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NASA Workshop on Collectives Ames Lab, 6 August 2002 Complex System Management: Hoping for the Best by Coping with the Worst Hoping for the best . . . but coping with the worst Neil F. Johnson n.johnson@physics.ox.ac.uk Department of Physics, Oxford University, U.K. Collaborators on several of the projects discussed: P. Jefferies, D. Lamper, M. Hart, R. Kay, P.M. Hui & Damien Challet Outline • Collectives & complex systems: design issues global outcomes: best-case vs. OK-case vs. worst-case static vs. dynamical • Dynamical collectives: multi-agent models generalized binary games with time-dependent global resources deterministic vs. stochastic formalism undesirable outcomes -- system crashes & their control -- fate, or just bad luck ? immunization of a complex system/collective • Static collectives: optimal collectives of autonomous defects near-perfect combinations of defective components defective component + defective component = working device • Winning by losing: optimal collectives of autonomous games successful combinations of unsuccessful games lose + lose = win, failure + failure = success, unsafe + unsafe = safe • Topology of collectives: network-based multi-agent games • Risk management in collectives & complex systems Topic • Collectives & complex systems: design issues global outcomes: best-case vs. OK-case vs. worst-case static vs. dynamical • Dynamical collectives: multi-agent models generalized binary games with time-dependent global resources deterministic vs. stochastic formalism undesirable outcomes -- system crashes & their control -- fate, or just bad luck ? immunization of a complex system/collective • Static collectives: optimal collectives of autonomous defects near-perfect combinations of defective components defective component + defective component = working device • Winning by losing: optimal collectives of autonomous games successful combinations of unsuccessful games lose + lose = win, failure + failure = success, unsafe + unsafe = safe • Topology of collectives: network-based multi-agent games • Risk management in collectives & complex systems • Complex Systems • Many degrees of freedom with internal frustration, feedback, history-dependence, adaptation, evolution, non-stationarity, non-equilibrium, memory, single realization, exogenous effects • Collectives, multi-agent systems, forward and inverse problems • Mix of deterministic and stochastic behavior • The Right Stuff System’s evolution can be optimized, controlled, managed. Robust • The Wrong Stuff System has a bad day . . . Heads down wrong path, leading to dangerous values, fluctuations, crashes. Endogenous and exogenous factors. Instabilities. Spontaneous secondary mission. Brittle • The Good Stuff System behaves OK, not great but not bad Avoids bad scenarios, e.g. system crash PLAN B may be ‘best’ e.g. lowest risk Consider the global performance S(t) of a collective/complex system Examples [Workshop website, Tumer & Wolpert]: throughput in a data network total scientific information gathered by a constellation of deployable instruments GDP growth in a human economy percentage of available free energy exploited by an ecosystem • The Right Stuff: optimize/maximize global performance S(t) mission successful • The Good Stuff: S(t) less/more than Scritical for all time t, or time-window T <S(t)> less/more than Scritical for all time t, or time-window T Var[ S(t) ] less/more than critical for all time t, or time-window T < [ S(t) ]n > less/more than X for any n etc…. mission reasonably successful … not a disaster – mission not a disaster ! real-world static system system’s time evolution S (t) ideal response L(t) = L actual response L + time …+1 …+2 …+3 …+4 …+5 e.g. minimize error by adjusting initial ‘quenched disorder’ real-world dynamical system system’s time evolution S (t) global resource level L(t) deterministic vs. stochastic continuous vs. discrete known vs. unknown …+1 …+2 …+3 …+4 …+5 endogenous vs. exogenous killer app: ‘designer system’ I system’s time evolution S (t) L(t) = L …+1 …+2 …+3 …+4 …+5 e.g. minimize ‘noise’, typical fluctuation size, hence optimize winnings, efficiency, use of global resource killer app: ‘designer system’ II system’s time evolution S (t) time …+1 …+2 …+3 …+4 …+5 e.g. avoid ‘dangerous’ large changes Complex Systems: Tails of the Unexpected Example of Fat Tails x2 C Typically Levy-like 1 px p x Sits somewhere between Lorentzian 2 2 2 2 (Lorentzian) e x (Gaussian) (0.1) x C 2 2 x 2 and Gaussian, but hard to tell since Distribution of increments of S (t ) • finite dataset 14 • non-stationarity probability density p[ S/S ] 12 10 Fat tails etc. are ‘obvious’ from statistics but … 8 temporal correlationsGaussian 6 do not show up! (e.g. system crashes) Lorentzian 4 2 0 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 re turn S/S ….but the variance (and hence volatilit y) is infinite for the Lorentzian big problem for standard risk analysis Topic • Collectives & complex systems: design issues global outcomes: best-case vs. OK-case vs. worst-case static vs. dynamical • Dynamical collectives: multi-agent models generalized binary games with time-dependent global resources deterministic vs. stochastic formalism undesirable outcomes -- system crashes & their control -- fate, or just bad luck ? immunization of a complex system/collective • Static collectives: optimal collectives of autonomous defects near-perfect combinations of defective components defective component + defective component = working device • Winning by losing: optimal collectives of autonomous games successful combinations of unsuccessful games lose + lose = win, failure + failure = success, unsafe + unsafe = safe • Topology of collectives: network-based multi-agent games • Risk management in collectives & complex systems In general, define w(t) according to the game of interest don’t enter the game Binary game f n t,n t ;n t 1,n t resource levelXt m wt Challet & Zhang at time t 1; ;Lt, limited global 2 histories 1 0 1 0 SO . . . . WHAT’S THE GAME ? 11 10 01 00 0 0 0 0 exces s demand 0 0 0 1 S n0 0 Dt n1 t n0 t . . . . e.g. sell ? V t,Dt wt 0 0 1 1 wt 0 1 wins 1 . . . . 0 1 1 1 n1 wt 0 0 wins . . . . e.g. buy N 1 1 1 1 n 0 t n 1 t V t N 2m 2 history at time t strategies ....10 history at time t +1 ....01 agent memory m = 2 Binary version of El Farol Game with time-dependent resource level (i.e. seating capacity) L(t) correlation between L(t) and A(t) system ‘learns’ frequency w system ‘confused’ attendance A (t) L(t)=L+L0 sin w t time t time t Deterministic map of binary game evolution Binary games behave as a stochastically perturbed deterministic system Global information (t) for m=2 3 2 1 0 32 64 96 12 8 16 0 19 2 Stochastic perturbations from coin-tossing agents Periods of entirely deterministic behaviour Replace stochastic term from coin-tossing agents by its mean Jefferies, Hart & NFJ Phys. Rev. E 65, 016105 (2002) Deterministic map of binary game evolution ‘ attendance ’ = ‘ demand ’ A ( t ) = n1 (t) - n0 (t) = D ( t ) [not always true!] ‘ volume ’ V ( t ) = n1 (t) + n0 (t) S (t) strategy score vector [ PRE 65, 016105 (2002) ] r confidence level (t) global information {0,1,..P-1} P = 2m a ( t ) response of strategies to ( t ) ; aR {-1,1} symmetrized strategy allocation tensor strategy R 3 0 4 0 0 5 2 0 0 3 Deterministic game defined by mapping equations: s=2 0 4 0 7 0 0 7 6 0 0 1 1 0 0 7 0 0 4 0 0 Binary El Farol Game: w(t) = L(t) V(t) - n1 (t) 0 0 0 0 8 0 3 2 0 0 MG: L(t)=0.5 w(t) > 0 1 wins w (t) < 0 0 wins 1 5 0 0 0 4 0 0 0 4 strategy R’ 0 4 0 7 0 6 0 3 0 0 3 0 0 0 1 0 0 1 0 7 0 0 1 0 3 0 0 2 3 0 0 2 0 3 0 2 7 0 4 0 4 0 7 0 4 3 0 3 0 0 random matrix initial strategy allocation quenched disorder 1 2 T In general, success & payoff may not be so simple to define w(t) complicated functional form Crowd - Anticrowd effect J. Phys. A: Math. Gen. 32, L427 (1999) Physica A 298, 537 (2001) large crowds >> 0 wastage but 0 for e.g. MG • stochastic strategy use • mixed-ability populations crowd - anticrowd coin-toss pairs execute volatility uncorrelated random walks sum of variances … also works for memory m generalized games Memory m 2m+1 << N.s 2m+1 ~ N.s 2m+1 >> N.s Crowd large medium ~1 size Anticrowd small medium ~0 size Net crowd Ğ anticrowd large small small walk step-size pair size >> 1 ~1 # crowd - ~ 2m ~ 2m < 2m anticrowd # of walks pairs << N <N ~N GCMG m =3 GCMG m =10 Jefferies & NFJ cond-mat/0207523 Design of generalized binary games $G11 m =3 $G11 m =10 dynamical properties very sensitive to game’s microstructure $G13 m =3 $G13 m =10 Jefferies & NFJ, cond-mat/0201540 Lamper & NFJ, PRL 017902 (2002) Jefferies & NFJ, cond-mat/0201540 Lamper & NFJ, PRL 017902 (2002) Anatomy of a system crash During persistence Dt sgnSR r sgnSR r demand described by: R a 1 R R a 1 R time during crash Assume: SR a 1 ~ N S 1, R crash length: ma x S 1, * c * * * participating ‘crash’ nodes Expected demand (and volume) during crash are thus given by: Dt V t c r c r erf 2 erf 2 2 c r c r erf erf 2 2 Hart & NFJ cond-mat/0207588 Physica A (2002) in press Convergence of ‘parallel-world’ trajectories prior to crash system’s evolution 2 j i 1 ( x x ) P( x j )P( xi ) 2 i j : spread of paths indicates role of ‘fate’ vs. ‘bad luck’ Hart & NFJ cond-mat/0207588 Physica A (2002) in press Immunizing against system crash Protecting the system Can reduce chances of system crash, by forcing earlier down-movements system gets immunized Topic • Collectives & complex systems: design issues global outcomes: best-case vs. OK-case vs. worst-case static vs. dynamical • Dynamical collectives: multi-agent models generalized binary games with time-dependent global resources deterministic vs. stochastic formalism undesirable outcomes -- system crashes & their control -- fate, or just bad luck ? immunization of a complex system/collective • Static collectives: optimal collectives of autonomous defects near-perfect combinations of defective components defective component + defective component = working device • Winning by losing: optimal collectives of autonomous games successful combinations of unsuccessful games lose + lose = win, failure + failure = success, unsafe + unsafe = safe • Topology of collectives: network-based multi-agent games • Risk management in collectives & complex systems Optimal collectives of autonomous defects e.g. nanodevice output, robot action, . . . Challet & NFJ, PRL 89, 028701 (2002) Tumer and Wolpert (2002) output ideal output L(t) = L actual output L + time …+1 …+2 …+3 …+4 …+5 N defective devices with a distribution of errors Combine a subset M < N to form high performance (i.e. low-error) collective: unconstrained, analog constrained, analog unconstrained, binary constrained, binary Optimal collectives of autonomous defects e.g. nanodevice output, robot action, . . . error over average Challet & NFJ, PRL 89, 028701 (2002) all components Tumer and Wolpert (2002) <> med N = 10 N = 20 random cost approach <> unconstrained, analog constrained, analog N devices N devices Optimal collectives of autonomous defects e.g. nanodevice output, robot action, . . . [Challet & NFJ, PRL (2002)] <> <> MG with agents accounting for their impact unconstrained, analog 2 strategies per agent N devices Optimal collectives of autonomous defects e.g. nanodevice output, robot action, . . . [Challet & NFJ, PRL (2002)] N binary components 0.2 Each component has I input bits simple enumeration Can perform F different logical 0.25 operations, hence P = F 2I transformations 0.3 & sorting f = probability that component i f systematically gives wrong output = fraction of component sets with at least one perfect subset unconstrained, binary N devices Optimal collectives of autonomous defects e.g. component sets = fraction ofnanodevice output, robot action, . . . Optimum: average [Challet & with at least one perfect subsetNFJ, PRL (2002)] over 10,000 samples 0.2 simple enumeration 0.25 Majority Game: average over 300 samples, 0.3 & sorting 500P iterations f 2 components/agent Majority Game constrains the system to M=N/2 Possible improvement with Grand Canonical Majority Game unconstrained, binary GCMajG ? N devices Topic • Collectives & complex systems: design issues global outcomes: best-case vs. OK-case vs. worst-case static vs. dynamical • Dynamical collectives: multi-agent models generalized binary games with time-dependent global resources deterministic vs. stochastic formalism undesirable outcomes -- system crashes & their control -- fate, or just bad luck ? immunization of a complex system/collective • Static collectives: optimal collectives of autonomous defects near-perfect combinations of defective components defective component + defective component = working device • Winning by losing: optimal collectives of autonomous games successful combinations of unsuccessful games lose + lose = win, failure + failure = success, unsafe + unsafe = safe • Topology of collectives: network-based multi-agent games • Risk management in collectives & complex systems Winning by losing losing game + losing game = winning game unsafe + unsafe = safe GAME A 2 win rotate randomly by n pwin 1 3 0.5 3 GAME B lose 2 rotate randomly by m pwin 3 7 0.5 7 Randomly playing Games A and B pwin 11 21 0.5 Colin Nash Colin A B A B A (2, 2) (-2, 9/4) A (-1/2, -1/2) (1, -1) Rose Rose B (9/4, -2) (-1, -1) B (-1, 1) (0, 0) Switching randomly Pareto between 2 ‘losing’ games gives ‘winning’ game Colin A B A (3/4, 3/4) (-1/2, 5/8) Rose B (5/8, -1/2) (-1/2, -1/2) Game A Game B J. Parrondo et al. PRL (1999) +1 state 1 p1 change in capital at ( -1, -1 ) Generalization to timestep t 1 p1 2 history-dependent +1 if win -1 -1 if lose games: +1 R. Kay & NFJ state 2 p2 cond-mat/0207386 +1 ( -1, +1 ) 1 p2 p +1 or -1 -1 1 p Application to +1 ( -1, -1 ) -1 state 3 p3 quantum computing: ( -1, +1 ) ( +1, -1 ) 1 p3 C.F. Lee & NFJ ( +1, -1 ) -1 quant-ph/0203043 ( +1, +1 ) +1 state 4 p4 ( +1, +1 ) 1 p4 -1 +1 change in capital in p1 -1 p2 previous two outcomes ( t-2, t-1) 1 p1 -1 -1 +1 +1 p4 p 3 1 p 2 1 p 3 1 p 4 +1 -1 Topic • Collectives & complex systems: design issues global outcomes: best-case vs. OK-case vs. worst-case static vs. dynamical • Dynamical collectives: multi-agent models generalized binary games with time-dependent global resources deterministic vs. stochastic formalism undesirable outcomes -- system crashes & their control -- fate, or just bad luck ? immunization of a complex system/collective • Static collectives: optimal collectives of autonomous defects near-perfect combinations of defective components defective component + defective component = working device • Winning by losing: optimal collectives of autonomous games successful combinations of unsuccessful games lose + lose = win, failure + failure = success, unsafe + unsafe = safe • Topology of collectives: network-based multi-agent games • Risk management in collectives & complex systems Network games Eguiluz & Zimmerman, PRL 85, 5659 (2001) power-law tails Zheng, NFJ et al., Eur. Phys. J. B 27, 213 (2002) Analytics using generating function tune power-law exponent Herding like-minded agents form clusters power-law distribution of cluster sizes & signal S(t) Topic • Collectives & complex systems: design issues global outcomes: best-case vs. OK-case vs. worst-case static vs. dynamical • Dynamical collectives: multi-agent models generalized binary games with time-dependent global resources deterministic vs. stochastic formalism undesirable outcomes -- system crashes & their control -- fate, or just bad luck ? immunization of a complex system/collective • Static collectives: optimal collectives of autonomous defects near-perfect combinations of defective components defective component + defective component = working device • Winning by losing: optimal collectives of autonomous games successful combinations of unsuccessful games lose + lose = win, failure + failure = success, unsafe + unsafe = safe • Topology of collectives: network-based multi-agent games • Risk management in collectives & complex systems Risk management in collectives borrow terminology from finance [c.f. Hogg, Huberman] avoid standard local-in-time stochastic p.d.e. approach allow for non-Gaussian, non-stationary distributions, temporal correlations include friction due to communication/intervention costs variation of global `wealth’: T / i 0 WT C0 T i [Si ] Si 1 Si i apply ‘no free lunch’ WT PRISK varWT minimize the ‘risk’ by choosing a suitable risk-management strategy varWT 0 S,t * no risk management mission mission unsuccessful successful 0.12 0.1 0.08 probability Prob 0.06 0.04 0.02 0 -204 -181 -158 -136 -113 -90 -67 -45 -22 1 Change in Wealth change in ‘wealth’ of system risk management … but assume no friction i.e. it ‘costs’ nothing to intervene 30 30 re-hedges interventions 3 interventions 3 re-hedges 0.25 0.07 0.2 0.06 probability 0.05 0.15 Prob Prob 0.04 0.1 0.03 0.02 0.05 0.01 0 0.00 -17.2 -14.6 -12.1 -9.5 -7.0 -4.4 -1.9 0.7 3.3 5.8 8.4 -61.1 -53.8 -46.4 -39.1 -31.8 -24.5 -17.2 -9.9 -2.6 4.8 12.1 change in ‘wealth’ of system Change in Wealth Change in of change in ‘wealth’ Wealthsystem standard deviation of ‘wealth’ distribution Standard Deviation 10.00 8.00 St. Dev. 6.00 4.00 2.00 0.00 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 time between interventions Trading time risk management … and friction i.e. it ‘costs’ something to intervene interventions 30 30 re-hedges 3 interventions 3 re-hedges 0.04 0.07 0.03 0.06 probability 0.03 0.05 Prob Prob 0.02 0.04 0.02 0.03 0.01 0.02 0.01 0.01 0.00 0.00 -87.1 -80.3 -73.5 -66.7 -59.9 -53.1 -46.3 -39.5 -32.8 -26.0 -19.2 -68.5 -62.2 -55.9 -49.6 -43.3 -37.0 -30.6 -24.3 -18.0 -11.7 -5.4 change in ‘wealth’ of system Change in Wealth change in ‘wealth’ of system Change in Wealth standard deviation of ‘wealth’ distribution Standard Deviation 14 12 there is an 10 ‘optimal’ St. Dev. 8 6 time-delay 4 between 2 0 interventions 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 time between interventions Trading time