Exploring evolutionary models with Lsd

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Exploring evolutionary models with Lsd

• PhD Eurolab on Simulation of Economic Evolution
  (SIME)
• University of Strasbourg, April 2004
• Revised 8 April 2004
• Esben Sloth Andersen
• DRUID and IKE, Aalborg University, Denmark

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KISS and TAMAS Conflicting principles?

• KISS Keep It Simple, Stupid!
• A slogan from the US army during World War II
• Generally acknowledged by scientific modellers
• TAMAS Take A Model, Add Something!
• Variant for Lsd modellers
  TAMAM Take A Model, Add Marco!
• Principle for cumulative modelling
• KISS TAMAS?
• Not when the initial model is complex and ill
  structured!
• In this case we need a new principle!
• TAMAKISS Take A Model And Keep It
  Simpler, Stupid!

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TAMAKISS with a Nelson-Winter model
Capital accumulation
Numi
Technical change
fi
Short-run process
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Simplifying the Nelson-Winter modelShort-term
and capital accumulation

• Reuse our model of replicator dynamics!
• Replicator equation
• NtNt-1(1a(f-af)/af)

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Lsd code for replicator dynamics

• EQUATION("af")
• / Average fitness /
• v00 v10
• CYCLE(cur, "Species")
• v0VLS(cur,"Num",1)
• v1VLS(cur,"Num",1)VS(cur,"f")
• RESULT(v1/v0)
• //EQUATION("f")
• / Unchanged fitness of Species. This version was
  replaced by the next equation /
• //RESULT( VL("f",1) )
• EQUATION("f")
• / Fitness of Species changed through random walk
  /
• RESULT( VL("f",1)UNIFORM(-.2,.2) )

• EQUATION("Num")
• / Replicator dynamicsNtNt-1(1a(f-af)/af)
  /
• v0VL("Num",1)
• v1V("a")
• v2V("f")
• v3V("af")
• v4v0(1v1(v2-v3)/v3)
• RESULT(v4)

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Start by copying the model

• Find your original modelin Browse Model window
• Edit/Copy
• Edit/Paste
• Write RepDyn2004 inmodel name
• Write rd2004 in dir name
• OK
• Write a description
• Save description
• Goto model equations

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Introducing a control variable for change of
fitnesses

• Case 1 Fixed fitnesses (productivities)
• ft ft-1
• Case 2 Random walk of fitnesses
• ft ft-1 UNIFORM(-.2,.2)
• Allowing for both cases
• if RandWalk0 then ft ft-1
• if RandWalk1 then ft ft-1
  UNIFORM(-.2,.2)

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Specifying the regimes

• RandWalk Change of fitnesses?
• 0 no change 1 random walk 2 define
  your own regime
• Later we add
• Fissions Change in number of firms?
• 0 no change 1 fission of large species
• 2 define your own change rule

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Rewrite equation for fitness (f)

• EQUATION("f")
• /
• Calculation of fitness
• If RandWalk 0, then fixed fitnesses
• If RandWalk 1, then random walk of fitnesses
• /
• v0V("RandWalk")
• if (v00) v1VL("f",1)
• if (v01) v1VL("f",1)UNIFORM(-.2,.2)
• RESULT(v1)

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Start model, load config and add parameter

• Load Sim1.lsd configuration file
• Goto Population and add parameter RandWalk
• Initialise RandWalk to 0
• Goto Species, Initial values, Set all to 10 incr.
  by -.5
• Run the model, then reload the config
• Set RandWalk 1 and rerun model. Then kill it!

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Add simple statistics

• Size of total population
• TotNum Sum(Num)
• Population shares of species
• s Num/TotNum
• Inverse Herfindahl index
• Standard concentration indicator in industrial
  economics
• InvHerf 1/Sum(s2)
• Between 1 and the number of species

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Implement simple statistics

• EQUATION("TotNum")
• / Total number of members of the population /
• v00
• CYCLE(cur, "Species")
• v0VLS(cur,"Num",1)
• RESULT(v0)
• EQUATION("s")
• / Population share /
• v0V("Num")
• v1V("TotNum")
• RESULT(v0/v1)

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Change model structure

• Add TotNum to Population (with save)
• Add s with time lag 1 to species (with save)
• Initialise s for all species to 0.1
• Reset RandWalk 0
• Run the model and check that it works correctly!
• Check what happens to s when RandWalk 1

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Add concentration index

• EQUATION("InvHerf")
• / Inverse Herfindahl index 1/SUM(s2) /
• v00
• CYCLE(cur,"Species")
• v1VS(cur,"s")
• v0v0v1v1
• RESULT(1/v0)
• Change model structure and check concentration
• dynamics. Then kill the model

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The logic of fissions of species

• Large species encounter varied pressures
• They tend to split into different species
• Large firms have conflicts and split
• I model fissions as a fixed propensity to split
• If population share is above 25
• Then the species will on average split once every
  40 periods
• Modelled as a Poisson process
• Result of fission Concentration is kept lower

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Introduce fissions of species

• EQUATION("Fission")
• / Fissions of species take on average place once
  every 40 periods
• if its population share is larger than 25. /
• V("Repro") // Ensure that reproduction
  coefficient is calculated
• v0 V("s")
• v1 V("Num")
• v2 V("Fissions")
• v3 RND-0.5
• if (v0gt0.25 v21 poisson(0.05v3)gt0)
• curp-gtup
• curADDOBJS_EX(cur,"Species",p)
• WRITELS(cur,"Num",0.4v1,t)
• WRITELS(cur,"s",0.4v0,t)
• WRITELS(p,"Num",0.6v1,t)
• WRITELS(p,"s",0.6v0,t)
• RESULT(v2)

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Change the model structure and check

• Add parameter Fissions to Population
• Initialise Fission 0 and RandWalk 0
• Add variable Fission to Species
• Run the model and check that nothing has changed
• Change Fission 1, and study the results
• Why is there no fissions at the end of the
  simulation?
• Change Fission 1 and RandWalk 1
• Study the results? What happens?
• Kill the model before proceeding

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Introducing fission in replicator dynamics
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Dynamics of the Herfindahl index
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Defining and calculating statistics

• Population information for two points of time
• Initial population share of each species
• Reproduction coefficient of each species
• Fitness of each species and its change
• Simple statistics
• Mean reproduction coefficient
• Change in mean fitness
• Variance of fitnesses
• Covariance of reproduction coefficients and
  fitnesses
• Regression of reproduction coefficients on
  fitnesses

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Prices partitioning of evolutionary change

• Total evolutionary change ?
  Selection effect Innovation effect

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The meaning of Prices equation

• The innovation effect is the creative part
• It takes place within the units, e.g. the firms
• It may be due to innovation, imitation, learning,
  
• It may also be due to intra-firm selection, e.g.
  of plants
• The selection effect means that some entities are
  promoted while other entities shrink
• It represents Schumpeters creative destruction
• Firms may try to avoid selection by imitation and
  learning
• The selection pressure sets the agenda for firms
• The Price equation ignores ecological effects
• Thus it is a form of short-term evolutionary
  analysis
• But short-term evolution is the starting point!

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Prices statistics reproduction coefficients

• EQUATION("Repro")
• / Repro Numt/Numt-1The reproduction
  coefficient of the species /
• RESULT(V("Num")/VL("Num",1))
• EQUATION("ReproMean")
• / Weighted mean of the species' reproduction
  coefficients /
• v00
• CYCLE(cur,"Species")
• v1 VLS(cur,"s",1)
• v2 VS(cur,"Repro")
• v0 v0v1v2
• RESULT(v0)

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Prices statistics selection as covariance

• EQUATION("Covar")
• / Cov(Repro,A) SUM st-1(Reprot-1-ReproMe
  ant-1)(ft-1-aft-1)) Covariance between
  species' reproduction coefficients and fitnesses
  /
• v00
• v3 V("ReproMean")
• v5 V("af")
• CYCLE(cur, "Species")
• v1 VLS(cur,"s",1)
• v2 VS(cur,"Repro")
• v4 VLS(cur,"f",1)
• v0 v0 v1(v2-v3)(v4-v5)
• RESULT(v0)

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Covariance in simple replicator dynamics
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Covariance in repdyn random walk
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Covariance in repdyn and fissions
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Covariance in randwalk repdyn and fissions
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Prices statistics the innovation effect

• EQUATION("InnoEffect")
• / E(st(ft-ft-1)) / ReproMean
• The innovation effect as defined by George
  Price's equation. /
• v00
• v10 V("ReproMean")
• CYCLE(cur, "Species")
• v1 VS(cur,"s")
• v2 VS(cur,"f")
• v3 VLS(cur,"f",1)
• v0 v0 v1(v2-v3)
• RESULT(v0/v10)

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Innovation effect in simple repdyn
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Innovation effect in randwalk repdyn