Power utility (CRRA) HARA (incr/decreasing absolute/relative risk aversion) u(y) = 1 η η ( ) η ay 1 η + b Ask enough questions to identify the parameters Why it might fail? Investor must commit to global structure Ignores how confident we are in the final choice of parameters 22 E. Delage Model Ambiguity in the Expected Utility Framework

37 Preference Robust Optimization Common Utility Function Estimation Techniques II Non-parametric approach: Identify the certainty equivalents of a list of lotteries. By answering: What is the smallest amount c i of money you would take instead of playing a lottery L i? Find a piecewise linear utility function that satisfies these certainty equivalents E[u(L i )] = u(c i ) i 23 E. Delage Model Ambiguity in the Expected Utility Framework

38 Preference Robust Optimization Common Utility Function Estimation Techniques II Non-parametric approach: Identify the certainty equivalents of a list of lotteries. By answering: What is the smallest amount c i of money you would take instead of playing a lottery L i? Find a piecewise linear utility function that satisfies these certainty equivalents E[u(L i )] = u(c i ) i Why it might fail? Expresses risk neutrality between breakpoints Ignores how confident we are in the final choice of u( ) 23 E. Delage Model Ambiguity in the Expected Utility Framework

39 Preference Robust Optimization Applying the Robust Optimization Framework Information can be used to characterize a set U of plausible utility functions. I.e., any u( ) such that: Global info: Risk aversion : u( ) is concave Prudence : u ( ) is convex S-shaped : u( ) convex-concave Local info : E[u(W k )] E[u(Y k )] k 24 E. Delage Model Ambiguity in the Expected Utility Framework

40 Preference Robust Optimization Applying the Robust Optimization Framework Information can be used to characterize a set U of plausible utility functions. I.e., any u( ) such that: Global info: Risk aversion : u( ) is concave Prudence : u ( ) is convex S-shaped : u( ) convex-concave Local info : E[u(W k )] E[u(Y k )] k Unfortunately, a direct application of RO is meaningless maximize x X inf u U E [u(h(x, ξ))] = 24 E. Delage Model Ambiguity in the Expected Utility Framework

41 Preference Robust Optimization Applying the Robust Optimization Framework Information can be used to characterize a set U of plausible utility functions. I.e., any u( ) such that: Global info: Risk aversion : u( ) is concave Prudence : u ( ) is convex S-shaped : u( ) convex-concave Local info : E[u(W k )] E[u(Y k )] k Unfortunately, a direct application of RO is meaningless maximize x X inf u U E [u(h(x, ξ))] = Even if we force u(0) = 0 u(1) = 1, model promotes risk neutrality. 24 E. Delage Model Ambiguity in the Expected Utility Framework

42 Preference Robust Optimization Ambiguity about the Certainty Equivalent Given a decision x, we can start by defining an interval [CE (h(x, ξ)), CE + (h(x, ξ))] of plausible minimum certain return that decision maker would prefer to random profit h(x, ξ) [ ] CE - (h(x, )) CE + (h(x, )) Cash amount 25 E. Delage Model Ambiguity in the Expected Utility Framework

43 Preference Robust Optimization Robust Certainty Equivalent Approach Theorem Identifying the decision that maximizes the lowest perceived CE max. x X CE (h(x, ξ)), a.k.a. max. x X inf CE u(h(x, ξ)), u U can be done efficiently. 26 E. Delage Model Ambiguity in the Expected Utility Framework

44 Preference Robust Optimization Robust Certainty Equivalent Approach Theorem Identifying the decision that maximizes the lowest perceived CE max. x X CE (h(x, ξ)), a.k.a. max. x X inf CE u(h(x, ξ)), u U can be done efficiently. Proof: Objective is quasiconcave and reduces to max. x,t t s.t. CE u (h(x, ξ)) t u U or equiv. max t t s.t. max x X inf u U E[u(h(x, ξ))] u(t) 0 Infimum over u U can be reduced to finite dimensional program so that duality can be applied details 26 E. Delage Model Ambiguity in the Expected Utility Framework

45 Preference Robust Optimization A Tool for Interacting with Investors 27 E. Delage Model Ambiguity in the Expected Utility Framework

46 Preference Robust Optimization Numerical experiments Obtained from Yahoo! Finance historical stock returns for 350 companies from 1993 to 2011 Ran extensive amount of trials using last 50 weekly returns to decide investment among 10 assets for next week In each experiment, the investor has an unknown risk averse utility function and compares up to 80 pairs of gambles 28 E. Delage Model Ambiguity in the Expected Utility Framework

47 Preference Robust Optimization Experimental Results Function Portfolio s true CE value (in perc. point) optimized 5 questions 20 questions 80 questions Exponential Fitted PWL Worst-case Worst-case prudent True «u (y)y» u (y) = 20 y 29 E. Delage Model Ambiguity in the Expected Utility Framework

48 Preference Robust Optimization Experimental Results Function Portfolio s true CE value (in perc. point) optimized 5 questions 20 questions 80 questions Exponential Fitted PWL Worst-case Worst-case prudent True «u (y)y» Observations: u (y) = 20 y Using wrong utility function can mislead the choice of portfolio Robust approach makes good use of available preference info Quality of portfolio is improved as more information is provided 29 E. Delage Model Ambiguity in the Expected Utility Framework

49 Preference Robust Optimization Accounting for elicitation errors Since to err is human, we should account for mislabelling of the compared lotteries Hence, that for some rand perception noise ɛ, we have that E[u(W k )] + ɛ k E[u(Y k )] 30 E. Delage Model Ambiguity in the Expected Utility Framework

50 Preference Robust Optimization Accounting for elicitation errors Since to err is human, we should account for mislabelling of the compared lotteries Hence, that for some rand perception noise ɛ, we have that E[u(W k )] + ɛ k E[u(Y k )] In that case, one could consider u( ) plausible as long as δ 0 such that k δ k Γ and that E[u(W k )] + δ k E[u(Y k )] k This can easily be incorporated to the model 30 E. Delage Model Ambiguity in the Expected Utility Framework

51 Outline Conclusion 1 Introduction 2 Distributionally Robust Optimization 3 Preference Robust Optimization 4 Conclusion 31 E. Delage Model Ambiguity in the Expected Utility Framework

52 Conclusion Conclusion & Future Work There is some wisdom in accounting for ambiguity about the expected utility model Disregarding it can be misleading 32 E. Delage Model Ambiguity in the Expected Utility Framework

53 Conclusion Conclusion & Future Work There is some wisdom in accounting for ambiguity about the expected utility model Disregarding it can be misleading Accounting for distribution ambiguity or ambiguity in risk preferences isn t computationally demanding It remains to further study whether both ambiguities can be accounted for jointly (see Haskell et al. 2014) 32 E. Delage Model Ambiguity in the Expected Utility Framework

54 Conclusion Conclusion & Future Work There is some wisdom in accounting for ambiguity about the expected utility model Disregarding it can be misleading Accounting for distribution ambiguity or ambiguity in risk preferences isn t computationally demanding It remains to further study whether both ambiguities can be accounted for jointly (see Haskell et al. 2014) Studying the sensitivity of optimal solution with respect to modelled ambiguity can be helpful Value of stochastic modelling Guidance for risk tolerance assessment 32 E. Delage Model Ambiguity in the Expected Utility Framework

55 Bibliography I Conclusion Ardestani-Jaafari, A., E. Delage Robust optimization of sums of piecewise linear functions with application to inventory problems. Working draft. Armbruster, B., E. Delage Decision making under uncertainty when preference information is incomplete. Management Science 61(1) Ben-Tal, A., A. Nemirovski Robust convex optimization. Mathematics of Operations Research 23(4) Bertsimas, D., X. V. Doan, K. Natarajan, C. P. Teo Models for minimax stochastic linear optimization problems with risk aversion. Mathematics of Operations Research 35(3) Carlsson, J. G., E. Delage Robust partitioning for stochastic multi-vehicle routing. Operations Research 61(3) Cheng, J., E. Delage, A. Lisser Distributionally robust stochastic knapsack problem. Journal on Optimization 24(3) Delage, E., S. Arroyo, Y. Ye The value of stochastic modeling in two-stage stochastic programs with cost uncertainty. Operations Research 62(6) Delage, E., Y. Ye Distributionally robust optimization under moment uncertainty with application to data-driven problems. Operations Research 58(3) E. Delage Model Ambiguity in the Expected Utility Framework

56 Bibliography II Conclusion Ellsberg, E Risk, ambiguity, and the savage axioms. Quarterly Journal of Economics Gilboa, I., D. Schmeidler Maxmin Expected Utility with Non-Unique Prior. Journal of Math. Economics 18(2) Goh, J., M. Sim Distributionally robust optimization and its tractable approximations. Operations Research Grable, J., R. H. Lytton Financial risk tolerance revisited: the development of a risk assessment instrument. Financial Services Review Haskell, W. B., L. Fu, M. Dessouky Ambiguity in risk preferences in robust stochastic optimization. Working draft. Popescu, I Robust mean-covariance solutions for stochastic optimization. Operations Research 55(1) von Neumann, J., O. Morgenstern Theory of Games and Economic Behavior. Princeton University Press. Wiesemann, W., D. Kuhn, M. Sim Distributionally robust convex optimization. Operations Research E. Delage Model Ambiguity in the Expected Utility Framework

57 Conclusion Questions & Comments Thank you! 35 E. Delage Model Ambiguity in the Expected Utility Framework

58 Dealing with model ambiguity: Ellsberg Paradox Consider an urn with 30 blue balls and 60 other balls that are either red or green (you don t know how many are red or green). 36 E. Delage Model Ambiguity in the Expected Utility Framework

59 Dealing with model ambiguity: Ellsberg Paradox Consider an urn with 30 blue balls and 60 other balls that are either red or green (you don t know how many are red or green). Experiment 1: Choose among the following two gambles Gamble A: If you draw a blue ball, then you win 100$ Gamble B: If you draw a red ball, then you win 100$ 36 E. Delage Model Ambiguity in the Expected Utility Framework

60 Dealing with model ambiguity: Ellsberg Paradox Consider an urn with 30 blue balls and 60 other balls that are either red or green (you don t know how many are red or green). Experiment 1: Choose among the following two gambles Gamble A: If you draw a blue ball, then you win 100$ Gamble B: If you draw a red ball, then you win 100$ Experiment 2: Choose among the following two gambles Gamble C: If you draw blue or green ball, then you win 100$ Gamble D: If you draw red or green ball, then you win 100$ 36 E. Delage Model Ambiguity in the Expected Utility Framework

61 Dealing with model ambiguity: Ellsberg Paradox Consider an urn with 30 blue balls and 60 other balls that are either red or green (you don t know how many are red or green). Experiment 1: Choose among the following two gambles Gamble A: If you draw a blue ball, then you win 100$ Gamble B: If you draw a red ball, then you win 100$ Experiment 2: Choose among the following two gambles Gamble C: If you draw blue or green ball, then you win 100$ Gamble D: If you draw red or green ball, then you win 100$ If you clearly prefer Gamble A & D, then you are averse to model ambiguity 36 E. Delage Model Ambiguity in the Expected Utility Framework

62 Outline Distributionally Robust Partitioning 5 Distributionally Robust Partitioning 6 Value of Stochastic Modelling in Fleet Composition 7 Robust Certainty Equivalent 37 E. Delage Model Ambiguity in the Expected Utility Framework

63 Distributionally Robust Partitioning Multi-Vehicle Routing on a Planar Region Divide a planar region into K subregions, each serviced by a different vehicle, so that the total workload be most evenly distributed among the fleet 38 E. Delage Model Ambiguity in the Expected Utility Framework

64 Distributionally Robust Partitioning Multi-Vehicle Routing on a Planar Region Divide a planar region into K subregions, each serviced by a different vehicle, so that the total workload be most evenly distributed among the fleet 39 E. Delage Model Ambiguity in the Expected Utility Framework

65 Distributionally Robust Partitioning Multi-Vehicle Routing on a Planar Region Divide a planar region into K subregions, each serviced by a different vehicle, so that the total workload be most evenly distributed among the fleet 40 E. Delage Model Ambiguity in the Expected Utility Framework

66 Distributionally Robust Partitioning Multi-Vehicle Routing on a Planar Region Divide a planar region into K subregions, each serviced by a different vehicle, so that the total workload be most evenly distributed among the fleet 41 E. Delage Model Ambiguity in the Expected Utility Framework

67 Distributionally Robust Partitioning Distributionally Robust Partitioning Given D, we partition so that the largest workload over the worst distribution of demand points is as small as possible { } min. max E[TSP({ξ 1, ξ 2,..., ξ N } R i )] {R 1,R 2,...,R K } i sup F D, 42 E. Delage Model Ambiguity in the Expected Utility Framework

68 Distributionally Robust Partitioning Distributionally Robust Partitioning Given D, we partition so that the largest workload over the worst distribution of demand points is as small as possible { } min. max E[TSP({ξ 1, ξ 2,..., ξ N } R i )] {R 1,R 2,...,R K } i sup F D, A side product is to characterize for any partition what is a worst-case distribution of demand locations 42 E. Delage Model Ambiguity in the Expected Utility Framework

69 Distributionally Robust Partitioning Distributionally Robust Partitioning We simulated three partition schemes on a set of randomly generated parcel delivery problems where the territory needed to be divided into two regions and the demand is drawn from a mixture of truncated Gaussian distribution Cumulative probobality CMF of largest workload 0.4 Sample based 0.2 Gaussian based Robust data driven Largest workload (in %) 43 E. Delage Model Ambiguity in the Expected Utility Framework

70 Distributionally Robust Partitioning Border Patrol Workload Partitioning Robust partitions of the USA-Mexico border obtained using our branch & bound algorithm. Back to DRO applications 44 E. Delage Model Ambiguity in the Expected Utility Framework

71 Value of Stochastic Modelling in Fleet Composition Outline 5 Distributionally Robust Partitioning 6 Value of Stochastic Modelling in Fleet Composition 7 Robust Certainty Equivalent 45 E. Delage Model Ambiguity in the Expected Utility Framework

72 Value of Stochastic Modelling in Fleet Composition The Robustness of the Deterministic Solution If we are risk neutral we might not even need distribution information Theorem The solution of is optimal with respect to maximize x X E[h(x, µ)] maximize x X inf F D(µ,Ψ) E F [h(x, ξ)], for any set of convex functions Ψ with { D(µ, Ψ) = F E[ξ] = µ E[ψ(ξ)] 0, ψ Ψ }. 46 E. Delage Model Ambiguity in the Expected Utility Framework

73 Value of Stochastic Modelling in Fleet Composition The Value of Stochastic Modelling Consider the situation: 1 We know of a set D such that F D 2 We have a candidate solution x 1 in mind 3 Is it worth developing a stochastic model: D F? (a) If yes, then develop a model & solve it (b) Otherwise, implement x 1 47 E. Delage Model Ambiguity in the Expected Utility Framework

74 Value of Stochastic Modelling in Fleet Composition The Value of Stochastic Modelling Consider the situation: 1 We know of a set D such that F D 2 We have a candidate solution x 1 in mind 3 Is it worth developing a stochastic model: D F? (a) If yes, then develop a model & solve it (b) Otherwise, implement x 1 The Value of Stochastic Modelling (VSM) gives an optimistic estimate of the value of obtaining perfect information about F. { } VSM(x 1 ) := sup F D max E F [h(x 2, ξ)] E F [h(x 1, ξ)] x 2 47 E. Delage Model Ambiguity in the Expected Utility Framework

75 Value of Stochastic Modelling in Fleet Composition The Value of Stochastic Modelling Consider the situation: 1 We know of a set D such that F D 2 We have a candidate solution x 1 in mind 3 Is it worth developing a stochastic model: D F? (a) If yes, then develop a model & solve it (b) Otherwise, implement x 1 The Value of Stochastic Modelling (VSM) gives an optimistic estimate of the value of obtaining perfect information about F. { } VSM(x 1 ) := sup F D max E F [h(x 2, ξ)] E F [h(x 1, ξ)] x 2 Theorem Unfortunately, evaluating VSM(x 1 ) exactly is NP-hard in general. 47 E. Delage Model Ambiguity in the Expected Utility Framework

76 Value of Stochastic Modelling in Fleet Composition Bounding the Value of Stochastic Modelling Theorem If S {ξ ξ 1 ρ}, an upper bound can be evaluated in O(d d T DCP ) using: UB(x 1, ȳ 1 ) := min s,q s + µ T q s.t. s α(ρe i ) ρe T i q, i {1,..., d} where α(ξ) = max x 2 h(x 2, ξ) h(x 1, ξ; ȳ 1 ). s α( ρe i ) + ρe T i q, i {1,..., d}, 48 E. Delage Model Ambiguity in the Expected Utility Framework

77 Value of Stochastic Modelling in Fleet Composition Are Airlines Adventurous in their Fleet Acquisition? Fleet composition is a difficult decision problem: Fleet contracts are signed 10 to 20 years ahead of schedule. Many factors are still unknown at that time: e.g., passenger demand, fuel prices, etc. 49 E. Delage Model Ambiguity in the Expected Utility Framework

78 Value of Stochastic Modelling in Fleet Composition Are Airlines Adventurous in their Fleet Acquisition? Fleet composition is a difficult decision problem: Fleet contracts are signed 10 to 20 years ahead of schedule. Many factors are still unknown at that time: e.g., passenger demand, fuel prices, etc. Yet, most airline companies sign these contracts based on a single scenario of what the future may be. 49 E. Delage Model Ambiguity in the Expected Utility Framework

79 Value of Stochastic Modelling in Fleet Composition Are Airlines Adventurous in their Fleet Acquisition? Fleet composition is a difficult decision problem: Fleet contracts are signed 10 to 20 years ahead of schedule. Many factors are still unknown at that time: e.g., passenger demand, fuel prices, etc. Yet, most airline companies sign these contracts based on a single scenario of what the future may be. Are airlines companies being neglectful? 49 E. Delage Model Ambiguity in the Expected Utility Framework

80 Value of Stochastic Modelling in Fleet Composition Mathematical formulation for Fleet Mix Problem The fleet composition problem is a stochastic mixed integer LP max. x E [ }{{} o T x + h(x, p, c, L) ], }{{} ownership cost future profits 50 E. Delage Model Ambiguity in the Expected Utility Framework

81 Value of Stochastic Modelling in Fleet Composition Mathematical formulation for Fleet Mix Problem The fleet composition problem is a stochastic mixed integer LP max. x with h(x, p, c, L) := max z 0,y 0,w ( i k E [ }{{} o T x + h(x, p, c, L) ], }{{} ownership cost future profits flight profit {}}{ p k i w k i rental cost lease revenue {}}{{}}{ c k (z k x k ) + + L k (x k z k ) + ) s.t. w k i {0, 1}, k, i & k w k i = 1, i } Cover y k g in(v) + z k = wi k i arr(v) v {v time(v)=0} = y k g out(v) + (y k g in(v) + i arr(v) w k i dep(v) w k i, k, v } Balance i ), k } Count 50 E. Delage Model Ambiguity in the Expected Utility Framework

82 Value of Stochastic Modelling in Fleet Composition Experiments in Fleet Mix Optimization We experimented with three test cases : 1 3 types of aircraft, 84 flights, σ pi /µ pi [4%, 53%] 2 4 types of aircraft, 240 flights, σ pi /µ pi [2%, 20%] 3 13 types of aircraft, 535 flights, σ pi /µ pi [2%, 58%] 51 E. Delage Model Ambiguity in the Expected Utility Framework

83 Value of Stochastic Modelling in Fleet Composition Experiments in Fleet Mix Optimization We experimented with three test cases : 1 3 types of aircraft, 84 flights, σ pi /µ pi [4%, 53%] 2 4 types of aircraft, 240 flights, σ pi /µ pi [2%, 20%] 3 13 types of aircraft, 535 flights, σ pi /µ pi [2%, 58%] Results: Test CPU Time DRO sub-optimality cases DRO SP with ˆF Under ˆF F D #1 0.6 s 3 min 0.001% < 6% #2 1 s 14 min 0.001% < 1% #3 5 s 21 h 0.003% < 7% 51 E. Delage Model Ambiguity in the Expected Utility Framework

84 Value of Stochastic Modelling in Fleet Composition Experiments in Fleet Mix Optimization We experimented with three test cases : 1 3 types of aircraft, 84 flights, σ pi /µ pi [4%, 53%] 2 4 types of aircraft, 240 flights, σ pi /µ pi [2%, 20%] 3 13 types of aircraft, 535 flights, σ pi /µ pi [2%, 58%] Results: Test CPU Time DRO sub-optimality cases DRO SP with ˆF Under ˆF F D #1 0.6 s 3 min 0.001% < 6% #2 1 s 14 min 0.001% < 1% #3 5 s 21 h 0.003% < 7% Conclusions: It s wasteful to invest more than 7% of profits in extra info Back to DRO applications 51 E. Delage Model Ambiguity in the Expected Utility Framework

85 Outline Robust Certainty Equivalent 5 Distributionally Robust Partitioning 6 Value of Stochastic Modelling in Fleet Composition 7 Robust Certainty Equivalent 52 E. Delage Model Ambiguity in the Expected Utility Framework

86 Robust Certainty Equivalent Constructing the Worst-case Utility I Define S = {y 1, y 2,..., y N } contains support of W k and Y k, and t. Define the values α i := u(y i ) y 1 y 2 y 3 y 4 53 E. Delage Model Ambiguity in the Expected Utility Framework

87 Robust Certainty Equivalent Constructing the Worst-case Utility I Define S = {y 1, y 2,..., y N } contains support of W k and Y k, and t. Define the values α i := u(y i ) (y 3,α 3 ) (y 4,α 4 ) (y 2,α 2 ) (y 1,α 1 ) y 1 y 2 y 3 y 4 54 E. Delage Model Ambiguity in the Expected Utility Framework

88 Robust Certainty Equivalent Constructing the Worst-case Utility I Define S = {y 1, y 2,..., y N } contains support of W k and Y k, and t. Define the values α i := u(y i ) (y 3,α 3 ) (y 4,α 4 ) (y 2,α 2 ) (y 1,α 1 ) β 1 y 1 y 2 y 3 y 4 55 E. Delage Model Ambiguity in the Expected Utility Framework

89 Robust Certainty Equivalent Constructing the Worst-case Utility II Once all (y i, u(y i )) are fixed, identify the worst-case utility value for u(h(x, ξ)). (, )) (y 3,α 3 ) (y 4,α 4 ) (y 2,α 2 ) (y 1,α 1 ) β 1 y 1 y 2 y 3 y 4 56 E. Delage Model Ambiguity in the Expected Utility Framework

90 Robust Certainty Equivalent Constructing the Worst-case Utility II Once all (y i, u(y i )) are fixed, identify the worst-case utility value for u(h(x, ξ)). (, )) u (y 1,α 1 ) β 1 y 1 y 2 y 3 y 4 57 E. Delage Model Ambiguity in the Expected Utility Framework

91 Robust Certainty Equivalent Constructing the Worst-case Utility II Once all (y i, u(y i )) are fixed, identify the worst-case utility value for u(h(x, ξ)). (, )) v w u (y 1,α 1 ) β 1 y 1 y 2 y 3 y 4 58 E. Delage Model Ambiguity in the Expected Utility Framework

92 Robust Certainty Equivalent LP reformulation of inf u U E[u(h(x, ξ))] u(t) We wish to find an x s.t. the following finite dimensional LP has a positive optimal value: p i (v i h(x, ξ i ) + w i ) α t min. α,β,v,w i s.t. v i y i + w i α j i, j (Risk aversion at h(x, ξ i )) P(W k = y j )α j P(Y k = y j )α j k (Local pref s) j j α j+1 α j + β j (y j+1 y j ) j (Risk aversion at y j s) α j 1 α j + β j (y j 1 y j ) j v 0, β 0 (Monotonicity) After taking the dual of this LP, we can join the maximization with x X Back to talk 59 E. Delage Model Ambiguity in the Expected Utility Framework