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2.4. SHORTESTPATHS
131
Exercises
Exercise1 Showthat,foranylinearsubspaceS
n
,wehaveS\S
?=f
0g
Exercise2 LetP=X(X
0
X)
1
X
0
andletP. ShowthatPandMarebothidempotent
andsymmetric.Canyougiveanyintuitionastowhytheyshouldbeidempotent?
Solutions
Solutionnotebook
2.4 ShortestPaths
Contents
• ShortestPaths
– Overview
– OutlineoftheProblem
– FindingLeast-CostPaths
– SolvingforJ
– Exercises
– Solutions
Overview
Theshortestpathproblemisaclassicprobleminmathematicsandcomputersciencewithappli-
cationsin
• Economics(sequentialdecisionmaking,analysisofsocialnetworks,etc.)
• Operationsresearchandtransportation
• Roboticsandartificialintelligence
• Telecommunicationnetworkdesignandrouting
• Etc.,etc.
Forus, theshortestpathproblemalsoprovidesasimpleintroductiontothelogicofdynamic
programming,whichisoneofourkeytopics
Variationsofthemethodswediscussareusedmillionsoftimeseveryday,inapplicationssuchas
GoogleMaps
T
HOMAS
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ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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2.4. SHORTESTPATHS
132
OutlineoftheProblem
Theshortestpathproblemisoneoffindinghowtotraverseagraphfromonespecifiednodeto
anotheratminimumcost
Considerthefollowinggraph
Wewishtotravelfromnode(vertex)AtonodeGatminimumcost
• Arrows(edges)indicatethemovementswecantake
• Numbersnexttoedgesindicatethecostoftravelingthatedge
Possibleinterpretationsofthegraphinclude
• Minimumcostforsuppliertoreachadestination
• Routingofpacketsontheinternet(minimizetime)
• Etc.,etc.
Forthissimplegraph,aquickscanoftheedgesshowsthattheoptimalpathsare
• A,C,F,Gatcost8
• A,D,F,Gatcost8
FindingLeast-CostPaths
Forlargegraphsweneedasystematicsolution
T
HOMAS
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ARGENTAND
J
OHN
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TACHURSKI
April20,2016
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2.4. SHORTESTPATHS
133
LetJ(v)denotetheminimumcost-to-gofromnodev,understoodasthetotalcostfromvifwe
takethebestroute
SupposethatweknowJ(v)foreachnodev,asshownbelowforthegraphfromthepreceding
example
NotethatJ(G)=0
Intuitively,thebestpathcannowbefoundasfollows
• StartatA
• Fromnodev,movetoanynodethatsolves
min
w2F
v
fc(v,w)+J(w)g
(2.16)
where
• F
v
isthesetofnodesthatcanbereachedfromvinonestep
• c(v,w)isthecostoftravelingfromvtow
Hence,ifweknowthefunctionJ,thenfindingthebestpathisalmosttrivial
ButhowtofindJ?
Somethoughtwillconvinceyouthat,foreverynodev,thefunctionJsatisfies
J(v)=min
w2F
v
fc(v,w)+J(w)g
(2.17)
ThisisknownastheBellmanequation
• Thatis,JisthesolutiontotheBellmanequation
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HOMAS
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ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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2.5. SCHELLING’SSEGREGATIONMODEL
134
• Therearealgorithmsforcomputingtheminimumcost-to-gofunctionJ
Solvingfor J
ThestandardalgorithmforfindingJistostartwith
J
0
(v)=Mifv6= destination,elseJ
0
(v)=0
(2.18)
whereMissomelargenumber
Nowweusethefollowingalgorithm
1. Setn=0
2. SetJ
n+1
(v)=min
w2F
v
fc(v,w)+J
n
(w)gforallv
3. IfJ
n+1
andJ
n
arenotequalthenincrementn,goto2
Ingeneral,thissequenceconvergestoJ—theproofisomitted
Exercises
Exercise1 Usethealgorithmgivenabovetofindtheoptimalpath(anditscost)forthis graph
Herethelinenode0, node1 0.04, node8 11.11, node14 72.21meansthatfromnode0wecan
goto
• node1atcost0.04
• node8atcost11.11
• node14atcost72.21
andsoon
Accordingtoourcalculations,theoptimalpathanditscostarelikethis
Yourcodeshouldreplicatethisresult
Solutions
Solutionnotebook
2.5 Schelling’sSegregationModel
T
HOMAS
S
ARGENTAND
J
OHN
S
TACHURSKI
April20,2016
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2.5. SCHELLING’SSEGREGATIONMODEL
135
Contents
• Schelling’sSegregationModel
– Outline
– TheModel
– Results
– Exercises
– Solutions
Outline
In1969,ThomasC.Schellingdevelopedasimplebutstrikingmodelofracialsegregation[Sch69]
Hismodelstudiesthedynamicsofraciallymixedneighborhoods
LikemuchofSchelling’swork,themodelshowshowlocalinteractionscanleadtosurprising
aggregatestructure
Inparticular, itshowsthatrelativelymildpreferencefor neighborsofsimilarracecanleadin
aggregatetothecollapseofmixedneighborhoods,andhighlevelsofsegregation
Inrecognitionofthisandotherresearch,Schellingwasawardedthe2005NobelPrizeinEconomic
Sciences(jointwithRobertAumann)
Inthislecturewe(infactyou)willbuildandrunaversionofSchelling’smodel
TheModel
WewillcoveravariationofSchelling’smodelthatiseasytoprogramandcapturesthemainidea
SetUp Supposewehavetwotypesofpeople:orangepeopleandgreenpeople
Forthepurposeofthislecture,wewillassumethereare250ofeachtype
Theseagentsallliveonasingleunitsquare
Thelocationofanagentisjustapoint(x,y),where0<x,y<1
Preferences Wewillsaythatanagentishappyifhalformoreofher10nearestneighborsareof
thesametype
Here‘nearest’isintermsofEuclideandistance
Anagentwhoisnothappyiscalledunhappy
Animportantpointhereisthatagentsarenotaversetolivinginmixedareas
Theyareperfectlyhappyifhalftheirneighborsareoftheothercolor
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HOMAS
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ARGENTAND
J
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TACHURSKI
April20,2016
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2.5. SCHELLING’SSEGREGATIONMODEL
136
Behavior Initially,agentsaremixedtogether(integrated)
Inparticular,theinitiallocationofeachagentisanindependentdrawfromabivariateuniform
distributiononS=(0,1)
2
Now,cyclingthroughthesetofallagents,eachagentisnowgiventhechancetostayormove
Weassumethateachagentwillstayputiftheyarehappyandmoveifunhappy
Thealgorithmformovingisasfollows
1. DrawarandomlocationinS
2. Ifhappyatnewlocation,movethere
3. Else,gotostep1
Inthisway,wecyclecontinuouslythroughtheagents,movingasrequired
Wecontinuetocycleuntilnoonewishestomove
Results
Let’shavealookattheresultswegotwhenwecodedandranthismodel
Asdiscussedabove,agentsareinitiallymixedrandomlytogether
Butafterseveralcyclestheybecomesegregatedintodistinctregions
Inthisinstance,theprogramterminatedafter4cyclesthroughthesetofagents,indicatingthatall
agentshadreachedastateofhappiness
Whatisstrikingaboutthepicturesishowrapidlyracialintegrationbreaksdown
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HOMAS
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ARGENTAND
J
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TACHURSKI
April20,2016
2.5. SCHELLING’SSEGREGATIONMODEL
137
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HOMAS
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ARGENTAND
J
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TACHURSKI
April20,2016
2.5. SCHELLING’SSEGREGATIONMODEL
138
Thisisdespitethefactthatpeopleinthemodeldon’tactuallymindlivingmixedwiththeother
type
Evenwiththesepreferences,theoutcomeisahighdegreeofsegregation
Exercises
Ratherthanshowyoutheprogramthatgeneratedthesefigures,we’llnowaskyoutowriteyour
ownversion
Youcanseeourprogramattheend,whenyoulookatthesolution
Exercise1 Implementandrunthissimulationforyourself
Considerthefollowingstructureforyourprogram
Agentsaremodeledasobjects
(Havealookatthislectureifyou’veforgottenhowtobuildyourownobjects)
Here’sanindicationofhowtheymightlook
* Data:
* type (green or orange)
* location
* Methods:
* Determine e whether r happy or not given locations of other agents
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HOMAS
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ARGENTAND
J
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TACHURSKI
April20,2016
2.6. LLNANDCLT
139
* If not happy, move
* find a a new w location where happy
Andhere’ssomepseudocodeforthemainloop
while agents are still moving
for agent in agents
give agent the opportunity to move
end
end
Use250agentsofeachtype
Solutions
Solutionnotebook
2.6 LLNandCLT
Contents
• LLNandCLT
– Overview
– Relationships
– LLN
– CLT
– Exercises
– Solutions
Overview
Thislectureillustratestwoofthemostimportanttheoremsofprobabilityandstatistics: Thelaw
oflargenumbers(LLN)andthecentrallimittheorem(CLT)
Thesebeautifultheoremsliebehindmanyofthemostfundamentalresultsineconometricsand
quantitativeeconomicmodeling
ThelectureisbasedaroundsimulationsthatshowtheLLNandCLTinaction
WealsodemonstratehowtheLLNandCLTbreakdownwhentheassumptionstheyarebasedon
donothold
Inaddition,weexamineseveralusefulextensionsoftheclassicaltheorems,suchas
• Thedeltamethod,forsmoothfunctionsofrandomvariables
• Themultivariatecase
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TACHURSKI
April20,2016
2.6. LLNANDCLT
140
Someoftheseextensionsarepresentedasexercises
Relationships
TheCLTrefinestheLLN
TheLLNgivesconditionsunder whichsamplemomentsconvergetopopulationmomentsas
samplesizeincreases
TheCLTprovidesinformationabouttherateatwhichsamplemomentsconvergetopopulation
momentsassamplesizeincreases
LLN
Webeginwiththelawoflargenumbers,whichtellsuswhensampleaverageswillconvergeto
theirpopulationmeans
TheClassicalLLN
Theclassicallawoflargenumbersconcernsindependentandidentically
distributed(IID)randomvariables
HereisthestrongestversionoftheclassicalLLN,knownasKolmogorov’sstronglaw
LetX
1
,...,X
n
beindependentandidenticallydistributedscalarrandomvariables,withcommon
distributionF
Whenitexists,letmdenotethecommonmeanofthissample:
m:=EX=
Z
xF(dx)
Inaddition,let
¯
X
n
:=
1
n
n
å
i=1
X
i
Kolmogorov’sstronglawstatesthat,ifEjXjisfinite,then
Pf
¯
X
n
!masn!¥g=1
(2.19)
Whatdoesthislastexpressionmean?
Let’sthinkaboutitfromasimulationperspective,imaginingforamomentthatourcomputercan
generateperfectrandomsamples(whichofcourseitcan’t)
Let’salsoimaginethatwecangenerateinfinitesequences,sothatthestatement
¯
X
n
!mcanbe
evaluated
Inthissetting,(2.19)shouldbeinterpretedasmeaningthattheprobabilityofthecomputerpro-
ducingasequencewhere
¯
X
n
!mfailstooccuriszero
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HOMAS
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ARGENTAND
J
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TACHURSKI
April20,2016
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