(* Content-type: application/vnd.wolfram.mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 10.4' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] NotebookDataLength[ 260507, 5951] NotebookOptionsPosition[ 258915, 5895] NotebookOutlinePosition[ 259259, 5910] CellTagsIndexPosition[ 259216, 5907] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Winner-Take-All Distortion", "Title", CellChangeTimes->{{3.776600831050974*^9, 3.7766008318453026`*^9}, { 3.7770585040903263`*^9, 3.777058516681843*^9}}], Cell["Adam Rumpf, 11/26/2016", "Text", CellChangeTimes->{{3.7766008347881403`*^9, 3.776600838290375*^9}, { 3.7770584392258434`*^9, 3.777058441473523*^9}}], Cell[CellGroupData[{ Cell["Introduction", "Section", CellChangeTimes->{{3.7766008459498987`*^9, 3.776600848547045*^9}}], Cell[TextData[{ "This Notebook defines a set of functions for demonstrating how the results \ of elections can be distored when evaluated as a collection of aggregated \ winner-take-all races instead of simply counting every individual vote. \ Specifically, we have a ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["2", "n"], "\[Times]", SuperscriptBox["2", "n"]}], TraditionalForm]], FormatType->"TraditionalForm"], " array of voters, each of which is affiliated with one party. Taken \ individually, this can be imagined as an array of ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["2", "n"], "\[Times]", SuperscriptBox["2", "n"]}], TraditionalForm]], FormatType->"TraditionalForm"], " districts, each of size ", Cell[BoxData[ FormBox[ RowBox[{"1", "\[Times]", "1"}], TraditionalForm]], FormatType->"TraditionalForm"], "." }], "Text", CellChangeTimes->{{3.776600856235587*^9, 3.776600860481224*^9}, { 3.7770626363556356`*^9, 3.777062641195082*^9}, {3.7770655128029575`*^9, 3.7770655223472586`*^9}, {3.777065611660858*^9, 3.7770657861992826`*^9}, { 3.777065827581188*^9, 3.7770658464521666`*^9}, {3.7770658819916897`*^9, 3.7770659128778276`*^9}}], Cell[TextData[{ "Next, we try grouping people into ", Cell[BoxData[ FormBox[ RowBox[{"2", "\[Times]", "2"}], TraditionalForm]], FormatType->"TraditionalForm"], " districts, resulting in a ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["2", RowBox[{"n", "-", "1"}]], "\[Times]", SuperscriptBox["2", RowBox[{"n", "-", "1"}]]}], TraditionalForm]], FormatType->"TraditionalForm"], " array of districts, and instead of giving each person a direct vote, we \ instead hold a vote within each individual district and evaluate all of the \ district-wide results. Within a district, the candidate with the most votes \ wins the entire district (with ties being broken arbitrarily). This could be \ thought of as introducing a congress of ", Cell[BoxData[ FormBox[ SuperscriptBox["2", "n"], TraditionalForm]], FormatType->"TraditionalForm"], " representatives and allowing each district to elect one representative." }], "Text", CellChangeTimes->{{3.7770657940237646`*^9, 3.777065807825654*^9}, { 3.7770658530893116`*^9, 3.7770658784408197`*^9}, {3.777065920253215*^9, 3.7770661073567085`*^9}, {3.777068822623542*^9, 3.7770688774318705`*^9}}], Cell[TextData[{ "After that we group people into ", Cell[BoxData[ FormBox[ RowBox[{"4", "\[Times]", "4"}], TraditionalForm]], FormatType->"TraditionalForm"], " districts in a ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["2", RowBox[{"n", "-", "1"}]], "\[Times]", SuperscriptBox["2", RowBox[{"n", "-", "1"}]]}], TraditionalForm]], FormatType->"TraditionalForm"], " array, and then ", Cell[BoxData[ FormBox[ RowBox[{"8", "\[Times]", "8"}], TraditionalForm]], FormatType->"TraditionalForm"], " districts in a ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["2", RowBox[{"n", "-", "2"}]], "\[Times]", SuperscriptBox["2", RowBox[{"n", "-", "2"}]]}], TraditionalForm]], FormatType->"TraditionalForm"], " array, and so on, ending when we have a single ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["2", "n"], "\[Times]", SuperscriptBox["2", "n"]}], TraditionalForm]], FormatType->"TraditionalForm"], " district in a ", Cell[BoxData[ FormBox[ RowBox[{"1", "\[Times]", "1"}], TraditionalForm]], FormatType->"TraditionalForm"], " array which yields only a single result for the entire population, which \ is a single-winner election like a presidential election." }], "Text", CellChangeTimes->{{3.7770661235504627`*^9, 3.7770662130632477`*^9}, { 3.77706889897327*^9, 3.7770689234279265`*^9}}], Cell[TextData[{ "For each collection of districts we display a pie chart of the \ congressional makeup as well as a \[OpenCurlyDoubleQuote]misrepresentation \ score\[CloseCurlyDoubleQuote], calculated as the sum over all parties of the \ difference between their individual-level support and their overall \ district-level support (in percentage points, \[OpenCurlyDoubleQuote]", Cell[BoxData[ FormBox["pp", TraditionalForm]], FormatType->"TraditionalForm"], "\[CloseCurlyDoubleQuote]). For example, if we begin with red and blue both \ having exactly ", Cell[BoxData[ FormBox[ RowBox[{"50", "%"}], TraditionalForm]], FormatType->"TraditionalForm"], " of the popular vote, but we district them in a way that gives red ", Cell[BoxData[ FormBox[ RowBox[{"60", "%"}], TraditionalForm]], FormatType->"TraditionalForm"], " of congress and blue ", Cell[BoxData[ FormBox[ RowBox[{"40", "%"}], TraditionalForm]], FormatType->"TraditionalForm"], ", then the misrepresentation score would be ", Cell[BoxData[ FormBox[ RowBox[{"20", "pp"}], TraditionalForm]], FormatType->"TraditionalForm"], ": the sum of red\[CloseCurlyQuote]s 10 pp overrepresentation and blue\ \[CloseCurlyQuote]s ", Cell[BoxData[ FormBox[ RowBox[{"10", "pp"}], TraditionalForm]], FormatType->"TraditionalForm"], " underrepresentation. By definition, the initial ", Cell[BoxData[ FormBox[ RowBox[{"1", "\[Times]", "1"}], TraditionalForm]], FormatType->"TraditionalForm"], " district case must always have a misrepresentation score of ", Cell[BoxData[ FormBox[ RowBox[{"0", "pp"}], TraditionalForm]], FormatType->"TraditionalForm"], " since this is direct representation. We also display a plot of the \ misrepresentation score versus the log of the number of districts." }], "Text", CellChangeTimes->{{3.777066217856719*^9, 3.7770665013717422`*^9}, { 3.7770665358157563`*^9, 3.777066573825621*^9}, {3.777067328821601*^9, 3.77706734173143*^9}, {3.7770687835929136`*^9, 3.777068783719892*^9}, { 3.777068817988021*^9, 3.777068819124673*^9}, {3.7770689370684757`*^9, 3.777068962032734*^9}, {3.7770689963299065`*^9, 3.777069006656398*^9}}], Cell[TextData[{ "The main function defined below is ", StyleBox["districttable[]", "Code"], ", which accepts three optional arguments:" }], "Text", CellChangeTimes->{{3.7770665909933376`*^9, 3.7770667134478984`*^9}, { 3.777066793081395*^9, 3.777066794403993*^9}, {3.7770668472768693`*^9, 3.7770669334752564`*^9}, {3.777066995328183*^9, 3.777066995662079*^9}}], Cell[CellGroupData[{ Cell[TextData[{ Cell[BoxData[ FormBox["n", TraditionalForm]], FormatType->"TraditionalForm"], " (default 3): Determines the size of the initial ", Cell[BoxData[ FormBox[ RowBox[{ SuperscriptBox["2", "n"], "\[Times]", SuperscriptBox["2", "n"]}], TraditionalForm]], FormatType->"TraditionalForm"], " array, and also implicitly the number of plots generated (which is ", Cell[BoxData[ FormBox[ RowBox[{"n", "+", "1"}], TraditionalForm]], FormatType->"TraditionalForm"], ")." }], "ItemNumbered", CellChangeTimes->{{3.777066944946657*^9, 3.7770669758074117`*^9}}], Cell[TextData[{ Cell[BoxData[ FormBox["w", TraditionalForm]], FormatType->"TraditionalForm"], " (default ", Cell[BoxData[ FormBox[ RowBox[{"{", RowBox[{"1", ",", "1"}], "}"}], TraditionalForm]], FormatType->"TraditionalForm"], "): A list of party support weights. The initial array is generated by \ randomly assigning each voter to a party, and the weight list determines the \ relative likelihood that any given voter belongs to each party. The length of \ this list implicitly determines the number of parties. Since the array is \ randomly generated, evaluating the function repeatedly with the same inputs \ will generally yield different results." }], "ItemNumbered", CellChangeTimes->{{3.777066944946657*^9, 3.7770669897009797`*^9}, { 3.777067253237111*^9, 3.777067276166293*^9}, {3.777069070452222*^9, 3.777069087532137*^9}}], Cell[TextData[{ Cell[BoxData[ FormBox["mode", TraditionalForm]], FormatType->"TraditionalForm"], " (default 0): If set to 0, displays only an array, pie chart, and \ misrepresentation score as described above. 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