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1 Catastrophe Theory Real Time Strategy & Decision Support Tuesday 4 December 2001, DSTL J Q Smith University of Warwick Coventry, CV4 7AL

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2 Contents of Talk The Rise & Fall of Catastrophe Theory The Development of Bayesian Decision Theory Theoretical links between Catastrophe Theory & Decision Theory M.U.I.A. Non-Linear Forecasting An automated Decision Support System Links with Game Theory Roles for Catastrophe Theory in Real Time Strategy Formation

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3 Catastrophe Theory Classification Theorem (1972)Maths Elegant local, generic classification of smooth families of potential functions in high dimensions parametered by finite low dimensional parameters Explanation of Morphogenesis ( ) Applications Descriptions of dynamic processes Examples from natural phenomena (breaking waves) biology (heart function) finance (stock market) psychology (behaviour of drivers) sociology (censorship) decision-making (prison riots)

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4 The Cusp Catastrophe

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6 Some Essential Features of Catastrophe Models Dynamic Smooth underlying potential Real time Discontinuous behaviour arising from conflicting objectives/information

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7 But What Rôle Catastrophe Theory? Certainly appears to describe common phenomena Appears purely descriptive Questions asked in the late 70s early 80s? Where is the potential? Why is it smooth? Why should we expect a local classification to be global? Often simple games Li & Yorke (1975), Rand (1977) do not exhibit catastrophes, but chaos Without a background theory it was difficult to justify

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8 Rational Behaviour & Bayesian Decision Theory Savage (1950), Lindley (1998), De Groot (1970), De Finetti (1972) To be rational choose an act maximising expected utility is a rational potential determining best actions. Q1. Do standard dynamic decision problems exhibit catastrophes? Q2. Can we assert that the local classification is valid globally for wide classes of these problems?

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9 1) Utilities: usually 1 dimensional and nearly always concave (so objects are e.g. expectations) No conflict of objectives single local global maximum 2) Usable forecasting/control models [e.g. K.Fs] were second order or Gaussian No conflict of information encoded Either of these model descriptions Catastrophe Theory does not apply Problems Then

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10 Smith, (1978) Theoretical Thesis & Subsequent Papers with Harrison & Zeeman If was bounded and if (i) was a mixture or was a mixture (Smith, 1979) (ii) the prior or likelihood had thick tails (Smith, 1979) (iii) had a mean/variance link (Harrison & Smith, 1979) then the classification of Catastrophe Theory was valid and global - often exhibiting cusp or butterfly catastrophes Problems 1) These types of models were hardly every used 2) When appropriate, few suitable calculation techniques were available to make these dynamic and real time

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11 New Developments 1) [M.U.I.A.] Keeney & Reiffa (1976), von Winterfeldt & Edwards (1986), French (1989), Clemens (1990), Keeney (1992) Developed a practical and operational Bayesian Decision Analysis Utilities need several attributes [conflicting objectives] Attributes often utility independent so where are bounded in [0,1] = attributes Note In most of these problems was not dynamic

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12 New Developments 2) Dynamic MCMC & Particle Filters & Control Gordon (1993), Shepherd & Pitt (1997), Aquila & West (2000), Tanner (1994), Doucet et al (2000), Santos & Smith (2001) Military problems (bearings only problem [Gordon] missile hit/loss trade off [Tanner]) Financial problems (dynamic portfolio choice, Aquila & West [2000], Santos & Smith [2001]) Features Applications typically have thick tailed likelihoods or priors Exhibiting catastrophes globally Quick new algorithms, coping with intrinsic conflict of information

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13 New Developments 3) Decision Support Systems RODOS-DAONEM ( ) Decision making through M.U.I.A. Dynamic processes of complex non-linear spatial time series On line accommodation of (sometimes conflicting) disparate sources of information Real time support

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14 Statistics Game Theory Use opponents past acts Appeal to rationality of to predict future opponents to predict their future acts Problems with Statistical Models Why should opponent be consistent with past? Particularly if we change our own strategy? Problems with Game Theory Models How can we take account of the rationality of opponent when we dont know their objectives, information, what they believe? Why should we believe our opponent rational? Kadane & Larkley, 1983 The Stress between Statistical Models & Game Theory [Smith, 1996]

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15 Choose to maximise where is chosen so as not to contradict conclusions from rationality i.e. make sure is not obviously wrong! Link with Catastrophe Models Away from current, opponents reaction less certain = increased variance on our prediction of opponents response = larger spread in (see Moffat & Larkley, 2000) Reconciliation of Statistics & Game Theory, (Smith 1996)

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16 1) Conflicting Objectives Attributes cannot be simultaneously satisfied Cost to self, cost to enemy, public response……. With two attributes Normal Factor attribute weights Splitting Factor distance between good strategies for individual attributes e.g. Either attack or retreat DONT attack in limited way Three Basic Models of Conflict

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17 2) Conflict of Information Science predicts what should be happening Early data observations is not consistent with this Normal Factor relative belief in outliers proneness reliability of model -v- observations (modelled through tail distributions of likelihood and prior) Splitting Factor measure of discrepancy between predictions of model and data indication Note Also mixture models for different explanations, Draper (1997) Three Basic Models of Conflict (Contd)

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18 3) Rationality -v- Past Action Some, (e.g. current) strategies provoke a predictable response from adversary Other strategies will produce a response which is unpredictable Normal Factor relative gain of speculative strategies/ increased uncertainty Splitting Factor potential gain, potential uncertainty Three Basic Models of Conflict (Contd)

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19 1) Its constructive [unlike Chaos], evocative and easy to appreciate from a technical view point 2) It is now properly justified e.g. through recent development in M.U.I.A. and Bayesian non-linear dynamic models 3) It is possible to use models which exhibit the non-linear dynamics it classifies. 4) It is feasible to operationalise within a decision support system. Why its timely to reconsider Catastrophe Theory

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20 Help to construct interesting scenarios for emergency training, that standard models avoid Produce evocative diagnostics [normal/splitting factors] for feedback support to real time decision-makers Classify types of strategies: contrast efficacy of different types/focus on, classes of decision worth checking Some Potential Uses of Catastrophe Theory

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