Read e-book online Basic Geometry of Voting PDF

By Donald G. Saari

ISBN-10: 3540600647

ISBN-13: 9783540600640

ISBN-10: 3642577482

ISBN-13: 9783642577482

A shock is how the complexities of balloting concept could be defined and resolved with the comfy geometry of our 3-dimensional global. This publication is directed towards scholars and others wishing to profit approximately balloting, specialists will observe formerly unpublished effects. for example, a brand new profile decomposition speedy resolves centuries previous controversies of Condorcet and Borda, demonstrates, that the ratings of pairwise and different tools fluctuate simply because they depend upon varied details, casts sequence doubt at the reliability of a Condorcet winner as a customary for the sector, makes the recognized Arrow`s Theorem predictable, and simplifies the development of examples. The geometry unifies doubtless disparate themes as manipulation, monotonicity, or even the apportionment problems with the U.S. ideal Court.

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Extra resources for Basic Geometry of Voting

Example text

Fig. 8. The universal threshold method = b. This threshold method becomes what method if 6 2 ~? What happens if 62 2: t? c. Find the relationship between 62 and 63 to ensure that an election outcome always defined. IS 8. Recall: the rational points in the representational triangle are dense. This means that if q' E Si(3) has components with irrational values, then arbitrarily close to q' are choices of q E Si(3) where all the components are rational numbers. Equivalentlj, there exists a sequence of points in Si(3) with rational components, {qn}~I' such that qn --+ q' as n -+ 00.

14~S9 ... SOO. 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.

For the three threshold methods from the Exercises of Sect. 21. a. Find a profile PI so that C! is top-ranked with the Universal but not with the Distinct or Quasi transitive threshold methods. b. Find a profile P2 so that C2 is top-ranked with the Distinct, but not with the Universal or Quasi-transitive threshold methods. c. Can a profile be found where Ca is top-ranked with the Quasi-transitive threshold method but not with the other two? Give a geometric one for tallying the plurality vote in terms of the entries near the vertex.

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Basic Geometry of Voting by Donald G. Saari

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