Basic Geometry of VotingA surprise is how the complexities of voting theory can be explained and resolved with the comfortable geometry of our three-dimensional world. This book is directed toward students and others wishing to learn about voting, experts will discover previously unpublished results. As an example, a new profile decomposition quickly resolves two centuries old controversies of Condorcet and Borda, demonstrates, that the rankings of pairwise and other methods differ because they rely on different information, casts series doubt on the reliability of a Condorcet winner as a standard for the field, makes the famous Arrow`s Theorem predictable, and simplifies the construction of examples. The geometry unifies seemingly disparate topics as manipulation, monotonicity, and even the apportionment issues of the US Supreme Court. |
Contents
I | 1 |
II | 2 |
III | 4 |
IV | 6 |
V | 7 |
VI | 8 |
VII | 9 |
VIII | 10 |
LXXVI | 136 |
LXXVII | 137 |
LXXVIII | 139 |
LXXX | 140 |
LXXXI | 141 |
LXXXII | 143 |
LXXXIII | 145 |
LXXXIV | 148 |
IX | 11 |
X | 12 |
XI | 15 |
XIII | 16 |
XIV | 17 |
XV | 18 |
XVI | 20 |
XVII | 21 |
XVIII | 22 |
XIX | 25 |
XX | 27 |
XXI | 29 |
XXII | 30 |
XXIII | 31 |
XXIV | 33 |
XXV | 35 |
XXVI | 38 |
XXVII | 40 |
XXVIII | 42 |
XXIX | 44 |
XXX | 45 |
XXXI | 47 |
XXXII | 49 |
XXXIII | 51 |
XXXIV | 54 |
XXXV | 56 |
XXXVI | 60 |
XXXVII | 63 |
XXXVIII | 65 |
XXXIX | 67 |
XL | 71 |
XLII | 72 |
XLIII | 75 |
XLIV | 79 |
XLV | 82 |
XLVI | 83 |
XLVIII | 84 |
XLIX | 85 |
L | 87 |
LII | 89 |
LIII | 92 |
LIV | 94 |
LV | 96 |
LVI | 98 |
LVII | 101 |
LIX | 102 |
LX | 104 |
LXI | 107 |
LXII | 108 |
LXIII | 110 |
LXV | 112 |
LXVI | 113 |
LXVII | 115 |
LXVIII | 118 |
LXIX | 121 |
LXX | 122 |
LXXI | 124 |
LXXII | 125 |
LXXIV | 127 |
LXXV | 131 |
LXXXV | 150 |
LXXXVI | 153 |
LXXXVII | 154 |
LXXXVIII | 156 |
LXXXIX | 159 |
XC | 163 |
XCI | 165 |
XCIII | 167 |
XCIV | 168 |
XCV | 170 |
XCVI | 172 |
XCVII | 174 |
XCVIII | 177 |
XCIX | 178 |
CI | 179 |
CII | 182 |
CIII | 185 |
CIV | 186 |
CV | 188 |
CVI | 189 |
CVII | 190 |
CIX | 192 |
CX | 193 |
CXI | 195 |
CXII | 196 |
CXIII | 197 |
CXIV | 201 |
CXVI | 204 |
CXVII | 205 |
CXVIII | 207 |
CXIX | 208 |
CXX | 209 |
CXXI | 212 |
CXXII | 214 |
CXXIII | 215 |
CXXIV | 219 |
CXXV | 224 |
CXXVI | 225 |
CXXVII | 228 |
CXXVIII | 230 |
CXXIX | 231 |
CXXX | 233 |
CXXXI | 242 |
CXXXIII | 244 |
CXXXIV | 246 |
CXXXV | 250 |
CXXXVI | 257 |
CXXXVII | 259 |
CXXXVIII | 263 |
CXL | 267 |
CXLI | 270 |
CXLII | 273 |
CXLIII | 275 |
CXLIV | 279 |
CXLV | 281 |
CXLVI | 284 |
CXLVII | 285 |
| 291 | |
| 297 | |
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Common terms and phrases
agenda Alabama Paradox antiplurality Arrow's Theorem assertion assignment triangle ballot base cube basic profile BC outcome BC ranking beverage example Borda Borda Count bottom-ranked boundary c₁ c2 c₂ choice compute Condorcet loser Condorcet portion Condorcet winner confused voters convex hull convex set Copeland's method cycle cyclic region defined determine e₁ election outcomes election ranking election tally equations geometry Gibbard-Satterthwaite Theorem illustrate image set indifference line inner normal instance integer line segment manipulated orthant p₁ pairwise election pairwise rankings pairwise vote plurality outcome plurality ranking plurality vote population positional methods positively involved preferences problems procedure line profile line profile set proof ranking c₁ ranking regions reduced profile relative ranking representation cube representation triangle runoff seats Sect shaded region specified Sup(p T₁ Theorem three candidates top-ranked candidate transitive type-one unanimity profiles vertex vertices voter types voting vector W₁ weakly consistent wins
Popular passages
Page 295 - Votjng methods in context: The development of a science of voting in French scientific institutions, 1699-1803, Stevens Institute of Technology preprint (1994).