Daniel Webster (1782-1852) proposed a method similar to Jefferson\u2019s in 1832. It was adopted by Congress in 1842, but replaced by Hamilton\u2019s method in 1852. It was then adopted again in 1901. The difference is that Webster rounds the quotas to the nearest whole number rather than dropping the decimal parts. If that doesn\u2019t produce the desired results at the beginning, he says, like Jefferson, to adjust the divisor until it does. (In Jefferson\u2019s case, at least the first adjustment will always be to make the divisor smaller. That is not always the case with Webster\u2019s method.)<\/p>\r\n We\u2019ll return to Delaware and apply Webster\u2019s method. As a reminder, the state of Delaware has three counties: Kent, New Castle, and Sussex. The Delaware state House of Representatives has [latex]41[\/latex] members.<\/p>\r\n The populations of the counties are as follows (from the 2010 Census):<\/p>\r\n\r\n [latex]\\begin{array}{lr} \\text { County } & \\text { Population } \\\\ \\hline \\text { Kent } & 162,310 \\\\ \\text { New Castle } & 538,479 \\\\ \\text { Sussex } & 197,145 \\\\ \\textbf{ Total } & \\bf{ 897,934 }\\end{array}[\/latex]<\/p> [reveal-answer q=\"4331\"]Show Solution[\/reveal-answer] We begin, as we did with Hamilton\u2019s method, by finding the quotas with the original divisor, [latex]21,900.82927[\/latex].<\/p>\r\n\r\n [latex]\\begin{array}{lrrc} \\text { County } & \\text { Population } & \\text{ Quota } \\\\ \\hline \\text { Kent } & 162,310 & 7.4111 \\\\ \\text { Newport } & 538,479 & 24.5872 \\\\ \\text { Providence } & 197,145 & 9.0017 \\\\ \\textbf{ Total } & \\bf{ 897,934 } & \\end{array} [\/latex]<\/p>\r\n\r\n Rounding the quotas up we get:<\/p>\r\n\r\n [latex]\\begin{array}{lrrc} \\text { County } & \\text { Population } & \\text{ Quota } & \\text{ Initial } \\\\ \\hline \\text { Kent } & 166,310 & 7.4111 & 7 \\\\ \\text { New Castle } & 538,479 & 24.5872 & 25 \\\\ \\text { Sussex } & 197,145 & 9.0017 & 9 \\\\ \\textbf{ Total } & \\bf{ 897,934 } & & \\bf{ 41 }\\end{array} [\/latex]<\/p>\r\n\r\n This gives the required total, so we\u2019re done.<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n [ohm2_question hide_question_numbers=1]13232[\/ohm2_question]<\/p>\r\n<\/section>\r\n Like Jefferson\u2019s method, Webster\u2019s method carries a bias in favor of states with large populations, but rounding the quotas to the nearest whole number greatly reduces this bias. Also like Jefferson\u2019s method, Webster\u2019s method does not always follow the quota rule, but it follows the quota rule much more often than Jefferson\u2019s method does. (In fact, if Webster\u2019s method had been applied to every apportionment of Congress in all of American history, it would have followed the quota rule every single time.)<\/p>\r\n In 1980, two mathematicians, Peyton Young and Mike Balinski, proved what we now call the Balinski-Young Impossibility Theorem<\/strong>.<\/p>\r\n Daniel Webster (1782-1852) proposed a method similar to Jefferson\u2019s in 1832. It was adopted by Congress in 1842, but replaced by Hamilton\u2019s method in 1852. It was then adopted again in 1901. The difference is that Webster rounds the quotas to the nearest whole number rather than dropping the decimal parts. If that doesn\u2019t produce the desired results at the beginning, he says, like Jefferson, to adjust the divisor until it does. (In Jefferson\u2019s case, at least the first adjustment will always be to make the divisor smaller. That is not always the case with Webster\u2019s method.)<\/p>\n We\u2019ll return to Delaware and apply Webster\u2019s method. As a reminder, the state of Delaware has three counties: Kent, New Castle, and Sussex. The Delaware state House of Representatives has [latex]41[\/latex] members.<\/p>\n The populations of the counties are as follows (from the 2010 Census):<\/p>\n [latex]\\begin{array}{lr} \\text { County } & \\text { Population } \\\\ \\hline \\text { Kent } & 162,310 \\\\ \\text { New Castle } & 538,479 \\\\ \\text { Sussex } & 197,145 \\\\ \\textbf{ Total } & \\bf{ 897,934 }\\end{array}[\/latex]<\/p>\nWebster\u2019s method<\/h3>\r\n
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\r\n[hidden-answer a=\"4331\"]\r\n\r\nBalinski-Young Impossibility Theorem<\/h3>\r\n\r\nThe Balinski-Young Impossibility Theorem<\/strong> shows that any apportionment method which always follows the quota rule will be subject to the possibility of paradoxes like the Alabama, New States, or Population paradoxes. In other words, we can choose a method that avoids those paradoxes, but only if we are willing to give up the guarantee of following the quota rule.<\/div>\r\n<\/section>","rendered":"
Webster\u2019s Method<\/h2>\n
Webster\u2019s method<\/h3>\n
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