Abstract
This chapter deals with the concept of manipulation, understood as preference misrepresentation, in the light of the main theoretical results focusing on their practical significance (This chapter is largely based on Nurmi (Transactions on computational collective intelligence XXIII. Springer, Berlin, pp. 149–161, 2016)). Manipulability is a pervasive property among choice rules. However, its practical importance hinges on several things. The information requirements of successful misrepresentation can be very demanding and suitable situations may not be common. We also review some indices measuring the degree of manipulability of choice functions. Moreover, the results on complexity of manipulation as well as on safe manipulability are briefly touched upon.
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Notes
- 1.
A Nash equilibrium in pure strategies is an n-tuple of strategies (one for each individual) such that no individual is better off deviating from his/her strategy in this n-tuple, provided that the others stick to their strategy choices. So, in Nash equilibrium unilateral deviations do not bring benefit to the deviator. The same idea extends to mixed strategies, i.e. probability distributions over pure strategies. In this work we focus on pure strategies only.
- 2.
With sincere voting the outcome is the worst for the 2 voters, with strategic voting their most preferred alternative wins.
- 3.
It should be mentioned that in the profiles constructed in this work, the number of voters having each preference ranking can be multiplied by any integer without changing the outcomes as long as the same integer is used as the multiplier in all preference rankings. This follows from the homogeneity of nearly all procedures discussed here. The only exception is Dodgson’s rule, but its non-homogeneity is not relevant in the examples discussed in this book.
- 4.
\(K = \sum _{j} J_{j}\).
- 5.
In 3-alternative contexts the threshold rule ranks alternative x ahead of y if and only if the number of individuals giving y the lowest grade is strictly larger than the number of individuals assigning x the lowest grade. In case x and y are given the lowest grade by the same number of individuals, their collective ranking is determined by the number of individuals assigning x and y the middle grade.
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de Almeida, A.T., Morais, D.C., Nurmi, H. (2019). Strategic Aspects. In: Systems, Procedures and Voting Rules in Context . Advances in Group Decision and Negotiation, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-030-30955-8_5
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