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Notebook[{

Cell[CellGroupData[{
Cell["Voting Theory with Representation Triangles and Cubes", "Title",
  TextAlignment->Center,
  FontColor->RGBColor[1, 0, 0]],

Cell["\<\
Alexander Tabarrok
Department of Economics
Ball State University
Muncie, Indiana, 47306
Email: 00ATTabarrok@BSU.Edu\
\>", "Subsubtitle",
  TextAlignment->Center],

Cell[CellGroupData[{

Cell["Introduction", "Section",
  Evaluatable->False],

Cell[TextData[{
  "\tVoting seems straightforward but on closer inspection even simple voting \
rules have mysterious and odd properties which often challenge our conception \
of democracy.  Some of these paradoxes have bothered social scientists for \
hundreds of years but until recently little theory existed which explained \
why and when paradoxes might be expected to occur.  In his book, \"The \
Geometry of Voting,\" the mathematician Donald Saari has shed considerable \
light on why voting 'paradoxes' occur and how they can be predicted.  The \
package voting.m uses ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "'s capabilities to implement many of the functions described in Saari's \
book \"The Geometry of Voting,\" Springer-Verlag, 1994.  The package allows \
us to experiment with positional voting systems, pairwise voting, approval \
voting and cumulative voting.  In this article I will focus on positional and \
pairwise voting, other package capabilities are explained at length in the \
electronic supplement. "
}], "Text",
  Evaluatable->False],

Cell[TextData[{
  "\tTo load the Voting.m package place it in a directory which you know is \
on Mathematica's search path.  Voting.m also needs a number of standard  \
packages which will be loaded automatically.  The non-standard package  \
VertexEnum is also used in some functions.  The VertexEnum package can be \
downloaded  from Wolfram Research's ",
  StyleBox["MathSource",
    FontSlant->"Italic"],
  " web site at http://www.mathsource.com/.  When you have downloaded the \
package rename it VertexEnumeration and place it in the standard package \
directory Geometry.  Open the package in a text editor and change the \
BeginPackage[\"VertexEnum`\"] command to \
BeginPackage[\"Geometry`VertexEnumeration`\"].  Finally, to avoid some error \
messages, search for the phrase \"\\dev\\null\" and replace it with the \
phrase \"Null\".   If you are using Windows 3.1 you will have to shorten the \
package title to VertexEn as above and also in the BeginPackage command in \
voting.m  When evaluating the following command using MMA 3.0 a message will \
appear indicating that CrossProduct is now autoloaded you can ignore this \
message."
}], "Text"],

Cell["<<Voting.m", "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
This package is Copyright 1997, Alexander Tabarrok.  If you use this package \
in academic work please cite, Tabarrok, Alexander. Voting Theory with \
Mathematica - Computer Software, Ball State University, 1997.  Thank you.\
\>", "Section",
  GeneratedCell->True,
  CellAutoOverwrite->True],

Cell[BoxData[
    \(TraditionalForm
    \`Cross::"obslt" \( : \ \) 
      "Cross functionality is now autoloaded.  The package \
LinearAlgebra`CrossProduct` is obsolete."\)], "Message"],

Cell[BoxData[
    FormBox[
      RowBox[{
      \(Needs::"nocont"\), \( : \ \), 
        "\<\"Context \
\\!\\(TraditionalForm\\`\\\"LinearAlgebra`CrossProduct`\\\"\\) was not \
created when Needs was evaluated.\"\>"}], TraditionalForm]], "Message"],

Cell[TextData[{
  "\tThe voting package works predominantly with elections where there are \
three or more candidates or policy options labeled C1,C2,C3.  There are as \
many voters as we like but every voter is assumed to have strict, complete, \
and transitive preferences.  This means that for any two candidates voters \
prefer one candidate or the other (we ignore ties) and, letting > indicate \
'is preferred to', if ",
  Cell[BoxData[
      \(TraditionalForm\`C1 > C2\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`C2 > C3\)]],
  " then ",
  Cell[BoxData[
      \(TraditionalForm\`C1 > C3\)]],
  ".  Thus, if a voter prefers chocolate to strawberry ice cream and \
strawberry to vanilla then she must also prefer chocolate to vanilla.  (One \
of the most famous paradoxes of voting shows that groups do not necessarily \
have transitive preferences.)  These assumptions imply that every voter is \
one of six possible types which can be shown using the command VoterTypes[].  \
"
}], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(VoterTypes[]\)], "Input",
  CellHorizontalScrolling->False],

Cell[BoxData[
    FormBox[
      TagBox[GridBox[{
            {"\<\"0\"\>", "\<\"Type 1\"\>", "\<\"C1\"\>", "\<\">\"\>", 
              "\<\"C2\"\>", "\<\">\"\>", "\<\"C3\"\>"},
            {"\<\"0\"\>", "\<\"Type 2\"\>", "\<\"C1\"\>", "\<\">\"\>", 
              "\<\"C3\"\>", "\<\">\"\>", "\<\"C2\"\>"},
            {"\<\"0\"\>", "\<\"Type 3\"\>", "\<\"C3\"\>", "\<\">\"\>", 
              "\<\"C1\"\>", "\<\">\"\>", "\<\"C2\"\>"},
            {"\<\"0\"\>", "\<\"Type 4\"\>", "\<\"C3\"\>", "\<\">\"\>", 
              "\<\"C2\"\>", "\<\">\"\>", "\<\"C1\"\>"},
            {"\<\"0\"\>", "\<\"Type 5\"\>", "\<\"C2\"\>", "\<\">\"\>", 
              "\<\"C3\"\>", "\<\">\"\>", "\<\"C1\"\>"},
            {"\<\"0\"\>", "\<\"Type 6\"\>", "\<\"C2\"\>", "\<\">\"\>", 
              "\<\"C1\"\>", "\<\">\"\>", "\<\"C3\"\>"}
            },
          RowSpacings->1,
          ColumnSpacings->3,
          RowAlignments->Baseline,
          ColumnAlignments->{Left}],
        (TableForm[ #]&)], TraditionalForm]], "Print"]
}, Open  ]],

Cell[TextData[{
  "\tThe above indicates that at present there are 0 voters of type one, 0 of \
type two and so forth.  To define some voters we write a list ",
  Cell[BoxData[
      \(TraditionalForm\`aprofile = {p1, p2, p3, p4, p5, p6}\)]],
  " where p1 is the number of voters of type one, p2 the number of voters of \
type two and so forth.  We call aprofile a ",
  StyleBox["profile vector",
    FontSlant->"Italic"],
  ".  For example:"
}], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(aprofile = {1, 2, 3, 4, 5, 6}\)], "Input",
  CellHorizontalScrolling->False],

Cell[BoxData[
    \(TraditionalForm\`{1, 2, 3, 4, 5, 6}\)], "Output"]
}, Open  ]],

Cell["\tNow evaluate VoterTypes[aprofile].", "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(VoterTypes[aprofile]\)], "Input",
  CellHorizontalScrolling->False],

Cell[BoxData[
    FormBox[
      TagBox[GridBox[{
            {"\<\"0.04762\"\>", "\<\"Type 1\"\>", "\<\"C1\"\>", "\<\">\"\>", 
              "\<\"C2\"\>", "\<\">\"\>", "\<\"C3\"\>"},
            {"\<\"0.09524\"\>", "\<\"Type 2\"\>", "\<\"C1\"\>", "\<\">\"\>", 
              "\<\"C3\"\>", "\<\">\"\>", "\<\"C2\"\>"},
            {"\<\"0.1429\"\>", "\<\"Type 3\"\>", "\<\"C3\"\>", "\<\">\"\>", 
              "\<\"C1\"\>", "\<\">\"\>", "\<\"C2\"\>"},
            {"\<\"0.1905\"\>", "\<\"Type 4\"\>", "\<\"C3\"\>", "\<\">\"\>", 
              "\<\"C2\"\>", "\<\">\"\>", "\<\"C1\"\>"},
            {"\<\"0.2381\"\>", "\<\"Type 5\"\>", "\<\"C2\"\>", "\<\">\"\>", 
              "\<\"C3\"\>", "\<\">\"\>", "\<\"C1\"\>"},
            {"\<\"0.2857\"\>", "\<\"Type 6\"\>", "\<\"C2\"\>", "\<\">\"\>", 
              "\<\"C1\"\>", "\<\">\"\>", "\<\"C3\"\>"}
            },
          RowSpacings->1,
          ColumnSpacings->3,
          RowAlignments->Baseline,
          ColumnAlignments->{Left}],
        (TableForm[ #]&)], TraditionalForm]], "Print"]
}, Open  ]],

Cell["\<\
\tVoter types returns the proportion of voters of each type.  There are \
twenty one voters in all so .047 are of type one, .142 are of type three, \
etc.\
\>", "Text",
  Evaluatable->False],

Cell[TextData[{
  "\tThere are many different types of voting schemes.  We will analyze \
positional voting systems first.  Plurality rule, for example, assigns one \
point to a voter's top ranked candidate and zero points to a voter's second \
and third ranked candidates.  The Borda count, named after the eighteenth \
century French mathematician Jean Charles Borda assigns two points to a \
voter's top ranked candidate, 1 point to a second ranked candidate and zero \
points to the least favored candidate.  (So when the voters enter the voting \
booth they write down the three candidates from most to least favored and the \
registrars tally the points.)  Using a normalization it is not to hard to \
show that every possible positional vote scheme can be represented by a \
single number ",
  Cell[BoxData[
      \(TraditionalForm\`s\ \[Epsilon]\ [0, 1\/2]\)]],
  ".",
  ButtonBox["\[SelectionPlaceholder]",
    ButtonData:>"EndNote 1",
    ButtonStyle->"Hyperlink"],
  ButtonBox["1",
    ButtonData:>"EndNote 1",
    ButtonStyle->"Hyperlink"],
  "   Using this normalization 0 corresponds to plurality rule, the Borda \
count corresponds with ",
  Cell[BoxData[
      \(TraditionalForm\`s = 1\/3\)]],
  " and anti-plurality rule is represented by ",
  Cell[BoxData[
      \(TraditionalForm\`s = 1\/2\)]],
  ".  Anti-plurality rule assigns 1 point to each of a voter's top-two ranked \
candidates, this is equivalent to assigning a vote of negative one to the \
least favored candidate and choosing the candidate with the least negative \
votes (hence the name anti-plurality rule)."
}], "Text",
  Evaluatable->False,
  CellTags->"Return 1"],

Cell["\<\
\tUsing the normalization we can represent every positional voting system by \
the following matrix:\
\>", "Text"],

Cell[TextData[{
  StyleBox["The Standardized Positional Vote Matrix\n\n",
    FontVariations->{"Underline"->True}],
  Cell[BoxData[
      RowBox[{"(", GridBox[{
            {"", 
              StyleBox["abc",
                FontSlant->"Italic"], 
              StyleBox["acb",
                FontSlant->"Italic"], 
              StyleBox["cab",
                FontSlant->"Italic"], 
              StyleBox["cba",
                FontSlant->"Italic"], 
              StyleBox["bca",
                FontSlant->"Italic"], 
              StyleBox["bac",
                FontSlant->"Italic"]},
            {"a", \(1 - s\), \(1 - s\), "s", "0", "0", "s"},
            {"b", "s", "0", "0", "s", \(1 - s\), \(1 - s\)},
            {"c", "0", "s", \(1 - s\), \(1 - s\), "s", "0"}
            },
          GridFrame->True,
          RowLines->True,
          ColumnLines->True], ")"}]]]
}], "Text",
  TextAlignment->Center,
  FontWeight->"Bold"],

Cell[TextData[{
  "\tThe vote matrix can be read in two ways. Reading down a particular \
column we see the number of points given to each candidate from a voter with \
the ranking indicated by that column. A voter with ranking ",
  StyleBox["cba",
    FontSlant->"Italic"],
  ", for example, gives points of 0, s, and 1-s to candidates a, b, and c  \
respectively. Reading across the rows we see where a candidate's votes come \
from. Candidate b, for example, gets s votes from each voter of type",
  StyleBox[" abc",
    FontSlant->"Italic"],
  " or ",
  StyleBox["cba",
    FontSlant->"Italic"],
  ", zero points from each voter of type ",
  StyleBox["acb",
    FontSlant->"Italic"],
  " or ",
  StyleBox["cab",
    FontSlant->"Italic"],
  " and 1-s points from each voter of type ",
  StyleBox["bca",
    FontSlant->"Italic"],
  " or",
  StyleBox[" bac",
    FontSlant->"Italic"],
  ".  Note that s indicates the weight placed on a voter's second ranking.  \
Plurality rule, ",
  Cell[BoxData[
      \(TraditionalForm\`s = 0\)]],
  ", counts second rankings for nothing.  The Borda Count, ",
  Cell[BoxData[
      FormBox[
        FormBox[\(s = 1\/3\),
          "TraditionalForm"], TraditionalForm]]],
  ", counts second rankings for half as much as first rankings, and \
anti-plurality rule, ",
  Cell[BoxData[
      FormBox[
        FormBox[\(s = 1\/2\),
          "TraditionalForm"], TraditionalForm]]],
  ", counts second rankings as highly as first rankings."
}], "Text"],

Cell["\<\
\tThe standardized positional vote system can therefore be written as:\
\>", "Text"],

Cell[BoxData[{
    FormBox[
      RowBox[{
        StyleBox[\(The\ Standardized\ Positional\ Voting\ System\),
          FontVariations->{"Underline"->True}], "\n"}], TextForm], 
    FormBox[
      RowBox[{
        RowBox[{
          RowBox[{"(", GridBox[{
                {\(1 - s\), \(1 - s\), "s", "0", "0", "s"},
                {"s", "0", "0", "s", \(1 - s\), \(1 - s\)},
                {"0", "s", \(1 - s\), \(1 - s\), "s", "0"}
                },
              GridFrame->True,
              RowLines->True,
              ColumnLines->True], ")"}], 
          RowBox[{"(", GridBox[{
                {"p1"},
                {"p2"},
                {"p3"},
                {"p4"},
                {"p5"},
                {"p6"}
                },
              GridFrame->True,
              RowLines->True,
              ColumnLines->True], ")"}]}], "=", 
        RowBox[{"(", GridBox[{
              {\(p1 + p2 + \((\(-p1\) - p2 + p3 + p6)\)*s\)},
              {\(p6 + p5 + \((p4 - p5 + p1 - p6)\)*s\)},
              {\(p3 + p4 + \((p2 - p3 - p4 + p5)\)*s\)}
              },
            GridFrame->True,
            RowLines->True,
            ColumnLines->True], ")"}]}], TextForm]}], "Text",
  TextAlignment->Center,
  FontWeight->"Bold"],

Cell[TextData[{
  "\tMultiplying the vote matrix by the profile vector gives us the vote \
outcome on the right hand side.  The voting package always converts profile \
vectors into proportions so the right hand side will give us the ",
  StyleBox["share",
    FontSlant->"Italic"],
  " of the vote going to candidates a, b, and c, respectively.",
  ButtonBox["\[SelectionPlaceholder]",
    ButtonData:>"EndNote 2",
    ButtonStyle->"Hyperlink"],
  ButtonBox["2",
    ButtonData:>"EndNote 2",
    ButtonStyle->"Hyperlink"]
}], "Text",
  CellTags->"Return 2"],

Cell[TextData[{
  "\tGiven a profile vector and a voting system, s, the function \
PositionalVoteTally calculates the voting outcome.  For example, the outcome \
when ",
  Cell[BoxData[
      \(TraditionalForm\`aprofile = {1, 2, 3, 4, 5, 6}\)]],
  " and the voting system is plurality rule is given by:"
}], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(PositionalVoteTally[aprofile, 0]\)], "Input",
  CellHorizontalScrolling->False],

Cell[BoxData[
    FormBox[
      InterpretationBox[GridBox[{
            {\({11\/21, 1\/3, 1\/7}\)},
            {\({"C2", "C3", "C1"}\)}
            },
          GridBaseline->{Baseline, {1, 1}},
          ColumnAlignments->{Left}],
        ColumnForm[ {{
          Rational[ 11, 21], 
          Rational[ 1, 3], 
          Rational[ 1, 7]}, {"C2", "C3", "C1"}}],
        Editable->False], TraditionalForm]], "Output"]
}, Open  ]],

Cell[TextData[{
  "\tThe vote tallies are given in proportions from highest to lowest.  The \
above indicates, for example, that C2 wins the election with ",
  Cell[BoxData[
      \(TraditionalForm\`11/21 = 52.3  %\)]],
  " of the vote.  C3 comes in second with ",
  Cell[BoxData[
      \(TraditionalForm\`1/3 = 33  %\)]],
  " of the vote and C1 comes in last with ",
  Cell[BoxData[
      \(TraditionalForm\`1/7 = 14.2  %\)]],
  " of the vote."
}], "Text",
  Evaluatable->False],

Cell[TextData[{
  "\tAs another example let ",
  Cell[BoxData[
      \(TraditionalForm\`aprofile = {0, 0.39, 0,  .244, 0.3, 0}\)]],
  " and compare the results of plurality rule with the Borda count.  (This is \
a normalized profile vector which indicates that 39% of the voters are of \
type 2 etc.)"
}], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(aprofile = {0, 0.39, 0,  .244, 0.3, 0}\)], "Input",
  CellHorizontalScrolling->False],

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            StyleBoxAutoDelete->True,
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          StyleBox["0.243999999999999994`",
            StyleBoxAutoDelete->True,
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Cell[BoxData[
    \(PositionalVoteTally[aprofile, 0]\)], "Input",
  CellHorizontalScrolling->False],

Cell[BoxData[
    FormBox[
      InterpretationBox[GridBox[{
            {
              \({0.417558886509636018`, 0.321199143468950776`, 
                0.261241970021413294`}\)},
            {\({"C1", "C2", "C3"}\)}
            },
          GridBaseline->{Baseline, {1, 1}},
          ColumnAlignments->{Left}],
        ColumnForm[ {{.41755888650963602, .32119914346895079,
           .26124197002141331}, {"C1", "C2", "C3"}}],
        Editable->False], TraditionalForm]], "Output"]
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Cell[CellGroupData[{

Cell[BoxData[
    \(PositionalVoteTally[aprofile, 1\/3]\)], "Input",
  CellHorizontalScrolling->False],

Cell[BoxData[
    FormBox[
      InterpretationBox[GridBox[{
            {
              \({0.420413990007137705`, 0.301213418986438252`, 
                0.278372591006424041`}\)},
            {\({"C3", "C2", "C1"}\)}
            },
          GridBaseline->{Baseline, {1, 1}},
          ColumnAlignments->{Left}],
        ColumnForm[ {{.42041399000713775, .30121341898643827,
           .27837259100642403}, {"C3", "C2", "C1"}}],
        Editable->False], TraditionalForm]], "Output"]
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Cell[TextData[{
  "\tNotice that plurality rule ranks the candidates ",
  Cell[BoxData[
      \(TraditionalForm\`C1 > C2 > C3\)]],
  " but the Borda Count ranks them in the completely opposite manner ",
  Cell[BoxData[
      \(TraditionalForm\`C3 > C2 > C1\)]],
  ".  This is quite disturbing!  It's not surprising, of course, that the \
voting rule changes the outcome somewhat but we might have hoped that given \
the same preferences different voting schemes would give approximately the \
same outcomes.  Now we are stuck in the difficult position of having to \
decide which outcome is better or more democratic.  Do the voter's think C3 \
is the best candidate or the worst candidate?  How can the results of any \
voting scheme be considered 'correct' or 'good' if some other voting scheme \
which appears equally legitimate gives entirely different results?"
}], "Text",
  Evaluatable->False],

Cell[TextData[{
  "\tSo far we have looked at only two voting systems, plurality rule and the \
Borda count.  But why stop there?  Why not also examine a system which gives \
the top candidate 20 points the second ranked candidate 6 points and the \
bottom ranked candidate 3 points?  Using the normalization this corresponds \
to ",
  Cell[BoxData[
      \(TraditionalForm\`s =  .15\)]],
  "."
}], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(PositionalVoteTally[aprofile, 0.15]\)], "Input",
  CellHorizontalScrolling->False],

Cell[BoxData[
    FormBox[
      InterpretationBox[GridBox[{
            {
              \({0.35492505353319057`, 0.332869379014989297`, 
                0.312205567451820176`}\)},
            {\({"C1", "C3", "C2"}\)}
            },
          GridBaseline->{Baseline, {1, 1}},
          ColumnAlignments->{Left}],
        ColumnForm[ {{.35492505353319059, .33286937901498931,
           .31220556745182015}, {"C1", "C3", "C2"}}],
        Editable->False], TraditionalForm]], "Output"]
}, Open  ]],

Cell[TextData[{
  "\tYet another ranking!  If all that matters is the winner of the election, \
the ",
  Cell[BoxData[
      \(TraditionalForm\`s = 0.15\)]],
  " system gives the same results as plurality rule but often the ranking of \
second and third placed candidates also matters.  In many contests, such as \
the Miss USA Pageant or the World BodyBuilding Championship, the second \
placed candidate receives a larger prize than the third placed candidate and \
may replace the winner if the winner is disqualified.  In politics, not all \
elections are all or nothing affairs.  A candidate ranked second in the New \
Hampshire Presidential primary could go on to eventually win the election \
while a third ranked candidate might find his campaign funds quickly \
declining as supporters jumped ship to 'stronger' candidates.  In Russian \
Presidential elections, French Parliamentary elections, and many state and \
local elections in the United States the top n (often 2) candidates go into a \
second round 'runoff' election if the top candidate receives less than x \
percent (usually 50%) of the vote.  In each of these cases the entire ranking \
matters and thus ",
  StyleBox["acb",
    FontSlant->"Italic"],
  " is a significantly different outcome than ",
  StyleBox["abc",
    FontSlant->"Italic"],
  ". "
}], "Text",
  Evaluatable->False],

Cell[TextData[{
  "\tIt would be interesting to see the election outcome for ",
  StyleBox["every",
    FontSlant->"Italic"],
  " possible method of positional voting.  We can do this using the \
representation triangle and the procedure line."
}], "Text"]
}, Open  ]],

Cell[CellGroupData[{

Cell["The Representation Triangle and the Procedure Line", "Section",
  Evaluatable->False],

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Cell[BoxData[
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        False,
        Editable->False], TraditionalForm]], "Output"]
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Cell[TextData[{
  "\tThe representation triangle is used to show all possible voting \
outcomes.   The x coordinate of any point in the triangle tells us the \
proportion of the vote going to C1.  The y coordinate tells us the proportion \
of the vote going to C2.  Since proportions must sum to one we can easily \
calculate C3 vote share as, ",
  Cell[BoxData[
      \(TraditionalForm\`C3 = 1 - C1 - C2\)]],
  ".  Notice, for example, that the origin is labelled C3, this is because if \
",
  Cell[BoxData[
      \(TraditionalForm\`C1 = 0\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`C2 = 0\)]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm\`C3 = 1\)]],
  " (C3 wins by unanimous vote).  \n\tThe representation triangle is split \
into the six possible strict voting outcomes (ties occur at the borders of \
the regions).  Region 1, for example, covers the entire area where ",
  Cell[BoxData[
      \(TraditionalForm\`C1 > C2 > C3\)]],
  ".  To see this more clearly consider the 45 degree line which runs from \
the origin to the triangle's hypoteneuse.  Along this line ",
  Cell[BoxData[
      \(TraditionalForm\`C1 = C2\)]],
  " so all points to the right or below the line have ",
  Cell[BoxData[
      \(TraditionalForm\`C1 > C2\)]],
  " and all points to left or above the line have ",
  Cell[BoxData[
      \(TraditionalForm\`C2 > C1\)]],
  ".   Similarly, along the line from the C2 vertex to the midpoint of ",
  Cell[BoxData[
      \(TraditionalForm\`C3 - C1\)]],
  ", ",
  Cell[BoxData[
      \(TraditionalForm\`C1 = C3\)]],
  ".  All points above or to the right of this line thus have coordinates \
such that ",
  Cell[BoxData[
      \(TraditionalForm\`C1 > C3\)]],
  ".  Finally, the line from the C1 vertex to the midpont of ",
  Cell[BoxData[
      \(TraditionalForm\`C3 - C2\)]],
  " indicates all the points where ",
  Cell[BoxData[
      \(TraditionalForm\`C2 = C3\)]],
  ".  Points above or to the right of this line have ",
  Cell[BoxData[
      \(TraditionalForm\`C2 > C3\)]],
  ".  The region where ",
  Cell[BoxData[
      \(TraditionalForm\`C1 > C2\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`C1 > C3\)]],
  " and ",
  Cell[BoxData[
      \(TraditionalForm\`C2 > C3\)]],
  " is thus Region 1.  (As a quick reminder of which regions mean what we can \
always jump back to the top of this notebook or simply reenter VoterTypes[].)\
\n     The function PlotPL[aprofile] will plot all possible positional voting \
election outcomes for the profile aprofile, ie. it will plot the election \
outcome as s ranges from 0 to ",
  Cell[BoxData[
      \(TraditionalForm\`1/2\)]],
  ".  The plurality vote outcome is indicated with a red point, the Borda \
Count outcome with a green point and the anti-plurality vote with a blue \
point.  (PL, by the way, stands for Procedure Line.)"
}], "Text",
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Cell[BoxData[
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Cell[TextData[{
  "\tFrom above we know that given ",
  Cell[BoxData[
      \(TraditionalForm\`aprofile = {0, 0.39, 0,  .244, 0.3, 0}\)]],
  " and ",
  Cell[BoxData[
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  " (plurality rule) the outcome is in Region 1, the exact outcome is \
indicated by the red point.  The Borda count outcome (",
  Cell[BoxData[
      \(TraditionalForm\`s = 1\/3\)]],
  ") is indicated with a green point and the left most point in blue is the \
anti-plurality (",
  Cell[BoxData[
      \(TraditionalForm\`s = 1\/2\)]],
  ") outcome.  The pointsizes are optimized for a hi-res monitor.  If they \
appear too small or to big PlotPL[profile,size,options] also takes a size \
option in the second slot.  Default pointsize is .025.  PlotPL[profile,.0105] \
will print smaller points, for example.\n\tNotice that the procedure line \
goes through regions 1,2,3, and 4 plus it intersects three boundaries so this \
single profile can lead to 7 different outcomes in all.  Obviously a single \
profile can generate a lot of different outcomes depending on the voting \
system!  This problem gets much worse as the number of candidates increases.  \
Saari (1992) shows that with only 10 candidates a single profile can generate \
over 3 million strict rankings and over 8 million rankings including ties!   \
"
}], "Text",
  Evaluatable->False],

Cell[TextData[{
  "\tWhat we have just shown is that with the single profile, ",
  Cell[BoxData[
      \(TraditionalForm\`aprofile = {0, 0.39, 0,  .244, 0.3, 0}\)]],
  ", the voting outcome can range from ",
  Cell[BoxData[
      \(TraditionalForm\`C1 > C2 > C3\)]],
  " to",
  Cell[BoxData[
      \(TraditionalForm\`\(\ C1 > C3 > C2\ \)\)]],
  "to",
  Cell[BoxData[
      \(TraditionalForm\`\(\ C3 > C1 > C2\)\)]],
  " and even ",
  Cell[BoxData[
      \(TraditionalForm\`C3 > C2 > C1\)]],
  "!  Which of these is the best outcome?  Which of these is the most \
democratic outcome?   The answers to these questions are not obvious.  \
Regardless of our answers to these questions our results show that any theory \
of democracy which speaks of a \"will of the majority\" is incoherent.  How \
can the majority be said to have a will when the same preferences may \
generate the ranking ",
  Cell[BoxData[
      \(TraditionalForm\`C1 > C2 > C3\)]],
  " or the reverse ranking ",
  Cell[BoxData[
      \(TraditionalForm\`C3 > C2 > C1\)]],
  "?  When there are ten candidates, a single profile may generate over 8 \
million rankings which of these rankings represents the majority's will?"
}], "Text",
  Evaluatable->False],

Cell[TextData[{
  "\tThe following command generates 10 random voter profiles and their \
respective procedure lines.",
  ButtonBox["\[SelectionPlaceholder]",
    ButtonData:>"EndNote 3",
    ButtonStyle->"Hyperlink"],
  ButtonBox["3",
    ButtonData:>"EndNote 3",
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  "  "
}], "Text",
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Cell[BoxData[
    FormBox[
      TagBox[\(\[SkeletonIndicator]  Graphics  \[SkeletonIndicator]\),
        False,
        Editable->False], TraditionalForm]], "Output"]
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Cell[TextData[{
  "\tRandom plots are always different of course but in the typical run of 10 \
plots it is often the case that most procedure lines cross at least one \
boundary and many cross more than one.  Thus,  most profiles result in more \
than one outcome when the voting system changes.  In fact, using Monte Carlo \
methods with ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " or analytically using the normal function as an approximation (Saari and \
Tataru, 1997) it's possible to show that as the number of voters gets large \
(say 100 or more) the probability of a single profile generating i different \
outcomes is:"
}], "Text",
  Evaluatable->False,
  TextAlignment->Left,
  TextJustification->0,
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  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[TextData[{
  StyleBox["Probability of a Single Profile Generating i Outcomes\n",
    FontVariations->{"Underline"->True}],
  "\n",
  Cell[BoxData[
      RowBox[{"(", GridBox[{
            {\(No . of\ Different\ Outcomes\), "Probability"},
            {"7", ".06"},
            {"5", ".19"},
            {"3", ".44"},
            {"1", ".31"}
            },
          GridFrame->True,
          RowLines->True,
          ColumnLines->True], ")"}]]]
}], "Text",
  Evaluatable->False,
  TextAlignment->Center,
  TextJustification->0,
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  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell["\<\
\tThe table contains no probabilities of even numbers of outcomes ocurring \
because an even number of outcomes requires at least one tie and the \
probability of a tie vote goes to zero as the number of voters increases.\
\>", "Text"],

Cell[TextData[{
  "\tMany non-obvious theorems about voting follow from from the fact that \
the procedure line is linear and is bracketed by the plurality and \
anti-plurality voting systems.  Some examples: If both the plurality and \
anti-plurality rankings are the same then ",
  StyleBox["all ",
    FontSlant->"Italic"],
  "positional vote systems give the same rankings.  A single profile can give \
rise to at most 4 different strict rankings or 7 different rankings including \
rankings with ties.  If a profile gives rise to an all-tied vote under ",
  StyleBox["any",
    FontSlant->"Italic"],
  " system then at most the profile gives rise to two different strict \
rankings.  You can probably find other theorems like these simply by thinking \
about ways in which straight lines can be placed within the RTriangle.  Or \
see Saari (1994) for more examples. "
}], "Text",
  Evaluatable->False]
}, Open  ]],

Cell[CellGroupData[{

Cell["Some Machiavellian Musings", "Section",
  Evaluatable->False],

Cell[TextData[{
  "\tA modern Machiavelli has a number of different strategies he can pursue \
in order to win an election.  First, he can estimate the voter profile and \
plot the procedure line.  If he is going to lose the election under the \
current electoral system perhaps there is a different electoral system under \
which he will win.   Such Machiavellianism is not uncommon.  In 1986, for \
example, Francois Mitterand arranged a change in the French electoral system \
in order to benefit his party.  In an effort to lessen the impact of black \
voters, many cities in the American South adjusted the geographic basis of \
their electoral systems shortly after the Civil Rights Act was passed.   And \
the Greek electoral system has changed in almost every election since World \
War II!  \n\tA second strategy is to find a profile such that the candidate \
will win under the current electoral system and then try to move voters \
towards that profile.  Suppose, for example, that we wanted to find all the \
voter profiles such that under plurality rule the outcome was ",
  Cell[BoxData[
      \(TraditionalForm\`{ .42,  .3,  .28}\)]],
  ".  This problem is easily solved using ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  "'s Solve function.  The function, f, is the core function underlying \
PositionalVoteTally it computes the vote tally and returns it in the form \
{C1,C2,C3}. "
}], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell["genericprofile={p1,p2,p3,p4,p5,p6}", "Input"],

Cell[BoxData[
    \(TraditionalForm\`{p1, p2, p3, p4, p5, p6}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell["sol=Solve[f[genericprofile,0]=={.42,.3,.28}]", "Input"],

Cell[BoxData[
    \(TraditionalForm
    \`Solve::"svars" \( : \ \) 
      "Equations may not give solutions for all \"solve\" variables."\)], 
  "Message"],

Cell[BoxData[
    \(TraditionalForm
    \`{{p1 \[Rule] \(0.420000000000000017`\[InvisibleSpace]\) - 1.`\ p2, 
        p3 \[Rule] \(0.280000000000000035`\[InvisibleSpace]\) - 1.`\ p4, 
        p5 \[Rule] \(0.3`\[InvisibleSpace]\) - 1.`\ p6}}\)], "Output"]
}, Open  ]],

Cell["\<\
\tAny sol