What (is ACE)? Giorgio Fagiolo. Sant Anna School of Advanced Studies, Pisa (Italy)

Transcription

1 Acknowledgment: Some of the slides below are taken from lecture slides by prof. Troitzsch, Prof. Gilbert and Prof. Tesfatsion. What (is ACE)? Giorgio Fagiolo Sant Anna School of Advanced Studies, Pisa (Italy)

2 What is ACE? ACE ACE is the computational study of economic processes modelled as dynamic systems of interacting agents (Tesfatsion, 2006, p. 835) Stress on: Computational study Economic processes Dynamic systems of Interacting Agents (complex systems) In other words ACE is the computational study of economies as complex evolving systems

3 Agent-Based Models A tool to model economies where agents are boundedly rational entities directly interact in non trivial networks might be persistently heterogeneous and State of the economy is not necessary an equilibrium A bottom-up approach Modeling agents behaviors and their interactions first Statistical analysis of models output Matching with empirical data

4 The Structure of Agent-Based Models Main ingredients (to cook an ABM) Bottom-up (agent-based) Philosophy (Tesfatsion, 1997) Agents live in complex systems evolving through time (Kirman, 1998) Agents might be heterogeneous in almost all their characteristics Hyper-rationality not viable (Dosi et al., 1996) Agents as boundedly rational entities with adaptive expectations True dynamics: Systems are typically non-reversible Agents interact directly, networks change over time (Fagiolo, 1997) Endogenous and persistent novelty: open-ended spaces Selection-based market mechanisms (Nelson & Winter, 1982)

5 Agent

6 The Economy and its Dynamics Time t-1 Micro Vars F1 Time t Time t Micro Vars F2 Macro Pars F4 F3 F6 Macro Pars Time t+1 F5 Time t-1 Macro Vars Time t Macro Vars

7 The Structure of Agent-Based Models Time t = 0, 1, 2,, (T) Discrete Sets of Agents It = {1, 2,,Nt } Often Nt =N Sets of Micro States i xi,t Firm s output Vectors of Micro-Parameters i i Res. Wage Vector of Macro-Parameters m Opportunities Interaction Structures Gt ( It ) Networks Micro Decision Rules Ri,t ( ) Innovation rule Aggregate variables Xt = f ( x1,t,, xnt,t ) GNP

8 Examples (Increasing Sophistication) Cellular Automata Game of life Schelling Segregation Model Sand-Pile Models Dynamic Local-Interaction Games ABM Growth Models

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12 N5'O940'574/5*(,&-7 8 &'((7,*++*56'9,*(/56,*,(45' 8 P)('7,9'I'59,/5(<,/5,:'((7,./,.A',('3.,*59,+46A. start with one on cell in the middle Rule is: cell becomes on if Generation left and centre is on, right is off or left is on, centre and right are off or left is off, centre and right are on or left and centre are off, right is on otherwise cell goes off!"#!"#"$$% &'(()(*+,-)./0*.*!M

13 Rule 90 for 1D CA: Details

14 !"#$%&'#"(&)"*+,-.%/,01 Example: Rule 90 (Wolfram,1983)

15 .%,01 1D CA in nature? (Wolfram, 1983)

16 -,6+49,/3,:'((7!"#!"#"$$% &'(()(*+,-)./0*.* M

17 Q'46A;/)+A//97 8 F/5,Q')0*55,5'46A;/)+A//9 8 R//+',5'46A;/)+A//9 North East South West North North-east East South-east South South-west West North-west!"#!"#"$$% &'(()(*+,-)./0*.*!!

18 HA',)54F'+7' Right neighbour is left edge cell Bottom neighbour is top edge cell!"#!"#"$$% &'(()(*+,-)./0*.*!"

19 2%,01 Entire 2D image replaced each time step.

20 Re-parameterizing CA Rules Introducing the Lambda Parameter One of the states of the automaton---state number 0, represented by the color white by default---is said to be "dead. Every other state is considered to be "alive." The Lambda parameter is simply the fraction of all the rules that lead to "alive" states. When Lambda is small, most rules lead to "death," and all the cells tend to die off quickly. When Lambda is large, most rules lead to "life," but the result tends to be "chaos," a meaningless jumble of color. The nicest-looking pictures are supposed to occur somewhere between these two extremes, on the "edge of chaos." Example

21 Cellular Automata Classes The various types of CAs fall into 4 Classes (defined by Stephen Wolfram) Class 1 - point attractors. CAs in this class eventually evolve to a homogeneous stationary arrangement, with every cell in the same state. Class 2 - limit cycles. CAs in this class form periodic structures that endlessly cycle through a fixed number of states. Class 3 - chaotic. CAs in this class form aperiodic random-like patterns that resemble static white noise on a bad T.V. channel, and are sensitive to initial conditions. (NOTE: All finite CAs eventually have to repeat themselves.) Class 4 - structured. CAs in this class (e.g., Game of Life) form complex patterns with localized structures that move through space over time. For finite CAs, the patterns eventually become homogeneous, as in Class 1, or periodic, as in Class 2.

22 2D CA Example: The Game of Life

23 !"#$%&'#$()$*+)#$, Best known 2D CA is John Horton Conway's "Game of Life" invented 1970 at U of Cambridge. Objective: To make a CA 'game' as unpredictable as possible using the simplest possible CA rule. 2D grid of squares on a (possibly infinite) plane. Each square can be white (unoccupied) or black (occupied).

24 %&'#$()$*+)#, Each square has 8 neighbors (pasted edges assumed), 4 adjacent orthogonally and 4 adjacent diagonally. So Game of Life assumes Moore Neighborhoods.

25 Conway s Game-of-Life Basic Rules Survivals Every alive cell with two or three neighboring alive cells survives for the next generation Avoid over- or under-populated environments Deaths Each alive cell with four or more neighbors dies from overpopulation Every alive cell with one neighbor or none dies from isolation Births Each empty cell adjacent to exactly three neighbors (no more, no fewer) becomes an alive cell

26 Visual Depiction of Rule Number of Black Neighbors White Black Current State New State

27 Dynamic Rule-Driven Behavior

28 -&..#/0$1'#/2#03#4$, AT RIGHT: A 'pentomino'. Simple starting state on a blank background => immense complexity over 1000 steps before it settles. Cccccccccccc c Cccccccccccc c Cccccccccccc c Cccccccccccc c Cccccccccccc c Cccccccccccc c Cccccccccccc c Cccccccccccc c cccccccccccc c Pentomino

29 -&..#/0$1'#/2#03#4 AT RIGHT: a 'Glider. On a clear background, after 4 time steps, Glider pattern will 'move' to the North East one square diagonally. Glider Cccccccccccccc c cccccccccccccc cccccccccccccc cccccccccccccc cccccccccccccc cccccccccccccc c Pentomino Each square does not actually 'move', but the pattern of black squares can be seen by an observer as a glider travelling across the background.

30 An ecology of emergent behaviors WWW resources Example Try with examples in repository GLIDERS6.LIF RPENTO.LIF [ ]

31 -&..#/0$1'#/2#03#4$, This pattern movement can be observed as the movement of a glider, even though no glider was mentioned in the rule. 'Emergent' behaviour at a higher level of description, emerging from a simple low-level rule. Emergence = emergence-in-the-eye-ofthe-beholder? (controversial definition?)

32 Emergence in Complex Adaptive Systems Emergence Central notion in complex adaptive system theory Not yet a fully satisfactory theory of emergence Individual, localized behavior aggregates into global behavior that is somehow disconnected from its origins 1+1=2 but to understand 2 it is not enough to know anything about 1 and + ; 2 has some properties that are not derivable from the knowledge of 1 and + Emergence: Mosaic picture metaphor (Miller and Page, 2007, p. 45)

33 %&'#$()$*+)#5$6('#$7'89+3&.+(0: Basic Complex System Paradigm: Many interacting units; Parallel (distributed) actions; Locally determined (bottom up) actions. Complex global system behavior arising from ( emergent from ) simple rules of unit behavior. Emergent Patterns: Gliders, pentominos, blocks, traffic lights, blinkers, glider-guns, eaters, puffertrains...

34 Game of Life and 2D-CAs GoL is a particular 2D-CA Given k=2 states for a cell and a (8-dim) Moore neighborhood there are in principle: 2^[2^(8+1) ]= = 1.3! possible rules, that is a number that is considered to be greater than the size of the known universe (in atoms) Therefore GoL is just one of all such rules: The fact that Conway has discovered is amazing! Like-Life Rules See Like-Life Rules (LLR) are other 2D-CA rules that can be compared to GoL in terms of complexity of the ensuing patterns Despite many LLR have been discovered, the beauty and simplicity of the original GoL is still unsurpassed

35 Game of Life: Implications (cont d) Looking for a synthesis GoL is a deterministic system evolving through fixed rules from configuration A at t=0 to configuration B at t=t Configurations can be associated to a measure of their complexity in terms of the minimal amount of bits needed to codify them B can be much more complex than A, but A (and the rule) contains (hidden in them) all the info to generate B Fascinating research line: Given a complex or beautiful observed configuration B, find the simplest configuration A that is able to generate it (synthesis) Gardens of Eden A Garden of Eden is a configuration that is never reachable starting from any other initial configuration; the only way to see them in a GoL is to manually turning on its active cells Gardens of Eden s existence has been proved and many researchers have been trying to identify them

36 Game of Life: Implications (cont d) GoL as a Turing Machine See Wolfram s book (A New Kind of Science) It has been proved that GoL can be a universal Turing Machine That is: GoL is a universal computer operating on initial configuration and memory state: This is true in general for some classes of CA GoL can be shown to operate through logical ports (AND, OR, NOT) and a memory where to save the data of the operations The Church-Turing thesis states that no system may have computational capabilities greater than those of universal computers like the GoL! Unpredictability in GoL The behavior of a universal computer given a particular input may be determined only in a time of the same order as the time required to run that universal computer explicitly In general the behavior of a universal computer is unpredictable and can be determined only by a procedure equivalent to observing the universal computer itself: the values of particular sites after a long time depend on an ever-increasing number of initial sites

37 The Schelling Segregation Model Note: T.C. Schelling was a co-recipient of the 2005 Nobel Prize in Economics

38 Illustrative Segregation Pattern Two Classes of Households: Blue and Red Black Squares = Empty

39 Schelling Segregation Model: The Basic Setup N agents located on a 2-dimensional grid (torus) of L!L cells. Types: Each agent can be either RED or GREEN Only a percentage p of cells is occupied: N < LxL Agents are initially located on the grid at random In each time period, agents may be happy or unhappy Agent cares about the proportion q of other agents of its same colour in its Moore neighbourhood of radius 1 Agents are unhappy if q is below a certain critical threshold (which is a parameter of the model) and happy otherwise In each iteration of the model one unhappy agent is randomly selected to move to a random empty cell in the lattice

40 Playing with the basic model Main Result: City can tip into high segregation even if agents have only mild preferences for living with agents of their own type!

41 Schelling Segregation Model: Robustness Pancs and Vriend (2007, JPE): Are Schelling s results robust to assumptions Lattice structure Percentage of empty cells Sequence of choices Preferences?? Fagiolo, Valente and Vriend (2007, JEBO): Does segregation occur also in social networks?

42 Self-Organized Criticality Per Bak et al. (1987) introduced CA s showing the phenomenon of selforganized criticality (SOC). Systems are able to sustain a limited amount of stress. If stress exceeds locally a certain critical threshold, the system relaxes locally to an unstressed state and the stress is distributed to the neighborhood ( chain reaction ).

43 SOC and Sand Piles Bak s original example was a sand pile - the phenomenon of SOC was studied both with CA models and experimentally with real sand piles. In the experimental case a sand pile is constructed by repeatedly dribbling grains of sand on an existing heap. The experimenter then observes the size of the avalanches that are generated when the slope becomes unstable.

44 Based on Bak et al. (1988): Self-Organized Criticality Agents: i = 1, 2,..., N Time: t = 0, 1, 2,... Choices: A={0,1,2,... } Level of use of a technology Quality of technology employed Number (variety) of technologies employed Interactions: Agents located on a line Agents are ordered: Agent 1 > Agent 2 >... > Agent N Think of a layered-economy, layers are input-output sectors or leaders-followers in a technological ladder. Each pair of agents i,i+1 cares about the difference between their level of use z it = a it a i+1,t

45 Individual Dynamics: Agents want to keep difference z it = a it a i+1,t as small as possible due to costs induced by technological dissimilarity As soon as this difference become larger than a given threshold z c then the leader decreases his level by 1 and the follower increases his level by 1 (or equivalently: leader passes by one unit) Therefore:!! "! # $!!! # $!!! % " %! %!! % %! % %! %! % %! %!! %! % %!! % % %! % % & ' & ' & ' & ' & ' & ' & ' & ' & ' & ' & ' & ' & ' & ' & ' & ' ( & ' ) * * ) ) * * ) ) * * ) * * * * ) )

46 Aggregate Dynamics - Suppose that the leader i=1 introduces random increases in the level of use of the technology - Studying how the system (a 1t,..., a N t ) evolves. Results - Condition for stability is z i <= z c i=1,...,n. - There are (z c+1) N stable states (z c is integer!) - Suppose we are in the state z i = z c i=1,...,n : minimally stable state (MSS) - No matter how random perturbations (addition of one unity) are communicated to the system, the system always relax to the MSS - When the MSS is reached, any additional perturbation is communicated through the entire system but the latter goes quickly back to the MSS. Avalanches are white noise!

47 a k+3 k+2 k+1 z=1 z=1 z=1 a k+3 k+2 k+1 z=0 z=2 z=1 k k i-1 i i+1 i+2 i-1 i i+1 i+2 a k+3 k+2 k+1 z=0 z=2 z=1 a k+3 k+2 k+1 z=1 z=0 z=2 k k i-1 i i+1 i+2 i-1 i i+1 i+2

48 2-Dimensional Lattices Suppose now agents are located on a 2-dim lattice in the following way: o o o y1 o o o y o o o y+1 x1 x x+1

49 Notice that interaction structures are asymmetric! - (x,y) affects downstream agents (x,y+1), (x+1, y) - (x,y) is affected by upstream agents (x,y1), (x1, y) Define: z(x,y) = z hor + z vert = = [a(x,y) a(x,y+1)] + [a(x,y) a(x+1,y)] = = 2 a(x,y) a(x,y+1) a(x+1,y) Individual Dynamics: Same as before dep. on z(x,y)>zc Aggregate Dynamics: Studying how the lattice-system evolves. Define the MSS: z(x,y)=z c

50 Results: 1. The MSS is stable but the system never reaches it! Its basin of attraction is empty. 2. Why? A perturbation starting from (x,y) always spreads across the system but it amplifies itself. This is because each site depends on more than a chain of reaction! In fact, if a unit is added at (x,y) this will affect all 4 neighbors. Perturbations are propagated back and forth across the system

51 Aerial extent (domain) for several different avalanches in a SOC model. Each avalanche was triggered by the addition of a single grain. Avalanches have orders of magnitude difference in their sizes (Bak et al., 1988).

52 Results: 1. The MSS is stable but the system never reaches it! Its basin of attraction is empty. 2. Why? A perturbation starting from (x,y) always spread across the system but it amplifies itself. This is because each site depends on more than a chain of reaction! In fact, if a unit is added at (x,y) this will affect all 4 neighbors. Perturbation are propagated back and forth across the system. 3. The system will relax to a self-organized critical state (SOCS) different from the MSS which has two important properties: a. Small perturbations may lead to avalanches of any size b. The frequency distribution of avalanche size (# of sites affected) follows a power law 1/f, 1.37 NB: Why b. is important? Same property followed by: City Sizes: Log[City Rank] = 1 Log[City Size] Earthquakes: Log[EQ Freq] = 1 Log[EQ Size]

53 Bak et al. (1988): Log-log plot of frequency of occurrence D(s) of avalanche of size s versus size s of avalanche for 200 avalanches. Avalanches exhibit a power law distribution (D(s) ~ s -1 ).

54 Example: Dynamic Games (1/7) Studying equilibrium selection in coordination games Old problem: selection among multiple equilibria Coordination game: Two pure-strategy equilibria, possibly Pareto ranked; Inefficiency may arise What happens in players do not play games with anyone else? Model N players arranged on 1-dimensional lattice Play games with nearest neighbors, care about total payoff from plays At each t, one is drawn at random and plays with nearest neighbors Goal Studying long-run coordination in the aggregate Likelihood of Pareto efficient equilibria?

55 Example: Dynamic Games (2/7) r (i)=1 Time t = 0, 1, 2, r (h)=1 Sets of Agents I = {1, 2,,N} Players Sets of Micro States i {-1,+1} Pure strategies Interaction Structures Gt = 1-Dim Lattice Circle Micro-Parameters r(i) Interaction Radius Vector of Macro-Parameters a b c d Stage-Game Payoffs Micro Decision Rules BR Rule Strategy Updating Aggregate variables Average Action Coordination Level r(k) =2

56 Example: Dynamic Games (3/7) Time t = 0, 1, 2, Sets of Agents I = {1, 2,,N} Players Sets of Micro States i s(i) {-1,+1} Pure strategies Strategic Problem: Overall Coordination out of 2-person games ( a > 1) Pareto-Efficient Strategy Risk-Efficient Strategy if a< a EU(+1)=2a!+0! = a EU(-1)=3!+2! = 2.5

57 Example: Dynamic Games (4/7) Interaction Structures Gt = 1-Dim Lattice Circle Each agent i interacts with neighbors closer than r(i) r =1 r =1 V ( i ) { j : i j r ( i )} r =2

58 Example: Dynamic Games (5/7) Micro-Parameters r(i) Interaction Radius Macro-Parameter a Stage-Game payoff of (+1,+1) Micro Decision Rules and Dynamics - At t=0 random draw of strategies - At each t>0 one agent is chosen at random - Chooses st(i) s.t. max total payoffs given neighbors choices at t-1 s * t ( i ) arg max u ( s ; s t1( s { 1, 1 } j V ( i ) j ))

59 Example: Dynamic Games (6/7) Aggregate Variable: LR Coordination Level c 1 N N i1 s ( i T ) [ 1, 1 ] Choosing T large enough (stability/convergence of moments) Goal: Studying MC distributions of LR coordination levels as a function of 1) Aggregate Parameter (a) 2) Micro Parameters (e.g. average radius)

60 Flexibility of ACE/EV Paradigm (1/5) Micro Decision rules Deterministic Myopic BR: At each t, agents only know the state of the system they are allowed to observe at t-1 and choose among the available actions the one that maximizes current payoffs (assuming that tomorrow e.t. will remain the same Stochastic Routines: BR w/ mistakes: Linear Prob: Logistic Prob: Pay-Back Rule: Adjust current variables using a deterministic rule (function) Apply Deterministic BR and flips it Choose actions prop to payoffs Choose actions prop to logistics of payoffs Algorithmic Updating rules cannot be summarized by some function

61 Flexibility of ACE/EV Paradigm (2/5) Dynamics of Micro Decision Rules Fixed Agents always employ the same rules (with constant parameters) Exogenously Changing Micro decision rules change due to exogenous shocks that change e.g. the parameters (payback parameter, willingness to explore, ) mutations, technological innovation, demand shocks, Endogenously Adapting Micro decision rules endogenously change because agents are able to select (or are selected against) among a pool of different rules: learning over the space of rules

62 Flexibility of ACE/EV Paradigm (3/5) Expectations Myopic/Adaptive/Econometric Agents employ the past to form expectations about the future in a naïve way: tomorrow expectations is a simple (linear) function of past observations x(t+1) = x(t) x(t+1) = f ( x(t), x(t-1) ),, x(t-k) ) x(t+1) x(t), x(t-1),, x(t-k) AI-Based Neural Networks

63 Flexibility of ACE/EV Paradigm (4/5) Interactions What does it mean direct interactions? Interaction structure: who interacts with whom at each point in time Interaction structure described by a graph Nodes and Edges

64 Flexibility of ACE/EV Paradigm (4/5) Interactions What does it mean direct interactions? Interaction structure: who interacts with whom at each point in time Interaction structure described by a graph Undirected vs. Directed

65 Flexibility of ACE/EV Paradigm (4/5) Interactions What does it mean direct interactions? Interaction structure: who interacts with whom at each point in time Interaction structure described by a graph Unweighted vs. Weighted w 4 w 1 w 2 w 3

66 Flexibility of ACE/EV Paradigm (4/5) Interactions Interaction structure described by a graph Lattices

67 Flexibility of ACE/EV Paradigm (4/5) Interactions Lattice useful to describe local (spatial) interactions Playing with different neighborhood structures

68 Flexibility of ACE/EV Paradigm (4/5) Interactions Interaction structure described by a graph Regular Graphs, Small-World Nets, Generic Graphs

69 Flexibility of ACE/EV Paradigm (5/5) Dynamics of Interaction Structures Static Interaction Structures are fixed across time Exogenously Evolving Interaction Structures change due to exogenous shocks (e.g. after the system has converged to some stable state) Endogenously Evolving Agents are able to choose whom to interact with (interaction structures become micro-variables and are updated in a strategic way thanks to properly defined decision rules)

70 A Large Set of Models Evolutionary-Games (P. Young, Kandori et al., Blume, Ellison ) (Local) Interaction Models (Kirman, Weisbuch, Lux, Topol, IPD Models ) Endogenous Network Formation Models (Vega-Redondo, Goyal, Jackson-Watts ) Polya-Urn Schemes (Arthur, Dosi, Kaniovski, Lane, ).. Industry-Dynamics Models (Nelson + Winter tradition, History-Friendly Models) Evolutionary Growth Models (Silverberg, Verspagen, Dosi et al., ) ACE Models of Market Dynamics (Axtell, Epstein, Tesfatsion, Vriend, )

71 A more micro-founded model: Motivations Building a dynamic model of growth that Is able (as a plausibility check) to reproduce the fundamental statistical properties of GDP time series Allows one to disentangle the role of the basic sources of growth on the technological side Growth as the result of exploration-exploitation trade-off driven by Technological opportunities Path dependency in technological accumulation Degree of locality / globality of information diffusion Increasing returns to knowledge base exploitation Willingness to explore/exploit

72 The Islands Metaphor Technological Space Technology Output Firms Production Technological Search Innovation Technological Diffusion Imitation Technological Difference Notionally Unbounded Sea Island ( mine ) Homogeneous Good Stylized Entrepreneurs Mining/Extracting the Good Exploration of the Sea Discovering a new island Spreading knowledge from islands Traveling between already known islands Distance between Islands

73 The Model (1/2) Basic ingredients Time is Discrete Finite, constant population of stylized firms I={1,2,...,N} Notionally endless, discrete set of technologies (islands) Homogeneous good Islands Stochastically distributed on a bi-dimensional lattice Each node of the lattice can be an island with probability ( sea with probability 1-) Each island (x,y) is characterized by a productivity coefficient s(x,y)= x + y

74 The Model (2/2) Initial Conditions Set of initially known islands (exploited technologies) All N firms mining on them (randomly allocated) Each firm working in island (x,y) produces output s.t q(x, y) = s(x, y)! n(x, y)!"1 where n(x,y) number of firms currently working on (x,y) α>1 increasing returns-to-scale coefficient

75 Example: 3 initial islands, 10 firms 5 3 2

76 Example: 3 initial islands, 10 firms 5 3 2

77 Dynamics (1/4) Exploration In each t, a miner becomes explorer with probability Constant willingness to explore Explorers move around randomly in each period

78 Dynamics (2/4) Innovation In each exploration period, explorers find a new island with a probability The productivity of the newly discovered island is s* = s(x*,,y*) = (1+W) { [ x* + y* ] + q i, + } Distance from Poisson () Random Variable (Low probability high jumps) the Origin Cumulative Learning Effect: Agents Zero-Mean Random Variable (High Probability Low Jumps) carry with them their previous skills

79 Example: Exploration 5 3 2

80 Example: Exploration 4 3 2

81 Example: Exploration

82 Example: Exploration

83 Dynamics (3/4) Imitation In each t, from any currently exploited island (with at least one miner on it) a signal about current island s productivity is released Any miner currently working on (x,y) receives and follows the signal with a probability proportional to: the productivity of the island the signal comes from the exp of minus the distance between island and miner q ( x, y ) exp{ d(island,miner)} The higher (smaller) the more global (local) is information and knowledge diffusion Imitators move toward the imitated island following the shortest path leading to it (one step per period)

84 Example: Imitation

85 Example: Imitation

86 Example: Imitation

87 Dynamics (4/4) Dynamics of agents states: summing up In probability, if reached by information on a more productive island Miners Reaching an island With prob. i = Imitators Explorers In probability, if reached by information on a more productive island

88 Timing and Aggregate Variables Miners update output Miners become explorers Explorers look around Imitators approach islands Information diffusion Miners and explorers collect signals Imitation decisions Time t1 Time t Given t1 micro & macro variables Update time t micro & macro variables; next iteration starts Focus on Aggregate output (sum of firms output) and growth rates Number of explorers, imitators, miners

89 Timing and Aggregate Variables Model s parameters : globality of information diffusion : path-dependency in learning : likelihood of radical innovations : baseline opportunity conditions : increasing returns to scale in exploitation : willingness to explore N :population size T : time horizon