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On the manipulability of approval voting and related scoring rules

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  • Hans Peters

    ()

  • Souvik Roy

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  • Ton Storcken

    ()

Abstract

We characterize all preference profiles at which the approval (voting) rule is manipulable, under three extensions of preferences to sets of alternatives: by comparison of worstalternatives, best alternatives, or by comparison based on stochastic dominance. We perform a similar exercise for $k$-approval rules, where voters approve of a fixed number $k$ of alternatives. These results can be used to compare ($k$-)approval rules with respect to their manipulability. Analytical results are obtained for the case of two voters, specifically, the values of $k$ for which the $k$-approval rule is minimally manipulable -- has the smallest number of manipulable preference profiles -- under the various preference extensions are determined. For the number of voters going to infinity, an asymptotic result is that the $k$-approval rule with $k$ around half the number of alternatives is minimally manipulable among all scoring rules. Further results are obtained by simulation and indicate that $k$-approval rules may improve on the approval rule as far as manipulability is concerned.
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Suggested Citation

  • Hans Peters & Souvik Roy & Ton Storcken, 2012. "On the manipulability of approval voting and related scoring rules," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 39(2), pages 399-429, July.
  • Handle: RePEc:spr:sochwe:v:39:y:2012:i:2:p:399-429
    DOI: 10.1007/s00355-011-0621-7
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    References listed on IDEAS

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    1. Donald Campbell & Jerry Kelly, 2009. "Gains from manipulating social choice rules," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 40(3), pages 349-371, September.
    2. Mostapha Diss & Vincent Merlin & Fabrice Valognes, 2010. "On the Condorcet efficiency of approval voting and extended scoring rules for three alternatives," Post-Print halshs-00533124, HAL.
    3. Satterthwaite, Mark Allen, 1975. "Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions," Journal of Economic Theory, Elsevier, vol. 10(2), pages 187-217, April.
    4. Gehrlein, William V. & Lepelley, Dominique, 1998. "The Condorcet efficiency of approval voting and the probability of electing the Condorcet loser," Journal of Mathematical Economics, Elsevier, vol. 29(3), pages 271-283, April.
    5. Pritchard, Geoffrey & Wilson, Mark C., 2009. "Asymptotics of the minimum manipulating coalition size for positional voting rules under impartial culture behaviour," Mathematical Social Sciences, Elsevier, vol. 58(1), pages 35-57, July.
    6. Maus, Stefan & Peters, Hans & Storcken, Ton, 2007. "Minimal manipulability: Unanimity and nondictatorship," Journal of Mathematical Economics, Elsevier, vol. 43(6), pages 675-691, August.
    7. Peter Fristrup & Hans Keiding, 1998. "Minimal manipulability and interjacency for two-person social choice functions," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 15(3), pages 455-467.
    8. Saari, Donald G, 1990. "Susceptibility to Manipulation," Public Choice, Springer, vol. 64(1), pages 21-41, January.
    9. Gibbard, Allan, 1973. "Manipulation of Voting Schemes: A General Result," Econometrica, Econometric Society, vol. 41(4), pages 587-601, July.
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    Cited by:

    1. Duddy, Conal, 2014. "Electing a representative committee by approval ballot: An impossibility result," Economics Letters, Elsevier, vol. 124(1), pages 14-16.
    2. Conal Duddy & Ashley Piggins & William Zwicker, 2016. "Aggregation of binary evaluations: a Borda-like approach," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 46(2), pages 301-333, February.
    3. Yuliya A. Veselova, 2016. "Does Incomplete Information Reduce Manipulability?," HSE Working papers WP BRP 152/EC/2016, National Research University Higher School of Economics.

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