* M. Larbani and **P. L. Yu * IIUM University, Kuala Lumpur, Malaysia

1. Decision Making and Optimization in Changeable Spaces, A Paradigm Shift *M. Larbani and **P. L. Yu *IIUM University, Kuala Lumpur, Malaysia **National Chiao Tung University Taiwan and Kansas University , USA

2. Presentation Outline • 1. Traditional Models of Decision Making and their Limitations • 2. Habitual Domain Theory • 3. Decision Making and Optimization in Changeable Spaces • 4. Applications • 5. Conclusion

3. 1.Traditional DM Models • There are two approaches to decision making: qualitative and quantitative approaches. • The quantitative approach to decision making went through two major stages. • Stage 1. Deterministic Approach. A fixed set X of decisions or alternatives is exactly known and unchangeable, and a utility (or value) function (Von Neumann) u(.): X R, that represents the preferences of the DMs is exactly determined.

4. To each decision x in X, a value u(x) is associated; a decision x is preferred to a decision y if u(y) < u(x). • Then the problem of finding the best or optimal decision is formulated as an optimization problem, Maxx  Xu(x) (1)

5. Numerous resolution methods were developed and implemented as linear programming (Dantzig) and nonlinear programming (Kuhn-Tucker). • (1) was extended to multiple criteria decision making (Yu, Steuer) Maxx  X u1(x), u2(x),…, um(x) , (2) and games: < {1,2,..n}, ui(x1 , x2 ,…, xn), Xi , i =1,2,…,n > (3) (Von Neumann, Nash, Aumann, Shaplay)

6. Stage 2. Incorporation of Structured Uncertainty (1)-(3) were extended to models incorporating partially or completely probabilistic or fuzzy or fuzzy-probabilistic inputs, (1) became Maxŷ  Ŷû(ŷ) whereû(ŷ) and Ŷ may be stochastic or fuzzy or fuzzy-stochastic (Zadeh, Bellman)

7. 1.2 Limitations of Traditional Models • Structured uncertainty. In probabilistic and/or fuzzy models it is assumed that the uncertain parameters vary within a certain range and have well known shapes of probability distributions or fuzzy membership functions. Whereas, often real decision making involve parameters the dimensions, ranges and shapes of which may not be easily predetermined or assumed as in probabilistic and/or fuzzy models (Unstructured Uncertainty).

8. Changeable Parameters. Often, in real-life decision making problems the parameters like the set of alternatives, the criteria and the DMs, change as situations and psychological states of the DMs change.

9. Example 1. Horse Race • A retiring corporate chairman invited to his ranch two finalists (A and B) from whom he would select his replacement using a horse race. A and B, equally skillful in horseback riding, were given a black and white horse, respectively. The chairman laid out the course for the horse race and said, “Starting at the same time now, whoever’s horse is slowerin completing the course will be selected as the next chairman!”

10. Here, we have a discovering problem; the two finalists were puzzled at the beginning, it was not immediately apparent what strategy was needed to win the race, they had to discover winning strategy themselves. • This decision making problem cannot be formulated within the traditional framework, let alone solve it, because • The set of strategies X of each candidate is a changeable space for the other player and for the candidate himself.

11. Example 2. Game of silence • Silence fell over a young couple after a family quarrel. They did not talk to each other for two days. The situation became uneasy for husband and wife. No one of them wanted to break the silence first because of the fear of losing face.

12. Game of Silence • This problem cannot be formulated in the framework of traditional mathematical programming because the utility function of the couple, u(.), and its set of alternatives, are not known. The couple has to generate solutions.

13. Here, the couple faces a discovering-covering (dis/covering for short) problem. They want to return to normal life (covering a target), but they don’t know how to achieve this target (discovering solutions). • Thus, despite the tremendous theoretical and practical results achieved, some important aspects of decision making have not been incorporated in the existing models as unstructured uncertainty, idea generation and human psychology.

14. The new model we propose, Decision Making and Optimization in Changeable Spaces (DMCS) is a paradigm shift from utility function –structured uncertainty based model to a model that incorporates human psychology and creativity, and can handle unstructured uncertainty. In this presentation, we will focus on Covering and Discovering Problems, which encompass most of decision making problem.

15. 2. Habitual Domain (P.L.Yu) • The collection of ideas and actions (including ways of perceiving, thinking, responding, acting, and memory) in our brain together with their formation, dynamics, and basis in experience and knowledge, is called our Habitual Domain (HD). • Over time, unless extraordinary event occurs or purposeful effort is exerted, our HD will become stabilized within a certain domain.

16. Elements of a Habitual Domain • Potential Domain (PD) - the collection of ideas and actions that can potentially be activated. • Actual Domain (AD) - the set of ideas and actions that are actually activated. • Activation Probabilities(AP) - the probabilities that ideas and actions in PD also belong to AD. • Reachable Domain (RD)- the set of ideas and actions that can be attained from a given set in AD.

17. HD(t) = { PD(t), AD(t), AP(t), RD(t)} (4) AD e AP(e) e PD RD

18. Eight hypotheses H1-H8 of HD • H1. Circuit Pattern Hypothesis. • Thoughts or concepts or ideas are represented by circuit patterns of the brain. • H2. Unlimited Capacity Hypothesis. • Practically every normal brain has capacity to encode and store all thoughts, and concepts that one intends to. • H3. Efficient Restructuring Hypothesis. • According to the dictation of attention our thoughts and memories are continuously restructured so that relevant ones can be efficiently retrieved to release charge.

19. H4. Analogy and Association Hypothesis. • The perception of new events, or ideas can be learned primarily by analogy and association with what is already known. • H5. Goal Setting and State Evaluation. • Each one of us has a set of goal functions and for each goal function we have an ideal state or equilibrium to reach and maintain (goal setting). We continuously monitor, where we are relative to the ideal state or equilibrium point (state evaluation). Goal setting and state evaluation are dynamic, interactive and are subject to change…. Goal’s Ideal state

20. H6. Charge Structure and Attention Allocation Hypothesis. • When there is an unfavorable deviation of perceived value from the ideal, each goal function will produce various levels of charge. The totality of the charges created by all goal functions is called the charge structure. At any point in time, our attention will be paid to the event which has the most influence on our charge structure. Ideal Perceived

21. Fifth level ( 5). The charge is extremely intense and the allowable time for discharge is very short. • Forth level (4). The charge is intense with a short allowable time for discharge. • Third level (3). The charge is less intense, but still strong, with a little longer allowable time for discharge. • Second level (2). The charge is mild and the allowable time for discharge is fairly long. This level, again, can be produced by almost all goal functions. • First level (1). The charge is slight with no specific discharge time. Again this level can be produced by almost all goal functions.

22. H7. Discharge Hypothesis：the least resistance principle. • To release charges, we tend to select the action which yields the lowest remaining charge (the remaining charge is the resistance to the total discharge). • H8. Information Input Hypothesis. • Humans have innate needs to gather external information. Unless attention is paid, external information may not be processed.

23. (1) (11) Internal Information Processing Center (14) Monitoring Physiological (3) (2) Goal State H5 : Goal Set ＆ States Evaluation Setting Evaluation H1: Circuit Patterns H2: Unlimited H3 : Restructure H4 : Association Problem Solving Avoidance Justification (4) (13) Compilation (5) H6 : Charge Structures ＆ Attention Allocation Unsolicited Information CHARGE STRUCTURE (10) (6) H8 : Information Input (7) Attention Allocation suggestion Self- (8) Solution Information Solicited (12) Obtained? (9) Extern Actions/ Discharges Inf. Experience/ Being H7:Dischange:Least Resistance

24. 3. Model Construction • Competence Set. when a DM faces a Problem E at some time t, he has a perceivedcompetence set CS t (E) to solve it. On the other hand there is the actual or true competence setCS(E) required to solve the problem.

25. Decision Blind Decision Blinds CSt (E) CS(E) E can be solved some at time t if CS (E)⊂CSt (E), that is, when CSt (E) covers CS (E). CS (E)\CSt (E) is the decision blindsNote, the larger the decision blind is, the morelikely it is that the DMs might make important mistakes.

26. Decision Trap • A DM is in Decision Trap if his competence setCSt (E) is trapped in some area and cannot expand to fully cover the targeted competence setCS(E). CS(E) Decision Trap CSt (E)

27. Discovering • Discoveringis getting out of a decision trap. Decision Trap CSt(E)

28. The resolution of challenging problems generally involve covering and discovering or dis/covering. Discovering requires a target to cover, while the covering process requires discovering when it falls in a decision trap.

29. The 7-8-9 principles for Dis/covering The Seven empowering operators. • M1. Everyone is a priceless living entity. We are all unique creations who carry the spark of the divine. • M2. Clear, specific and challenging goals produce energy for our lives. I am totally committed to doing and learning with confidence. This is the only way I can reach the goals.

30. M3. There are reasons for everything that occurs. One major reason is to help us grow and develop. • M4. Every task is part of my life mission. I have the enthusiasm and confidence to accomplish this mission. • M5. I am the master of my living domain. I take responsibility for everything that happens in it.

31. M6. Be appreciative and grateful and don’t forget to give back to society. • M7. Our remaining lifetime is our most valuable asset. I will enjoy it fully and make a 100 percent contribution to society in each moment of my remaining life.

32. Eight methods for expanding and enriching HDs. • M8. Learning Actively • M9. Projecting from a Higher Position • M10. Active Association • M11. Changing the Relevant Parameters • M12. Changing the Environment • M13. Brainstorming • M14. Retreating in Order to Advance • M15. Praying or Meditating

33. Nine principles of deep knowledge. • M16. The Deep and Down Principle • M17. The Alternating Principle • M18. The Contrasting and Complementing Principle • M19. The Revolving and Cycling Principle • M20. The Inner Connection Principle • M21. The Changing and Transforming Principle • M22. The Contradiction Principle • M23. The Cracking and Ripping Principle • M24. The Void Principle

34. Let M= {M1, M2,…, M24 } be the set of the 7-8-9 principles and CM= {H/ H=Ms(1)o Ms(2),…oMs(e) , Ms(i) M} be the set of finite sequences resulting from combination of elements of M. H: Ω Ω1 Ω= The set of all knowledge humanity has reached so far. Ω1 = set of all knowledge humanity can reach in the future.

35. Formulation of Dis/covering as OCS • Given a target competence set CS(E) and an initial competence set CS0 (E) of the DMs, the dis/covering problem is formulated as the Optimization in Changeable Space (OCS) problem find H in CM such that (5) CS (E)⊂H(CS0(E))

36. When time is emphasized, Mint(H) (6) CS (E)⊂H(CS0(E)) , H CM Where t(H)= period for finding H and implementing it When Cost is emphasized Minc(H) (7) CS (E)⊂H(CS0(E)) , H CM Where c(H)= cost for implementing H

37. When charge (state of mind) is emphasized Minch(H) (8) CS (E)⊂H(CS0(E)) , H CM Where ch(H)= the resulting charge after implementing H

38. Note. The operator Min in the models (6)-(8) should be understood in the sense of satisfaction not in the sense of absolute minimum.

39. Application • Example 1. Horse Race. Formulation as OCS from A’s perspective: • Stage 1. The candidates were in a decision trap. Initial competence set: CS0(A)= traditional knowledge about the rules of horse races and the individual skills of riding a horse.

40. Target competence set: CS= set of skills and knowledge needed to make sure one’s horse crosses the finishing line last. Candidate A faces a Dis/covering problem • The OCS problem of candidate A is Mint(H) (9) CS ⊂H(CS0(A)) , H CM

41. Resolution: After a puzzling period, A jumped on B’s horse and rode as fast as he could to the finishing line, while leaving his own horse behind. By the time B realized what was going on, it was already too late! Naturally, A became the new chairman. Let us formulate this solution in terms of operators from CM.

42. Step 1. A analyzed the situation using the “deep and down principle”, M16 and the “projection from higher position” M9 to generate ideas, he narrowed down his focus to the most important elements of the race: (A, HR1) and (B,HR2), A and his horse and B and his horse: M9 o M16 (CS0(A))= CS1(A) (10) where CS1(A) is the resulting competence set.

43. Mathematically, assume that CS0(A)={a1, a2,…, an } , and let , P0(aj), be the initial activation probabilities of aj , j =1,2,…, n, from his potential domain PD to his actual domain AD. Generally speaking, the activation probability can be seen as a time dependent function with domain as the potential domain and range [0,1], that is, Pt(.): PD [0,1].

44. Then the operation (10) reduces to the transformation of the initial activation probabilities P0(aj), j =1,2,…, n to a new set of probabilities Pt(aj), j =1,2,…, n such that Pt((A, HR1)) = Pt((B, HR2)) =1, where t is the duration of operation (10). • This transformation can be expressed as a function S: {P0(aj) , j =1,2,…, n }[0,1], such that S(P0(aj))= Pt(aj) , j =1,2,…, n .

45. Step 2. The problem now is how to change the situation so that it becomes: the one who crosses the finishing line will be the winner? The solution is to use the “alternating principle” , M17by “jumping on B’s horse and riding it to the finishing line as soon as possible”. This operation can be formulated as follows.

46. The alternating principle changed the pairs (A, HR1) and (B,HR2) into the pairs (A, HR2) and (B,HR1), • Mathematically, denote A, HR1, B, HR2 by 1,2,3,4 respectively and d=(1,2,3,4), the alternating of horses corresponds to the following operation

47. d’= F d Where d’=(1,4,3,2) and

48. Thus the solution of the OCS (9) can be expressed as: H*= (M16 o M9) o M17 and Min t(H)= t* = duration of operation H* H* corresponds to the mathematical operation FoS.

49. DMCS has a great potential of application in Artificial Intelligence, Management, conflict resolution and Scientific Discovery. • 5. Conclusion. We hope that this presentation will trigger more research on this new area. • For more details please read the paper: Larbani M., Yu P.L., Decision Making and Optimization in Changeable Spaces, a New Paradigm, Journal of Optimization Theory and Application (JOTA), Vol.155, No.3.

50. References. • Yu, P.L., 1990, Forming Winning Strategies, An Integrated Theory of Habitual Domains, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo. • Yu, P.L.,2002, Habitual Domains and Forming Winning Strategies, NCTU Press, Hsinchu, Taiwan. • Yu P.L. and Larbani M.,2009, Two-Person Second-Order Games, Part 1: Formulation and Transition Anatomy, Journal of Optimization Theory and Application (JOTA). • Larbani M. and Yu P.L., 2009, Two-Person Second-Order Games, Part 2: Restructuring Operations to Reach a Win-Win Profile. • Larbani M. and Yu P.L.,2011, Second Order Games: A Paradigm Shift in Game Theory, JOTA, Vol. 149, 447-473.