Abstract
Among many other topics, Hannu Nurmi has worked on voting paradoxes and how to deal with them. In his work he often uses a geometric approach developed by Don Saari for the analysis of paradoxes of preference aggregation such as the Condorcet paradox or Arrow’s general possibility theorem. In this chapter this approach is extended to other paradoxes analysed by Nurmi and the recent work in judgment aggregation. In particular we use Saari’s representation cubes to provide a geometric representation of profiles and majority outcomes.
An earlier version of this chapter has been published in Homo Oecomicus 26(3/4): Essays in Honor of Hannu Nurmi: Volume I, edited by Manfred J. Holler and Mika Widgren, 2009.
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Notes
- 1.
A “Condorcet portion” is a multiple of the set of individuals that has the following preferences over 3 alternatives \(a,b,c\): \(a\succ _{1}b\succ _{1}c\), \(c\succ _{2}a\succ _{2}b\), \(b\succ _{3}c\succ _{3}a\) leading to the the majority cycle \(a\succ b\succ c\succ a\).
- 2.
Equivalently we could say that they are each of Hamming distance 1 from their common neighbor.
- 3.
See e.g. Saari (2008) for a very brief discussion of the link of his geometric approach to judgment aggregation.
- 4.
For a more elaborated discussion on majority voting on restricted domains see also Dietrich and List (2010).
- 5.
See Gehrlein (2006) for a general discussion of the impartial anonymous culture.
- 6.
In social choice theory, aggregation rules based on such restrictions are often called Condorcet extensions.
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Eckert, D., Klamler, C. (2013). A Geometric Approach to Paradoxes of Majority Voting: From Anscombe’s Paradox to the Discursive Dilemma with Saari and Nurmi. In: Holler, M., Nurmi, H. (eds) Power, Voting, and Voting Power: 30 Years After. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35929-3_33
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