Agent-based models applied to Biology and Economics.

Transcription

1 Agent-based models applied to Biology and Economics. Gur Yaari Phd student, under the supervision of Prof. Sorin Solomon Racah Institute of Physics in the Hebrew University of Jerusalem, Israel Multi-Agent Systems Division. ISI, Torino, Italy

2 G.L.V Generalized Lotka Volterra Stochastic Lotka-Volterra Systems of Competing Auto-catalytic Agents lead Generically to Truncated Pareto Power Wealth Distribution, Truncated Levy Distribution of Market Returns, Clustered Volatility, Booms and Crashes. S. Solomon, in Decision Technologies for Computational Finance, edited by A.-P. Refenes, A. N. Burgess, and J. E. Moody (Kluwer Academic Publishers, 1998) Theoretical analysis and simulations of the generalized Lotka-Volterra model Ofer Malcai,1 Ofer Biham,1 Peter Richmond,2 and Sorin Solomon Phys. Rev. E 66, (2002) URL: doi: /physreve Generalized Lotka Volterra (GLV) Models of Stock Markets S. Solomon, pp in "Applications of Simulation to Social Sciences", Eds: G Ballot and G Weisbuch; Hermes Science Publications Power laws in cities population, financimarkets and internet sites (scaling in systems with a variable number of components) Aharon Blank and Sorin Solomon Physica A 287 (1-2) (2000) pp Power-law distributions and L'evy-stable intermittent fluctuations in stochastic systems of many autocatalytic elements. O. Malcai O. Biham and S. Solomon, Phys. Rev. E, 60, 1299 (1999).

3 AB Model The importance of being discrete: Life always wins on the surface Nadav M. Shnerb, Yoram Louzoun, Eldad Bettelheim, and Sorin Solomon Proc. Natl. Acad. Sci. USA, Vol. 97, Issue 19, , September 12, Adaptation of Autocatalytic Fluctuations to Diffusive Noise N. M. Shnerb, E. Bettelheim, Y. Louzoun, O. Agam, S. Solomon Phys Rev E Vol 63, No 2, 2001; HIV time hierarchy: winning the war while, loosing all the battles Uri Hershberg, Yoram Louzoun, Henri Atlan and Sorin Solomon Physica A: 289 (1-2) (2001) pp ; [pdf] The Emergence of Spatial Complexity in the immune System. Louzoun, Y, Solomon. S., Atlan. H., Cohen.I.R. (2001) Physica A, 297 (1-2) pp Proliferation and Competition in Discrete Biological Systems Y Louzoun S Solomon, H Atlan and I R. Cohend Bulletin of Mathematical Biology Volume 65, Issue 3, May 2003, P

4 Issues that had been explained (at least qualitatively) by these two models and their extensions: - The distribution of individuals in colonies (cities population) - The distribution of individuals/companies wealth - The distribution of number of individuals for various species - The distribution of word's frequency in a language -Genetic evolution The origin of life (??) The H.I.V war against our Immune system -Patches of green vegetation in the desert - Waiting for new Ideas...

5 G.L.V Generalized Lotka Volterra Thomas Robert Malthus ( ) 1798 autocatalitic proliferation: dx = a x dt with a =birth rate - death rate exponential solution: X(t) = X(0)ea t contemporary estimations= doubling of the population every 30yrs

6 Pierre François Verhulst ( ) way out exponential explosion: dx/dt = a X c X Solution: exponential ========== saturation at X= a / c

7 For humans data at the time could not discriminate between: 1. exponential growth of Malthus 2. logistic growth of Verhulst But data fit on animal population: sheep in Tasmania - exponential in the first 20 years after their introduction and completely saturated after about half a century. ==> Verhulst

8 G.L.V Generalized Lotka Volterra Alfred James Lotka (March 2, December 5, 1949) Vito Volterra (May 3, October 11, 1940) proposed independently (Lotka-1925, Volterra-1926) The Lotka Volterra Equations. Describes Predator-Prey Relations: dx = x y dt dy = y x dt

9 Taking into account self competition (limited resources) we have: dx = x 1 x 12 y dt dy = y 1 y 21 x dt Results: Cycles in time in the number of Predator's-Prey's Number of individuals

10 G.L.V Generalized Lotka Volterra General Frame-work for various processes in Economics and Biology. The Model N agents (fix number) : N individuals (Economics) N spieces (Biology)

11 Update: each time step Choose individual i randomly from 1..N w j t 1 =w j t i=1.. N ;i j N N j=1 j=1 w i t 1 =w i t 1 t a i, j w j t c i, j w j t w i t Stochastic Term! t Is a stochastic variable drawn from a distribution with : D t t 2 2

12 In order To Kis (keep it simple) it: one can choose uniform interaction: a a c i, j= ai, j= N N And then the process can be described by: w i t 1 =w i t 1 t a w t c w i t w t N where: 1 w t = w i t N j=1

13 Another possibility is: w i t 1 =w i t 1 t With the restriction: wi t c w t

14 Xi (t+τ) Xi (t) = λi (t) Xi (t) + a X (t) c(x.,t) Xi (t) admits a few practical interpretations Xi (t) = the individual wealth of the agent i then λi (t) = the random part of the returns that its capital Xi (t) produces during the time between t and t+τ a = the autocatalytic property of wealth at the social level = the wealth that individuals receive as members of the society in subsidies, services and social benefits. This is the reason it is proportional to the average wealth This term prevents the individual wealth falling below a certain minimum fraction of the average. c(x.,t) parametrizes the general state of the economy: large and positive correspond = boom periods negative =recessions

15 A different interpretation: a set of companies i = 1,, N Xi (t)= shares prices ~ capitalization of the company i ~ total wealth of all the market shares of the company λi (t) = fluctuations in the market worth of the company ~ relative changes in individual share prices (typically fractions of the nominal share price) ax = correlation between Xi and the market index w c(x.,t) usually of the form c X represents competition Time variations in global resources may lead to lower or higher values of c increases or decreases in the total X

16 Yet another interpretation: investors herding behavior (also cities) Xi (t)= number of traders adopting a similar investment policy or position. they comprise herd i one assumes that the sizes of these sets vary autocatalytically according to the random factor λi (t) This can be justied by the fact that the visibility and social connections of a herd are proportional to its size ax represents the diffusion of traders between the herds c(x.,t) = popularity of the stock market as a whole competition between various herds in attracting individuals

17 Important Results: When N is very large and t, The wealth (w's) distribution approaches a stationary distribution with the well common empirical feature of a power-law tail. For the first case one gets: P x =x t xi t =w i t i=1,...n w For the second 1 exp x 1 2 a =1 D case one gets: P x ~x 1 1 = 1 c

18 Computer's Simulations

19 Fits the well known empirical Pareto's law Or the Zipf's law Vilfredo Pareto ( ) Economics Income distribution P(x) ~ x 1-α d x George Kingsley Zipf, ( ) Statistics Word's distribution Cities size's distribution P(n) ~ n-α d n n- rank

20 red circles: Pareto for 400 richest people is USA blue squares: simulation results α Zipf plot of the wealths of the investors in the Forbes 400 of 2003 vs. their ranks. The corresponding simulation results are shown in the inset. the average wealth history and model fit

21 In addition, the average of the wealth is basically a result of a random walk with steps taken from a power-law distribution truncated Levy distribution: w t w t r = w t L r=0 P w t = w t ~ A strong prediction of this model is that the exponents of the wealth distribution and the returns of the stock market (the fluctuations of the average wealth ) are connected through a simple connection: 1 1 1

22 1/β And indeed it was found empirically: Prediction of The LotkaVolterra- model: The relative probability of the price being the same as a function of the time interval α β M. Levy S.S

23 The AB Model Two types of agents: A B Three types of processes: 1. Diffusion Locate ourselves in concrete space until now d-dimensional square lattice 2. Birth -with rate λ : B(r) + A(r) B(r) + B(r) + A(r) 3. Death - with rate µ : B(r) Ø 4. Competition -with rate σ : B(r) + B(r) B(r) -with rate σ <B>r : -with rate σ : B(r) + B(r) B(r) B(r) + B(r) B(r)+B(r+i)

24 Differential Equation: da x,t = D a 2 A x, t dt db x,t = Db 2 B x,t A x,t B x, t dt

25 Differential Equation: da x,t =D a 2 A x,t dt Competition db x,t = Db 2 B x,t A x,t B x, t dt 2 B r B r B r

26 Differential Equation: Naïve approach: when t is very large: A r n A db x,t 2 = Db B x,t A x,t B x, t dt db x,t = A x,t B x,t dt Naïve prediction n A 0 n A 0 Exponential Growth Exponential Decay Malthus

27 Important Results: If d 2 Life always win! For all parameter's values In higher dimensions, λ/da > 1-Pd Computer's Simulations + R.G (Renormalization Group) calculation:

28 Simulations + R.G calculation: outcome of the discreteness nature of the agents, Surprize!

29 Only Practical Effect: Islands are of limited height

30 Island's formation Movie 1

31 In Complexity it is very popular to talk about : Emergent Collective Dynamics: B-islands search, follow, adapt to, and exploit fortuitous fluctuations in A density. This is in apparent contradiction to the fundamental laws where individual B don t follow anybody

32 B's Island's search for food.. Movie2

33 Interpretations in Various Fields Finance: sites= individuals, companies, B = capital A= wealth generating conditions jobs, good location, good managers, customers, education Desert / Vegetation: - B = plants, - A= water / light / nutrients - patches- patterns= stripes, oases Desert <-> Patchy <-> Full cover (contact to Judean Desert-Jerusalem mountains studies) ; Reclaim; PRL 90, (2003)

34 Genetic Evolution: - Sites: various genomic configurations. - B= individuals; Jumps of B= mutations. - A= advantaged niches - emergent adaptive patches= species why are there not a continuum of creatures between snails and salamanders (both are partenogenetic).

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36 Polish spatial economic map since 89 (Andrzej Nowak ).

37 The analogy between these two models w i t 1 =w i t 1 t a w t c w i t w t dwi t =w i t t a w t c w i t w t dt Stochastic noise Discrete A's db x, t = Db 2 B x,t A x,t B x, t B x, t B x, t dt da x,t =D a 2 A x,t dt

38 Major differences dwi t =w i t t a w t c w i t w t dt db x,t = Db 2 B x,t A x,t B x, t B x, t B x,t dt

39 Major differences dw i t =w i t t a w t c w i t w t dt db x,t = Db 2 B x,t A x,t B x, t B x, t B x,t dt But... If And a w t a w i t We get similar results for the GLV N.O.N 1 a w i t B r = B r e j N.O.N j=1 2

40 Major differences And i,t j,t' = i, j t,t ' A r,t A r ',t ' But it turns out that the results of the AB Model are still valid even for uncorrelated noise

41 Future research Combine these two models in a more quantitative manner Find more empirical evidences for these type of processes Adjust the model to the specific case studied

42 NAGYON KÖSZÖNÖM Grazie mille Thank You very much dunke schoen תודה רבה Merci beaucoup