{VERSION 4 0 "APPLE_PPC_MAC" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Arial" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Arial" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 3 0 3 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 " Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Dash Item" -1 16 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 16 3 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 256 1 {CSTYLE "" -1 -1 "Ar ial" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Dash Item" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 16 3 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 80 "From random walks to \nA rthur's competing technology models: \nA hands-on approach" }}{PARA 19 "" 0 "" {TEXT -1 42 "Esben Sloth Andersen\nRevision: 25 Apr 2001" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 4 "" 0 "" {TEXT -1 15 "1. Introduction" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 192 "This note suggests a hands-on simulation appro ach to the study of models with increasing returns and network effects . As soon as we move away from verbal introductions such as Shapiro/Va rian: " }{TEXT 257 17 "Information Rules" }{TEXT -1 623 "(1999), we fi nd out that we are dealing with a highly formalised field. If we want \+ a deeper understanding, we need to start with simple models like the o nes presented in the present note. These models have analytical soluti ons, so why should we try simulate them? For two reasons: First, the s imulation approach gives a deeper understanding and control over the f ormal models. Second, when we have implemented the simple models, it i s easy to develop them further. Thereby, we may quickly move toward fr ontiers that can be understood even though we don't immediately go for (potentially non-existing) analytical solutions.\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 769 "You don't have to use the hands-on approach si nce many simple examples are included in the present note. So you can \+ just read it. However, although the note looks like a text document, i t is actually a \"worksheet\" for the mathematical programming package Maple, which is available for many operation systems (Unix, Windows, \+ Macintosh, etc.). Maple exists in different versions. The present work sheet is for Maple 6, but after small modifications if can be used for previous versions of Maple. If the worksheet is loaded into Maple, th en all Maple input areas can be loaded into Maple's mathematical engin e. Some Maple inputs are stored for later use (eg. program procedures \+ and values of variables). Other inputs give an automatic response. If \+ we after a Maple prompt (" }{TEXT 286 1 ">" }{TEXT -1 8 ") write " } {TEXT 284 11 "2+2;" }{TEXT -1 16 ", Maple answers " }{TEXT 285 1 "4" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 401 "\nIf you don't want to use Maple (which is available as site licences at many \+ universities), you can more or less easily reprogram the procedures fo r other programming systems Ð both the general ones like Pascal and C+ + and the specific ones like Mathematica. Later the present worksheet \+ will be used for programs in the programming system Lsd (Laboratrory f or Simulation Development) Ð available from " }{URLLINK 17 "http://www .business.auc.dk/lsd/" 4 "http://www.business.auc.dk/lsd/" "" }{TEXT -1 3 ". \n" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 35 "2. Random walks and Polya processes" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 26 "\n2.1. Stocha stic processes" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 372 "The random wal k model is a convenient starting point that traces back to statistical mechanics with its Brownian movements of molecules, but we can just a s well think of a drunkards walk with small and erratic steps in two d irections. In the case of network economics we may think of the moveme nt of users from an old system/technology to two new and alternative s ystems, " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 147 ". Let us assume that new users come one by one and adopt each system with probability 0.5. We may also sa y we have two types of potential users:\n¥ " }{XPPEDIT 18 0 "R;" "6#% \"RG" }{TEXT -1 13 "-type users: " }{XPPEDIT 18 0 "u(B) < u(A);" "6#2- %\"uG6#%\"BG-F%6#%\"AG" }{TEXT -1 4 " \n¥ " }{XPPEDIT 18 0 "S;" "6#%\" SG" }{TEXT -1 13 "-type users: " }{XPPEDIT 18 0 "u(A) < u(B);" "6#2-% \"uG6#%\"AG-F%6#%\"BG" }{TEXT -1 286 "\nUsers choose the best, and we \+ assume that the choice is irreversible Ð ie. that the user sticks to t he chosen new system \"forever\". Given these assumptions, we have a s tochastic process that generates a random sequence of the two systems. Here is such a random sequence of 30 choices:\n" }{XPPEDIT 18 0 "A,B, A,A,B,B,A,B,A,A,A,A,A,B,B,A,B,A,A,A,B,B,B,A,A,A,B,B,A,B;" "6@%\"AG%\"B GF#F#F$F$F#F$F#F#F#F#F#F$F$F#F$F#F#F#F$F$F$F#F#F#F$F$F#F$" }{TEXT -1 139 "\nSuch a sequence of random choices can be transformed into a ran dom walk by the studying the difference between the number of adoption s of " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 34 ". The sequence of differences is:\n" } {XPPEDIT 18 0 "1,0,1,2,1,0,1,0,1,2,3,4,5,4,3,4,3,4,5,6,5,4,3,4,5,6,5,4 ,5,4;" "6@\"\"\"\"\"!F#\"\"#F#F$F#F$F#F%\"\"$\"\"%\"\"&F'F&F'F&F'F(\" \"'F(F'F&F'F(F)F(F'F(F'" }{TEXT -1 46 "\nIn period 30 we have a instal lation share of " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 10 " equal t o " }{XPPEDIT 18 0 "(15+4)/30 = .63;" "6#/*&,&\"#:\"\"\"\"\"%F'F'\"#I! \"\"$\"#j!\"#" }{TEXT -1 40 ". What about the market share of system \+ " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 826 " in the longer run? It \+ is not difficult to see that due to the law of large numbers this mark et share will be very close to 0.5.\n\nThe crucial assumption underlyi ng the random walk is that the existing market shares do not influence the adoption decision. This is not realistic. In a simple diffusion m odel of the systems, we may assume that each previous adopter helps to recruit new adopters. This may lead to a Polya process (referring to \+ the mathematician George Polya, 1887-1985). Here the probability that \+ a new adopter applies a particular system is proportional to its marke t share. We may interpret this in the following way: All potential use rs are homogeneous with respect to their lack of information of the ut ilities of the two systems. To obtain information they select by chanc e one existing user:\n¥ If it is an " }{XPPEDIT 18 0 "A;" "6#%\"AG" } {TEXT -1 50 "-user, then expectations of the new user are that " } {XPPEDIT 18 0 "u(B) < u(A);" "6#2-%\"uG6#%\"BG-F%6#%\"AG" }{TEXT -1 15 "\n¥ If it is an " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 50 "-use r, then expectations of the new user are that " }{XPPEDIT 18 0 "u(A) < u(B);" "6#2-%\"uG6#%\"AG-F%6#%\"BG" }{TEXT -1 117 "\nIn this case the long-term behaviour of the market shares is totally different from th e random walk case. Actually, " }{TEXT 258 3 "any" }{TEXT -1 728 " mar ket share can represent a long-term equilibrium. The only thing we kno w (from Polya's proof) is that with probability 1, one of the infinite number of equilibria will be found for any particular process of adop tion. The background is again the law of large numbers. In the beginni ng there are only a small number of adopters that try to recruit new o nes. Therefore, the choices behave randomly. But these initial random \+ events move the system towards more and more stable market shares.\n\n Even the Polya process seems much too simplistic. For instance, it doe s not take into account the possibility of a \"corner solution\" in wh ich one of the market shares reaches 1. This case can easily be obtain ed: Assume that that system " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 152 " have different utility characteristics and that these characteristics are fully know n to all adopters. In that case all adopters chose the best system." } }}{EXCHG {PARA 5 "" 0 "" {TEXT -1 91 "\n2.2. Preparing for the simulat ions (plotting procedure - only relevant when running Maple)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 423 "In the next section we shall see \+ how to implement the stochastic processes in the mathematical programm ing package Maple (version 6, but with small changes previous systems \+ can be used). However, these programs just product a lot of data, so w e need a tool for an easy visialisation of them. This is the following plotting procedure that uses a lot of Maple programming and plotting \+ tricks. So, don't try to understand it. " }{TEXT 259 136 "If you are r unning the programe, you need to load the DataPlot procedure by placin g the cursor in the field below and pressing .\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1830 "DataPlot := proc(varlist,t,T,varm in,varmax,ymarks,colours,Type)\nlocal colourlist,colourseq,DATA,i,j,Me rgeLists,n;\noption `E.S. Andersen, 11 Feb 2001`;\n\nMergeLists := pro c(list1, list2)\nglobal LookUpItem;\nlocal i;\n LookUpItem := proc(li st1, list2, seriesnumber)\n local listnumber;\n if type(seriesnumbe r, even) then\n listnumber := seriesnumber/2;\n RETURN(list1[lis tnumber]);\n else\n listnumber := (seriesnumber-1)/2;\n RETURN (list2[listnumber]);\n end if;\n end proc;\nRETURN([seq(LookUpItem( list1, list2, i), \n i = 2..nops(list1)*2 + 1)]); \nend proc: \ncolourlist := [[1.0, 0, 0], [0, 1.0, 0],[1.0, 1.0, 0],\n[.439, .859, .576], [0, 0, 0],[0, 0, 1.0],\n[.647, .165, .165], [.310, .184, .310 ], [.137, .137, .557], \n[1.0, .498, 0], [0, 1.0, 1.00], [.8, .498, .1 96], \n[.753, .753, .753], [.624, .624, .373], [1.0, 0, 1.00], \n[.557 , .137, .420], [.8, .196, .196], [.737, .561, .561], \n[.918, .678, .9 18], [.557, .420, .137], [.859, .576, .439], \n[.678, .918, .918], [.8 47, .847, .749]];\ncolourseq := NULL;\nDATA:= table();\nDATA['Title'] \+ := NULL;\nif Type=dt then DATA['Title'] := `Difference between install bases of A and B`; else DATA['Title'] := `Market share for system A`; end if;\nDATA['Plotdata'] := NULL;\nif colours=black then\nfor n from 1 to nops(varlist) do\n colourseq := colourseq, 0,0,0; \nend do:\nel se for n from 1 to nops(varlist) do\n colourseq := colourseq, colourl ist[n][];\nend do;\nend if;\nfor n from 1 to nops(varlist) do\n DATA[ 'Plotdata'] := DATA['Plotdata'], \n [seq([i,varli st[n][i]], i=t..T)];\nend do;\nDATA['Plotdata'] := 'CURVES'(DATA['Plot data'],\n 'COLOUR'('RGB', colourseq)), TITLE(DATA['Title']),\n 'VIEW'(1..T+1,varmin..varmax), \n 'AXESSTYLE'(NORMAL) ,\n 'AXESTICKS'(4,ymarks, 'FONT'(COURIER, DEFAULT, 9));\nPLOT( DATA['Plotdata']);\nend proc:" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 57 " \n2.3. Maple programs for random walks and Polya processes" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "Now we are ready for developing the prog rams. We start with the random walk program that can be invoked by a p rocedure call with two parameters: " }{TEXT 260 13 "walk(T, seed)" } {TEXT -1 23 ". The first parameter (" }{TEXT 261 1 "T" }{TEXT -1 74 ") is simply the integer number of periods for the simulation. The secon d (" }{TEXT 262 4 "seed" }{TEXT -1 152 ") is a little more complicated to understand: it is an integer that determines a unique sequence of \+ random numbers. Thus if you call the procedure with " }{TEXT 263 11 "w alk(100,1)" }{TEXT -1 0 "" }{TEXT 264 1 ";" }{TEXT -1 121 " the result is a random walk for 100 periods (ie. with 100 adoption decisions). I f you call the procedure once more with " }{TEXT 265 12 "walk(1000,1) " }{TEXT -1 0 "" }{TEXT 266 1 ";" }{TEXT -1 102 " the first 100 steps \+ will be exactly the same as before. However, if you call the procedure with with " }{TEXT 267 11 "walk(100,2)" }{TEXT -1 0 "" }{TEXT 268 1 " ;" }{TEXT -1 144 " you will see a totally different sequence of random numbers. Any seed produces its own unique sequence.\n\nThe real progr am is found in the loop " }{TEXT 269 25 "for t to T do ... end do;" } {TEXT -1 60 " Here you see that we call a random number generator call ed " }{TEXT 270 5 "coin " }{TEXT -1 76 "that gives values 0 and 1 with equal probability. If the value of 1, system " }{XPPEDIT 18 0 "A;" "6 #%\"AG" }{TEXT -1 29 " is chosen; otherwise system " }{XPPEDIT 18 0 "B ;" "6#%\"BG" }{TEXT -1 122 " is chosen. Finally, we call the DataPlot \+ procedure to produce two time plots: one of the movement of the market share of " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 62 " and one of th e difference between the number of adoptions of " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 2 ".\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 624 "walk := proc(T,s eed)\nglobal _seed;\nlocal A_share, choice, coin, d, d_max, d_min, n_ A, n_B, t;\noption `E.S. Andersen, 10 Feb 2001`;\n\n_seed := seed;\nd \+ := array(0..T);\nd[0] := 0;\nd_max := 0; d_min := 0; \nA_share := arra y(1..T);\nA_share := 0; \nn_A := 0; n_B := 0;\ncoin := rand(0..1);\n\n for t from 1 to T do\nchoice := coin();\nif choice=1 then n_A := n_A + 1;\n else n_B := n_B + 1;\n end if;\nd[t] := n_A - n_B; \nd_max := \+ max(d[t], d_max); \nd_min := min(d[t], d_min);\nA_share[t] := n_A/t;\n end do;\nprint(`\\n`);\nprint(DataPlot([A_share],1,T,0,1,5,black,ms)); \nprint(`\\n\\n`);\nprint(DataPlot([d],1,T,d_min,d_max,5,black,dt));\n end proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "The program for Poly a processes is called by " }{TEXT 271 15 "polya(T,N,seed)" }{TEXT -1 19 ". The new thing is " }{TEXT 272 1 "N" }{TEXT -1 323 ", the number \+ of simulations that you want to have plotted in a single figure. To ma ke several runs, the program uses some Maple tricks that make it a lit tle more difficult to interpret. The important thing is that we create a random number generator that produces uniformly distributed numbers between 0 and 1. In the loop " }{TEXT 273 25 "for t to T do ... end d o;" }{TEXT -1 72 " a random number is produced. If it is smaller than \+ the market share of " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 25 ", th en the person adopts " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 2 ".\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 533 "polya := proc(T,N,seed )\nglobal _seed;\nlocal A_share, chance, n, n_A, n_B, RandGenerator, \+ t;\noption `E.S. Andersen, 10 Feb 2001`;\n\nRandGenerator := stats[ran dom,uniform[0,1]]('generator');\n_seed := seed;\n\nfor n from 1 to N d o\nA_share||n[0] := 1/2; \nn_A := 1; n_B := 1;\nfor t from 1 to T do\n chance := RandGenerator();\nif chance() < A_share||n[t-1] then n_A := \+ n_A + 1;\n else n_B := n_B + 1;\n end if;\nA_share||n[t] := n_A/(n_A +n_B);\nend do;\nend do;\nprint(`\\n`);\nprint(DataPlot([seq(A_share|| n,n=1..N)],0,T,0,1,5,black,ms));\nend proc:" }}}{EXCHG {PARA 5 "" 0 " " {TEXT -1 34 "\n2.4. Simulating with the programs" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "As an example of a random walk we take 500 perio ds based on the seed value 1. In the first plot we see that the moveme nts of the market share of " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 205 " gradually becomes smaller and (as predicted) approaches 0.5. In \+ the second plot we see the random walk of the difference between the n umber of adopters of the two systems. However, during the 500 periods \+ " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 37 " has all the time more a dopters than " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 110 ". This is \+ not in accordance with our predictions that the walk should also shown an overweight of adopters of " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 95 ". To check that everything is in order, we extend the number of periods in the next simulation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "walk(500,1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 164 "We now make a simulation for 5000 periods with the same seed (and thu s the same first 500 adoptions). Now we see the expected behaviour of \+ both the market share of " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 55 " and the difference between the number of adoptions of " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "walk(5000 ,1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 216 "As mentioned in section \+ 2.2, random walks are not only pretty boring but also unrealistic. As \+ a first step towards more \"realism\" about the diffusion process, we \+ turn to the Polya process. We make the procedure call " }{TEXT 274 15 "polya(600,8,1);" }{TEXT -1 729 " ie. we simulate for 8 different runs for 600 periods. You may ask why this gives 8 different stochastic pr ocessen, since we have set the seed to 1. However, the seed is only se t once by us. Maple has its own internal seed value and this is differ ent for each of the simulation runs. However, if you run the procedure once more with the same seen and the same number of runs, you will se e exactly the same pucture.\n\nThe Polya procedure generates the expec ted results (see section 2.2). In the beginning there are large fluctu ations in market shares, but very quickly the system settles down to t he predicted pattern: due to the initial path-dependency the system is brought to equilibria and these equilibria seems to be stable. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "polya(500,8,1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 309 "We would like to extend the number of pe riods of the above simulation plot for two reasons. First to make sure that each of the different paths are really an equilibrium path. Seco nd, from the plot it is not obvious that all the real numbers between \+ 0 and 1 can emerge as equilibrium market shares. \n\nAs our " }{TEXT 278 5 "polya" }{TEXT -1 954 " procedure is constructed, this is not po ssible. The reason is the following. The random number generaor produ ces a single sequence of uniformly distributed numbers based on the gi ven seed value. In the case of 8 different simulations for 500 periods , there is produced 4000 random numbers. In the case of 8 runs for 200 0 periods, the 4000 numbers are used for the two simulations. When the program starts the second simulation it starts with a different rando m number than was used in the 500-period simulation. \n\nThis, of cour se, does not mean that we cannot study the properties of Polya process es. The questions of the stability of the equilibrium and the even dis tribution of the equilibria over the interval can be confronted by mak ing many simulation experiments and by studying the underlying mathema tics. In the present note we shall just consider one set of 8 simulati on Ð each for 2000 periods. The plot below corresponds to our expectat ions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "polya(2000,8,1);" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 30 "\n2.5. Empirical considerations " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 491 "When we see a time series pr oduced by an unknown mechanism, it is not easy to find out whether the patterns of the series are due to a random walk (or a Polya process). One tool for determining the question is spectral analytis (Fourier a nalysis) of the time series. A very simple introduction to this topic \+ is found in Peak and Frame (1994, pp. 187-204). Here we should just no te a few points of how the analysis proceeds. First, we define the len gth of the time interval (the \"frequency\", " }{XPPEDIT 18 0 "f;" "6# %\"fG" }{TEXT -1 304 ") for the study. Second, we study the change of \+ the variable that take place for each time interval. Third, we calcula te the number of changes that fall into different size catagories (\"i ntensities of change\"). Finally, we study the functional dependence b etween frequency and intensity. If we find that " }{XPPEDIT 18 0 "inte nsity = c/(frequency^2)+e;" "6#/%*intensityG,&*&%\"cG\"\"\"*$%*frequen cyG\"\"#!\"\"F(%\"eGF(" }{TEXT -1 18 " (with a constant " }{XPPEDIT 18 0 "c;" "6#%\"cG" }{TEXT -1 19 " and an error term " }{XPPEDIT 18 0 "e;" "6#%\"eG" }{TEXT -1 140 "), then we have a random walk (also call ed a Brownian motion) and we say that the data is characterised by Bro wnian noise or that we have a " }{XPPEDIT 18 0 "1/(f^2);" "6#*&\"\"\"F $*$%\"fG\"\"#!\"\"" }{TEXT -1 294 " spectrum.\n The reason is found in the incremental nature of random walks. Large change within a small t ime interval presuppose the unlikely case of many random moves in the \+ same direction.\n\nSpectral analysis can also be used to find other un derlying mechanisms. If we, for instance, find that " }{XPPEDIT 18 0 " intensity = c+e;" "6#/%*intensityG,&%\"cG\"\"\"%\"eGF'" }{TEXT -1 106 ", then we have total randomness \"white noise\") Ð but this is not ea sy to discern from deterministic chaos." }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 17 "\n2.6. Conclusions" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 573 "The study of random walks and Polya processes helps to the unders tanding of several concepts and problems. Among these are:\n\n¥ dynami cal systems whose movement is partly determined stochastically\n¥ the \+ law of large numbers as giving equilibria although invividual decision s are random\n¥ multiple equilibria and a selection between them becau se of small historical events in the beginning of the (eg. Polya) proc ess\n¥ the possibility of working backwards from time series data gene rated by an unknown mechanism to the underlying process Ð eg. by means of spectral analysis.\n" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 38 "3. Ar thur's competing technology model" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 26 "3.1. Arthur's contribution" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 1463 "The Polya process demonstrates a simple case of path dependence where the effects of decisions by earl ier adopters on the decisions of later adopters determines ultimate eq uilibrium outcome. Such an influence of early events is often signific ant in network markets and other areas of technology. But there is a n eed of adding a little more structure to the model before interesting \+ problems arise. The main issue is that of increasing returns and posit ive feedback. These issues have especially been emphasised by Paul Dav id (1985) and Brian Arthur (1989). They emphasise that path dependence induces a potential inefficiency arising from small differences in in itial conditions which lead to outcomes that are likely to be costly t o change. The case of the Qwerty keyboard is a well-known example of t he path-dependence problem. The current dominance of the Qwerty keyboa rd today is not thought to be due to its superiority for typing but be cause it was invented earlier than the Dvorak keyboard. Although Liebo witz and Margolis (1994) have challenged the importance of the path de pendency problem, it seems to be a basic ingredient of network economi cs.\n\nBecause of Arthur's emphasis of basic problems and a relationsh ip to the proder issues of increasing returns in technological develop ment (partly in connection with the Santa Fe Institute of complexity s tudies), he is a good starting point for entering the debate. His main papers are reprinted in Arthur: " }{TEXT 256 53 "Increasing Returns a nd Path Dependence in the Economy" }{TEXT -1 526 " (1994). The papers \+ in this book are relatively easy to access because they combine stylis ed models of core issues with important background materials. The best model to start with was originally published as Arthur (1989). A some what extended version is found in Arthur (1994, pp. 13Ð32). This paper is used for the following reconstruction and exploration by means of \+ simulation.\n\nThe starting point is an understanding of random walks \+ and Polya processes. We have already seen how the adoption decision be tween two systems " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 699 " takes place here. Arthur makes a slight addition, namely that the utility of a system is not o nly determined by the intrinsic preferences of each potential user but also by the number of users that have already adopted the system. Her e Arthur is especially interested in the increasing returns case where the utility of a system for a new user increases with the number of u sers that have already adopted the system.\n\nTo keep things simple Ar thur concentrates on systems that are not controlled and exploited by \+ a particular firm (like the Windows operation whose diffusion is influ enced by the strategy of Microsoft). Instead he thinks of the Qwerty k eyboard which is just one of the appoximately " }{XPPEDIT 18 0 "10^40; " "6#*$\"#5\"#S" }{TEXT -1 389 " ways of organising the keys of a keyb oard (although Dworak took a parent of a better layout). If we further assume that the systems costs the same, then we have a pure user-side selection process. This is the topic of Arthur's paper and other cont ributions comes from Farell and Saloner. Katz and Shapiro are among th e authors who emphasise the decision making of firm-controlled systems ." }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 30 "3.2. The elements of the model" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "We model the evolution of at the adoptions of two non-sponsored standards or systems, " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 67 " (like in an abstract Qwerty/Dworak story). At each point of time, " }{XPPEDIT 18 0 "t = 1 \+ .. T;" "6#/%\"tG;\"\"\"%\"TG" }{TEXT -1 154 " , one new buyer enters t he game and select s once-and-for-all-times one system. The buyer is r andomly selected between two infinite sets of buyer types, " } {XPPEDIT 18 0 "R;" "6#%\"RG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "S;" " 6#%\"SG" }{TEXT -1 64 ". If we ignore a network effect, the former typ e prefers system " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 33 " while \+ the latter prefers system " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 151 ". However, there is a network externality from the stock of adopt ers of a system (its install base) which influences the decision of th e next adopter.\n" }}{PARA 0 "" 0 "" {TEXT -1 76 "Let us summarise the notation:\n¥ two non-sponsored and unchanging standards " }{XPPEDIT 18 0 "i = A,B;" "6$/%\"iG%\"AG%\"BG" }{TEXT -1 26 "\n¥ two types of co nsumers " }{XPPEDIT 18 0 "j = R,S;" "6$/%\"jG%\"RG%\"SG" }{TEXT -1 48 "\n\nThe core of the model is the utility functions" }}{PARA 0 "" 0 " " {TEXT -1 50 " " } {XPPEDIT 18 0 "u[j] = a[ij]+b[i]*n[it];" "6#/&%\"uG6#%\"jG,&&%\"aG6#%# ijG\"\"\"*&&%\"bG6#%\"iGF-&%\"nG6#%#itGF-F-" }{TEXT -1 7 ", where" }} {PARA 0 "" 0 "" {TEXT -1 3 " ¥ " }{XPPEDIT 18 0 "n[it];" "6#&%\"nG6#%# itG" }{TEXT -1 39 " is the number of adopters of standard " }{XPPEDIT 18 0 "i;" "6#%\"iG" }{TEXT -1 11 " in period " }{XPPEDIT 18 0 "t;" "6# %\"tG" }}{PARA 0 "" 0 "" {TEXT -1 3 " ¥ " }{XPPEDIT 18 0 "a[ij];" "6#& %\"aG6#%#ijG" }{TEXT -1 51 " is the intrinsic preference of a consumer of type " }{XPPEDIT 18 0 "j;" "6#%\"jG" }{TEXT -1 20 " for system of \+ type " }{XPPEDIT 18 0 "i;" "6#%\"iG" }}{PARA 0 "" 0 "" {TEXT -1 3 " ¥ \+ " }{XPPEDIT 18 0 "a[BR] < a[AR] and a[AS] < a[BS];" "6#32&%\"aG6#%#BRG &F&6#%#ARG2&F&6#%#ASG&F&6#%#BSG" }{TEXT -1 34 " (to make the choices n on-trivial)" }}{PARA 0 "" 0 "" {TEXT -1 2 "¥ " }{XPPEDIT 18 0 "b[j];" "6#&%\"bG6#%\"jG" }{TEXT -1 18 " is the size of a " }{XPPEDIT 18 0 "j; " "6#%\"jG" }{TEXT -1 90 "-type buyer's evaluation of the effect of e ach of the adopter's effect on his own utility" }}{PARA 0 "" 0 "" {TEXT -1 2 "¥ " }{XPPEDIT 18 0 "b[j];" "6#&%\"bG6#%\"jG" }{TEXT -1 62 " reflects the character of network externalities\n " } {XPPEDIT 18 0 "0 < b[j];" "6#2\"\"!&%\"bG6#%\"jG" }{TEXT -1 45 ": incr easing returns to the scale of adoption" }}{PARA 0 "" 0 "" {TEXT -1 13 " " }{XPPEDIT 18 0 "b[j] = 0;" "6#/&%\"bG6#%\"jG\"\"!" }{TEXT -1 27 ": constant returns to scale" }}{PARA 0 "" 0 "" {TEXT -1 13 " " }{XPPEDIT 18 0 "b[j] < 0;" "6#2&%\"bG6#%\"jG\"\"!" }{TEXT -1 65 ": decreasing returns to scale\nIn each round one consume r of type " }{XPPEDIT 18 0 "j = \{R, S\};" "6#/%\"jG<$%\"RG%\"SG" } {TEXT -1 139 "\013comes with equal probability to the market and buy o ne unit. After the purchase the consumer is locked-in to the chosen st andard for ever." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 5 " " 0 "" {TEXT -1 19 "3.3. Model analysis" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Since one system is chosen in each period, the total numb er of systems in period " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 11 " is simply " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 14 ". Thus system " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 2 "'s" }{MPLTEXT 1 0 0 "" } {TEXT -1 29 " installation share at time " }{XPPEDIT 18 0 "t;" "6#%\" tG" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "n[At]/t;" "6#*&&%\"nG6#%#AtG\" \"\"%\"tG!\"\"" }{TEXT -1 160 ". Is is, however, more operational to \+ discuss the dynamics of the stochastic process in terms of the differe nce in install base between the two standards, ie. " }{XPPEDIT 18 0 "d [t] = n[At]-n[Bt];" "6#/&%\"dG6#%\"tG,&&%\"nG6#%#AtG\"\"\"&F*6#%#BtG! \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "R;" "6#%\"RG" }{TEXT -1 46 "-type agents maximise their utility by buying " } {XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 4 " if:" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "a[BR]+b[R]*n[Bt] <= a[AR]+b[R]*n[At ];" "6#1,&&%\"aG6#%#BRG\"\"\"*&&%\"bG6#%\"RGF)&%\"nG6#%#BtGF)F),&&F&6# %#ARGF)*&&F,6#F.F)&F06#%#AtGF)F)" }{TEXT -1 10 ", and thus" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "(a[BR]-a[AR])/b[R] <= n[At] -n[Bt];" "6#1*&,&&%\"aG6#%#BRG\"\"\"&F'6#%#ARG!\"\"F*&%\"bG6#%\"RGF.,& &%\"nG6#%#AtGF*&F56#%#BtGF." }{TEXT -1 28 ", or, in abreviated form: \+ " }{XPPEDIT 18 0 "Delta[R] <= d[t];" "6#1&%&DeltaG6#%\"RG&%\"dG6#%\"t G" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "S;" "6#%\"SG" }{TEXT -1 17 "-type \+ agents buy " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 4 " if:" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "n[At]-n[Bt] <= (a[BS]-a[A S])/b[S];" "6#1,&&%\"nG6#%#AtG\"\"\"&F&6#%#BtG!\"\"*&,&&%\"aG6#%#BSGF) &F16#%#ASGF-F)&%\"bG6#%\"SGF-" }{TEXT -1 28 ", or, in abreviated form : " }{XPPEDIT 18 0 "d[t] <= Delta[S];" "6#1&%\"dG6#%\"tG&%&DeltaG6#% \"SG" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "We thus se e that all agents choose their preferred standards as long as " } {XPPEDIT 18 0 "d[t];" "6#&%\"dG6#%\"tG" }{TEXT -1 25 " belongs to the \+ interval " }{XPPEDIT 18 0 "[Delta[R], Delta[S]]" "6#7$&%&DeltaG6#%\"RG &F%6#%\"SG" }{TEXT -1 101 ". The size of this interval is determined b y the parameters. Let us study the decision of agent type " }{XPPEDIT 18 0 "R;" "6#%\"RG" }{TEXT -1 9 ". \n\n¥ If " }{XPPEDIT 18 0 "b[R];" " 6#&%\"bG6#%\"RG" }{TEXT -1 57 " approaches 0, then the interval approa ches infinity and " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 65 " will \+ always be bought. This is the constant returns case. \n¥ If " } {XPPEDIT 18 0 "b[R];" "6#&%\"bG6#%\"RG" }{TEXT -1 19 " is negative, th en " }{XPPEDIT 18 0 "Delta[R];" "6#&%&DeltaG6#%\"RG" }{TEXT -1 46 " re presents the highest possible dominance of " }{XPPEDIT 18 0 "A;" "6#% \"AG" }{TEXT -1 144 " that will motivate its adoption. In this decreas ing returns case we thus have an interval with \"reflecting\" boundari es. In this case have that " }{XPPEDIT 18 0 "Delta[S] < Delta[R];" "6# 2&%&DeltaG6#%\"SG&F%6#%\"RG" }{TEXT -1 7 ".\n¥ If " }{XPPEDIT 18 0 "b[ R];" "6#&%\"bG6#%\"RG" }{TEXT -1 19 " is positive, then " }{XPPEDIT 18 0 "Delta[R];" "6#&%&DeltaG6#%\"RG" }{TEXT -1 44 " represents the lo west possible weakness of " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 60 " that will motivate its adoption, and beoound this boundary " } {XPPEDIT 18 0 "R;" "6#%\"RG" }{TEXT -1 29 "-type agents will always bu y " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 28 ". In this case we have that " }{XPPEDIT 18 0 "Delta[R] < Delta[S];" "6#2&%&DeltaG6#%\"RG&F%6 #%\"SG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 50 "The cases are d epicted in the following figure:\n\n\n" }{METAFILE 470 344 344 1 ":::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::>tn\\:FZRGZPU>?rySjyyyyyY:B:j:Tg:FZR?:<:t[;b;BBHZBZ:::F:Z:F:BJ=bZx=hFnVbujx[S=MJ:\\SwW:xIoYeyylF?:V:>mvuvyy;[J`yyyA\\WAyy Y:By<:li;ryN:VY>MH:yI`uQ:jw;H:^YEryyAB:IfyHJFkx;S@nnya;xiR@:yQ:JxG:ya< vYpi:V:Z:>M:V:Z:>MM:_B`J:::NyyY:byDJF=:p[:NqRly?:xIyAkF`:o?wArMvYwQ?\\C>:oghNYEZR\\x=[yyEr[T@Wiwyy;NqRlyoY:;JFyIcyyIBmkfuWAkF`J::oQyw\\Rpwh=ryr[M_y]yeyTKFs[k^yE:\\CZtPQ TtyA:wqwAM:oghnNbyP:yAwq:Byjxw:\\Cbyh:JF;NmRMvYwQ?:;JWeEoq Zy]K:>mR<;:NYwqE:Zy]AHOZyeyT;XRFsyvYvY>mymykwyy<:diC:>MnxyI=Zy=::byjxw[ROuHxyY;J:nYRvy i;SpJxYSDMWEyyA[:vY>MvY:cQ:TPxCKgyyM:B`RxjxO;\\CxI\\sxg;xIxIZte:Ze:vY: :>mymhyIyA:[B>eLLyy:byP:>:wq:Byjx[y]y]RhY]J:xIxI\\sysyrx:NYM:B`bwyAjy; ::nYRyX;jy;>MvY[B>eLv;MFYRyd:k>yAkVyp:jy;NY?:V?Z y=:ZRxYpGxI:mwU>gyy:xI::Z x=uqE:xIZR\\yE>[RO\\IJ:nYRv^S:B:yAk>yAJt;:ZR\\s=uQ?B`ryB`uq@:xIZte:Ze: vY::>mymhky;JB:xId i;nw>gYiryyay=::by:m:ry::pwdZWyylN@gyy=bur=yyyvY :JtwZJ>\\WEm^sqyyC:vYFYRy`:jyKxk:vyuex>:sAmryyay=::Zuu;yyyu@`TMyy::xIdi;nWv<:wADIZl:t HxZymy;:cABm::yAmApiC:vY^YF:hitM:^YF=ry:::rxX:u@`TMC:ti?:;JyG:li;jxWYA :liuu:xIdiW::yA::Jy[wU>vYvY:JxO:NYE:byJs;::ry :Zo=::ryRxjxG;Zy]vU:VYbw@:sAhiS:vY:::kAnNry\\C>:wAlIfY=ZuU:^Yrx`:jykwG :TiC:vYNYu::xI::Zx=uqBryry:Zve:bxP:Jy[l=::jy;:eA::jykv[wU<:yAsq:rxJuG: lIVYm:ry:::\\IBmyQUiwye:yI`>MyI:tI^YRyJF]Cyyy^YuZRoyyEZR\\xU:VX?B`TPyyY;XaApx yi\\:r[G=ry:XrNSwnf:vjyqw[F^qoXMZy]y]su=BmbcyA\\Ckvyby:=Zw=kF`vyyMxg=:ry:\\C:bx`:yAyAoqHr[GM:ZAHOZR\\y ]t\\xuAFOjyKFkyky;^SNYEZR\\xU:VX::eZApFZy=r[M_o=I>j>ryryRxjN;yAJxg;\\C xI\\Cdi;Jxg?Zy=o?ry::B`k>BmbcZRLFyItIj:RyJF]Cyyy^Yu::xIZR<:diCjykyKwg= XRm>:r[M?B`rybhbyx;M?vY>MvYvY:S@oq:oq>Z::di:ByDjyCyZwe:xiv]xE:kfy;b;yAyQFsmwE:di;B:sAV:ByDkyyY:vY::kVyZc\\FqnyaY\\SyyItI^Y=Zw =L:M:>:FO^st]wE :knyA:cA@mxyYnYBy@:uA^:>:sQ:j?:yQ:JFs:B`>ZRPYM:<:bxByD:kFyZwe:Byby< ZxAZ@:yQFs:;JxO;Z:ZR`YeAsQ:XAZ::`ItY:h:vY:sAuq ;JFqi:bx@Z:^Yj;Zve>>:yA:JFqIRdBmSxyll:yykyy:RZDtyA:nky=DDJA;DD^oyEH=vy AA@b`xy?JWk:oKZ`<:W>bc:^OPm:yyk;B:MWy>m;xyH<::::BeyqF@ryAryAB::R:xY:`; FJ::j::RZ::Z;:j:;::=:Z;<::@::=>::F::@B::R::FJ::j::RZ::Z;:j:;::=:Z;<::@ ::=>::F::@B::R::FJ::j::RZ::Z;:j:;::=:Z;<::@::=>::F::@B::R::FJ::j::RZ:: Z;:j:;::=:Z;<::@::=>::F:Vjvyy;nWwq::\\Ckvc yY:>ryI:Tbpj[;:BuJn[;=:JZ[J=o@\\fmyEyD:SXy:>Mbxyyx@ZRxyy?:yI:rxmyE:ByZR`YoAZmv=:?:sq>:<:hi:ByH:lF=Z[=I?HF::lIBmV:ByDjySyD:JF[t];: :M:vy:ZuE:li=ZFqY=Z\\=xYF=:fYA:k^Zw=yQ:vYwq;\\SquhxAZDZyE:dUWcryY:nYaB:;ryB`::tiY>mhqW;:J:NqR:<:hi:ByH:lF=Z^yy;>:oq=B`jyuIZ:bhbyjx[t@MbyDZ:byJvk>Z:B`:fYr y:JF;:nYy:<:Z:bhk>JWsyrytiD:MnY=Zw=j yoYAB`byd:xi:bhRr:sAyQFqIxiuU:j;J`qIJyw:B:xIZR`YVYZxYy;>:o?wApINqR:wA\\I`:>:k>Zw=yQ:vYwq;nWtiDZyE:dUWcrJyW\\:: \\C:ZxuAJ::>:oG`:dEyAyQyO;J:vJJyG;ZR\\xU:fY:xixu:k>wQ?jy?:o?=AByry\\sx kyWY=:H:lvx;tiA:;jy;>mv]u=N:^YM:B:rx\\OCyyY:FYMZyE:\\CtI>Y?B:fyyiZ:ry:fYOB:xIxI\\CQZRxYxiJ><\\Osqyy:Jy_[R< ;JF]CnYy:<:Z:bhk>JW;xixeMnYZA:vYwq;nWtiDZyE:dELxyA:oqC:xI \\sysyYjxg::jxO;xIxI\\CQ:yayE>[ZZ=wq@Z:ry:hi[Zw]:vY;JFyIyAJwO:fY:ti;jx;ryB`V;vYxiJ>\\Osq:wQ@:;JF]CnYy:<:Z:bhk>JW; xixeMnYZA:vYwq;ZxeY jxg::jxO;xIxI\\CQ:yayE>[ZZ=wq@Z:ry:hi[Zw]:vY;JFyIyAJwO:fY:ti ;jx;ryB`V;vYxiJ>\\Osq:wQ@:;JF]CnYy:<:Z:bhk>JW;xixeMnYZA:vY wq;Zxe:>:kvyvYVY?:uq<:r;JB<;uA;ryMh:x IyQB:xIliG:vYu@`Tk[xu:jwW:[RODHNYe::qq:ByhInWV=:Mwyry<::ZtU:^YnY=Zw=jyKxg;Zy UqjNAH;:;JFkykwO:fYE:ZA:[B>fY>ZyE:\\sysy:diMh:xIyQBYBy@:oq:Ry`:\\CxIsq<:b:brdJ:vYkF :B:hi;jw[vU:VYEbF_;liPNQZy=:Zt=sq:bxjx;k>yavU;:N:FVUB:Zy=rx@:sqwW:::`t yky;:Jw[v]x=uyy:xIliG:vYu@`TlyY;<;:<::liYBy@:oApiCZR\\y Mxg::D:DHvYk>MvYli?:J;jjC:Zy=rx@:sqwW :::`tyky;:Jw[v]x=uyy:xIliG:vYu@`Tl;:<::liYBy@:oApiCZR\\ yMxg::D:DHvYk>MvYli?:J;jjC:Zy=rx@:sqw W:::`tyky;:Jw[v]x=uyy:xIliG:vYu@`tnyYwQ>>ZctyB`wq;xi:Byp:jyK`KkyYGJFC`yyy;JF]Cyyy:J xG?\\CZAmRwQ>RdxI\\ SysyByp:jyK`Kk;k^RxY>mRlyyI:^YM[RMtIFOZx=^YU:ryBmR r>:jy;VY=:@:qq<:b?ryry::bxJxG:dIfyy]yAMxyyCyP:yAN:^V:vY>mRwQ>RdxI\\SysyByp:jyK`Kk;k^RxY>mRlyyI:^YM[RMtIFOZx=^YU:ryBmRr>:jy;VY=:@:qq<:b?ryry::bxJxG:dIfyy]yAMxyyCyP :yAN:FWlXxyy:thxE;[bhyyykVyh:ryZc\\FqyyE`wQ;xiR`YIryZ:by<:dyyKx?ZRhy:> ZRty;Jy=J;:AryBy\\:jyGO^s\\yyvE:vYpi=ry\\sx_:>:sQ:j;JFmiMH:^YK:ZFIvyi<;Z::ZR@MZvUD>M:I>J`aDyAyA:>mtLw;P :kF`nYKZclyKyW:Byp:yAyAS@Cyy=buZt\\wu:uQ:\\c yA:wI:?:V:By\\:yAyqNK`EHjySyH:\\sx_:>:sQ:j;JFmiMH:^YK:ZFIV>ZyE::B`k>By`<:vJ:SXdvYvY:JFeE:P:kF`nYKZclyKyW:Byp:y AyAS@CAZt\\wu:uQ:\\cyA:wI:?:V:By\\:yAyqNK`EH jySyH:\\sx_:>:sQ:j;JFmiMH:^YK:ZFIV>ZyE:: B`k>By`<:vJ:SXdvYvY:JFeE:P:kF`nYKZclyKyW:Byp:yAyAS@Myy=rP=vywvyrcl:yAy A>mwE;B:li:RyH:\\SxO:`DY:vY;:JFcxJx;;B`xY:ny:N:\\CA:liBjy?:yqNC:SXZyi[ yY:vNZ::B`uQmve:FOx:ry<:ZRLw[v=::kVyL:MdI^YJ:\\cyMy=:\\CA:liB:;jyGO::FZy=RZ::Z;:j:;::=:Z;<::@::=>::F: :@B::R::FJ::j::RZ::Z;:j:;::=:Z;<::@::=>::F::@B::R::FJ::j::RZ::Z;ZXFjyI JmvyB_^aQyy;j:OwyAZ:Jiy y:ZdU::;:w@bt@:J:Ji[N:vuyi Y=vyQQyKp;xGrtZf=vUflyeGkxyuq:_WwA;yysq:yypgY^WJj;UA^WJjKEcG^Y=Nqyyxkj yywGjyiWsAWAFVjp;WAFVnlymfyywGJWQxKkyYwGjyywyYYnWJk;YAnWJkkEyY:T=oq:?x yYYVVmq:yyqOY>Xjk;YA>XjkkEyIoxymq:?XoACqvKrGy::oACA>XmANY^VvL:=nrGy:Jl ?Y=vy]A:^Vjr;:CAkN:T=kq:OXmAEyyiq:yyr?YNXjl;]ANXjlKFyIqxyiq:OXkA=Q:<:W quKs;=AnWJs;=Ac^NotyE:=nqwXGvy;jw?Z:NXoA_AcAW>qA_AcA=>VJNYNX^ X^LPVxyY:NYNX^XZI\\t]l]n]I\\u]l] n]>oA_AcAY>qA_AcAC>:wq::Jy;\\ILCwYyD:ZxU::by:kA[vy:ya:ry`yy=ZJd:;Zu]xU::byZ:>Y>L;B:oQ:NX^XfJJwg :ya:ry`IY>L;B:oQ:NX^XfJJwg:ya:ry`ILc; v`xyy<>:dIjyK:xyyC:xI;jyGYFlyQZ:jwKyG::nYJ:\\ITrycyxiy E:ya:>YZrM :@C>Zt=vY>ZyyYYZrM:DsyE:;bx:yA;ryya:ry>:yqvk=yYxOYAB:yyypyy]ZyyYyI:x iwE:;Rxb^ZrM:Dc:yqvk=;A:piB:yI:xiwE:`ILCi C:yIy:J>ZyA:yqx?:mAavy;Jykw[yiY:vY;ry>ZrM:Xb:>Zx]tU?jyK:vYZA\\x]tU?jyK:vY:y a:ry>:yqvKD_YV:Ry\\:vy:rypi:RxB\\yax]s]xU:JyGZy]=Z:>YVLnY;JwgKnYFYnY=:ti;:A:;Bxr^by V:Ry\\:vy:rypi:RxB\\byRxby@:nY=:H::ya:ry>:yqvk:;c>w AmAwq:ZxU:j;J:\\ILCti:bx:yA;jyC:xI;jyGYFJJDS]gxyflyE:wAmAwq:ZxU:j;J:\\ I@BZc]O\\xE:`Iti;:wq:Z=:kA=nh;MAe>wQ:FYnY=:ti;:A:\\I@bW]NLUTwy=Z;dtZW= `HbtZW=`HbtZW]N=No;u@UAqq:by:rxZ;dtZW]g]ueyPjy?Z:vYuQxyIi>=No;u@UAqAZy=r_RZd HRqRurx:vY:?`k;=No;u@UAsq;<:;RyD:;Rxr_RZdHRqRuBy::jEk:OWjhkpKx;:Z:yQx?:i>=No;u@UAwA:yQxK[CZgxy;j:OWjhKpoYwQyW:>ZxU:J:J:VMFJOAfU^WnYwq;ZxU::>:q> =No;u@SAwQyW:by@::;J]cGB:=No;u@FWJo;u@FWJo;u@c^p:=No;u@FWJo;u@FWJo;u@c >j:OWjh;MANWjh;MANWjhKD;FJOAfUjn;OAfUjn;OAfU^L:=No;u@FWJo;u@FWJo;u@c^t otym:OWjhKpgYmyynYNxyE:ca:>xyAZ:vqym:OWjhKp;nYNX^X>XvQFJOAfU^W:wA_AcA[ AOyyTReyI=NoC:fU^W;JyKw?:wA_AcA[Ay?=No;u@SAwAoAwA_AcA[Ay?=No;u@SAwAoAw A_AcA[AOq:LLeyI=No;u@SAwQ=vyymyoY=:hyyA:wQyW:byR:j;:H:;jxoY=J:tiZyIZfntym:OWjhKpgYKryyI:yyy;:wq; rypiZryyqy=jyyI;jy;::>Ztu:Jy?\\:vYxI:yQek:OWjhKp;vYJyW:Ry:yI:xI;jy;::>Zt=nY[:xI:ZyuynYA:lI:xY:vY:Zwe:RxZ:nYI:xIpi=Z:fTFJOAfU^WnYK:xiyMyW: By:ry;jy;:pi:yA::jv;?:wq@ryxY:xyyC:xiP];dtZW]f]xE<:yaxu :jx;xY:vYZxeypi>:B:\\ItIpixey@:wAPG@bd=pGlH:uQ=Z yMxk:Z=Z:nYwqx_::;JvKykxoYwq:byDZ:B:g<@bd=pGLvy?Z_yy:RyM:XxyA:sQ:FSFJOAfU^RvV:`F@bd=pGLFXH:cap: =No;u@FWJo;u@FWJo;u@c>j:OWjh;MANWjh;MANWjhKDCRgxyFJOA]Q:ZyyYoAgAG>IQ:fU;jNyI=>FX :JuK=kmkhkNk:;]A:nXnJvVfU^tyAZBqtyA:=>FXOB:xIya:vYwQxyy:bytixU:::nYuvy vVnU;JNyI=>FXZ:vYIAw@K?=>FXZ:vYIAw@a p:hLeyI=>FXSB:yqy=:xY:xIsQI:ya:ry>:yAJIyISq::ti;:::PiFXjy=:xiyMx;xIJ:vYSAtix];:::PIBoyY:WcoyI:@BRvZyA:vYyavE:lyyuJ:xIyI:xI>ZyUyPjysy:ya:ry>:ya:ryXyyE:\\tySZZkM:ry;:yqyCyZyUyZyE:w>UAry: ;jyC:xI;jyC:xiq=\\D@BRv>:xY:jywYli:Byyi=jy=Zy=;rypIryUAsqCZyAZyyYZyyyw= tILGyboya;<@I;Z:ry;:yaveAjy;UAsyyU:xY:vY:ya:ry:tCpHlyyq:vy:ry>:yqx;gAaxyd=FJjr?:wQOFJjrKy_:B:uAsAr;jyC:wq::j;ZX\\g=vYZ:ry >:ya:ry>:yqvKy?:K?=>FXnYC:lIoOnk:WxyAZ:nY>xym:GW^KD=NQj:WXnY>XFJMAS^l:o?F JaAwA[Ac^N;SNvkxGxyAZ:FJJp?:=nnK@;:=nsKpk:GW^K:j:WX^W^Ld?FJ?jyoYFXFJaQ y=JqyY:jn;?:jrk:WX:FWJ;:FXFJaAJD;FJ?:jrk:?X;jqyi:FJMAN:nYFXFJ[AYa:FJMA N:nYFXFJ[AYa:^LXOeyI=n:byBvya;Tvxym:GXZh]NLgZ;D:yYy;ruyAPItItH@bc=<::PItItH@bc=<::PItItHLC]doya;t:;Z:nwy=e Qy_Wjn;A::PI:`Hr::jt;^L\\>FJ;B:wAYyyj:OWjh;MANWjh;MANWjhKD;FJOAfUjn;OAfUjn;OAfU^L:=No;u@FWJo; u@FWJo;u@c>j:OWjh;MANWjh;MANWjhKD;FJOAfUjn;OAfUjn;OAfU^L:=No;u@FWJo;u@ FWJo;u@c>j:OWjh;MANWjh;MANWjhKD;FJOAfUjn;OAfUjn;OAfU^L:=No;u@FWJo;u@FW Jo;u@c>j:OWjh;MANWjh;MANWjhKD;FJOAfUjn;OAfUjn;OAfU^L:=No;u@FWJo;u@FWJo ;u@c>j:OWjh;MANWjh;MANWjhKD;FJOAfUjn;OAfUjn;OAfU^L:=No;u@FWJo;u@FWJo;u @c>j:OWjh;MANWjh;MANWjhKD;FJOAfUjn;OAfUjn;OAfU^L:=No;u@FWJo;u@FWJo;u@c >j:OWjh;MANWjh;MANWjhKD;FJOAfUjn;OAfUjn;OAfU^L:=No;u@FWJo;u@FWJo;u@c>j :OWjh;MANWjh;MANWjhKD;FJOAfUjn;OAfUjn;OAfU^L:=No;u@FWJo;u@FWJo;u@c>j:O Wjh;MANWjh;MANWjhKD;FJOAfUjn;OAfUjn;OAfU^L:=No;u@FWJo;u@FWJo;u@c>j:OWj h;MANWjh;MANWjhKD;FJOAfUjn;OAfUjn;OAfU^L:=No;u@FWJo;u@FWJo;u@c>j:OWjh; MANWjh;MANWjhKDcOj:OWjhkpyYy;wAovy;j:OWjhkp;nYNMFJOAfUfW:wAExyD@FJOAfU nWnYfW;JyKGk:OWjhKqKykpKyKGk:OWjhKqKykpKyk\\sZj:OWjhKqkwC:ti;>:ZuE:ti: b`RZdHRqburx@:wq::VY;Jy?:o>=No;u@WAqq:by@:ZuE:ti:RkflPE:@bd=pGtHliGZ:v y:vyyM:vYsQ:nY:wvy;j:OWjhKqKxg;ry;jyC:xivE:tIZX\\;dtZW]h]vU=jy=ZyM:vYs Q:nY:M`wotym:OWjhKqkwoYEjy?ZyyywEybyb`RZdHRqburx:vY:wAo>=No;u@WAqAZy=b yRkfIRZdHRqbuByH:ZW]B];:yQx;;JGkt;>VZ:nY=:Z::tCTItIhIZB]h]xE:hi:Br>ZX]<];ZB]h]xE:hi:B r>ZX]<]NT:choyaqyy:bybxyy:B:wQBZy]:vyyM:vYuqHryxIJ:vYZ=yY@Brw:ya:ry>:ya:ryXiZ=]NLgUwyswpi:vywAZxu:B:wq=Zy]wu:B:sQ:jCZyAZyyY:ya:ry>:ya:ryXi:ya:ry>:ya:ryX i:>yyA:wQ;Jyg:ry;bxyI:pi :Byp:jyyI;jygY?:gQxwrymqyyx?:uQyW:>Zxu;jykxW:>ZvE:r>jy=jyyI;jy[:vyyiYk AwQ:VRFJJvW:>ZrE:ti:yAYnY FYnY?:wq@J:xY:xyyC:xiwUAjy;:I@[yyAB:BxbyD:ti?:yI:oAuQ :^YU:ryya:rypi:ya:ry>:y a:ry`IB:XFXA:\\Iti:yqxg=ry:;jyC:xI ;jyC:xis=ZxU:: :mA>:J_Kr?:wAmAwQ;Jyg;ryxyyC:xiwUAjy;:yxyVZxU:::mA>:JkKyknKoKykw?:;Aw@YyynYFWnV;jx?:A@GQy?WFXnYvXnYfYwQyG:t IVY=Zxu;B:ya:nY=::RxZ::DHtI`HdHtIhi:BrbqRy@jWBy]` ]x]c]X]w]=]\\]x]c]d]x]vey@:N:xIwq:byjw[xeyR:::Zo=;BvyQ:ti:RtBtJx?:C@GA wAMAw@uAA@?AwAMAOAwAsQyG:b:vYti;Jy[u=wQyk::::eA][M:tI\\ikutyax]tu:B:wQBZyyYY:wAoAby<;vY>ZyyYwAMA;AZ>][]x]r=bybx:wQBZy]:vyyM:vYuAry:;jyC:xI;jyC:xiq=pi;nbmy Yti:Bxr[yax]c];dxyax]c]eyywE:pixu:Jyw:ryRyH:ZyyywExZyyy wExZx]rUvZx]c]; ZxE:`HXBtI\\ik=byRtBu:kQ:nY?:wq:yqxO:nXfyyIZvWxyMy?:M AeQyWY;Jj?Z:nUuQmyIwAMA=nvyYy?W;Jp?:wAmAwQ;Jyg;ry;ryya:rypiW:yAB:xI;jy C:xI;jyGYJ:vwyE:wAMQ:fXVY>VZyAZyyYVZyAZyyY< ZyUy:yAB:xI;jyC:xI;jyGYJ:fY=Ba_yywAvX;JykwKjC:yxyWvXnYVY;JjkiyIwQ :nVnYFWfWwqwKjKi;fW;JykuKykxKyG:tIVY=Zxu;jyC:wq:::`IB:R:xy:HPeyIiAkQyG :tiYF:b:byt i;Jy[uU:nYwA=:::fXnY;jlKnkqyIwAqQ:>VvuyMy?:C@o@uAAp::ya:ry>:ya:ryXiwukyaiyy:>Zr];:?:wQyG:;byjwG:tix]; :::PIti:BkBqyY:;Jx?:C@cP:vXwqv;;by<;vY>:ya:rypIjy;]N]qey`IB:wQBZy]:ry>:yqx;xIJ:v YZxu;jykx=j;J:lIr>jy=ZyM:vYB:yqxwxyA:wQ:>SvwyqYmA>ZxE>jyK:vY:vYfYA:;By<:H;vy:ry>:yAZxE>jyK:vYZvE :r>jy=ZyM:vYB:yqxwxyA:wQ:>Svwy=J:tiJ:yA;jyC:xiw=vYZ:ry>:ya:ry>:yqugY?@ cn:dTeyIeAmQ:nY?Z:nYE:xY:dIpi:Byp:jyyI;jygY?:eyynY;j]Kp_YV:byXJ:jykxW: >ZvE:r>jy=jyyI;jy[:vYuquyY:nY;j[?:=>fXFYnY?:wq:yA ZyAZt]wE:liG:vyyM:vYuQ;jtKy?:I@SAV:byXJ:jykxW:>ZvE:r>jy=jyyI;jy[:vYuqu yIwQ:VR^L\\VeyImAwAmAwQ;Jyg;>ZyAZyyY:ya:ry>:yQv?:>:I@Uyy FYnY?:wqFYnYFYnY?:wq@ZyAZyyY:ya:ry>:yQv;;j]kpkvKyO:nYE:xY:dIpi:By:vyyM:vYuQ;jtKy?:?@=>FYnYFY nY?:wq@ZyAZyyY:ya:ry>:yQv;;j]kpkvKyO:nYE:xY:dIpi:By:vyyM :vYuQ;jtKy?:?@c^lqT;jvoYgyynYfYwQyG:tIVYbyXJ:jyC:wq:::\\iwUkyakE:;byZX]NyY`ixewby RytixU:nYrx@:wq=ZyM:ti;::>Yuq\\krKy?:mQ:nY?:wq@ZyAZyyY:y a:ry>:yQv?:>:?@MAqQ:>V:ya:ry>:ya:ry\\i:Z:bjRtrxY:QAwQ:VY;JjkiyIwQ:^TvVfY;J?krK uKykxoYwq:byjwG:tiA:ya:nY=::Bx:hHti:rx:sQ:NKvuymx ?:E>QAkQyG:;byD:tixU:nYrx@:wQyk::::eAwQ:^TFWByb\\rqyaw]?\\e]r];:?:nY=Z x=qAtix];:::PIti:BoRtJxK?kiyIuAE>QAkA=:D:by@:wAhInYwA=:::fXnY?:=r>gxyF JOA=q::ya:rypIjy;:ya:r y>:ya:ryXIpyy=JvUwySZ:pG;BtlIH:nYI:xIpY:H:ZxU;B:yI:oAuAsq @:xyyC:xiwe:RwbyZx]CD:@BRuyax ]s]xe:bypJ:jywyyM:vYuqHZy=>:ya:ry>:ya:ry\\yy=::yqx;xIJ:vY:ya:ry>:ya:ry\\IB:pi:VyP]a]x]C\\;<;Rutip]x]weyti;Jy[u=wq=<: xI;by@::JvgY[xy;B:IAwQ:FKFJjpKuKykxKyG:tIVYbyX:vYYVY;bmgxyFYnYwqwG:ti::hIj::::EwyFJjgyY:mAwQy;ti::hIj::::E?=>VUfYfLFYnYwAby<:rx:=:::fNFJjgkxKZsrgxyFYnYA:p i@ryvYti:ByXK:xIyI:yyy>:ya:ry>:yQMyI=>fU=B:li:R_RxbyH:fYZyMy?:sqEZyA:y a:ry>:ya:ryTD@BRq:;jDkvKyW:Ry:yaxE:liQ:yI:xI;jyC:xI;jyoNFJjh;ZyUyH:fNFJ:MAmAwq;J:lID:ti:By<:ti?Z:vYuARcRZpGRt RxbyH:ZyUy:E?=nhKD?Zt:fY?:oAG? =>jnkvKyW:Z@:yaxE:lIpi=:uQ;:nNFJu@FWFYnYA:b;jycy:ya:ry>:yQMk:;FWFYnYA:pIjycy:ya:ry>: yQMk:gUjnkvKyW:Ry:yaxE:lIjy=ZyM:vYZxU::PD@BZc]s]xu:JxK;Jy?:qAtixu:B:wq::fNFJu@FWFYnYA:lID:ti:rxJyo YA:;by@:Z_\\;TqB_n@RZdHRqZc=dHRqZc=dHRqB_:@bd=pGRtZd=pGRtZd=pGLCZ;dtZW =`HbtZW=`HbtZW]NCRZdHbyyi:o_]yYRtZd=tI:RtZd=tI:LCZ;dtZxMSrr< :`HbtZx=:`HbtZx=ZNLgZEd:o?RZTZyQQyiFLxy=J:Rtr\\:RZT::Zc]EZ;d:dUStI_chEyyjn K@;:=N;::RtB]:Z;d:::c^];S^l:Ca:FJN@wq:de_E:RtB]:Z;d:::FW^K:j:O::ZNLmZG xy;NQwal:@b;NQsalbsZc]GfP]HT:^YD]`];DxRyxh:Zc]H<:RZ\\I: Rtb]::@br=ZNLmZHLS^Yde`];Dwj:oX:Zc]<Zx]t]p]A\\g eylHtIZXD:@BBvbybxbwr[Ru:tIZX\\;<:ya:ry:xC;buByp:vy:ry>:yQx?:wA:NNFJJr_[: Z:Zyuy^yyq=ry:;jyC:xI;jy;vM:>:vY yavyYliH:yI:xI>ZyUy;R;xiyE:y^:nWVYwqZyUyP:x i:vM:>ZyuyyqyCyyI:liB:yI:xiw=li=:y^:nW^YZw=`ItIZX\\;< :>ZyuyyqyCyyI:liB:yI:xiw=li=:y^:nW^YZw=`ItIZCmOTG@BBvlJ:J:xiyyYyav yyX:yAB:xI;jyC:xIZYL:tHtiI>:;jywy:ry>:yQx?:>:JIk:;[Q@<:>X;B:wQ:pIlInYI:xyy?ZxU::r:jIC:UAwQyW:by@ :qA>::NNFJJr?:wQ<:uAsAtiA:yyyZg]xeyH:nY=Zu=<::pf:n@RZ:pG;Zc =dHRqZc=dHRqB_n@RZdHRqZc=dHRqZc=dHRqB_:@bd=pGRtZd=pGRtZd=pGLCZ;dtZW=`H btZW=`HbtZW]N@bd=pGph:bybxbybvBwBvbiRZdHRqRuby bxbybvBwBvbiRZdHRqRubybxbybvBwBvRt@:RZdHRqRubyTZy]yyyxU:xYyWY=ZxeyH:nY F:r::AZ:Ryti;B:wQ;: D:by@:ZN];dtZW]g]xe;B:xixU:xIqAtixu:Jyk:Z=:V:RyR:<:?:nY=:HZ:vY=REgxyFJ OAfUfWuQ>vYyyy;:wAyyypiZryvy:vyyM:vY:::;bxH:nY[:xI:yqdk:OWjhkp;vYJyW:Ry:yI:xI;jy;::>Zt= nY]B:yyyxI:;jy[xu:JxoYcZymy=Zy=:fY?:mq;J:tiA:y AuI:A:;Bo>Z;dtZW]gyYtiB:yqycyH:^YZymy=Zy=:fY?:mA>Zxu;jykx=j;J:LG;RZdHR qRuyaxEZyU:wboya;dtZW]gyYtiB:yI :yaxu:jxG=ry;jyC:xI:ZvE:`i:>Zxe:>ZxU;jy=JwyY:fTFJOAfUfWnYK:yyyJyW:Ry:y I:xI;jy;:^YFYnY?:wq: yqdk:OWjhkpKy?;ZyMyW:Ry:yI:xI;jy;::FYJ;Jyg;ry;ryya:ryPG@bd=pGpHtiB:vYt i=:uAry;jyC:xI::Zs=D:tiH>:vy:vyyM:vY@j=TwySZdHRqRupixyy;B:uQ=:yQxoY=ry ya=Z:nYwqx_Z:vYJ:>Y;JykxoYwq:byJdC:=No;u@UAZwe;jyCyR:j;J:tixUyL:Z:Bxby RytixU:nYBoRZdHRqRu:fYG:xIsA=:H::kAwAuQyoY=Zxu:<:TOek:OWjhk \\klkx?:K@=No;u@E@EAuAK@=No;u@E@EAuAayy;RPg@=No;u@E@Iq:By:wQyk:::: ea:nY;JOk:OWjhkp[xeyR:::Zo]xE:dD@bd=pGpHnYwA=:::fXnY;jdSbj:OWjhKqg=ry: ;jyC:xI;jyC:xiqUy\\tySZdHRqbu:yAB:xI;jyC:xI;jywX:=No;u@WAry:;jyC:xI;jy C:xiq]MyY=?@bd=pGth:ByH;vy:ry>:yAZy UyZx]c\\;dtZW]h]v=ZyA:ya:rypiOFJOAfUnWZy=>:ya:ry>:ya:ry\\ItI\\D@bd=pG tHjy;:vY ZxU:::kAj:OWjhkp[xu;jyC:wq:::\\I@wyo@RZdHRqZc=dHRqZc=dHRqB_^DRZdHDbwEy ZxE:lIDITILbwexyY?jyK:xyyC:xI;jyWyyA:Y N:ZyyY:yqwKDsGuyyuAZy]:ry>:ya:ryhIZtE:LBti:rxD:pi;:K::ya:ryh i:R[bybxHZymx?[:vyyuy;jygY;jwKBC:wAqQ;jxG::>ZyM:di:R^bxP;vY>ZyyYoqDZy]:vyyM:vYZyA:yqx?:qA]>wAqQ;jx;:;ry>ZtE:PBtI diO:yA;jyC:xI;jyWY;jBKyKwW:Ry\\:vy:rypi:rxR^byrxD:pI:B:ya:NY;j:ya:ry>:yqwKC?:wAoq;ZwE:ya:ryhIDCtIdi=:uQ>ZyA:yqx?:qAG>JyG: by@:V:>ZtE:@Cti:bxP;vY>:ya:ry>:yqwKCKyKwW:Ry\\:vy:rypi:rxb[:ti;:wq:Z=Z :NY;JD?Z[IZyyyw=ti;:wq:byH:NYNLnY;JwgjnkC;nY=:ti;:A:dI@bW=`HHCZxU:JyG:r: Jwk:gU^LT=FJOAfUjn;OAfUjn;OAfU^L:=No;u@FWJo;u@FWJo;u@c>j:OWjh;MANWjh;M ANWjhKD;FJOAfUjn;OAfUjn;OAfU^L:=No;u@FWJo;u@FWJo;u@c>j:OWjh;MANWjh;MAN WjhKD;FJOAFVkq:gW;:OAFVjp;OAFV>MTMgk:oW;:=qv=:[Qv=:WAFVJr;WAFVvLT=mq:G XkAEqvKrGy:jr;EA>Xjr;EAi>j:GXmAAQwGJ[AFXjk;[AFXjkkEkfk:GXjkWYnWoI:FXjk ;WAFXjkKEcGVY=nqOy:Jk_Y=NqWYvWJk;WAvWJkKEKfk:wWqA=qxGJSQxkq;=A^Wjq;=Ae ^pjxGJUQxKjoy:FJSqxkp;;A^Wjp;;Ae^s:=npgYvU=B:@bdeyRuZY=NWjp;y@LCsso@:= nooYvUZdU:jo;y@bt:QAvUB_:FJQq:ZW=dHjo;pGbt:QARqB_v?>Z;tt:=N\\yi:jo;@BZ e=FJYQ: " 0 "" {MPLTEXT 1 0 887 "arthur89 := proc(T,seed)\nglobal a_AR, a_AS , a_BR, a_BS, b_R, b_S, _seed;\nlocal A_share, chance, d, d_max, d_mi n, \nn_A, n_B, RandGenerator, t;\noption `E.S. Andersen, 10 Feb 2001`; \n\nd := array(0..T);\nd[0] := 0;\nd_max := 0; d_min := 0; \nA_share : = array(1..T);\nA_share := 0; \nn_A := 0; n_B := 0;\nRandGenerator := \+ stats[random,uniform[0,1]]('generator');\n_seed := seed;\n\nfor t from 1 to T do\nchance := RandGenerator();\nif chance>1/2 then \n if a_BR + b_R*n_B <= a_AR + b_R*n_A then n_A := n_A + 1;\n else n_B := n_B + 1;\n end if;\nelse \n if a_BS + b_S*n_B >= a_AS + b_S*n_A then n_B \+ := n_B + 1;\n else n_A := n_A + 1;\n end if;\nend if;\nd[t] := n_A - n_B; \nd_max := max(d[t], d_max); \nd_min := min(d[t], d_min);\nA_sha re[t] := n_A/t;\nend do;\n#print(`\\n`);\nprint(DataPlot([A_share],1,T ,0,1,5,black,ms));\nprint(`\\n\\n`);\nprint(DataPlot([d],1,T,d_min,d_m ax,5,black,dt));\n#print(`\\n`);\nend proc:\n" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 51 "4.2. Simulation of con stant returns (no boundaries)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 177 "The first simulation starts by setting the values of parameters for t he constant returns case. Because our program does not divide with the scale factor, we can use the values: " }{XPPEDIT 18 0 "b[R] = 0;" "6# /&%\"bG6#%\"RG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b[S] = 0;" "6 #/&%\"bG6#%\"SG\"\"!" }{TEXT -1 84 ". In this case the exact value of \+ the other parameters does not matter Ð as long as " }{XPPEDIT 18 0 "a[ AS] < a[AR];" "6#2&%\"aG6#%#ASG&F%6#%#ARG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "a[BR] < a[BS];" "6#2&%\"aG6#%#BRG&F%6#%#BSG" }{TEXT -1 15 ". In this case " }{XPPEDIT 18 0 "R;" "6#%\"RG" }{TEXT -1 31 "-type agents will always chose " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 7 ", and " }{XPPEDIT 18 0 "S;" "6#%\"SG" }{TEXT -1 31 "-type agents wil l always chose " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 83 ". This is simply a random walk in the difference between the number of adopters of " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "a_AR := 10:\na_AS := 5:\na_BR := 5:\na_BS := 10:\nb_R := 0:\nb _S := 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "arthur89(5000,5 );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 236 "In the above plot of the m arket share of A we see that is, after an initial turbulent period com es closer and closer to 50%. The difference between the install bases \+ of the two systems is more voilative. After 5000 periods, the process \+ " }{TEXT 282 5 "seems" }{TEXT -1 302 " to be locked-in into a small ov erweight of system A. But sooner or later the process will move to an \+ overweight of system B. If we continue the sequence of random events t o cover 15000 periods, we see that the system moves towards an overwei ght of B. In the very long run the average should be zero. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "arthur89(15000,5);" }}} {EXCHG {PARA 5 "" 0 "" {TEXT -1 0 "" }}{PARA 5 "" 0 "" {TEXT -1 61 "4. 3. Simulation of decreasing returns (reflecting boundaries)" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "Now we turn to the decreasing ret urns case and thus to the story of negative feedback. This case is def ined by " }{XPPEDIT 18 0 "b[R] < 0;" "6#2&%\"bG6#%\"RG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b[S] < 0;" "6#2&%\"bG6#%\"SG\"\"!" } {TEXT -1 93 ", so now the exact value of the other parameters do matte s, but we of course still have that " }{XPPEDIT 18 0 "a[AS] < a[AR];" "6#2&%\"aG6#%#ASG&F%6#%#ARG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "a[BR] < a[BS];" "6#2&%\"aG6#%#BRG&F%6#%#BSG" }{TEXT -1 64 ". In the followi ng the values of the parameters are set so that " }{XPPEDIT 18 0 "R;" "6#%\"RG" }{TEXT -1 53 "-type agents have an intrinsic preference for \+ system " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 65 " that there is a \+ need for an overweight of 25 adopters of system " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 77 " in order to crowd them out from their preferr ed choice and switch to system " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 3 ". " }{XPPEDIT 18 0 "S;" "6#%\"SG" }{TEXT -1 80 "-type agents al so needs the negative effects of an overweight of 25 adopters of " } {XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 49 " to switch from their prefe rred choice of system " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 11 " t o system " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 73 ".\n \nAs long a s the process does not reach the boundaries of the interval " } {XPPEDIT 18 0 "[-25, 25];" "6#7$,$\"#D!\"\"F%" }{TEXT -1 472 " the pro cess goes on just like in the constant returns case. As we see in the \+ plot below, this is the case until period 275. Then the process hits t he border. It continues to move randomly with occational reflections f rom a border. In the very long run the process is expected to be just \+ as frequently in the A overweight as in the B overweight (but this is \+ not shown in the 5000 period simulation). It is, however, obvious that both install shares come very close to 50%." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "a_AR := 10:\na_AS := 5:\na_BR := 5:\na_BS := 10: \nb_R := -0.2:\nb_S := -0.2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "arthur89(5000,4);" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 0 "" }} {PARA 5 "" 0 "" {TEXT -1 60 "4.4. Simulation of increasing returns (ab sorbing boundaries)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 197 "Finally, \+ we come to Arthur's core case of increasing returns to adoption and th us to the positive feedback case. Given the previous analysis, this ca se is simple to handle. The case is defined by " }{XPPEDIT 18 0 "0 < \+ b[R];" "6#2\"\"!&%\"bG6#%\"RG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "0 < b[S];" "6#2\"\"!&%\"bG6#%\"SG" }{TEXT -1 35 ", and we of course still have that " }{XPPEDIT 18 0 "a[AS] < a[AR];" "6#2&%\"aG6#%#ASG&F%6#%#A RG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "a[BR] < a[BS];" "6#2&%\"aG6#%# BRG&F%6#%#BSG" }{TEXT -1 86 ". We follow the decreasing returns case i n setting the rest of the parameters so that " }{XPPEDIT 18 0 "R;" "6# %\"RG" }{TEXT -1 61 "-type agents need for an overweight of 25 adopter s of system " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 47 " in order to persuade them to switch to system " }{XPPEDIT 18 0 "B;" "6#%\"BG" } {TEXT -1 3 ". " }{XPPEDIT 18 0 "S;" "6#%\"SG" }{TEXT -1 56 "-type age nts also needs an overweight of 25 adopters of " }{XPPEDIT 18 0 "A;" " 6#%\"AG" }{TEXT -1 209 " to switch from their preferred choice.\n\nUnt il period 275 the process goes exactly as in the constant returns and \+ decreasing returns cases. Then it is absorbed, and each and every new \+ adaptation choose system " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 27 ". Thus the market share of " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 117 " moves towards zero, and the random walk in the difference of ado ption is changed to a infinite directed walk in the " }{XPPEDIT 18 0 " B;" "6#%\"BG" }{TEXT -1 11 " direction." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "a_AR := 10:\na_AS := 5:\na_BR := 5:\na_BS := 10:\nb_R := 0.2:\nb_S := 0.2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "ar thur89(5000,4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "It is more re levant to study the process just before the lock in in period 275. Thi s is shown in the following plot:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "arthur89(325,4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "We may also study the lock-in to system " }{XPPEDIT 18 0 "A;" "6#% \"AG" }{TEXT -1 29 " after nearly 1000 periods..." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 17 "arthur89(1000,1);" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 555 "\nWe do not need to make other simulations to demonstr ate that the stochastic process with increasing returns is \"tippy,\" \+ as we know from network markets with inherent instability untill they \+ may reach a lock-in (cf. Bensen and Farrell, 1994). One may observe an unstable coexistence of incompatible products such as the classic cas e in the videocassette recorder market, but they also refer to video e ncryption of cable television programs. Dominance of one technology to day does not guarantee co ntinued success forever since such instabili ty can happen." }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 22 "5. Further expe riments" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "\nAlthough there are \+ many possibilities, the Arthur model is obviously constructed for the \+ increasing returns case. Here we conclude:\n" }}{PARA 0 "" 0 "" {TEXT -1 264 "¥ we have a random walk process with potentially absorbing bar riers\n¥ the proceess is not predictable (with probability 1 one of th e barriers will be reached)\n¥ ex-post ineciency possible\n¥ small eve nts of history can be important under increasing returns to scale" }} {PARA 0 "" 0 "" {TEXT -1 66 "¥ it is easy to introduce expectations th at may quicken absorbtion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "It is obvious that the Arthur model is j ust a simple starting point. Some of the problems to be exploited in f urther experiments are\n" }}{PARA 0 "" 0 "" {TEXT -1 137 "¥ It is not \+ obvious that the evaluation of external effects is connected to user t ypes rather than to systems. In most cases a parameter " }{XPPEDIT 18 0 "b[A];" "6#&%\"bG6#%\"AG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b[B]; " "6#&%\"bG6#%\"BG" }{TEXT -1 44 " seem easier to interpret and more r elevant." }}{PARA 0 "" 0 "" {TEXT -1 253 "¥ The role of expectations i s not discussed in Arthur's paper. Rapid, random movement toward one s ystem may create expectations that this system is \"taking over\". It \+ is easy to include in the program.\n¥ More learning behaviour is found in Arthur (1993)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 10 "References" }}}{EXCHG {PARA 16 "" 0 "" {TEXT -1 146 "Arthur, W. Brian (1989), ÔCompeting Technologies, Increasing Retu rns, and Lock-In by Historical EventsÕ, Economic Journal, Vol. 99, pp. 116--131. " }}{PARA 16 "" 0 "" {TEXT -1 141 "Arthur, W. Brian (1993), ÔOn Designing Economic Agents that Behave Like Human AgentsÕ, Journal of Evolutionary Economics, Vol. 3, pp. 1--22. " }}{PARA 16 "" 0 "" {TEXT -1 127 "Arthur, W. Brian (1994), Increasing Returns and Path Dep endence in the Economy, University of Michigan Press, Ann Arbor, Mich. " }}{PARA 258 "" 0 "" {TEXT 283 172 "Bensen, Stanley M, and Farrell, \+ Joseph (1994), ÔChoosing How to Compete: Strategies and Tactics in Sta ndardizationÕ, Journal of Economic Perspectives, Vol. 8, pp. 117-310. \+ " }}{PARA 16 "" 0 "" {TEXT -1 131 "David, Paul A. (1985), ÔClio and th e Economics of QWERTYÕ, American Economic Review. Papers and Proceedin gs, Vol. 75, pp. 332--337." }}{PARA 16 "" 0 "" {TEXT -1 156 "Liebowitz , Stanley J., and Margolis, Stephen E. (1994), ÔNetwork Externality: A n Uncommon TragedyÕ, Journal of Economic Perspectives, Vol. 8, pp. 133 --150. " }}{PARA 16 "" 0 "" {TEXT -1 115 "Peak, David, and Frame, Mich ael (1994), Chaos Under Control: The Art and Science of Complexity, Fr eeman, New York. " }}{PARA 16 "" 0 "" {TEXT -1 148 "Shapiro, Carl, and Varian, Hal R. (1999), Information Rules: A Strategic Guide to the Ne twork Economy, Harvard Business School Press, Boston, Mass. " }}} {PARA 13 "" 1 "" {TEXT -1 0 "" }}}{MARK "0 1 0" 36 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 1 33 1 1 }