Polya urn model<\/a>. This is a ball-and-urn process where, instead of simple replacement, there is replacement with addition of a ball of the same colour. This gives one colour a dominant position in the long run.<\/p>\nI coded a script to run Polya urn simulations. It’s in JavaScript, and easily adaptable. I tested it in Chrome.<\/p>\n
[sourcecode language=”javascript”]
\n<html><body><script language="JavaScript">
\n\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/<\/p>\n
\/\/ Number of iterations
\nvar runs = 400;<\/p>\n
\/\/ Number of blue balls initially in urn
\nvar num_blue = 1;<\/p>\n
\/\/ Number of red balls initially in urn
\nvar num_red = 1;<\/p>\n
\/\/ Number of additional red\/blue balls added after red\/blue "wins"
\nvar extra_extra = 1;<\/p>\n
\/\/ Show which colour was picked etc
\nvar details = true;<\/p>\n
\/\/ Summarize the mean at the end
\nvar mean_details = true;<\/p>\n
\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/<\/p>\n
var initial_dist = num_red \/ (num_red + num_blue);
\nred_width = initial_dist*400;<\/p>\n
\/\/ Graph initial distribution
\ndocument.write("Initial Distribution<br \/><div style=’background-color:blue;height:10;width:400;’>");
\ndocument.write("<div style=’background-color:red;height:10;width:" + red_width +";’><\/div><\/div>");<\/p>\n
\/\/ Random number
\nvar r_n = 0;<\/p>\n
var red_share = new Array();
\nred_share[0] = num_red \/ (num_red + num_blue);<\/p>\n
for(i=1;i<runs;i++)
\n{<\/p>\n
\/\/ Random number on unit interval
\nr_n = Math.random();<\/p>\n
\tif (r_n > red_share[i-1])
\n\t{
\n\tnum_blue = num_blue + extra_extra;
\n\t\t\tif(details){document.write("Blue.");}
\n\t}<\/p>\n
\tif (r_n <= red_share[i-1])
\n\t{
\n\tnum_red = num_red + 1;
\n\t\t\tif(details){document.write("Red.");}
\n\t}<\/p>\n
red_share[i] = num_red \/ (num_red + num_blue);
\n\t\t\tif(details){document.write(" Red Share = " + num_red + "\/(" + num_red + " + " + num_blue + ") = " +<\/p>\n
red_share[i]);}<\/p>\n
document.write("<div style=’background-color:blue;height:10;width:400;’>");
\ndocument.write("<div style=’background-color:red;height:10;width:" + red_share[i]*400 + ";><\/div>");
\ndocument.write("<\/div><div style=’background-color:white;height:5;width:400;’><\/div>");
\n}<\/p>\n
var total = 0;
\nvar mean = 0;
\nfor(j=0;j<runs;j++)
\n{total = total+red_share[j]}
\nmean = total\/runs;
\nif(mean_details){document.write("Mean of red share: " + mean);}
\n<\/script><\/body><\/html>
\n[\/sourcecode]<\/p>\n","protected":false},"excerpt":{"rendered":"
I had to present on path dependence recently. Paul David’s famous QWERTY paper (American Economic Review P&P, 1985) was covered, as was Paul Pierson’s Increasing Returns, Path Dependence, and the Study of Politics (American Political…<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[],"class_list":["post-860","post","type-post","status-publish","format-standard","hentry","category-economics"],"_links":{"self":[{"href":"https:\/\/hargaden.com\/enda\/wp-json\/wp\/v2\/posts\/860","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/hargaden.com\/enda\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/hargaden.com\/enda\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/hargaden.com\/enda\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/hargaden.com\/enda\/wp-json\/wp\/v2\/comments?post=860"}],"version-history":[{"count":0,"href":"https:\/\/hargaden.com\/enda\/wp-json\/wp\/v2\/posts\/860\/revisions"}],"wp:attachment":[{"href":"https:\/\/hargaden.com\/enda\/wp-json\/wp\/v2\/media?parent=860"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/hargaden.com\/enda\/wp-json\/wp\/v2\/categories?post=860"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/hargaden.com\/enda\/wp-json\/wp\/v2\/tags?post=860"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}