Basic Geometry of Voting - download pdf or read online

By Donald G. Saari

ISBN-10: 3540600647

ISBN-13: 9783540600640

ISBN-10: 3642577482

ISBN-13: 9783642577482

A shock is how the complexities of balloting thought will be defined and resolved with the comfy geometry of our three-d global. This e-book is directed towards scholars and others wishing to profit approximately vote casting, specialists will notice formerly unpublished effects. to illustrate, a brand new profile decomposition speedy resolves centuries previous controversies of Condorcet and Borda, demonstrates, that the scores of pairwise and different equipment range simply because they depend on diverse details, casts sequence doubt at the reliability of a Condorcet winner as a typical for the sphere, makes the recognized Arrow`s Theorem predictable, and simplifies the development of examples. The geometry unifies possible disparate subject matters as manipulation, monotonicity, or even the apportionment problems with the united states superb Court.

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Extra resources for Basic Geometry of Voting

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For instance, (45,75,80) indicates that Cl receives 45 votes, C2 receives 75 votes, and Ca receives 80 votes leading to the election ranking Ca r C2 r CI. Because an ordinal ranking of the candidates depends only on the relative sizes of the coordinates of a vector, any positive multiple of the tally preserves the election ranking. 1 NORMALIZED ELECTION TALLY So, although a multiple of 1000 separates (3,5,2) from (3000,5000,2000), both totals define the same election ranking Cz ~ CI ~ Ca. There is nothing special about three candidates; with five candidates the election tally is given by (nl,n2,na,n4,n5) which is a point in the five dimensional space R5.

A. If the voters are of the respective types one, three, five, find the weighted election outcome. b. Express the profile of part a as a normalized profile in Si(6). c. Show for these weights that whatever voter-one wants, voter-one gets. Namely, voter-one is a dictator. Show that if voter-one does not vote, then voter-two becomes a dictator. d. To extend a voting vector from strict preferences to all preferences, average the number of votes cast over all ways indifference can be broken. In this manner, a voter with ranking CI "" C3 >- C2 would cast the ballot (~,O,~) ~([elh + [ella).

If both candidates, c;,cj, receive the same number of votes, their tied ranking is Ci '" Cj. 0 The vote totals for the candidates {Cl,c2,ca} define the array (nl,n2,na) where n j is the number of points assigned to candidate Cj, j = 1,2,3. Byexploiting our familiarity with the three-dimensional world, treat (nl, n2, na) as a point in the three-dimensional space R a = {(Xl,X2,Xa) I - 00 < Xj < 00, j = 1,2,3}. Namely, identify values on the Xj axis of R3 with the election tally for candidate Cj; j = 1,2,3.

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Basic Geometry of Voting by Donald G. Saari

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